Functions

Manipulation and Creation of States and Operators

Quantum States

basis(dimensions, n=None, offset=None, *, dtype=None)[source]

Generates the vector representation of a Fock state.

Parameters:

dimensionsint or list of ints: Number of Fock states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
nint or list of ints, optional (default 0 for all dimensions): Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match dimensions, e.g. if dimensions is a list, then n must either be omitted or a list of equal length.
offsetint or list of ints, optional (default 0 for all dimensions): The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
dtypetype or str: storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

statequtip.Qobj: Qobj representing the requested number state |n>.

Notes

A subtle incompatibility with the quantum optics toolbox: In QuTiP:

basis(N, 0) = ground state

but in the qotoolbox:

basis(N, 1) = ground state

Examples

>>> basis(5,2) 
Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket
Qobj data =
[[ 0.+0.j]
 [ 0.+0.j]
 [ 1.+0.j]
 [ 0.+0.j]
 [ 0.+0.j]]
>>> basis([2,2,2], [0,1,0]) 
Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
Qobj data =
[[0.]
 [0.]
 [1.]
 [0.]
 [0.]
 [0.]
 [0.]
 [0.]]

bell_state(state='00', *, dtype=None)[source]

Returns the selected Bell state:

\[\begin{split}\begin{aligned} \lvert B_{00}\rangle &= \frac1{\sqrt2}(\lvert00\rangle+\lvert11\rangle)\\ \lvert B_{01}\rangle &= \frac1{\sqrt2}(\lvert00\rangle-\lvert11\rangle)\\ \lvert B_{10}\rangle &= \frac1{\sqrt2}(\lvert01\rangle+\lvert10\rangle)\\ \lvert B_{11}\rangle &= \frac1{\sqrt2}(\lvert01\rangle-\lvert10\rangle)\\ \end{aligned}\end{split}\]

Parameters:

statestr [‘00’, ‘01’, 10, 11]: Which bell state to return
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

Bell_stateqobj: Bell state

bra(seq, dim=2, *, dtype=None)[source]

Produces a multiparticle bra state for a list or string, where each element stands for state of the respective particle.

Parameters:

seqstr / list of ints or characters

Each element defines state of the respective particle. (e.g. [1,1,0,1] or a string “1101”). For qubits it is also possible to use the following conventions:

‘g’/’e’ (ground and excited state)
‘u’/’d’ (spin up and down)
‘H’/’V’ (horizontal and vertical polarization)

Note: for dimension > 9 you need to use a list.

dimint (default: 2) / list of ints

Space dimension for each particle: int if there are the same, list if they are different.

dtypetype or str

Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

braqobj

Examples

>>> bra("10") 
Quantum object: dims = [[1, 1], [2, 2]], shape = [1, 4], type = bra
Qobj data =
[[ 0.  0.  1.  0.]]

>>> bra("Hue") 
Quantum object: dims = [[1, 1, 1], [2, 2, 2]], shape = [1, 8], type = bra
Qobj data =
[[ 0.  1.  0.  0.  0.  0.  0.  0.]]

>>> bra("12", 3) 
Quantum object: dims = [[1, 1], [3, 3]], shape = [1, 9], type = bra
Qobj data =
[[ 0.  0.  0.  0.  0.  1.  0.  0.  0.]]

>>> bra("31", [5, 2]) 
Quantum object: dims = [[1, 1], [5, 2]], shape = [1, 10], type = bra
Qobj data =
[[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]]

coherent(N, alpha, offset=0, method=None, *, dtype=None)[source]

Generates a coherent state with eigenvalue alpha.

Constructed using displacement operator on vacuum state.

Parameters:

Nint: Number of Fock states in Hilbert space.
alphafloat/complex: Eigenvalue of coherent state.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the state. Using a non-zero offset will make the default method ‘analytic’.
methodstring {‘operator’, ‘analytic’}: Method for generating coherent state.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

stateqobj: Qobj quantum object for coherent state

Notes

Select method ‘operator’ (default) or ‘analytic’. With the ‘operator’ method, the coherent state is generated by displacing the vacuum state using the displacement operator defined in the truncated Hilbert space of size ‘N’. This method guarantees that the resulting state is normalized. With ‘analytic’ method the coherent state is generated using the analytical formula for the coherent state coefficients in the Fock basis. This method does not guarantee that the state is normalized if truncated to a small number of Fock states, but would in that case give more accurate coefficients.

Examples

>>> coherent(5,0.25j) 
Quantum object: dims = [[5], [1]], shape = [5, 1], type = ket
Qobj data =
[[  9.69233235e-01+0.j        ]
 [  0.00000000e+00+0.24230831j]
 [ -4.28344935e-02+0.j        ]
 [  0.00000000e+00-0.00618204j]
 [  7.80904967e-04+0.j        ]]

coherent_dm(N, alpha, offset=0, method='operator', *, dtype=None)[source]

Density matrix representation of a coherent state.

Constructed via outer product of qutip.states.coherent

Parameters:

Nint: Number of Fock states in Hilbert space.
alphafloat/complex: Eigenvalue for coherent state.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the state.
methodstring {‘operator’, ‘analytic’}: Method for generating coherent density matrix.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

dmqobj: Density matrix representation of coherent state.

Notes

Select method ‘operator’ (default) or ‘analytic’. With the ‘operator’ method, the coherent density matrix is generated by displacing the vacuum state using the displacement operator defined in the truncated Hilbert space of size ‘N’. This method guarantees that the resulting density matrix is normalized. With ‘analytic’ method the coherent density matrix is generated using the analytical formula for the coherent state coefficients in the Fock basis. This method does not guarantee that the state is normalized if truncated to a small number of Fock states, but would in that case give more accurate coefficients.

Examples

>>> coherent_dm(3,0.25j) 
Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True
Qobj data =
[[ 0.93941695+0.j          0.00000000-0.23480733j -0.04216943+0.j        ]
 [ 0.00000000+0.23480733j  0.05869011+0.j          0.00000000-0.01054025j]
 [-0.04216943+0.j          0.00000000+0.01054025j  0.00189294+0.j        ]]

enr_fock(dims, excitations, state, *, dtype=None)[source]

Generate the Fock state representation in a excitation-number restricted state space. The dims argument is a list of integers that define the number of quantums states of each component of a composite quantum system, and the excitations specifies the maximum number of excitations for the basis states that are to be included in the state space. The state argument is a tuple of integers that specifies the state (in the number basis representation) for which to generate the Fock state representation.

Parameters:

dimslist: A list of the dimensions of each subsystem of a composite quantum system.
excitationsinteger: The maximum number of excitations that are to be included in the state space.
statelist of integers: The state in the number basis representation.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

ketQobj: A Qobj instance that represent a Fock state in the exication-number- restricted state space defined by dims and exciations.

enr_state_dictionaries(dims, excitations)[source]

Return the number of states, and lookup-dictionaries for translating a state tuple to a state index, and vice versa, for a system with a given number of components and maximum number of excitations.

Parameters:

dims: list: A list with the number of states in each sub-system.
excitationsinteger: The maximum numbers of dimension

Returns:

nstates, state2idx, idx2state: integer, dict, list: The number of states nstates, a dictionary for looking up state indices from a state tuple, and a list containing the state tuples ordered by state indices. state2idx and idx2state are reverses of each other, i.e., state2idx[idx2state[idx]] = idx and idx2state[state2idx[state]] = state.

enr_thermal_dm(dims, excitations, n, *, dtype=None)[source]

Generate the density operator for a thermal state in the excitation-number- restricted state space defined by the dims and exciations arguments. See the documentation for enr_fock for a more detailed description of these arguments. The temperature of each mode in dims is specified by the average number of excitatons n.

Parameters:

dimslist: A list of the dimensions of each subsystem of a composite quantum system.
excitationsinteger: The maximum number of excitations that are to be included in the state space.
ninteger: The average number of exciations in the thermal state. n can be a float (which then applies to each mode), or a list/array of the same length as dims, in which each element corresponds specifies the temperature of the corresponding mode.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

dmQobj: Thermal state density matrix.

fock(dimensions, n=None, offset=None, *, dtype=None)[source]

Bosonic Fock (number) state.

Same as qutip.states.basis.

Parameters:

dimensionsint or list of ints: Number of Fock states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
nint or list of ints, optional (default 0 for all dimensions): Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match dimensions, e.g. if dimensions is a list, then n must either be omitted or a list of equal length.
offsetint or list of ints, optional (default 0 for all dimensions): The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

Requested number state $\left|n\right>$.

Examples

>>> fock(4,3) 
Quantum object: dims = [[4], [1]], shape = [4, 1], type = ket
Qobj data =
[[ 0.+0.j]
 [ 0.+0.j]
 [ 0.+0.j]
 [ 1.+0.j]]

fock_dm(dimensions, n=None, offset=None, *, dtype=None)[source]

Density matrix representation of a Fock state

Constructed via outer product of qutip.states.fock.

Parameters:

dimensionsint or list of ints: Number of Fock states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
nint or list of ints, optional (default 0 for all dimensions): Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match dimensions, e.g. if dimensions is a list, then n must either be omitted or a list of equal length.
offsetint or list of ints, optional (default 0 for all dimensions): The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

dmqobj: Density matrix representation of Fock state.

Examples

>>> fock_dm(3,1) 
Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  1.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j  0.+0.j]]

ghz_state(N=3, *, dtype=None)[source]

Returns the N-qubit GHZ-state:: [ |00...00> + |11...11> ] / sqrt(2)

Parameters:

Nint (default=3): Number of qubits in state
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

Gqobj: N-qubit GHZ-state

ket(seq, dim=2, *, dtype=None)[source]

Produces a multiparticle ket state for a list or string, where each element stands for state of the respective particle.

Parameters:

seqstr / list of ints or characters: Each element defines state of the respective particle. (e.g. [1,1,0,1] or a string “1101”). For qubits it is also possible to use the following conventions: - ‘g’/’e’ (ground and excited state) - ‘u’/’d’ (spin up and down) - ‘H’/’V’ (horizontal and vertical polarization) Note: for dimension > 9 you need to use a list.
dimint (default: 2) / list of ints: Space dimension for each particle: int if there are the same, list if they are different.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

ketqobj

Examples

>>> ket("10") 
Quantum object: dims = [[2, 2], [1, 1]], shape = [4, 1], type = ket
Qobj data =
[[ 0.]
 [ 0.]
 [ 1.]
 [ 0.]]

>>> ket("Hue") 
Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket
Qobj data =
[[ 0.]
 [ 1.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]]

>>> ket("12", 3) 
Quantum object: dims = [[3, 3], [1, 1]], shape = [9, 1], type = ket
Qobj data =
[[ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 1.]
 [ 0.]
 [ 0.]
 [ 0.]]

>>> ket("31", [5, 2]) 
Quantum object: dims = [[5, 2], [1, 1]], shape = [10, 1], type = ket
Qobj data =
[[ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 1.]
 [ 0.]
 [ 0.]]

ket2dm(Q)[source]

Takes input ket or bra vector and returns density matrix formed by outer product. This is completely identical to calling Q.proj().

Parameters:

Qqobj: Ket or bra type quantum object.

Returns:

dmqobj: Density matrix formed by outer product of Q.

Examples

>>> x=basis(3,2)
>>> ket2dm(x) 
Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j  1.+0.j]]

maximally_mixed_dm(N, *, dtype=None)[source]

Returns the maximally mixed density matrix for a Hilbert space of dimension N.

Parameters:

Nint: Number of basis states in Hilbert space.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

dmqobj: Thermal state density matrix.

phase_basis(N, m, phi0=0, *, dtype=None)[source]

Basis vector for the mth phase of the Pegg-Barnett phase operator.

Parameters:

Nint: Number of basis vectors in Hilbert space.
mint: Integer corresponding to the mth discrete phase phi_m = phi0 + 2 * pi * m / N
phi0float (default=0): Reference phase angle.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

stateqobj: Ket vector for mth Pegg-Barnett phase operator basis state.

Notes

The Pegg-Barnett basis states form a complete set over the truncated Hilbert space.

projection(N, n, m, offset=None, *, dtype=None)[source]

The projection operator that projects state $\lvert m\rangle$ on state $\lvert n\rangle$.

Parameters:

Nint: Number of basis states in Hilbert space.
n, mfloat: The number states in the projection.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the projector.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Requested projection operator.

qutrit_basis(*, dtype=None)[source]

Basis states for a three level system (qutrit)

dtypetype or str: storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

qstatesarray: Array of qutrit basis vectors

singlet_state(*, dtype=None)[source]

Returns the two particle singlet-state:

\[\lvert S\rangle = \frac1{\sqrt2}(\lvert01\rangle-\lvert10\rangle)\]

that is identical to the fourth bell state.

Parameters:

dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

Bell_stateqobj: $\lvert B_{11}\rangle$ Bell state

spin_coherent(j, theta, phi, type='ket', *, dtype=None)[source]

Generate the coherent spin state $\lvert \theta, \phi\rangle$.

Parameters:

jfloat: The spin of the state.
thetafloat: Angle from z axis.
phifloat: Angle from x axis.
typestring {‘ket’, ‘bra’, ‘dm’}: Type of state to generate.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

stateqobj: Qobj quantum object for spin coherent state

spin_state(j, m, type='ket', *, dtype=None)[source]

Generates the spin state $\lvert j, m\rangle$, i.e. the eigenstate of the spin-j Sz operator with eigenvalue m.

Parameters:

jfloat: The spin of the state ().
mint: Eigenvalue of the spin-j Sz operator.
typestring {‘ket’, ‘bra’, ‘dm’}: Type of state to generate.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

stateqobj: Qobj quantum object for spin state

state_index_number(dims, index)[source]

Return a quantum number representation given a state index, for a system of composite structure defined by dims.

Example

>>> state_index_number([2, 2, 2], 6)
[1, 1, 0]

Parameters:

dimslist or array: The quantum state dimensions array, as it would appear in a Qobj.
indexinteger: The index of the state in standard enumeration ordering.

Returns:

statetuple: The state number tuple corresponding to index index in standard enumeration ordering.

state_number_enumerate(dims, excitations=None)[source]

An iterator that enumerates all the state number tuples (quantum numbers of the form (n1, n2, n3, …)) for a system with dimensions given by dims.

Example

>>> for state in state_number_enumerate([2,2]): 
>>>     print(state) 
( 0  0 )
( 0  1 )
( 1  0 )
( 1  1 )

Parameters:

dimslist or array: The quantum state dimensions array, as it would appear in a Qobj.
excitationsinteger (None): Restrict state space to states with excitation numbers below or equal to this value.

Returns:

state_numbertuple: Successive state number tuples that can be used in loops and other iterations, using standard state enumeration by definition.

state_number_index(dims, state)[source]

Return the index of a quantum state corresponding to state, given a system with dimensions given by dims.

Example

>>> state_number_index([2, 2, 2], [1, 1, 0])
6

Parameters:

dimslist or array: The quantum state dimensions array, as it would appear in a Qobj.
statelist: State number array.

Returns:

idxint: The index of the state given by state in standard enumeration ordering.

state_number_qobj(dims, state, *, dtype=None)[source]

Return a Qobj representation of a quantum state specified by the state array state.

Example

>>> state_number_qobj([2, 2, 2], [1, 0, 1]) 
Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket
Qobj data =
[[ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 0.]
 [ 1.]
 [ 0.]
 [ 0.]]

Parameters:

dimslist or array: The quantum state dimensions array, as it would appear in a Qobj.
statelist: State number array.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

statequtip.Qobj: The state as a qutip.Qobj instance.

Note

Deprecated in QuTiP 5.0, use basis instead.

thermal_dm(N, n, method='operator', *, dtype=None)[source]

Density matrix for a thermal state of n particles

Parameters:

Nint: Number of basis states in Hilbert space.
nfloat: Expectation value for number of particles in thermal state.
methodstring {‘operator’, ‘analytic’}: string that sets the method used to generate the thermal state probabilities
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

dmqobj: Thermal state density matrix.

Notes

The ‘operator’ method (default) generates the thermal state using the truncated number operator num(N). This is the method that should be used in computations. The ‘analytic’ method uses the analytic coefficients derived in an infinite Hilbert space. The analytic form is not necessarily normalized, if truncated too aggressively.

Examples

>>> thermal_dm(5, 1) 
Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isHerm = True
Qobj data =
[[ 0.51612903  0.          0.          0.          0.        ]
 [ 0.          0.25806452  0.          0.          0.        ]
 [ 0.          0.          0.12903226  0.          0.        ]
 [ 0.          0.          0.          0.06451613  0.        ]
 [ 0.          0.          0.          0.          0.03225806]]

>>> thermal_dm(5, 1, 'analytic') 
Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isHerm = True
Qobj data =
[[ 0.5      0.       0.       0.       0.     ]
 [ 0.       0.25     0.       0.       0.     ]
 [ 0.       0.       0.125    0.       0.     ]
 [ 0.       0.       0.       0.0625   0.     ]
 [ 0.       0.       0.       0.       0.03125]]

triplet_states(*, dtype=None)[source]

Returns a list of the two particle triplet-states:

\[\lvert T_1\rangle = \lvert11\rangle \lvert T_2\rangle = \frac1{\sqrt2}(\lvert01\rangle + \lvert10\rangle) \lvert T_3\rangle = \lvert00\rangle\]

Parameters:

dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

trip_stateslist: 2 particle triplet states

w_state(N=3, *, dtype=None)[source]

Returns the N-qubit W-state:: [ |100..0> + |010..0> + |001..0> + ... |000..1> ] / sqrt(n)

Parameters:

Nint (default=3): Number of qubits in state
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

WQobj: N-qubit W-state

zero_ket(N, dims=None, *, dtype=None)[source]

Creates the zero ket vector with shape Nx1 and dimensions dims.

Parameters:

Nint: Hilbert space dimensionality
dimslist: Optional dimensions if ket corresponds to a composite Hilbert space.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

zero_ketqobj: Zero ket on given Hilbert space.

Quantum Operators

This module contains functions for generating Qobj representation of a variety of commonly occuring quantum operators.

charge(Nmax, Nmin=None, frac=1, *, dtype=None)[source]

Generate the diagonal charge operator over charge states from Nmin to Nmax.

Parameters:

Nmaxint: Maximum charge state to consider.
Nminint (default = -Nmax): Lowest charge state to consider.
fracfloat (default = 1): Specify fractional charge if needed.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

CQobj: Charge operator over [Nmin, Nmax].

Notes

New in version 3.2.

commutator(A, B, kind='normal')[source]: Return the commutator of kind kind (normal, anti) of the two operators A and B.

create(N, offset=0, *, dtype=None)[source]

Creation (raising) operator.

Parameters:

Nint: Dimension of Hilbert space.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Qobj for raising operator.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.

Examples

>>> create(4) 
Quantum object: dims=[[4], [4]], shape=(4, 4), type='oper', isherm=False
Qobj data =
[[ 0.00000000+0.j  0.00000000+0.j  0.00000000+0.j  0.00000000+0.j]
 [ 1.00000000+0.j  0.00000000+0.j  0.00000000+0.j  0.00000000+0.j]
 [ 0.00000000+0.j  1.41421356+0.j  0.00000000+0.j  0.00000000+0.j]
 [ 0.00000000+0.j  0.00000000+0.j  1.73205081+0.j  0.00000000+0.j]]

destroy(N, offset=0, *, dtype=None)[source]

Destruction (lowering) operator.

Parameters:

Nint: Dimension of Hilbert space.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Qobj for lowering operator.

Examples

>>> destroy(4) 
Quantum object: dims=[[4], [4]], shape=(4, 4), type='oper', isherm=False
Qobj data =
[[ 0.00000000+0.j  1.00000000+0.j  0.00000000+0.j  0.00000000+0.j]
 [ 0.00000000+0.j  0.00000000+0.j  1.41421356+0.j  0.00000000+0.j]
 [ 0.00000000+0.j  0.00000000+0.j  0.00000000+0.j  1.73205081+0.j]
 [ 0.00000000+0.j  0.00000000+0.j  0.00000000+0.j  0.00000000+0.j]]

displace(N, alpha, offset=0, *, dtype=None)[source]

Single-mode displacement operator.

Parameters:

Nint: Dimension of Hilbert space.
alphafloat/complex: Displacement amplitude.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Displacement operator.

Examples

>>> displace(4,0.25) 
Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 0.96923323+0.j -0.24230859+0.j  0.04282883+0.j -0.00626025+0.j]
 [ 0.24230859+0.j  0.90866411+0.j -0.33183303+0.j  0.07418172+0.j]
 [ 0.04282883+0.j  0.33183303+0.j  0.84809499+0.j -0.41083747+0.j]
 [ 0.00626025+0.j  0.07418172+0.j  0.41083747+0.j  0.90866411+0.j]]

enr_destroy(dims, excitations, *, dtype=None)[source]

Generate annilation operators for modes in a excitation-number-restricted state space. For example, consider a system consisting of 4 modes, each with 5 states. The total hilbert space size is 5**4 = 625. If we are only interested in states that contain up to 2 excitations, we only need to include states such as

(0, 0, 0, 0) (0, 0, 0, 1) (0, 0, 0, 2) (0, 0, 1, 0) (0, 0, 1, 1) (0, 0, 2, 0) …

This function creates annihilation operators for the 4 modes that act within this state space:

a1, a2, a3, a4 = enr_destroy([5, 5, 5, 5], excitations=2)

From this point onwards, the annihiltion operators a1, …, a4 can be used to setup a Hamiltonian, collapse operators and expectation-value operators, etc., following the usual pattern.

Parameters:

dimslist: A list of the dimensions of each subsystem of a composite quantum system.
excitationsinteger: The maximum number of excitations that are to be included in the state space.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

a_opslist of qobj: A list of annihilation operators for each mode in the composite quantum system described by dims.

enr_identity(dims, excitations, *, dtype=None)[source]

Generate the identity operator for the excitation-number restricted state space defined by the dims and exciations arguments. See the docstring for enr_fock for a more detailed description of these arguments.

Parameters:

dimslist: A list of the dimensions of each subsystem of a composite quantum system.
excitationsinteger: The maximum number of excitations that are to be included in the state space.
statelist of integers: The state in the number basis representation.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opQobj: A Qobj instance that represent the identity operator in the exication-number-restricted state space defined by dims and exciations.

fcreate(n_sites, site, dtype=None)[source]

Fermionic creation operator. We use the Jordan-Wigner transformation, making use of the Jordan-Wigner ZZ..Z strings, to construct this as follows:

\[a_j = \sigma_z^{\otimes j} \otimes (frac{sigma_x - i sigma_y}{2}) \otimes I^{\otimes N-j-1}\]

Parameters:

n_sitesint: Number of sites in Fock space.
siteint: The site in Fock space to add a fermion to. Corresponds to j in the above JW transform.

Returns:

operqobj: Qobj for raising operator.

Examples

>>> fcreate(2) 
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4),     type = oper, isherm = False
Qobj data =
[[0. 0. 0. 0.]
[0. 0. 0. 0.]
[1. 0. 0. 0.]
[0. 1. 0. 0.]]

fdestroy(n_sites, site, dtype=None)[source]

Fermionic destruction operator. We use the Jordan-Wigner transformation, making use of the Jordan-Wigner ZZ..Z strings, to construct this as follows:

\[a_j = \sigma_z^{\otimes j} \otimes (\frac{\sigma_x + i \sigma_y}{2}) \otimes I^{\otimes N-j-1}\]

Parameters:

n_sitesint: Number of sites in Fock space.
siteint (default 0): The site in Fock space to add a fermion to. Corresponds to j in the above JW transform.

Returns:

operqobj: Qobj for destruction operator.

Examples

>>> fdestroy(2) 
Quantum object: dims=[[2 2], [2 2]], shape=(4, 4),     type='oper', isherm=False
Qobj data =
[[0. 0. 1. 0.]
[0. 0. 0. 1.]
[0. 0. 0. 0.]
[0. 0. 0. 0.]]

identity(dimensions, *, dtype=None)

Identity operator.

Parameters:

dimensions(int) or (list of int) or (list of list of int): Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Identity operator Qobj.

Examples

>>> qeye(3) 
Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True
Qobj data =
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]
>>> qeye([2,2]) 
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
Qobj data =
[[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]

jmat(j, which=None, *, dtype=None)[source]

Higher-order spin operators:

Parameters:

jfloat: Spin of operator
whichstr: Which operator to return ‘x’,’y’,’z’,’+’,’-‘. If no args given, then output is [‘x’,’y’,’z’]
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

jmatQobj or tuple of Qobj: qobj for requested spin operator(s).

Notes

If no ‘args’ input, then returns array of [‘x’,’y’,’z’] operators.

Examples

>>> jmat(1) 
[ Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True
Qobj data =
[[ 0.          0.70710678  0.        ]
 [ 0.70710678  0.          0.70710678]
 [ 0.          0.70710678  0.        ]]
 Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j          0.-0.70710678j  0.+0.j        ]
 [ 0.+0.70710678j  0.+0.j          0.-0.70710678j]
 [ 0.+0.j          0.+0.70710678j  0.+0.j        ]]
 Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True
Qobj data =
[[ 1.  0.  0.]
 [ 0.  0.  0.]
 [ 0.  0. -1.]]]

momentum(N, offset=0, *, dtype=None)[source]

Momentum operator p=-1j/sqrt(2)*(a-a.dag())

Parameters:

Nint: Number of Fock states in Hilbert space.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Momentum operator as Qobj.

num(N, offset=0, *, dtype=None)[source]

Quantum object for number operator.

Parameters:

Nint: The dimension of the Hilbert space.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

oper: qobj: Qobj for number operator.

Examples

>>> num(4) 
Quantum object: dims=[[4], [4]], shape=(4, 4), type='oper', isherm=True
Qobj data =
[[0 0 0 0]
 [0 1 0 0]
 [0 0 2 0]
 [0 0 0 3]]

phase(N, phi0=0, *, dtype=None)[source]

Single-mode Pegg-Barnett phase operator.

Parameters:

Nint: Number of basis states in Hilbert space.
phi0float: Reference phase.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Phase operator with respect to reference phase.

Notes

The Pegg-Barnett phase operator is Hermitian on a truncated Hilbert space.

position(N, offset=0, *, dtype=None)[source]

Position operator x=1/sqrt(2)*(a+a.dag())

Parameters:

Nint: Number of Fock states in Hilbert space.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Position operator as Qobj.

qdiags(diagonals, offsets=None, dims=None, shape=None, *, dtype=None)[source]

Constructs an operator from an array of diagonals.

Parameters:

diagonalssequence of array_like

Array of elements to place along the selected diagonals.

offsetssequence of ints, optional

Sequence for diagonals to be set:

k=0 main diagonal
k>0 kth upper diagonal
k<0 kth lower diagonal

dimslist, optional

Dimensions for operator

shapelist, tuple, optional

Shape of operator. If omitted, a square operator large enough to contain the diagonals is generated.

dtypetype or str

Storage representation. Any data-layer known to qutip.data.to is accepted.

Examples

>>> qdiags(sqrt(range(1, 4)), 1) 
Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isherm = False
Qobj data =
[[ 0.          1.          0.          0.        ]
 [ 0.          0.          1.41421356  0.        ]
 [ 0.          0.          0.          1.73205081]
 [ 0.          0.          0.          0.        ]]

qeye(dimensions, *, dtype=None)[source]

Identity operator.

Parameters:

dimensions(int) or (list of int) or (list of list of int): Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Identity operator Qobj.

Examples

>>> qeye(3) 
Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True
Qobj data =
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]
>>> qeye([2,2]) 
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
Qobj data =
[[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]

qutrit_ops(*, dtype=None)[source]

Operators for a three level system (qutrit).

Parameters:

dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opers: array: array of qutrit operators.

qzero(dimensions, *, dtype=None)[source]

Zero operator.

Parameters:

dimensions(int) or (list of int) or (list of list of int): Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

qzeroqobj: Zero operator Qobj.

sigmam()[source]

Annihilation operator for Pauli spins.

Examples

>>> sigmam() 
Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False
Qobj data =
[[ 0.  0.]
 [ 1.  0.]]

sigmap()[source]

Creation operator for Pauli spins.

Examples

>>> sigmap() 
Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False
Qobj data =
[[ 0.  1.]
 [ 0.  0.]]

sigmax()[source]

Pauli spin 1/2 sigma-x operator

Examples

>>> sigmax() 
Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]

sigmay()[source]

Pauli spin 1/2 sigma-y operator.

Examples

>>> sigmay() 
Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j  0.-1.j]
 [ 0.+1.j  0.+0.j]]

sigmaz()[source]

Pauli spin 1/2 sigma-z operator.

Examples

>>> sigmaz() 
Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True
Qobj data =
[[ 1.  0.]
 [ 0. -1.]]

spin_Jm(j, *, dtype=None)[source]

Spin-j annihilation operator

Parameters:

jfloat: Spin of operator
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opQobj: qobj representation of the operator.

spin_Jp(j, *, dtype=None)[source]

Spin-j creation operator

Parameters:

jfloat: Spin of operator
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opQobj: qobj representation of the operator.

spin_Jx(j, *, dtype=None)[source]

Spin-j x operator

Parameters:

jfloat: Spin of operator
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opQobj: qobj representation of the operator.

spin_Jy(j, *, dtype=None)[source]

Spin-j y operator

Parameters:

jfloat: Spin of operator
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opQobj: qobj representation of the operator.

spin_Jz(j, *, dtype=None)[source]

Spin-j z operator

Parameters:

jfloat: Spin of operator
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

opQobj: qobj representation of the operator.

squeeze(N, z, offset=0, *, dtype=None)[source]

Single-mode squeezing operator.

Parameters:

Nint: Dimension of hilbert space.
zfloat/complex: Squeezing parameter.
offsetint (default 0): The lowest number state that is included in the finite number state representation of the operator.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqutip.Qobj: Squeezing operator.

Examples

>>> squeeze(4, 0.25) 
Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 0.98441565+0.j  0.00000000+0.j  0.17585742+0.j  0.00000000+0.j]
 [ 0.00000000+0.j  0.95349007+0.j  0.00000000+0.j  0.30142443+0.j]
 [-0.17585742+0.j  0.00000000+0.j  0.98441565+0.j  0.00000000+0.j]
 [ 0.00000000+0.j -0.30142443+0.j  0.00000000+0.j  0.95349007+0.j]]

squeezing(a1, a2, z)[source]

Generalized squeezing operator.

\[S(z) = \exp\left(\frac{1}{2}\left(z^*a_1a_2 - za_1^\dagger a_2^\dagger\right)\right)\]

Parameters:

a1qutip.Qobj: Operator 1.
a2qutip.Qobj: Operator 2.
zfloat/complex: Squeezing parameter.

Returns:

operqutip.Qobj: Squeezing operator.

tunneling(N, m=1, *, dtype=None)[source]

Tunneling operator with elements of the form $\\sum |N><N+m| + |N+m><N|$.

Parameters:

Nint: Number of basis states in Hilbert space.
mint (default = 1): Number of excitations in tunneling event.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

TQobj: Tunneling operator.

Notes

New in version 3.2.

Quantum Objects

The Quantum Object (Qobj) class, for representing quantum states and operators, and related functions.

ptrace(Q, sel)[source]

Partial trace of the Qobj with selected components remaining.

Parameters:

Qqutip.Qobj: Composite quantum object.
selint/list: An int or list of components to keep after partial trace.

Returns:

operqutip.Qobj: Quantum object representing partial trace with selected components remaining.

Notes

This function is for legacy compatibility only. It is recommended to use the ptrace() Qobj method.

Random Operators and States

This module is a collection of random state and operator generators.

rand_dm(dimensions, density=0.75, distribution='ginibre', *, eigenvalues=(), rank=None, seed=None, dtype=None)[source]

Creates a random density matrix of the desired dimensions.

Parameters:

dimensions(int) or (list of int) or (list of list of int)

Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.

densityfloat

Density between [0,1] of output density matrix. Used by the “pure”, “eigen” and “herm”.

distributionstr {“ginibre”, “hs”, “pure”, “eigen”, “uniform”}

Method used to obtain the density matrices.

“ginibre” : Ginibre random density operator of rank rank by using the algorithm of [BCSZ08].
“hs” : Hilbert-Schmidt ensemble, equivalent to a full rank ginibre operator.
“pure” : Density matrix created from a random ket.
“eigen” : A density matrix with the given eigenvalues.
“herm” : Build from a random hermitian matrix using rand_herm.

eigenvaluesarray_like, optional

Eigenvalues of the output Hermitian matrix. The len must match the shape of the matrix.

rankint, optional

When using the “ginibre” distribution, rank of the density matrix. Will default to a full rank operator when not provided.

seedint, SeedSequence, Generator, optional

Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.

dtypetype or str

Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Density matrix quantum operator.

rand_herm(dimensions, density=0.3, distribution='fill', *, eigenvalues=(), seed=None, dtype=None)[source]

Creates a random sparse Hermitian quantum object.

Parameters:

dimensions(int) or (list of int) or (list of list of int)

Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.

densityfloat, [0.30]

Density between [0,1] of output Hermitian operator.

distributionstr {“fill”, “pos_def”, “eigen”}

Method used to obtain the density matrices.

“fill” : Uses $H=0.5*(X+X^{+})$ where $X$ is a randomly generated quantum operator with elements uniformly distributed between [-1, 1] + [-1j, 1j].
“eigen” : A density matrix with the given eigenvalues. It uses random complex Jacobi rotations to shuffle the operator.
“pos_def” : Return a positive semi-definite matrix by diagonal dominance.

eigenvaluesarray_like, optional

Eigenvalues of the output Hermitian matrix. The len must match the shape of the matrix.

seedint, SeedSequence, Generator, optional

Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.

dtypetype or str

Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Hermitian quantum operator.

Notes

If given a list of eigenvalues the object is created using complex Jacobi rotations. While this method is fast for small matrices, it should not be repeatedly used for generating matrices larger than ~1000x1000.

rand_ket(dimensions, density=1, distribution='haar', *, seed=None, dtype=None)[source]

Creates a random ket vector.

Parameters:

dimensions(int) or (list of int) or (list of list of int)

Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.

densityfloat, [1]

Density between [0,1] of output ket state when using the fill method.

distributionstr {“haar”, “fill”}

Method used to obtain the kets.

haar : Haar random pure state obtained by applying a Haar random unitary to a fixed pure state.
fill : Fill the ket with uniformly distributed random complex number.

seedint, SeedSequence, Generator, optional

Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.

dtypetype or str

Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Ket quantum state vector.

rand_stochastic(dimensions, density=0.75, kind='left', *, seed=None, dtype=None)[source]

Generates a random stochastic matrix.

Parameters:

dimensions(int) or (list of int) or (list of list of int): Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
densityfloat, [0.75]: Density between [0,1] of output density matrix.
kindstr (Default = ‘left’): Generate ‘left’ or ‘right’ stochastic matrix.
seedint, SeedSequence, Generator, optional: Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Quantum operator form of stochastic matrix.

rand_super(dimensions, *, superrep='super', seed=None, dtype=None)[source]

Returns a randomly drawn superoperator acting on operators acting on N dimensions.

Parameters:

dimensions(int) or (list of int) or (list of list of int): Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
superropstr, optional, {“super”}: Representation of the super operator
seedint, SeedSequence, Generator, optional: Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

rand_super_bcsz(dimensions, enforce_tp=True, rank=None, *, superrep='super', seed=None, dtype=None)[source]

Returns a random superoperator drawn from the Bruzda et al ensemble for CPTP maps [BCSZ08]. Note that due to finite numerical precision, for ranks less than full-rank, zero eigenvalues may become slightly negative, such that the returned operator is not actually completely positive.

Parameters:

dimensions(int) or (list of int) or (list of list of int): Dimension of Hilbert space. If an int is provided, it is understood as the Square root of the dimension of the superoperator to be returned, with the corresponding dims as [[[N],[N]], [[N],[N]]]. If provided as a list of ints, then the dimensions is understood as the space of density matrices this superoperator is applied to: dimensions=[2,2] dims=[[[2,2],[2,2]], [[2,2],[2,2]]].
enforce_tpbool: If True, the trace-preserving condition of [BCSZ08] is enforced; otherwise only complete positivity is enforced.
rankint or None: Rank of the sampled superoperator. If None, a full-rank superoperator is generated.
seedint, SeedSequence, Generator, optional: Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
superropstr, optional, {“super”}: representation of the
dtypetype or str: Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

rhoQobj: A superoperator acting on vectorized dim × dim density operators, sampled from the BCSZ distribution.

rand_unitary(dimensions, density=1, distribution='haar', *, seed=None, dtype=None)[source]

Creates a random sparse unitary quantum object.

Parameters:

dimensions(int) or (list of int) or (list of list of int)

Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the dims property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.

densityfloat, [1]

Density between [0,1] of output unitary operator.

distribution[“haar”, “exp”]

Method used to obtain the unitary matrices.

haar : Haar random unitary matrix using the algorithm of [Mez07].
exp : Uses $\exp(-iH)$, where H is a randomly generated Hermitian operator.

seedint, SeedSequence, Generator, optional

Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.

dtypetype or str

Storage representation. Any data-layer known to qutip.data.to is accepted.

Returns:

operqobj: Unitary quantum operator.

Superoperators and Liouvillians

lindblad_dissipator(a, b=None, data_only=False, chi=None)[source]

Lindblad dissipator (generalized) for a single pair of collapse operators (a, b), or for a single collapse operator (a) when b is not specified:

\[\mathcal{D}[a,b]\rho = a \rho b^\dagger - \frac{1}{2}a^\dagger b\rho - \frac{1}{2}\rho a^\dagger b\]

Parameters:

aQobj or QobjEvo: Left part of collapse operator.
bQobj or QobjEvo (optional): Right part of collapse operator. If not specified, b defaults to a.
chifloat [None]: In some systems it is possible to determine the statistical moments (mean, variance, etc) of the probability distribution of the occupation numbers of states by numerically evaluating the derivatives of the steady state occupation probability as a function of an artificial phase parameter chi which multiplies the a \rho a^dagger term of the dissipator by e ^ (i * chi). The factor e ^ (i * chi) is introduced via the generating function of the statistical moments. For examples of the technique, see Full counting statistics of nano-electromechanical systems and Photon-mediated electron transport in hybrid circuit-QED. This parameter is deprecated and may be removed in QuTiP 5.
data_onlybool [False]: Return the data object instead of a Qobj

Returns:

Dqobj, QobjEvo: Lindblad dissipator superoperator.

liouvillian(H=None, c_ops=None, data_only=False, chi=None)[source]

Assembles the Liouvillian superoperator from a Hamiltonian and a list of collapse operators.

Parameters:

HQobj or QobjEvo (optional): System Hamiltonian or Hamiltonian component of a Liouvillian. Considered 0 if not given.
c_opsarray_like of Qobj or QobjEvo: A list or array of collapse operators.
data_onlybool [False]: Return the data object instead of a Qobj
chiarray_like of float [None]: In some systems it is possible to determine the statistical moments (mean, variance, etc) of the probability distributions of occupation of various states by numerically evaluating the derivatives of the steady state occupation probability as a function of artificial phase parameters chi which are included in the lindblad_dissipator for each collapse operator. See the documentation of lindblad_dissipator for references and further details. This parameter is deprecated and may be removed in QuTiP 5.

Returns:

LQobj or QobjEvo: Liouvillian superoperator.

operator_to_vector(op)[source]

Create a vector representation given a quantum operator in matrix form. The passed object should have a Qobj.type of ‘oper’ or ‘super’; this function is not designed for general-purpose matrix reshaping.

Parameters:

opQobj or QobjEvo: Quantum operator in matrix form. This must have a type of ‘oper’ or ‘super’.

Returns:

Qobj or QobjEvo: The same object, but re-cast into a column-stacked-vector form of type ‘operator-ket’. The output is the same type as the passed object.

spost(A)[source]

Superoperator formed from post-multiplication by operator A

Parameters:

AQobj or QobjEvo: Quantum operator for post multiplication.

Returns:

superQobj or QobjEvo: Superoperator formed from input qauntum object.

spre(A)[source]

Superoperator formed from pre-multiplication by operator A.

Parameters:

AQobj or QobjEvo: Quantum operator for pre-multiplication.

Returns:

super :Qobj or QobjEvo: Superoperator formed from input quantum object.

sprepost(A, B)[source]

Superoperator formed from pre-multiplication by A and post-multiplication by B.

Parameters:

AQobj or QobjEvo: Quantum operator for pre-multiplication.
BQobj or QobjEvo: Quantum operator for post-multiplication.

Returns:

superQobj or QobjEvo: Superoperator formed from input quantum objects.

vector_to_operator(op)[source]

Create a matrix representation given a quantum operator in vector form. The passed object should have a Qobj.type of ‘operator-ket’; this function is not designed for general-purpose matrix reshaping.

Parameters:

opQobj or QobjEvo: Quantum operator in column-stacked-vector form. This must have a type of ‘operator-ket’.

Returns:

Qobj or QobjEvo: The same object, but re-cast into “standard” operator form. The output is the same type as the passed object.

Superoperator Representations

This module implements transformations between superoperator representations, including supermatrix, Kraus, Choi and Chi (process) matrix formalisms.

kraus_to_choi(kraus_list)[source]: Take a list of Kraus operators and returns the Choi matrix for the channel represented by the Kraus operators in kraus_list

kraus_to_super(kraus_list)[source]: Convert a list of Kraus operators to a superoperator.

to_chi(q_oper)[source]

Converts a Qobj representing a quantum map to a representation as a chi (process) matrix in the Pauli basis, such that the trace of the returned operator is equal to the dimension of the system.

Parameters:

q_operQobj: Superoperator to be converted to Chi representation. If q_oper is type="oper", then it is taken to act by conjugation, such that to_chi(A) == to_chi(sprepost(A, A.dag())).

Returns:

chiQobj: A quantum object representing the same map as q_oper, such that chi.superrep == "chi".

Raises:

TypeError: if the given quantum object is not a map, or cannot be converted: to Chi representation.

to_choi(q_oper)[source]

Converts a Qobj representing a quantum map to the Choi representation, such that the trace of the returned operator is equal to the dimension of the system.

Parameters:

q_operQobj: Superoperator to be converted to Choi representation. If q_oper is type="oper", then it is taken to act by conjugation, such that to_choi(A) == to_choi(sprepost(A, A.dag())).

Returns:

choiQobj: A quantum object representing the same map as q_oper, such that choi.superrep == "choi".

Raises:

TypeError: if the given quantum object is not a map, or cannot be converted: to Choi representation.

to_kraus(q_oper, tol=1e-09)[source]

Converts a Qobj representing a quantum map to a list of quantum objects, each representing an operator in the Kraus decomposition of the given map.

Parameters:

q_operQobj: Superoperator to be converted to Kraus representation. If q_oper is type="oper", then it is taken to act by conjugation, such that to_kraus(A) == to_kraus(sprepost(A, A.dag())) == [A].
tolFloat: Optional threshold parameter for eigenvalues/Kraus ops to be discarded. The default is to=1e-9.

Returns:

kraus_opslist of Qobj: A list of quantum objects, each representing a Kraus operator in the decomposition of q_oper.

Raises:

TypeError: if the given quantum object is not a map, or cannot be: decomposed into Kraus operators.

to_stinespring(q_oper, threshold=1e-10)[source]

Converts a Qobj representing a quantum map $Lambda$ to a pair of partial isometries $A$ and $B$ such that $Lambda(X) = Tr_2(A X B^dagger)$ for all inputs $X$, where the partial trace is taken over a a new index on the output dimensions of $A$ and $B$.

For completely positive inputs, $A$ will always equal $B$ up to precision errors.

Parameters:

q_operQobj: Superoperator to be converted to a Stinespring pair.

Returns:

A, BQobj: Quantum objects representing each of the Stinespring matrices for the input Qobj.

to_super(q_oper)[source]

Converts a Qobj representing a quantum map to the supermatrix (Liouville) representation.

Parameters:

q_operQobj: Superoperator to be converted to supermatrix representation. If q_oper is type="oper", then it is taken to act by conjugation, such that to_super(A) == sprepost(A, A.dag()).

Returns:

superopQobj: A quantum object representing the same map as q_oper, such that superop.superrep == "super".

Raises:

TypeError: If the given quantum object is not a map, or cannot be converted to supermatrix representation.

Operators and Superoperator Dimensions

Internal use module for manipulating dims specifications.

collapse_dims_oper(dims)[source]

Given the dimensions specifications for a ket-, bra- or oper-type Qobj, returns a dimensions specification describing the same shape by collapsing all composite systems. For instance, the bra-type dimensions specification [[2, 3], [1]] collapses to [[6], [1]].

Parameters:

dimslist of lists of ints: Dimensions specifications to be collapsed.

Returns:

collapsed_dimslist of lists of ints: Collapsed dimensions specification describing the same shape such that len(collapsed_dims[0]) == len(collapsed_dims[1]) == 1.

collapse_dims_super(dims)[source]

Given the dimensions specifications for an operator-ket-, operator-bra- or super-type Qobj, returns a dimensions specification describing the same shape by collapsing all composite systems. For instance, the super-type dimensions specification [[[2, 3], [2, 3]], [[2, 3], [2, 3]]] collapses to [[[6], [6]], [[6], [6]]].

Parameters:

dimslist of lists of ints: Dimensions specifications to be collapsed.

Returns:

collapsed_dimslist of lists of ints: Collapsed dimensions specification describing the same shape such that len(collapsed_dims[i][j]) == 1 for i and j in range(2).

deep_remove(l, *what)[source]

Removes scalars from all levels of a nested list.

Given a list containing a mix of scalars and lists, returns a list of the same structure, but where one or more scalars have been removed.

Examples

>>> deep_remove([[[[0, 1, 2]], [3, 4], [5], [6, 7]]], 0, 5) 
[[[[1, 2]], [3, 4], [], [6, 7]]]

dims_idxs_to_tensor_idxs(dims, indices)[source]

Given the dims of a Qobj instance, and some indices into dims, returns the corresponding tensor indices. This helps resolve, for instance, that column-stacking for superoperators, oper-ket and oper-bra implies that the input and output tensor indices are reversed from their order in dims.

Parameters:

dimslist: Dimensions specification for a Qobj.
indicesint, list or tuple: Indices to convert to tensor indices. Can be specified as a single index, or as a collection of indices. In the latter case, this can be nested arbitrarily deep. For instance, [0, [0, (2, 3)]].

Returns:

tens_indicesint, list or tuple: Container of the same structure as indices containing the tensor indices for each element of indices.

dims_to_tensor_perm(dims)[source]

Given the dims of a Qobj instance, returns a list representing a permutation from the flattening of that dims specification to the corresponding tensor indices.

Parameters:

dimslist: Dimensions specification for a Qobj.

Returns:

permlist: A list such that data[flatten(dims)[idx]] gives the index of the tensor data corresponding to the idx``th dimension of ``dims.

dims_to_tensor_shape(dims)[source]

Given the dims of a Qobj instance, returns the shape of the corresponding tensor. This helps, for instance, resolve the column-stacking convention for superoperators.

Parameters:

dimslist: Dimensions specification for a Qobj.

Returns:

tensor_shapetuple: NumPy shape of the corresponding tensor.

enumerate_flat(l)[source]

Labels the indices at which scalars occur in a flattened list.

Given a list containing a mix of scalars and lists, returns a list of the same structure, where each scalar has been replaced by an index into the flattened list.

Examples

>>> print(enumerate_flat([[[10], [20, 30]], 40])) 
[[[0], [1, 2]], 3]

flatten(l)[source]

Flattens a list of lists to the first level.

Given a list containing a mix of scalars and lists, flattens down to a list of the scalars within the original list.

Examples

>>> flatten([[[0], 1], 2]) 
[0, 1, 2]

is_scalar(dims)[source]: Returns True if a dims specification is effectively a scalar (has dimension 1).

unflatten(l, idxs)[source]

Unflattens a list by a given structure.

Given a list of scalars and a deep list of indices as produced by flatten, returns an “unflattened” form of the list. This perfectly inverts flatten.

Examples

>>> l = [[[10, 20, 30], [40, 50, 60]], [[70, 80, 90], [100, 110, 120]]] 
>>> idxs = enumerate_flat(l) 
>>> unflatten(flatten(l), idxs) == l 
True

Functions acting on states and operators

Expectation Values

expect(oper, state)[source]

Calculate the expectation value for operator(s) and state(s). The expectation of state k on operator A is defined as k.dag() @ A @ k, and for density matrix R on operator A it is trace(A @ R).

Parameters:

operqobj/array-like: A single or a list or operators for expectation value.
stateqobj/array-like: A single or a list of quantum states or density matrices.

Returns:

exptfloat/complex/array-like: Expectation value. real if oper is Hermitian, complex otherwise. A (nested) array of expectaction values of state or operator are arrays.

Examples

>>> expect(num(4), basis(4, 3)) == 3 
    True

variance(oper, state)[source]

Variance of an operator for the given state vector or density matrix.

Parameters:

operqobj: Operator for expectation value.
stateqobj/list: A single or list of quantum states or density matrices..

Returns:

varfloat: Variance of operator ‘oper’ for given state.

Tensor

Module for the creation of composite quantum objects via the tensor product.

composite(*args)[source]

Given two or more operators, kets or bras, returns the Qobj corresponding to a composite system over each argument. For ordinary operators and vectors, this is the tensor product, while for superoperators and vectorized operators, this is the column-reshuffled tensor product.

If a mix of Qobjs supported on Hilbert and Liouville spaces are passed in, the former are promoted. Ordinary operators are assumed to be unitaries, and are promoted using to_super, while kets and bras are promoted by taking their projectors and using operator_to_vector(ket2dm(arg)).

super_tensor(*args)[source]

Calculate the tensor product of input superoperators, by tensoring together the underlying Hilbert spaces on which each vectorized operator acts.

Parameters:

argsarray_like: list or array of quantum objects with type="super".

Returns:

objqobj: A composite quantum object.

tensor(*args)[source]

Calculates the tensor product of input operators.

Parameters:

argsarray_like: list or array of quantum objects for tensor product.

Returns:

objqobj: A composite quantum object.

Examples

>>> tensor([sigmax(), sigmax()]) 
Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j  0.+0.j  0.+0.j  1.+0.j]
 [ 0.+0.j  0.+0.j  1.+0.j  0.+0.j]
 [ 0.+0.j  1.+0.j  0.+0.j  0.+0.j]
 [ 1.+0.j  0.+0.j  0.+0.j  0.+0.j]]

tensor_contract(qobj, *pairs)[source]

Contracts a qobj along one or more index pairs. Note that this uses dense representations and thus should not be used for very large Qobjs.

Parameters:

pairstuple: One or more tuples (i, j) indicating that the i and j dimensions of the original qobj should be contracted.

Returns:

cqobjQobj: The original Qobj with all named index pairs contracted away.

Partial Transpose

partial_transpose(rho, mask, method='dense')[source]

Return the partial transpose of a Qobj instance rho, where mask is an array/list with length that equals the number of components of rho (that is, the length of rho.dims[0]), and the values in mask indicates whether or not the corresponding subsystem is to be transposed. The elements in mask can be boolean or integers 0 or 1, where True/1 indicates that the corresponding subsystem should be tranposed.

Parameters:

rhoqutip.qobj: A density matrix.
masklist / array: A mask that selects which subsystems should be transposed.
methodstr: choice of method, dense or sparse. The default method is dense. The sparse implementation can be faster for large and sparse systems (hundreds of quantum states).

Returns:

rho_pr: qutip.qobj: A density matrix with the selected subsystems transposed.

Entropy Functions

concurrence(rho)[source]

Calculate the concurrence entanglement measure for a two-qubit state.

Parameters:

stateqobj: Ket, bra, or density matrix for a two-qubit state.

Returns:

concurfloat: Concurrence

References

[1]

https://en.wikipedia.org/wiki/Concurrence_(quantum_computing)

entropy_conditional(rho, selB, base=2.718281828459045, sparse=False)[source]

Calculates the conditional entropy $S(A|B)=S(A,B)-S(B)$ of a selected density matrix component.

Parameters:

rhoqobj: Density matrix of composite object
selBint/list: Selected components for density matrix B
base{e,2}: Base of logarithm.
sparse{False,True}: Use sparse eigensolver.

Returns:

ent_condfloat: Value of conditional entropy

entropy_linear(rho)[source]

Linear entropy of a density matrix.

Parameters:

rhoqobj: sensity matrix or ket/bra vector.

Returns:

entropyfloat: Linear entropy of rho.

Examples

>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_linear(rho)
0.5

entropy_mutual(rho, selA, selB, base=2.718281828459045, sparse=False)[source]

Calculates the mutual information S(A:B) between selection components of a system density matrix.

Parameters:

rhoqobj: Density matrix for composite quantum systems
selAint/list: int or list of first selected density matrix components.
selBint/list: int or list of second selected density matrix components.
base{e,2}: Base of logarithm.
sparse{False,True}: Use sparse eigensolver.

Returns:

ent_mutfloat: Mutual information between selected components.

entropy_relative(rho, sigma, base=2.718281828459045, sparse=False, tol=1e-12)[source]

Calculates the relative entropy S(rho||sigma) between two density matrices.

Parameters:

rhoqutip.Qobj: First density matrix (or ket which will be converted to a density matrix).
sigmaqutip.Qobj: Second density matrix (or ket which will be converted to a density matrix).
base{e,2}: Base of logarithm. Defaults to e.
sparsebool: Flag to use sparse solver when determining the eigenvectors of the density matrices. Defaults to False.
tolfloat: Tolerance to use to detect 0 eigenvalues or dot producted between eigenvectors. Defaults to 1e-12.

Returns:

rel_entfloat: Value of relative entropy. Guaranteed to be greater than zero and should equal zero only when rho and sigma are identical.

References

See Nielsen & Chuang, “Quantum Computation and Quantum Information”, Section 11.3.1, pg. 511 for a detailed explanation of quantum relative entropy.

Examples

First we define two density matrices:

>>> rho = qutip.ket2dm(qutip.ket("00"))
>>> sigma = rho + qutip.ket2dm(qutip.ket("01"))
>>> sigma = sigma.unit()

Then we calculate their relative entropy using base 2 (i.e. log2) and base e (i.e. log).

>>> qutip.entropy_relative(rho, sigma, base=2)
1.0
>>> qutip.entropy_relative(rho, sigma)
0.6931471805599453

entropy_vn(rho, base=2.718281828459045, sparse=False)[source]

Von-Neumann entropy of density matrix

Parameters:

rhoqobj: Density matrix.
base{e,2}: Base of logarithm.
sparse{False,True}: Use sparse eigensolver.

Returns:

entropyfloat: Von-Neumann entropy of rho.

Examples

>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_vn(rho,2)
1.0

Density Matrix Metrics

This module contains a collection of functions for calculating metrics (distance measures) between states and operators.

average_gate_fidelity(oper, target=None)[source]

Returns the average gate fidelity of a quantum channel to the target channel, or to the identity channel if no target is given.

Parameters:

operqutip.Qobj/list: A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators
targetqutip.Qobj: A unitary operator

Returns:

fidfloat: Average gate fidelity between oper and target, or between oper and identity.

Notes

The average gate fidelity is defined for example in: A. Gilchrist, N.K. Langford, M.A. Nielsen, Phys. Rev. A 71, 062310 (2005). The definition of state fidelity that the average gate fidelity is based on is the one from R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994). It is the square of the fidelity implemented in qutip.core.metrics.fidelity which follows Nielsen & Chuang, “Quantum Computation and Quantum Information”

bures_angle(A, B)[source]

Returns the Bures Angle between two density matrices A & B.

The Bures angle ranges from 0, for states with unit fidelity, to pi/2.

Parameters:

Aqobj: Density matrix or state vector.
Bqobj: Density matrix or state vector with same dimensions as A.

Returns:

anglefloat: Bures angle between density matrices.

bures_dist(A, B)[source]

Returns the Bures distance between two density matrices A & B.

The Bures distance ranges from 0, for states with unit fidelity, to sqrt(2).

Parameters:

Aqobj: Density matrix or state vector.
Bqobj: Density matrix or state vector with same dimensions as A.

Returns:

distfloat: Bures distance between density matrices.

dnorm(A, B=None, solver='CVXOPT', verbose=False, force_solve=False, sparse=True)[source]

Calculates the diamond norm of the quantum map q_oper, using the simplified semidefinite program of [Wat13].

The diamond norm SDP is solved by using CVXPY.

Parameters:

AQobj: Quantum map to take the diamond norm of.
BQobj or None: If provided, the diamond norm of $A - B$ is taken instead.
solverstr: Solver to use with CVXPY. One of “CVXOPT” (default) or “SCS”. The latter tends to be significantly faster, but somewhat less accurate.
verbosebool: If True, prints additional information about the solution.
force_solvebool: If True, forces dnorm to solve the associated SDP, even if a special case is known for the argument.
sparsebool: Whether to use sparse matrices in the convex optimisation problem. Default True.

Returns:

dnfloat: Diamond norm of q_oper.

Raises:

ImportError: If CVXPY cannot be imported.

fidelity(A, B)[source]

Calculates the fidelity (pseudo-metric) between two density matrices. See: Nielsen & Chuang, “Quantum Computation and Quantum Information”

Parameters:

Aqobj: Density matrix or state vector.
Bqobj: Density matrix or state vector with same dimensions as A.

Returns:

fidfloat: Fidelity pseudo-metric between A and B.

Examples

>>> x = fock_dm(5,3)
>>> y = coherent_dm(5,1)
>>> np.testing.assert_almost_equal(fidelity(x,y), 0.24104350624628332)

hellinger_dist(A, B, sparse=False, tol=0)[source]

Calculates the quantum Hellinger distance between two density matrices.

Formula: hellinger_dist(A, B) = sqrt(2-2*Tr(sqrt(A)*sqrt(B)))

See: D. Spehner, F. Illuminati, M. Orszag, and W. Roga, “Geometric measures of quantum correlations with Bures and Hellinger distances” arXiv:1611.03449

Parameters:

Aqutip.Qobj: Density matrix or state vector.
Bqutip.Qobj: Density matrix or state vector with same dimensions as A.
tolfloat: Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse{False, True}: Use sparse eigensolver.

Returns:

hellinger_distfloat: Quantum Hellinger distance between A and B. Ranges from 0 to sqrt(2).

Examples

>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> np.testing.assert_almost_equal(hellinger_dist(x,y), 1.3725145002591095)

hilbert_dist(A, B)[source]

Returns the Hilbert-Schmidt distance between two density matrices A & B.

Parameters:

Aqobj: Density matrix or state vector.
Bqobj: Density matrix or state vector with same dimensions as A.

Returns:

distfloat: Hilbert-Schmidt distance between density matrices.

Notes

See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).

process_fidelity(oper, target=None)[source]

Returns the process fidelity of a quantum channel to the target channel, or to the identity channel if no target is given. The process fidelity between two channels is defined as the state fidelity between their normalized Choi matrices.

Parameters:

operqutip.Qobj/list: A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators
targetqutip.Qobj/list: A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators

Returns:

fidfloat: Process fidelity between oper and target, or between oper and identity.

Notes

Since Qutip 5.0, this function computes the process fidelity as defined for example in: A. Gilchrist, N.K. Langford, M.A. Nielsen, Phys. Rev. A 71, 062310 (2005). Previously, it computed a function that is now implemented in control.fidcomp.FidCompUnitary.get_fidelity. The definition of state fidelity that the process fidelity is based on is the one from R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994). It is the square of the one implemented in qutip.core.metrics.fidelity which follows Nielsen & Chuang, “Quantum Computation and Quantum Information”

tracedist(A, B, sparse=False, tol=0)[source]

Calculates the trace distance between two density matrices.. See: Nielsen & Chuang, “Quantum Computation and Quantum Information”

Parameters:

Aqobj: Density matrix or state vector.
Bqobj: Density matrix or state vector with same dimensions as A.
tolfloat: Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse{False, True}: Use sparse eigensolver.

Returns:

tracedistfloat: Trace distance between A and B.

Examples

>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> np.testing.assert_almost_equal(tracedist(x,y), 0.9705143161472971)

unitarity(oper)[source]

Returns the unitarity of a quantum map, defined as the Frobenius norm of the unital block of that map’s superoperator representation.

Parameters:

operQobj: Quantum map under consideration.

Returns:

ufloat: Unitarity of oper.

Continuous Variables

This module contains a collection functions for calculating continuous variable quantities from fock-basis representation of the state of multi-mode fields.

correlation_matrix(basis, rho=None)[source]

Given a basis set of operators $\{a\}_n$, calculate the correlation matrix:

\[C_{mn} = \langle a_m a_n \rangle\]

Parameters:

basislist: List of operators that defines the basis for the correlation matrix.
rhoQobj: Density matrix for which to calculate the correlation matrix. If rho is None, then a matrix of correlation matrix operators is returned instead of expectation values of those operators.

Returns:

corr_matndarray: A 2-dimensional array of correlation values or operators.

correlation_matrix_field(a1, a2, rho=None)[source]

Calculates the correlation matrix for given field operators $a_1$ and $a_2$. If a density matrix is given the expectation values are calculated, otherwise a matrix with operators is returned.

Parameters:

a1Qobj: Field operator for mode 1.
a2Qobj: Field operator for mode 2.
rhoQobj: Density matrix for which to calculate the covariance matrix.

Returns:

cov_matndarray: Array of complex numbers or Qobj’s A 2-dimensional array of covariance values, or, if rho=0, a matrix of operators.

correlation_matrix_quadrature(a1, a2, rho=None, g=1.4142135623730951)[source]

Calculate the quadrature correlation matrix with given field operators $a_1$ and $a_2$. If a density matrix is given the expectation values are calculated, otherwise a matrix with operators is returned.

Parameters:

a1Qobj: Field operator for mode 1.
a2Qobj: Field operator for mode 2.
rhoQobj: Density matrix for which to calculate the covariance matrix.
gfloat: Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g ** 2 giving the default value hbar=1.

Returns:

corr_matndarray: Array of complex numbers or Qobj’s A 2-dimensional array of covariance values for the field quadratures, or, if rho=0, a matrix of operators.

covariance_matrix(basis, rho, symmetrized=True)[source]

Given a basis set of operators $\{a\}_n$, calculate the covariance matrix:

\[V_{mn} = \frac{1}{2}\langle a_m a_n + a_n a_m \rangle - \langle a_m \rangle \langle a_n\rangle\]

or, if of the optional argument symmetrized=False,

\[V_{mn} = \langle a_m a_n\rangle - \langle a_m \rangle \langle a_n\rangle\]

Parameters:

basislist: List of operators that defines the basis for the covariance matrix.
rhoQobj: Density matrix for which to calculate the covariance matrix.
symmetrizedbool {True, False}: Flag indicating whether the symmetrized (default) or non-symmetrized correlation matrix is to be calculated.

Returns:

corr_matndarray: A 2-dimensional array of covariance values.

logarithmic_negativity(V, g=1.4142135623730951)[source]

Calculates the logarithmic negativity given a symmetrized covariance matrix, see qutip.continuous_variables.covariance_matrix. Note that the two-mode field state that is described by V must be Gaussian for this function to applicable.

Parameters:

V2d array: The covariance matrix.
gfloat: Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g ** 2 giving the default value hbar=1.

Returns:

Nfloat: The logarithmic negativity for the two-mode Gaussian state that is described by the the Wigner covariance matrix V.

wigner_covariance_matrix(a1=None, a2=None, R=None, rho=None, g=1.4142135623730951)[source]

Calculates the Wigner covariance matrix $V_{ij} = \frac{1}{2}(R_{ij} + R_{ji})$, given the quadrature correlation matrix $R_{ij} = \langle R_{i} R_{j}\rangle - \langle R_{i}\rangle \langle R_{j}\rangle$, where $R = (q_1, p_1, q_2, p_2)^T$ is the vector with quadrature operators for the two modes.

Alternatively, if R = None, and if annihilation operators a1 and a2 for the two modes are supplied instead, the quadrature correlation matrix is constructed from the annihilation operators before then the covariance matrix is calculated.

Parameters:

a1Qobj: Field operator for mode 1.
a2Qobj: Field operator for mode 2.
Rndarray: The quadrature correlation matrix.
rhoQobj: Density matrix for which to calculate the covariance matrix.
gfloat: Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g ** 2 giving the default value hbar=1.

Returns:

cov_matndarray: A 2-dimensional array of covariance values.

Measurement

Measurement of quantum states

Module for measuring quantum objects.

measure(state, ops)[source]

A dispatch method that provides measurement results handling both observable style measurements and projector style measurements (POVMs and PVMs).

For return signatures, please check:

measure_observable for observable measurements.
measure_povm for POVM measurements.

Parameters:

stateQobj

The ket or density matrix specifying the state to measure.

opsQobj or list of Qobj

measurement observable (Qobj); or
list of measurement operators $M_i$ or kets (list of Qobj) Either:
1. specifying a POVM s.t. $E_i = M_i^\dagger M_i$
2. projection operators if ops correspond to projectors (s.t. $E_i = M_i^\dagger = M_i$)
3. kets (transformed to projectors)

measure_observable(state, op)[source]

Perform a measurement specified by an operator on the given state.

This function simulates the classic quantum measurement described in many introductory texts on quantum mechanics. The measurement collapses the state to one of the eigenstates of the given operator and the result of the measurement is the corresponding eigenvalue.

Parameters:

stateQobj: The ket or density matrix specifying the state to measure.
opQobj: The measurement operator.

Returns:

measured_valuefloat: The result of the measurement (one of the eigenvalues of op).
stateQobj: The new state (a ket if a ket was given, otherwise a density matrix).

Examples

Measure the z-component of the spin of the spin-up basis state:

>>> measure_observable(basis(2, 0), sigmaz())
(1.0, Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[-1.]
 [ 0.]])

Since the spin-up basis is an eigenstate of sigmaz, this measurement always returns 1 as the measurement result (the eigenvalue of the spin-up basis) and the original state (up to a global phase).

Measure the x-component of the spin of the spin-down basis state:

>>> measure_observable(basis(2, 1), sigmax())
(-1.0, Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[-0.70710678]
 [ 0.70710678]])

This measurement returns 1 fifty percent of the time and -1 the other fifty percent of the time. The new state returned is the corresponding eigenstate of sigmax.

One may also perform a measurement on a density matrix. Below we perform the same measurement as above, but on the density matrix representing the pure spin-down state:

>>> measure_observable(ket2dm(basis(2, 1)), sigmax())
(-1.0, Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper
Qobj data =
[[ 0.5 -0.5]
 [-0.5  0.5]])

The measurement result is the same, but the new state is returned as a density matrix.

measure_povm(state, ops)[source]

Perform a measurement specified by list of POVMs.

This function simulates a POVM measurement. The measurement collapses the state to one of the resultant states of the measurement and returns the index of the operator corresponding to the collapsed state as well as the collapsed state.

Parameters:

stateQobj

The ket or density matrix specifying the state to measure.

opslist of Qobj

List of measurement operators $M_i$ or kets. Either:

specifying a POVM s.t. $E_i = M_i^\dagger M_i$
projection operators if ops correspond to projectors (s.t. $E_i = M_i^\dagger = M_i$)
kets (transformed to projectors)

Returns:

indexfloat: The resultant index of the measurement.
stateQobj: The new state (a ket if a ket was given, otherwise a density matrix).

measurement_statistics(state, ops)[source]

A dispatch method that provides measurement statistics handling both observable style measurements and projector style measurements(POVMs and PVMs).

For return signatures, please check:

measurement_statistics_observable for observable measurements.
measurement_statistics_povm for POVM measurements.

Parameters:

stateQobj

The ket or density matrix specifying the state to measure.

opsQobj or list of Qobj

measurement observable (:class:.Qobj); or
list of measurement operators $M_i$ or kets (list of Qobj) Either:
1. specifying a POVM s.t. $E_i = M_i^\dagger * M_i$
2. projection operators if ops correspond to projectors (s.t. $E_i = M_i^\dagger = M_i$)
3. kets (transformed to projectors)

measurement_statistics_observable(state, op)[source]

Return the measurement eigenvalues, eigenstates (or projectors) and measurement probabilities for the given state and measurement operator.

Parameters:

stateQobj: The ket or density matrix specifying the state to measure.
opQobj: The measurement operator.

Returns:

eigenvalues: list of float: The list of eigenvalues of the measurement operator.
eigenstates_or_projectors: list of Qobj: If the state was a ket, return the eigenstates of the measurement operator. Otherwise return the projectors onto the eigenstates.
probabilities: list of float: The probability of measuring the state as being in the corresponding eigenstate (and the measurement result being the corresponding eigenvalue).

measurement_statistics_povm(state, ops)[source]

Returns measurement statistics (resultant states and probabilities) for a measurement specified by a set of positive operator valued measurements on a specified ket or density matrix.

Parameters:

stateQobj

The ket or density matrix specifying the state to measure.

opslist of Qobj

List of measurement operators $M_i$ or kets. Either:

specifying a POVM s.t. $E_i = M_i^\dagger M_i$
projection operators if ops correspond to projectors (s.t. $E_i = M_i^\dagger = M_i$)
kets (transformed to projectors)

Returns:

collapsed_stateslist of Qobj: The collapsed states obtained after measuring the qubits and obtaining the qubit specified by the target in the state specified by the index.
probabilitieslist of floats: The probability of measuring a state in a the state specified by the index.

Dynamics and Time-Evolution

Schrödinger Equation

This module provides solvers for the unitary Schrodinger equation.

sesolve(H, psi0, tlist, e_ops=None, args=None, options=None)[source]

Schrodinger equation evolution of a state vector or unitary matrix for a given Hamiltonian.

Evolve the state vector (psi0) using a given Hamiltonian (H), by integrating the set of ordinary differential equations that define the system. Alternatively evolve a unitary matrix in solving the Schrodinger operator equation.

The output is either the state vector or unitary matrix at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values. e_ops cannot be used in conjunction with solving the Schrodinger operator equation

Time-dependent operators

For time-dependent problems, H and c_ops can be a QobjEvo or object that can be interpreted as QobjEvo such as a list of (Qobj, Coefficient) pairs or a function.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

System Hamiltonian as a Qobj or QobjEvo for time-dependent Hamiltonians. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

psi0qutip.qobj

initial state vector (ket) or initial unitary operator psi0 = U

tlistlist / array

list of times for $t$.

e_opsqutip.qobj, callable, or list.

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation.

argsNone / dictionary

dictionary of parameters for time-dependent Hamiltonians

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’} How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.] Which differential equation integration method to use.
atol, rtol : float Absolute and relative tolerance of the ODE integrator.
nsteps : Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, 0 Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.

Other options could be supported depending on the integration method, see Integrator.

Returns:

result: qutip.Result: An instance of the class qutip.Result, which contains a list of array result.expect of expectation values for the times specified by tlist, and/or a list result.states of state vectors or density matrices corresponding to the times in tlist [if e_ops is an empty list of store_states=True in options].

Master Equation

This module provides solvers for the Lindblad master equation and von Neumann equation.

mesolve(H, rho0, tlist, c_ops=None, e_ops=None, args=None, options=None)[source]

Master equation evolution of a density matrix for a given Hamiltonian and set of collapse operators, or a Liouvillian.

Evolve the state vector or density matrix (rho0) using a given Hamiltonian or Liouvillian (H) and an optional set of collapse operators (c_ops), by integrating the set of ordinary differential equations that define the system. In the absence of collapse operators the system is evolved according to the unitary evolution of the Hamiltonian.

The output is either the state vector at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values.

If either H or the Qobj elements in c_ops are superoperators, they will be treated as direct contributions to the total system Liouvillian. This allows the solution of master equations that are not in standard Lindblad form.

Time-dependent operators

For time-dependent problems, H and c_ops can be a QobjEvo or object that can be interpreted as QobjEvo such as a list of (Qobj, Coefficient) pairs or a function.

Additional options

Additional options to mesolve can be set via the options argument. Many ODE integration options can be set this way, and the store_states and store_final_state options can be used to store states even though expectation values are requested via the e_ops argument.

Note

When no collapse operator are given and the H is not a superoperator, it will defer to sesolve.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

rho0Qobj

initial density matrix or state vector (ket).

tlistlist / array

list of times for $t$.

c_opslist of (QobjEvo, QobjEvo compatible format)

Single collapse operator, or list of collapse operators, or a list of Liouvillian superoperators. None is equivalent to an empty list.

e_opslist of Qobj / callback function

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation.

argsNone / dictionary

dictionary of parameters for time-dependent Hamiltonians and collapse operators.

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’} How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.] Which differential equation integration method to use.
atol, rtol : float Absolute and relative tolerance of the ODE integrator.
nsteps : Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, 0 Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.

Other options could be supported depending on the integration method, see Integrator.

Returns:

result: qutip.Result: An instance of the class qutip.Result, which contains a list of array result.expect of expectation values for the times specified by tlist, and/or a list result.states of state vectors or density matrices corresponding to the times in tlist [if e_ops is an empty list of store_states=True in options].

Monte Carlo Evolution

mcsolve(H, state, tlist, c_ops=(), e_ops=None, ntraj=500, *, args=None, options=None, seeds=None, target_tol=None, timeout=None)[source]

Monte Carlo evolution of a state vector $|\psi \rangle$ for a given Hamiltonian and sets of collapse operators. Options for the underlying ODE solver are given by the Options class.

Parameters:

Hqutip.Qobj, qutip.QobjEvo, list, callable.

System Hamiltonian as a Qobj, QobjEvo. It can also be any input type that QobjEvo accepts (see qutip.QobjEvo’s documentation). H can also be a superoperator (liouvillian) if some collapse operators are to be treated deterministically.

statequtip.Qobj

Initial state vector.

tlistarray_like

Times at which results are recorded.

c_opslist

A list of collapse operators in any input type that QobjEvo accepts (see qutip.QobjEvo’s documentation). They must be operators even if H is a superoperator. If none are given, the solver will defer to sesolve or mesolve.

e_opslist, [optional]

A list of operator as Qobj, QobjEvo or callable with signature of (t, state: Qobj) for calculating expectation values. When no e_ops are given, the solver will default to save the states.

ntrajint

Maximum number of trajectories to run. Can be cut short if a time limit is passed with the timeout keyword or if the target tolerance is reached, see target_tol.

argsNone / dict

Arguments for time-dependent Hamiltonian and collapse operator terms.

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool, [False] Whether or not to store the final state of the evolution in the result class.
store_states : bool, NoneType, [None] Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, [‘text’] How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict, [{“chunk_size”: 10}] kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str {“adams”, “bdf”, “dop853”, “vern9”, etc.}, [“adams”] Which differential equation integration method to use.
keep_runs_results : bool, [False] Whether to store results from all trajectories or just store the averages.
map : str {“serial”, “parallel”, “loky”}, [“serial”] How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
job_timeout : NoneType, int, [None] Maximum time to compute one trajectory.
num_cpus : NoneType, int, [None] Number of cpus to use when running in parallel. None detect the number of available cpus.
norm_t_tol, norm_tol, norm_steps : float, float, int, [1e-6, 1e-4, 5] Parameters used to find the collapse location. norm_t_tol and norm_tol are the tolerance in time and norm respectively. An error will be raised if the collapse could not be found within norm_steps tries.
mc_corr_eps : float, [1e-10] Small number used to detect non-physical collapse caused by numerical imprecision.
atol, rtol : float, [1e-8, 1e-6] Absolute and relative tolerance of the ODE integrator.
nsteps : int [2500] Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, [0] Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
improved_sampling : Bool Whether to use the improved sampling algorithm from Abdelhafez et al. PRA (2019)

seedsint, SeedSequence, list, [optional]

Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:

seeds=prev_result.seeds

target_tolfloat, tuple, list, [optional]

Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by ntraj. The error is computed using jackknife resampling. target_tol can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops.

timeoutfloat, [optional]

Maximum time for the evolution in second. When reached, no more trajectories will be computed.

Returns:

resultsqutip.solver.McResult: Object storing all results from the simulation. Which results is saved depends on the presence of e_ops and the options used. collapse and photocurrent is available to Monte Carlo simulation results.

nm_mcsolve(H, state, tlist, ops_and_rates=(), e_ops=None, ntraj=500, *, args=None, options=None, seeds=None, target_tol=None, timeout=None)[source]

Monte-Carlo evolution corresponding to a Lindblad equation with “rates” that may be negative. Usage of this function is analogous to mcsolve, but the c_ops parameter is replaced by an ops_and_rates parameter to allow for negative rates. Options for the underlying ODE solver are given by the Options class.

Parameters:

Hqutip.Qobj, qutip.QobjEvo, list, callable.

System Hamiltonian as a Qobj, QobjEvo. It can also be any input type that QobjEvo accepts (see qutip.QobjEvo’s documentation). H can also be a superoperator (liouvillian) if some collapse operators are to be treated deterministically.

statequtip.Qobj

Initial state vector.

tlistarray_like

Times at which results are recorded.

ops_and_rateslist

A list of tuples (L, Gamma), where the Lindblad operator L is a qutip.Qobj and Gamma represents the corresponding rate, which is allowed to be negative. The Lindblad operators must be operators even if H is a superoperator. If none are given, the solver will defer to sesolve or mesolve. Each rate Gamma may be just a number (in the case of a constant rate) or, otherwise, specified using any format accepted by qutip.coefficient.

e_opslist, [optional]

A list of operator as Qobj, QobjEvo or callable with signature of (t, state: Qobj) for calculating expectation values. When no e_ops are given, the solver will default to save the states.

ntrajint

Maximum number of trajectories to run. Can be cut short if a time limit is passed with the timeout keyword or if the target tolerance is reached, see target_tol.

argsNone / dict

Arguments for time-dependent Hamiltonian and collapse operator terms.

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool, [False] Whether or not to store the final state of the evolution in the result class.
store_states : bool, NoneType, [None] Whether or not to store the state density matrices. On None the states will be saved if no expectation operators are given.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, [‘text’] How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict, [{“chunk_size”: 10}] kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str {“adams”, “bdf”, “dop853”, “vern9”, etc.}, [“adams”] Which differential equation integration method to use.
keep_runs_results : bool, [False] Whether to store results from all trajectories or just store the averages.
map : str {“serial”, “parallel”, “loky”}, [“serial”] How to run the trajectories. “parallel” uses concurrent module to run in parallel while “loky” use the module of the same name to do so.
job_timeout : NoneType, int, [None] Maximum time to compute one trajectory.
num_cpus : NoneType, int, [None] Number of cpus to use when running in parallel. None detect the number of available cpus.
norm_t_tol, norm_tol, norm_steps : float, float, int, [1e-6, 1e-4, 5] Parameters used to find the collapse location. norm_t_tol and norm_tol are the tolerance in time and norm respectively. An error will be raised if the collapse could not be found within norm_steps tries.
mc_corr_eps : float, [1e-10] Small number used to detect non-physical collapse caused by numerical imprecision.
atol, rtol : float, [1e-8, 1e-6] Absolute and relative tolerance of the ODE integrator.
nsteps : int [2500] Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, [0] Maximum length of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
completeness_rtol, completeness_atol : float, float, [1e-5, 1e-8] Parameters used in determining whether the given Lindblad operators satisfy a certain completeness relation. If they do not, an additional Lindblad operator is added automatically (with zero rate).
martingale_quad_limit : float or int, [100] An upper bound on the number of subintervals used in the adaptive integration of the martingale.

seedsint, SeedSequence, list, [optional]

Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:

seeds=prev_result.seeds

target_tolfloat, tuple, list, [optional]

Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by ntraj. The error is computed using jackknife resampling. target_tol can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops.

timeoutfloat, [optional]

Maximum time for the evolution in seconds. When reached, no more trajectories will be computed.

Returns:

resultsqutip.solver.NmmcResult: Object storing all results from the simulation. Compared to a result returned by mcsolve, this result contains the additional field trace (and runs_trace if store_final_state is set). Note that the states on the individual trajectories are not normalized. This field contains the average of their trace, which will converge to one in the limit of sufficiently many trajectories.

Krylov Subspace Solver

krylovsolve(H, psi0, tlist, krylov_dim, e_ops=None, args=None, options=None)[source]

Schrodinger equation evolution of a state vector for time independent Hamiltonians using Krylov method.

Evolve the state vector (“psi0”) finding an approximation for the time evolution operator of Hamiltonian (“H”) by obtaining the projection of the time evolution operator on a set of small dimensional Krylov subspaces (m << dim(H)).

The output is either the state vector or unitary matrix at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values. e_ops cannot be used in conjunction with solving the Schrodinger operator equation

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

System Hamiltonian as a Qobj or QobjEvo for time-dependent Hamiltonians. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

psi0qutip.qobj

initial state vector (ket) or initial unitary operator psi0 = U

tlistlist / array

list of times for $t$.

krylov_dim: int

Dimension of Krylov approximation subspaces used for the time evolution approximation.

e_opsqutip.qobj, callable, or list.

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation.

argsNone / dictionary

dictionary of parameters for time-dependent Hamiltonians

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool, [False] Whether or not to store the final state of the evolution in the result class.
store_states : bool, [None] Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool, [True] Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, [“text”] How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict, [{“chunk_size”: 10}] kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
atol: float [1e-7] Absolute and relative tolerance of the ODE integrator.
nsteps : int [100] Maximum number of (internally defined) steps allowed in one tlist step.
min_step, max_step : float, [1e-5, 1e5] Miniumum and maximum lenght of one internal step.
always_compute_step: bool [False] If True, the step lenght is computed each time a new Krylov subspace is computed. Otherwise it is computed only once when creating the integrator.
sub_system_tol: float, [1e-7] Tolerance to detect an happy breakdown. An happy breakdown happens when the initial ket is in a subspace of the Hamiltonian smaller than krylov_dim.

Returns:

result: qutip.Result: An instance of the class qutip.Result, which contains a list of array result.expect of expectation values for the times specified by tlist, and/or a list result.states of state vectors or density matrices corresponding to the times in tlist [if e_ops is an empty list of store_states=True in options].

Bloch-Redfield Master Equation

This module provides solvers for the Lindblad master equation and von Neumann equation.

brmesolve(H, psi0, tlist, a_ops=[], e_ops=[], c_ops=[], args={}, sec_cutoff=0.1, options=None)[source]

Solves for the dynamics of a system using the Bloch-Redfield master equation, given an input Hamiltonian, Hermitian bath-coupling terms and their associated spectral functions, as well as possible Lindblad collapse operators.

Parameters:

HQobj, QobjEvo

Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

psi0: Qobj

Initial density matrix or state vector (ket).

tlistarray_like

List of times for evaluating evolution

a_opslist of (a_op, spectra)

Nested list of system operators that couple to the environment, and the corresponding bath spectra.

a_opqutip.Qobj, qutip.QobjEvo

The operator coupling to the environment. Must be hermitian.

spectraCoefficient, str, func

The corresponding bath spectral responce. Can be a Coefficient using an ‘w’ args, a function of the frequence or a string. Coefficient build from a numpy array are understood as a function of w instead of t. Function are expected to be of the signature f(w) or f(t, w, **args).

The spectra function can depend on t if the corresponding a_op is a QobjEvo.

Example:

a_ops = [
    (a+a.dag(), ('w>0', args={"w": 0})),
    (QobjEvo(a+a.dag()), 'w > exp(-t)'),
    (QobjEvo([b+b.dag(), lambda t: ...]), lambda w: ...)),
    (c+c.dag(), SpectraCoefficient(coefficient(array, tlist=ws))),
]

e_opslist of Qobj / callback function

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation

c_opslist of (QobjEvo, QobjEvo compatible format)

List of collapse operators.

argsdict

Dictionary of parameters for time-dependent Hamiltonians and collapse operators. The key w is reserved for the spectra function.

sec_cutofffloat {0.1}

Cutoff for secular approximation. Use -1 if secular approximation is not used when evaluating bath-coupling terms.

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’} How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
tensor_type : str [‘sparse’, ‘dense’, ‘data’] Which data type to use when computing the brtensor. With a cutoff ‘sparse’ is usually the most efficient.
sparse_eigensolver : bool {False} Whether to use the sparse eigensolver
method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.] Which differential equation integration method to use.
atol, rtol : float Absolute and relative tolerance of the ODE integrator.
nsteps : Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, 0 Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.

Other options could be supported depending on the integration method, see Integrator.

Returns:

result: qutip.solver.Result: An instance of the class qutip.solver.Result, which contains either an array of expectation values, for operators given in e_ops, or a list of states for the times specified by tlist.

Floquet States and Floquet-Markov Master Equation

class FMESolver(floquet_basis, a_ops, w_th=0.0, *, kmax=5, nT=None, options=None)[source]

Solver for the Floquet-Markov master equation.

Note

Operators (c_ops and e_ops) are in the laboratory basis.

Parameters:

floquet_basisqutip.FloquetBasis: The system Hamiltonian wrapped in a FloquetBasis object. Choosing a different integrator for the floquet_basis than for the evolution of the floquet state can improve the performance.
a_opslist of tuple(qutip.Qobj, callable): List of collapse operators and the corresponding function for the noise power spectrum. The collapse operator must be a Qobj and cannot be time dependent. The spectrum function must take and return an numpy array.
w_thfloat: The temperature of the environment in units of Hamiltonian frequency.
kmaxint [5]: The truncation of the number of sidebands..
nTint [20*kmax]: The number of integration steps (for calculating X) within one period.
optionsdict, optional: Options for the solver, see FMESolver.options and Integrator for a list of all options.

resultclass: alias of FloquetResult

run(state0, tlist, *, floquet=False, args=None, e_ops=None)[source]

Calculate the evolution of the quantum system.

For a state0 at time tlist[0] do the evolution as directed by rhs and for each time in tlist store the state and/or expectation values in a Result. The evolution method and stored results are determined by options.

Parameters:

state0Qobj: Initial state of the evolution.
tlistlist of double: Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Each times of the list must be increasing, but does not need to be uniformy distributed.
floquetbool, optional {False}: Whether the initial state in the floquet basis or laboratory basis.
argsdict, optional {None}: Not supported
e_opslist {None}: List of Qobj, QobjEvo or callable to compute the expectation values. Function[s] must have the signature f(t : float, state : Qobj) -> expect.

Returns:

resultsqutip.solver.FloquetResult: Results of the evolution. States and/or expect will be saved. You can control the saved data in the options.

start(state0, t0, *, floquet=False)[source]

Set the initial state and time for a step evolution. options for the evolutions are read at this step.

Parameters:

state0Qobj: Initial state of the evolution.
t0double: Initial time of the evolution.
floquetbool, optional {False}: Whether the initial state is in the floquet basis or laboratory basis.

step(t, *, args=None, copy=True, floquet=False)[source]

Evolve the state to t and return the state as a Qobj.

Parameters:

tdouble: Time to evolve to, must be higher than the last call.
copybool, optional {True}: Whether to return a copy of the data or the data in the ODE solver.
floquetbool, optional {False}: Whether to return the state in the floquet basis or laboratory basis.
argsdict, optional {None}: Not supported
.. note::: The state must be initialized first by calling start or run. If run is called, step will continue from the last time and state obtained.

class FloquetBasis(H, T, args=None, options=None, sparse=False, sort=True, precompute=None)[source]

Utility to compute floquet modes and states.

Attributes:

UPropagator: The propagator of the Hamiltonian over one period.
evecsqutip.data.Data: Matrix where each column is an initial Floquet mode.
e_quasinp.ndarray[float]: The quasi energies of the Hamiltonian.

from_floquet_basis(floquet_basis, t=0)[source]

Transform a ket or density matrix from the Floquet basis at time t to the lab basis.

Parameters:

floquet_basisQobj, qutip.data.Data: Initial state in the Floquet basis at time t. May be either a ket or density matrix.
tfloat [0]: The time at which to evaluate the Floquet states.

Returns:

outputQobj, qutip.data.Data: The state in the lab basis. The return type is the same as the type of the input state.

mode(t, data=False)[source]

Calculate the Floquet modes at time t.

Parameters:

tfloat: The time for which to evaluate the Floquet mode.
databool [False]: Whether to return the states as a single data matrix or a list of ket states.

Returns:

outputlist[Qobj], qutip.data.Data: A list of Floquet states for the time t or the states as column in a single matrix.

state(t, data=False)[source]

Evaluate the floquet states at time t.

Parameters:

tfloat: The time for which to evaluate the Floquet states.
databool [False]: Whether to return the states as a single data matrix or a list of ket states.

Returns:

outputlist[Qobj], qutip.data.Data: A list of Floquet states for the time t or the states as column in a single matrix.

to_floquet_basis(lab_basis, t=0)[source]

Transform a ket or density matrix in the lab basis to the Floquet basis at time t.

Parameters:

lab_basisQobj, qutip.data.Data: Initial state in the lab basis.
tfloat [0]: The time at which to evaluate the Floquet states.

Returns:

outputQobj, qutip.data.Data: The state in the Floquet basis. The return type is the same as the type of the input state.

floquet_tensor(H, c_ops, spectra_cb, T=0, w_th=0.0, kmax=5, nT=100)[source]

Construct a tensor that represents the master equation in the floquet basis.

Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)]

Parameters:

HQobjEvo: Periodic Hamiltonian
Tfloat: The period of the time-dependence of the hamiltonian.
c_opslist of qutip.qobj: list of collapse operators.
spectra_cblist callback functions: List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in c_ops.
w_thfloat: The temperature in units of frequency.
kmaxint: The truncation of the number of sidebands (default 5).

Returns:

outputarray: The Floquet-Markov master equation tensor R.

fmmesolve(H, rho0, tlist, c_ops=None, e_ops=None, spectra_cb=None, T=0, w_th=0.0, args=None, options=None)[source]

Solve the dynamics for the system using the Floquet-Markov master equation.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

Periodic system Hamiltonian as QobjEvo. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

rho0 / psi0qutip.Qobj

Initial density matrix or state vector (ket).

tlistlist / array

List of times for $t$.

c_opslist of qutip.Qobj

List of collapse operators. Time dependent collapse operators are not supported.

e_opslist of qutip.Qobj / callback function

List of operators for which to evaluate expectation values. The states are reverted to the lab basis before applying the

spectra_cblist callback functions

List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in c_ops.

Tfloat

The period of the time-dependence of the hamiltonian. The default value ‘None’ indicates that the ‘tlist’ spans a single period of the driving.

w_thfloat

The temperature of the environment in units of frequency. For example, if the Hamiltonian written in units of 2pi GHz, and the temperature is given in K, use the following conversion:

temperature = 25e-3 # unit K h = 6.626e-34 kB = 1.38e-23 args[‘w_th’] = temperature * (kB / h) * 2 * pi * 1e-9

argsdictionary

Dictionary of parameters for time-dependent Hamiltonian

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
store_floquet_states : bool Whether or not to store the density matrices in the floquet basis in result.floquet_states.
normalize_output : bool Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’} How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.] Which differential equation integration method to use.
atol, rtol : float Absolute and relative tolerance of the ODE integrator.
nsteps : Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, 0 Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.

Other options could be supported depending on the integration method, see Integrator.

Returns:

result: qutip.Result: An instance of the class qutip.Result, which contains the expectation values for the times specified by tlist, and/or the state density matrices corresponding to the times.

fsesolve(H, psi0, tlist, e_ops=None, T=0.0, args=None, options=None)[source]

Solve the Schrodinger equation using the Floquet formalism.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

Periodic system Hamiltonian as QobjEvo. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

psi0qutip.qobj

Initial state vector (ket). If an operator is provided,

tlistlist / array

List of times for $t$.

e_opslist of qutip.qobj / callback function, optional

List of operators for which to evaluate expectation values. If this list is empty, the state vectors for each time in tlist will be returned instead of expectation values.

Tfloat, default=tlist[-1]

The period of the time-dependence of the hamiltonian.

argsdictionary, optional

Dictionary with variables required to evaluate H.

optionsdict, optional

Options for the results.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool Normalize output state to hide ODE numerical errors.

Returns:

outputqutip.solver.Result: An instance of the class qutip.solver.Result, which contains either an array of expectation values or an array of state vectors, for the times specified by tlist.

Stochastic Schrödinger Equation and Master Equation

smesolve(H, rho0, tlist, c_ops=(), sc_ops=(), heterodyne=False, *, e_ops=(), args={}, ntraj=500, options=None, seeds=None, target_tol=None, timeout=None)[source]

Solve stochastic master equation.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

System Hamiltonian as a Qobj or QobjEvo for time-dependent Hamiltonians. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

rho0qutip.Qobj

Initial density matrix or state vector (ket).

tlistlist / array

List of times for $t$.

c_opslist of (QobjEvo, QobjEvo compatible format)

Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.

sc_opslist of (QobjEvo, QobjEvo compatible format)

List of stochastic collapse operators.

e_ops: qutip.qobj, callable, or list.

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation.

argsNone / dictionary

Dictionary of parameters for time-dependent Hamiltonians and collapse operators.

ntrajint [500]

Number of trajectories to compute.

heterodynebool [False]

Whether to use heterodyne or homodyne detection.

seedsint, SeedSequence, list, [optional]

Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:

seeds=prev_result.seeds

When using a parallel map, the trajectories can be re-ordered.

target_tol{float, tuple, list}, optional

Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by ntraj. The error is computed using jackknife resampling. target_tol can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops.

timeoutfloat [optional]

Maximum time for the evolution in second. When reached, no more trajectories will be computed. Overwrite the option of the same name.

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool, [False] Whether or not to store the final state of the evolution in the result class.
store_states : bool, None, [None] Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
store_measurement: bool, [False] Whether to store the measurement and wiener process for each trajectories.
keep_runs_results : bool, [False] Whether to store results from all trajectories or just store the averages.
normalize_output : bool, [False] Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, [“text”] How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict, [{“chunk_size”: 10}] kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str, [“rouchon”] Which stochastic differential equation integration method to use. Main ones are {“euler”, “rouchon”, “platen”, “taylor1.5_imp”}
map : str {“serial”, “parallel”, “loky”}, [“serial”] How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
job_timeout : NoneType, int, [None] Maximum time to compute one trajectory.
num_cpus : NoneType, int, [None] Number of cpus to use when running in parallel. None detect the number of available cpus.
dt : float [0.001 ~ 0.0001] The finite steps lenght for the Stochastic integration method. Default change depending on the integrator.

Other options could be supported depending on the integration method, see SIntegrator.

Returns:

output: qutip.solver.Result: An instance of the class qutip.solver.Result.

ssesolve(H, psi0, tlist, sc_ops=(), heterodyne=False, *, e_ops=(), args={}, ntraj=500, options=None, seeds=None, target_tol=None, timeout=None)[source]

Solve stochastic Schrodinger equation.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.

System Hamiltonian as a Qobj or QobjEvo for time-dependent Hamiltonians. List of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

psi0qutip.Qobj

Initial state vector (ket).

tlistlist / array

List of times for $t$.

sc_opslist of (QobjEvo, QobjEvo compatible format)

List of stochastic collapse operators.

e_opsqutip.qobj, callable, or list.

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation.

argsNone / dictionary

Dictionary of parameters for time-dependent Hamiltonians and collapse operators.

ntrajint [500]

Number of trajectories to compute.

heterodynebool [False]

Whether to use heterodyne or homodyne detection.

seedsint, SeedSequence, list, [optional]

Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:

seeds=prev_result.seeds

target_tol{float, tuple, list}, optional

Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by ntraj. The error is computed using jackknife resampling. target_tol can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops.

timeoutfloat [optional]

Maximum time for the evolution in second. When reached, no more trajectories will be computed. Overwrite the option of the same name.

optionsNone / dict

Dictionary of options for the solver.

store_final_state : bool, [False] Whether or not to store the final state of the evolution in the result class.
store_states : bool, None, [None] Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
store_measurement: bool, [False] Whether to store the measurement and wiener process, or brownian noise for each trajectories.
keep_runs_results : bool, [False] Whether to store results from all trajectories or just store the averages.
normalize_output : bool, [False] Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, [“text”] How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict, [{“chunk_size”: 10}] kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
method : str, [“rouchon”] Which stochastic differential equation integration method to use. Main ones are {“euler”, “rouchon”, “platen”, “taylor1.5_imp”}
map : str {“serial”, “parallel”, “loky”}, [“serial”] How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
job_timeout : NoneType, int, [None] Maximum time to compute one trajectory.
num_cpus : NoneType, int, [None] Number of cpus to use when running in parallel. None detect the number of available cpus.
dt : float [0.001 ~ 0.0001] The finite steps lenght for the Stochastic integration method. Default change depending on the integrator.

Other options could be supported depending on the integration method, see SIntegrator.

Returns:

output: qutip.solver.Result: An instance of the class qutip.solver.Result.

Hierarchical Equations of Motion

This module provides solvers for system-bath evoluation using the HEOM (hierarchy equations of motion).

See https://en.wikipedia.org/wiki/Hierarchical_equations_of_motion for a very basic introduction to the technique.

The implementation is derived from the BoFiN library (see https://github.com/tehruhn/bofin) which was itself derived from an earlier implementation in QuTiP itself.

For backwards compatibility with QuTiP 4.6 and below, a new version of HSolverDL (the Drude-Lorentz specific HEOM solver) is provided. It is implemented on top of the new HEOMSolver but should largely be a drop-in replacement for the old HSolverDL.

heomsolve(H, bath, max_depth, state0, tlist, *, e_ops=None, args=None, options=None)[source]

Hierarchical Equations of Motion (HEOM) solver that supports multiple baths.

The baths must be all either bosonic or fermionic baths.

If you need to run many evolutions of the same system and bath, consider using HEOMSolver directly to avoid having to continually reconstruct the equation hierarchy for every evolution.

Parameters:

HQobj, QobjEvo

Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.

bathBath or list of Bath

A Bath containing the exponents of the expansion of the bath correlation funcion and their associated coefficients and coupling operators, or a list of baths.

If multiple baths are given, they must all be either fermionic or bosonic baths.

max_depthint

The maximum depth of the heirarchy (i.e. the maximum number of bath exponent “excitations” to retain).

state0Qobj or HierarchyADOsState or array-like

If rho0 is a Qobj the it is the initial state of the system (i.e. a Qobj density matrix).

If it is a HierarchyADOsState or array-like, then rho0 gives the initial state of all ADOs.

Usually the state of the ADOs would be determine from a previous call to .run(...) with the solver results option store_ados set to True. For example, result = solver.run(...) could be followed by solver.run(result.ado_states[-1], tlist).

If a numpy array-like is passed its shape must be (number_of_ados, n, n) where (n, n) is the system shape (i.e. shape of the system density matrix) and the ADOs must be in the same order as in .ados.labels.

tlistlist

An ordered list of times at which to return the value of the state.

e_opsQobj / QobjEvo / callable / list / dict / None, optional

A list or dictionary of operators as Qobj, QobjEvo and/or callable functions (they can be mixed) or a single operator or callable function. For an operator op, the result will be computed using (state * op).tr() and the state at each time t. For callable functions, f, the result is computed using f(t, ado_state). The values are stored in the expect and e_data attributes of the result (see the return section below).

argsdict, optional {None}

Change the args of the RHS for the evolution.

optionsdict, optional {None}

Generic solver options.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
store_ados : bool {False, True} Whether or not to store the HEOM ADOs.
normalize_output : bool Normalize output state to hide ODE numerical errors.
progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’} How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
progress_kwargs : dict kwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
state_data_type: str {‘dense’} Name of the data type of the state used during the ODE evolution. Use an empty string to keep the input state type. Many integrator can only work with Dense.
method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.] Which differential equation integration method to use.
atol, rtol : float Absolute and relative tolerance of the ODE integrator.
nsteps : Maximum number of (internally defined) steps allowed in one tlist step.
max_step : float, 0 Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.

Returns:

HEOMResult

The results of the simulation run, with the following important attributes:

times: the times t (i.e. the tlist).
states: the system state at each time t (only available if e_ops was None or if the solver option store_states was set to True).
ado_states: the full ADO state at each time (only available if the results option ado_return was set to True). Each element is an instance of HierarchyADOsState. The state of a particular ADO may be extracted from result.ado_states[i] by calling extract.
expect: a list containing the values of each e_ops at time t.
e_data: a dictionary containing the values of each e_ops at tme t. The keys are those given by e_ops if it was a dict, otherwise they are the indexes of the supplied e_ops.

See HEOMResult and Result for the complete list of attributes.

Correlation Functions

coherence_function_g1(H, state0, taulist, c_ops, a_op, solver='me', args={}, options={})[source]

Calculate the normalized first-order quantum coherence function:

\[g^{(1)}(\tau) = \frac{\langle A^\dagger(\tau)A(0)\rangle} {\sqrt{\langle A^\dagger(\tau)A(\tau)\rangle \langle A^\dagger(0)A(0)\rangle}}\]

using the quantum regression theorem and the evolution solver indicated by the solver parameter.

Parameters:

HQobj, QobjEvo: System Hamiltonian, may be time-dependent for solver choice of me.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$. If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steady-state’ is only implemented if c_ops are provided and the Hamiltonian is constant.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
c_opslist of {Qobj, QobjEvo}: List of collapse operators
a_opQobj, QobjEvo: Operator A.
solverstr {‘me’, ‘es’}: Choice of solver, me for master-equation, and es for exponential series. es is equivalent to me with options={"method": "diag"}.
optionsdict, optional: Options for the solver. Only used with me solver.

Returns:

g1, G1tuple: The normalized and unnormalized second-order coherence function.

coherence_function_g2(H, state0, taulist, c_ops, a_op, solver='me', args={}, options={})[source]

Calculate the normalized second-order quantum coherence function:

\[ g^{(2)}(\tau) = \frac{\langle A^\dagger(0)A^\dagger(\tau)A(\tau)A(0)\rangle} {\langle A^\dagger(\tau)A(\tau)\rangle \langle A^\dagger(0)A(0)\rangle}\]

using the quantum regression theorem and the evolution solver indicated by the solver parameter.

Parameters:

HQobj, QobjEvo: System Hamiltonian, may be time-dependent for solver choice of me.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$. If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steady-state’ is only implemented if c_ops are provided and the Hamiltonian is constant.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
c_opslist: List of collapse operators, may be time-dependent for solver choice of me.
a_opQobj: Operator A.
argsdict: Dictionary of arguments to be passed to solver.
solverstr {‘me’, ‘es’}: Choice of solver, me for master-equation, and es for exponential series. es is equivalent to me with options={"method": "diag"}.
optionsdict, optional: Options for the solver. Only used with me solver.

Returns:

g2, G2tuple: The normalized and unnormalized second-order coherence function.

correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options={})[source]

Calculate the two-operator one-time correlation function: $\left<A(\tau)B(0)\right>$ along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.

Parameters:

HQobj, QobjEvo: System Hamiltonian, may be time-dependent for solver choice of me.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$. If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steady-state’ is only implemented if c_ops are provided and the Hamiltonian is constant.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
c_opslist of {Qobj, QobjEvo}: List of collapse operators
a_opQobj, QobjEvo: Operator A.
b_opQobj, QobjEvo: Operator B.
reversebool {False}: If True, calculate $\left<A(t)B(t+\tau)\right>$ instead of $\left<A(t+\tau)B(t)\right>$.
solverstr {‘me’, ‘es’}: Choice of solver, me for master-equation, and es for exponential series. es is equivalent to me with options={"method": "diag"}.
optionsdict, optional: Options for the solver. Only used with me solver.

Returns:

corr_vecndarray: An array of correlation values for the times specified by taulist.

See also

correlation_3op: Similar function supporting various solver types.

References

See, Gardiner, Quantum Noise, Section 5.2.

correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options={})[source]

Calculate the two-operator two-time correlation function: $\left<A(t+\tau)B(t)\right>$ along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.

Parameters:

HQobj, QobjEvo: System Hamiltonian, may be time-dependent for solver choice of me.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$. If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steady-state’ is only implemented if c_ops are provided and the Hamiltonian is constant.
tlistarray_like: List of times for $t$. tlist must be positive and contain the element 0. When taking steady-steady correlations only one tlist value is necessary, i.e. when $t \rightarrow \infty$. If tlist is None, tlist=[0] is assumed.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
c_opslist of {Qobj, QobjEvo}: List of collapse operators
a_opQobj, QobjEvo: Operator A.
b_opQobj, QobjEvo: Operator B.
reversebool {False}: If True, calculate $\left<A(t)B(t+\tau)\right>$ instead of $\left<A(t+\tau)B(t)\right>$.
solverstr {‘me’, ‘es’}: Choice of solver, me for master-equation, and es for exponential series. es is equivalent to me with options={"method": "diag"}.
optionsdict, optional: Options for the solver. Only used with me solver.

Returns:

corr_matndarray: An 2-dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index).

See also

correlation_3op: Similar function supporting various solver types.

References

See, Gardiner, Quantum Noise, Section 5.2.

correlation_3op(solver, state0, tlist, taulist, A=None, B=None, C=None)[source]

Calculate the three-operator two-time correlation function:

$\left<A(t)B(t+\tau)C(t)\right>$.

from a open system Solver.

Note: it is not possible to calculate a physically meaningful correlation where $\tau<0$.

Parameters:

solverMESolver, BRSolver: Qutip solver for an open system.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$.
tlistarray_like: List of times for $t$. tlist must be positive and contain the element 0.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
A, B, C: :class:`Qobj`, :class:`QobjEvo`, optional, default=None: Operators A, B, C from the equation <A(t)B(t+\tau)C(t)> in the Schrodinger picture. They do not need to be all provided. For exemple, if A is not provided, <B(t+\tau)C(t)> is computed.

Returns:

corr_matarray: An 2-dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index). If tlist is None, then a 1-dimensional array of correlation values is returned instead.

correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op, solver='me', args={}, options={})[source]

Calculate the three-operator two-time correlation function: $\left<A(0)B(\tau)C(0)\right>$ along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.

Note: it is not possibly to calculate a physically meaningful correlation of this form where $\tau<0$.

Parameters:

HQobj, QobjEvo: System Hamiltonian, may be time-dependent for solver choice of me.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$. If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steady-state’ is only implemented if c_ops are provided and the Hamiltonian is constant.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
c_opslist of {Qobj, QobjEvo}: List of collapse operators
a_opQobj, QobjEvo: Operator A.
b_opQobj, QobjEvo: Operator B.
c_opQobj, QobjEvo: Operator C.
solverstr {‘me’, ‘es’}: Choice of solver, me for master-equation, and es for exponential series. es is equivalent to me with options={"method": "diag"}.
optionsdict, optional: Options for the solver. Only used with me solver.

Returns:

corr_vecarray: An array of correlation values for the times specified by taulist.

See also

correlation_3op: Similar function supporting various solver types.

References

See, Gardiner, Quantum Noise, Section 5.2.

correlation_3op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, solver='me', args={}, options={})[source]

Calculate the three-operator two-time correlation function: $\left<A(t)B(t+\tau)C(t)\right>$ along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.

Note: it is not possibly to calculate a physically meaningful correlation of this form where $\tau<0$.

Parameters:

HQobj, QobjEvo: System Hamiltonian, may be time-dependent for solver choice of me.
state0Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$. If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steady-state’ is only implemented if c_ops are provided and the Hamiltonian is constant.
tlistarray_like: List of times for $t$. tlist must be positive and contain the element 0. When taking steady-steady correlations only one tlist value is necessary, i.e. when $t \rightarrow \infty$. If tlist is None, tlist=[0] is assumed.
taulistarray_like: List of times for $\tau$. taulist must be positive and contain the element 0.
c_opslist of {Qobj, QobjEvo}: List of collapse operators
a_opQobj, QobjEvo: Operator A.
b_opQobj, QobjEvo: Operator B.
c_opQobj, QobjEvo: Operator C.
solverstr {‘me’, ‘es’}: Choice of solver, me for master-equation, and es for exponential series. es is equivalent to me with options={"method": "diag"}.
optionsdict, optional: Options for the solver. Only used with me solver.

Returns:

corr_matarray: An 2-dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index).

See also

correlation_3op: Similar function supporting various solver types.

References

See, Gardiner, Quantum Noise, Section 5.2.

spectrum(H, wlist, c_ops, a_op, b_op, solver='es')[source]

Calculate the spectrum of the correlation function $\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>$, i.e., the Fourier transform of the correlation function:

\[S(\omega) = \int_{-\infty}^{\infty} \lim_{t \to \infty} \left<A(t+\tau)B(t)\right> e^{-i\omega\tau} d\tau.\]

using the solver indicated by the solver parameter. Note: this spectrum is only defined for stationary statistics (uses steady state rho0)

Parameters:

Hqutip.qobj: system Hamiltonian.
wlistarray_like: List of frequencies for $\omega$.
c_opslist: List of collapse operators.
a_opQobj: Operator A.
b_opQobj: Operator B.
solverstr: Choice of solver (es for exponential series and pi for psuedo-inverse, solve for generic solver).

Returns:

spectrumarray: An array with spectrum $S(\omega)$ for the frequencies specified in wlist.

spectrum_correlation_fft(tlist, y, inverse=False)[source]

Calculate the power spectrum corresponding to a two-time correlation function using FFT.

Parameters:

tlistarray_like: list/array of times $t$ which the correlation function is given.
yarray_like: list/array of correlations corresponding to time delays $t$.
inverse: boolean: boolean parameter for using a positive exponent in the Fourier Transform instead. Default is False.

Returns:

w, Stuple: Returns an array of angular frequencies ‘w’ and the corresponding two-sided power spectrum ‘S(w)’.

Steady-state Solvers

pseudo_inverse(L, rhoss=None, w=None, method='splu', *, use_rcm=False, **kwargs)[source]

Compute the pseudo inverse for a Liouvillian superoperator, optionally given its steady state density matrix (which will be computed if not given).

Parameters:

LQobj

A Liouvillian superoperator for which to compute the pseudo inverse.

rhossQobj

A steadystate density matrix as Qobj instance, for the Liouvillian superoperator L.

wdouble

frequency at which to evaluate pseudo-inverse. Can be zero for dense systems and large sparse systems. Small sparse systems can fail for zero frequencies.

sparsebool

Flag that indicate whether to use sparse or dense matrix methods when computing the pseudo inverse.

methodstring

Method used to compte matrix inverse. Choice are ‘pinv’ to use scipy’s function of the same name, or a linear system solver. Default supported solver are:

“solve”, “lstsq” dense solver from numpy.linalg
“spsolve”, “gmres”, “lgmres”, “bicgstab”, “splu” sparse solver from scipy.sparse.linalg
“mkl_spsolve”, sparse solver by mkl.

Extension to qutip, such as qutip-tensorflow, can use come with their own solver. When L use these data backends, see the corresponding libraries linalg for available solver.

kwargsdictionary

Additional keyword arguments for setting parameters for solver methods.

Returns:

RQobj: Returns a Qobj instance representing the pseudo inverse of L.

Note

In general the inverse of a sparse matrix will be dense. If you are applying the inverse to a density matrix then it is better to cast the problem as an Ax=b type problem where the explicit calculation of the inverse is not required. See page 67 of “Electrons in nanostructures” C. Flindt, PhD Thesis available online: https://orbit.dtu.dk/fedora/objects/orbit:82314/datastreams/ file_4732600/content

Note also that the definition of the pseudo-inverse herein is different from numpys pinv() alone, as it includes pre and post projection onto the subspace defined by the projector Q.

steadystate(A, c_ops=[], *, method='direct', solver=None, **kwargs)[source]

Calculates the steady state for quantum evolution subject to the supplied Hamiltonian or Liouvillian operator and (if given a Hamiltonian) a list of collapse operators.

If the user passes a Hamiltonian then it, along with the list of collapse operators, will be converted into a Liouvillian operator in Lindblad form.

Parameters:

AQobj

A Hamiltonian or Liouvillian operator.

c_op_listlist

A list of collapse operators.

methodstr, default=’direct’

The allowed methods are composed of 2 parts, the steadystate method: - “direct”: Solving L(rho_ss) = 0 - “eigen” : Eigenvalue problem - “svd” : Singular value decomposition - “power” : Inverse-power method

solverstr, default=None

‘direct’ and ‘power’ methods only. Solver to use when solving the L(rho_ss) = 0 equation. Default supported solver are:

“solve”, “lstsq” dense solver from numpy.linalg
“spsolve”, “gmres”, “lgmres”, “bicgstab” sparse solver from scipy.sparse.linalg
“mkl_spsolve” sparse solver by mkl.

Extension to qutip, such as qutip-tensorflow, can use come with their own solver. When A and c_ops use these data backends, see the corresponding libraries linalg for available solver.

Extra options for these solver can be passed in **kw.

use_rcmbool, default False

Use reverse Cuthill-Mckee reordering to minimize fill-in in the LU factorization of the Liouvillian. Used with ‘direct’ or ‘power’ method.

use_wbmbool, default False

Use Weighted Bipartite Matching reordering to make the Liouvillian diagonally dominant. This is useful for iterative preconditioners only. Used with ‘direct’ or ‘power’ method.

weightfloat, optional

Sets the size of the elements used for adding the unity trace condition to the linear solvers. This is set to the average abs value of the Liouvillian elements if not specified by the user. Used with ‘direct’ method.

power_tolfloat, default 1e-12

Tolerance for the solution when using the ‘power’ method.

power_maxiterint, default 10

Maximum number of iteration to use when looking for a solution when using the ‘power’ method.

power_eps: double, default 1e-15

Small weight used in the “power” method.

sparse: bool

Whether to use the sparse eigen solver with the “eigen” method (default sparse). With “direct” and “power” method, when the solver is not specified, it is used to set whether “solve” or “spsolve” is used as default solver.

**kwargs

Extra options to pass to the linear system solver. See the documentation of the used solver in numpy.linalg or scipy.sparse.linalg to see what extra arguments are supported.

Returns:

dmqobj: Steady state density matrix.
infodict, optional: Dictionary containing solver-specific information about the solution.

Note

The SVD method works only for dense operators (i.e. small systems).

steadystate_floquet(H_0, c_ops, Op_t, w_d=1.0, n_it=3, sparse=False, solver=None, **kwargs)[source]

Calculates the effective steady state for a driven: system with a time-dependent cosinusoidal term:

\[\mathcal{\hat{H}}(t) = \hat{H}_0 + \mathcal{\hat{O}} \cos(\omega_d t)\]

Parameters:

H_0Qobj

A Hamiltonian or Liouvillian operator.

c_opslist

A list of collapse operators.

Op_tQobj

The the interaction operator which is multiplied by the cosine

w_dfloat, default 1.0

The frequency of the drive

n_itint, default 3

The number of iterations for the solver

sparsebool, default False

Solve for the steady state using sparse algorithms.

solverstr, default=None

Solver to use when solving the linear system. Default supported solver are:

“solve”, “lstsq” dense solver from numpy.linalg
“spsolve”, “gmres”, “lgmres”, “bicgstab” sparse solver from scipy.sparse.linalg
“mkl_spsolve” sparse solver by mkl.

Extensions to qutip, such as qutip-tensorflow, may provide their own solvers. When H_0 and c_ops use these data backends, see their documentation for the names and details of additional solvers they may provide.

**kwargs:

Extra options to pass to the linear system solver. See the documentation of the used solver in numpy.linalg or scipy.sparse.linalg to see what extra arguments are supported.

Returns:

dmqobj: Steady state density matrix.

Note

See: Sze Meng Tan, https://copilot.caltech.edu/documents/16743/qousersguide.pdf, Section (10.16)

Propagators

propagator(H, t, c_ops=(), args=None, options=None, **kwargs)[source]

Calculate the propagator U(t) for the density matrix or wave function such that $\psi(t) = U(t)\psi(0)$ or $\rho_{\mathrm vec}(t) = U(t) \rho_{\mathrm vec}(0)$ where $\rho_{\mathrm vec}$ is the vector representation of the density matrix.

Parameters:

HQobj, QobjEvo, QobjEvo compatible format.: Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [Qobj, Coefficient] or callable that can be made into QobjEvo are also accepted.
tfloat or array-like: Time or list of times for which to evaluate the propagator.
c_opslist, optional: List of Qobj or QobjEvo collapse operators.
argsdictionary, optional: Parameters to callback functions for time-dependent Hamiltonians and collapse operators.
optionsdict, optional: Options for the solver.
**kwargs: Extra parameters to use when creating the QobjEvo from a list format H.

Returns:

Uqobj, list: Instance representing the propagator(s) $U(t)$. Return a single Qobj when t is a number or a list when t is a list.

propagator_steadystate(U)[source]

Find the steady state for successive applications of the propagator $U$.

Parameters:

Uqobj: Operator representing the propagator.

Returns:

aqobj: Instance representing the steady-state density matrix.

Scattering in Quantum Optical Systems

Photon scattering in quantum optical systems

This module includes a collection of functions for numerically computing photon scattering in driven arbitrary systems coupled to some configuration of output waveguides. The implementation of these functions closely follows the mathematical treatment given in K.A. Fischer, et. al., Scattering of Coherent Pulses from Quantum Optical Systems (2017, arXiv:1710.02875).

scattering_probability(H, psi0, n_emissions, c_ops, tlist, system_zero_state=None, construct_effective_hamiltonian=True)[source]

Compute the integrated probability of scattering n photons in an arbitrary system. This function accepts a nonlinearly spaced array of times.

Parameters:

H:class: qutip.Qobj or list: System-waveguide(s) Hamiltonian or effective Hamiltonian in Qobj or list-callback format. If construct_effective_hamiltonian is not specified, an effective Hamiltonian is constructed from H and c_ops.
psi0:class: qutip.Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$.
n_emissionsint: Number of photons emitted by the system (into any combination of waveguides).
c_opslist: List of collapse operators for each waveguide; these are assumed to include spontaneous decay rates, e.g. $\sigma = \sqrt \gamma \cdot a$.
tlistarray_like: List of times for $\tau_i$. tlist should contain 0 and exceed the pulse duration / temporal region of interest; tlist need not be linearly spaced.
system_zero_state:class: qutip.Qobj: State representing zero excitations in the system. Defaults to basis(systemDims, 0).
construct_effective_hamiltonianbool: Whether an effective Hamiltonian should be constructed from H and c_ops: $H_{eff} = H - \frac{i}{2} \sum_n \sigma_n^\dagger \sigma_n$ Default: True.

Returns:

scattering_probfloat: The probability of scattering n photons from the system over the time range specified.

temporal_basis_vector(waveguide_emission_indices, n_time_bins, n_emissions=None)[source]

Generate a temporal basis vector for emissions at specified time bins into specified waveguides.

Parameters:

waveguide_emission_indiceslist or tuple: List of indices where photon emission occurs for each waveguide, e.g. [[t1_wg1], [t1_wg2, t2_wg2], [], [t1_wg4, t2_wg4, t3_wg4]].
n_time_binsint: Number of time bins; the range over which each index can vary.

Returns:

temporal_basis_vector:class: qutip.Qobj: A basis vector representing photon scattering at the specified indices. If there are W waveguides, T times, and N photon emissions, then the basis vector has dimensionality (W*T)^N.

temporal_scattered_state(H, psi0, n_emissions, c_ops, tlist, system_zero_state=None, construct_effective_hamiltonian=True)[source]

Compute the scattered n-photon state projected onto the temporal basis.

Parameters:

H:class: qutip.Qobj or list: System-waveguide(s) Hamiltonian or effective Hamiltonian in Qobj or list-callback format. If construct_effective_hamiltonian is not specified, an effective Hamiltonian is constructed from H and c_ops.
psi0:class: qutip.Qobj: Initial state density matrix $\rho(t_0)$ or state vector $\psi(t_0)$.
n_emissionsint: Number of photon emissions to calculate.
c_opslist: List of collapse operators for each waveguide; these are assumed to include spontaneous decay rates, e.g. $\sigma = \sqrt \gamma \cdot a$
tlistarray_like: List of times for $\tau_i$. tlist should contain 0 and exceed the pulse duration / temporal region of interest.
system_zero_state:class: qutip.Qobj: State representing zero excitations in the system. Defaults to $\psi(t_0)$
construct_effective_hamiltonianbool: Whether an effective Hamiltonian should be constructed from H and c_ops: $H_{eff} = H - \frac{i}{2} \sum_n \sigma_n^\dagger \sigma_n$ Default: True.

Returns:

phi_n:class: qutip.Qobj: The scattered bath state projected onto the temporal basis given by tlist. If there are W waveguides, T times, and N photon emissions, then the state is a tensor product state with dimensionality T^(W*N).

Permutational Invariance

Permutational Invariant Quantum Solver (PIQS)

This module calculates the Liouvillian for the dynamics of ensembles of identical two-level systems (TLS) in the presence of local and collective processes by exploiting permutational symmetry and using the Dicke basis. It also allows to characterize nonlinear functions of the density matrix.

am(j, m)[source]

Calculate the operator am used later.

The action of ap is given by: $J_{-}\lvert j,m\rangle = A_{-}(jm)\lvert j,m-1\rangle$

Parameters:

j: float: The value for j.
m: float: The value for m.

Returns:

a_minus: float: The value of $a_{-}$.

ap(j, m)[source]

Calculate the coefficient ap by applying $J_+\lvert j,m\rangle$.

The action of ap is given by: $J_{+}\lvert j, m\rangle = A_{+}(j, m) \lvert j, m+1\rangle$

Parameters:

j, m: float: The value for j and m in the dicke basis $\lvert j, m\rangle$.

Returns:

a_plus: float: The value of $a_{+}$.

block_matrix(N, elements='ones')[source]

Construct the block-diagonal matrix for the Dicke basis.

Parameters:

Nint: Number of two-level systems.
elementsstr {‘ones’ (default),’degeneracy’}

Returns:

block_matrndarray: A 2D block-diagonal matrix with dimension (nds,nds), where nds is the number of Dicke states for N two-level systems. Filled with ones or the value of degeneracy at each matrix element.

collapse_uncoupled(N, emission=0.0, dephasing=0.0, pumping=0.0, collective_emission=0.0, collective_dephasing=0.0, collective_pumping=0.0)[source]

Create the collapse operators (c_ops) of the Lindbladian in the uncoupled basis

These operators are in the uncoupled basis of the two-level system (TLS) SU(2) Pauli matrices.

Parameters:

N: int: The number of two-level systems.
emission: float: Incoherent emission coefficient (also nonradiative emission). default: 0.0
dephasing: float: Local dephasing coefficient. default: 0.0
pumping: float: Incoherent pumping coefficient. default: 0.0
collective_emission: float: Collective (superradiant) emmission coefficient. default: 0.0
collective_pumping: float: Collective pumping coefficient. default: 0.0
collective_dephasing: float: Collective dephasing coefficient. default: 0.0

Returns:

c_ops: list: The list of collapse operators as qutip.Qobj for the system.

Notes

The collapse operator list can be given to qutip.mesolve. Notice that the operators are placed in a Hilbert space of dimension $2^N$. Thus the method is suitable only for small N (of the order of 10).

css(N, x=0.7071067811865475, y=0.7071067811865475, basis='dicke', coordinates='cartesian')[source]

Generate the density matrix of the Coherent Spin State (CSS).

It can be defined as, $\lvert CSS\rangle = \prod_i^N(a\lvert1\rangle_i+b\lvert0\rangle_i)$ with $a = sin(\frac{\theta}{2})$, $b = e^{i \phi}\cos(\frac{\theta}{2})$. The default basis is that of Dicke space $\lvert j, m\rangle \langle j, m'\rvert$. The default state is the symmetric CSS, $\lvert CSS\rangle = \lvert+\rangle$.

Parameters:

N: int: The number of two-level systems.
x, y: float: The coefficients of the CSS state.
basis: str: The basis to use. Either “dicke” or “uncoupled”.
coordinates: str: Either “cartesian” or “polar”. If polar then the coefficients are constructed as sin(x/2), cos(x/2)e^(iy).

Returns:

rho: :class: qutip.Qobj: The CSS state density matrix.

dicke(N, j, m)[source]

Generate a Dicke state as a pure density matrix in the Dicke basis.

For instance, the superradiant state given by $\lvert j, m\rangle = \lvert 1, 0\rangle$ for N = 2, and the state is represented as a density matrix of size (nds, nds) or (4, 4), with the (1, 1) element set to 1.

Parameters:

N: int: The number of two-level systems.
j: float: The eigenvalue j of the Dicke state (j, m).
m: float: The eigenvalue m of the Dicke state (j, m).

Returns:

rho: :class: qutip.Qobj: The density matrix.

dicke_basis(N, jmm1=None)[source]

Initialize the density matrix of a Dicke state for several (j, m, m1).

This function can be used to build arbitrary states in the Dicke basis $\lvert j, m\rangle\langle j, m'\rvert$. We create coefficients for each (j, m, m1) value in the dictionary jmm1. The mapping for the (i, k) index of the density matrix to the $\lvert j, m\rangle$ values is given by the cythonized function jmm1_dictionary. A density matrix is created from the given dictionary of coefficients for each (j, m, m1).

Parameters:

N: int: The number of two-level systems.
jmm1: dict: A dictionary of {(j, m, m1): p} that gives a density p for the (j, m, m1) matrix element.

Returns:

rho: :class: qutip.Qobj: The density matrix in the Dicke basis.

dicke_blocks(rho)[source]

Create the list of blocks for block-diagonal density matrix in the Dicke basis.

Parameters:

rhoqutip.Qobj: A 2D block-diagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N two-level systems.

Returns:

square_blocks: list of np.array: Give back the blocks list.

dicke_blocks_full(rho)[source]

Give the full (2^N-dimensional) list of blocks for a Dicke-basis matrix.

Parameters:

rhoqutip.Qobj: A 2D block-diagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N two-level systems.

Returns:

full_blockslist: The list of blocks expanded in the 2^N space for N qubits.

dicke_function_trace(f, rho)[source]

Calculate the trace of a function on a Dicke density matrix. :param f: A Taylor-expandable function of rho. :type f: function :param rho: A density matrix in the Dicke basis. :type rho: qutip.Qobj

Returns:

resfloat: Trace of a nonlinear function on rho.

energy_degeneracy(N, m)[source]

Calculate the number of Dicke states with same energy.

The use of the Decimals class allows to explore N > 1000, unlike the built-in function scipy.special.binom

Parameters:

N: int: The number of two-level systems.
m: float: Total spin z-axis projection eigenvalue. This is proportional to the total energy.

Returns:

degeneracy: int: The energy degeneracy

entropy_vn_dicke(rho)[source]

Von Neumann Entropy of a Dicke-basis density matrix.

Parameters:

rhoqutip.Qobj: A 2D block-diagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N two-level systems.

Returns:

entropy_dm: float: Entropy. Use degeneracy to multiply each block.

excited(N, basis='dicke')[source]

Generate the density matrix for the excited state.

This state is given by (N/2, N/2) in the default Dicke basis. If the argument basis is “uncoupled” then it generates the state in a 2**N dim Hilbert space.

Parameters:

N: int: The number of two-level systems.
basis: str: The basis to use. Either “dicke” or “uncoupled”.

Returns:

state: :class: qutip.Qobj: The excited state density matrix in the requested basis.

ghz(N, basis='dicke')[source]

Generate the density matrix of the GHZ state.

If the argument basis is “uncoupled” then it generates the state in a $2^N$-dimensional Hilbert space.

Parameters:

N: int: The number of two-level systems.
basis: str: The basis to use. Either “dicke” or “uncoupled”.

Returns:

state: :class: qutip.Qobj: The GHZ state density matrix in the requested basis.

ground(N, basis='dicke')[source]

Generate the density matrix of the ground state.

This state is given by (N/2, -N/2) in the Dicke basis. If the argument basis is “uncoupled” then it generates the state in a $2^N$-dimensional Hilbert space.

Parameters:

N: int: The number of two-level systems.
basis: str: The basis to use. Either “dicke” or “uncoupled”

Returns:

state: :class: qutip.Qobj: The ground state density matrix in the requested basis.

identity_uncoupled(N)[source]

Generate the identity in a $2^N$-dimensional Hilbert space.

The identity matrix is formed from the tensor product of N TLSs.

Parameters:

N: int: The number of two-level systems.

Returns:

identity: :class: qutip.Qobj: The identity matrix.

isdiagonal(mat)[source]

Check if the input matrix is diagonal.

Parameters:

mat: ndarray/Qobj: A 2D numpy array

Returns:

diag: bool: True/False depending on whether the input matrix is diagonal.

jspin(N, op=None, basis='dicke')[source]

Calculate the list of collective operators of the total algebra.

The Dicke basis $\lvert j,m\rangle\langle j,m'\rvert$ is used by default. Otherwise with “uncoupled” the operators are in a $2^N$ space.

Parameters:

N: int: Number of two-level systems.
op: str: The operator to return ‘x’,’y’,’z’,’+’,’-‘. If no operator given, then output is the list of operators for [‘x’,’y’,’z’].
basis: str: The basis of the operators - “dicke” or “uncoupled” default: “dicke”.

Returns:

j_alg: list or :class: qutip.Qobj: A list of qutip.Qobj representing all the operators in the “dicke” or “uncoupled” basis or a single operator requested.

m_degeneracy(N, m)[source]

Calculate the number of Dicke states $\lvert j, m\rangle$ with same energy.

Parameters:

N: int: The number of two-level systems.
m: float: Total spin z-axis projection eigenvalue (proportional to the total energy).

Returns:

degeneracy: int: The m-degeneracy.

num_dicke_ladders(N)[source]

Calculate the total number of ladders in the Dicke space.

For a collection of N two-level systems it counts how many different “j” exist or the number of blocks in the block-diagonal matrix.

Parameters:

N: int: The number of two-level systems.

Returns:

Nj: int: The number of Dicke ladders.

num_dicke_states(N)[source]

Calculate the number of Dicke states.

Parameters:

N: int: The number of two-level systems.

Returns:

nds: int: The number of Dicke states.

num_tls(nds)[source]

Calculate the number of two-level systems.

Parameters:

nds: int: The number of Dicke states.

Returns:

N: int: The number of two-level systems.

purity_dicke(rho)[source]

Calculate purity of a density matrix in the Dicke basis. It accounts for the degenerate blocks in the density matrix.

Parameters:

rhoqutip.Qobj: Density matrix in the Dicke basis of qutip.piqs.jspin(N), for N spins.

Returns:

purityfloat: The purity of the quantum state. It’s 1 for pure states, 0<=purity<1 for mixed states.

spin_algebra(N, op=None)[source]

Create the list [sx, sy, sz] with the spin operators.

The operators are constructed for a collection of N two-level systems (TLSs). Each element of the list, i.e., sx, is a vector of qutip.Qobj objects (spin matrices), as it cointains the list of the SU(2) Pauli matrices for the N TLSs. Each TLS operator sx[i], with i = 0, …, (N-1), is placed in a $2^N$-dimensional Hilbert space.

Parameters:

N: int: The number of two-level systems.

Returns:

spin_operators: list or :class: qutip.Qobj: A list of qutip.Qobj operators - [sx, sy, sz] or the requested operator.

Notes

sx[i] is $\frac{\sigma_x}{2}$ in the composite Hilbert space.

state_degeneracy(N, j)[source]

Calculate the degeneracy of the Dicke state.

Each state $\lvert j, m\rangle$ includes D(N,j) irreducible representations $\lvert j, m, \alpha\rangle$.

Uses Decimals to calculate higher numerator and denominators numbers.

Parameters:

N: int: The number of two-level systems.
j: float: Total spin eigenvalue (cooperativity).

Returns:

degeneracy: int: The state degeneracy.

superradiant(N, basis='dicke')[source]

Generate the density matrix of the superradiant state.

This state is given by (N/2, 0) or (N/2, 0.5) in the Dicke basis. If the argument basis is “uncoupled” then it generates the state in a 2**N dim Hilbert space.

Parameters:

N: int: The number of two-level systems.
basis: str: The basis to use. Either “dicke” or “uncoupled”.

Returns:

state: :class: qutip.Qobj: The superradiant state density matrix in the requested basis.

tau_column(tau, k, j)[source]

Determine the column index for the non-zero elements of the matrix for a particular row k and the value of j from the Dicke space.

Parameters:

tau: str: The tau function to check for this k and j.
k: int: The row of the matrix M for which the non zero elements have to be calculated.
j: float: The value of j for this row.

Visualization

Pseudoprobability Functions

qfunc(state: Qobj, xvec, yvec, g: float = 1.4142135623730951, precompute_memory: float = 1024)[source]

Husimi-Q function of a given state vector or density matrix at phase-space points 0.5 * g * (xvec + i*yvec).

Parameters:

stateQobj: A state vector or density matrix. This cannot have tensor-product structure.
xvec, yvecarray_like: x- and y-coordinates at which to calculate the Husimi-Q function.
gfloat, default sqrt(2): Scaling factor for a = 0.5 * g * (x + iy). The value of g is related to the value of $\hbar$ in the commutation relation $[x,\,y] = i\hbar$ via $\hbar=2/g^2$, so the default corresponds to $\hbar=1$.
precompute_memoryreal, default 1024: Size in MB that may be used during calculations as working space when dealing with density-matrix inputs. This is ignored for state-vector inputs. The bound is not quite exact due to other, order-of-magnitude smaller, intermediaries being necessary, but is a good approximation. If you want to use the same iterative algorithm for density matrices that is used for single kets, set precompute_memory=None.

Returns:

ndarray: Values representing the Husimi-Q function calculated over the specified range [xvec, yvec].

See also

QFunc: a class-based version, more efficient if you want to calculate the Husimi-Q function for several states over the same coordinates.

spin_q_function(rho, theta, phi)[source]

The Husimi Q function for spins is defined as Q(theta, phi) = SCS.dag() * rho * SCS for the spin coherent state SCS = spin_coherent( j, theta, phi) where j is the spin length. The implementation here is more efficient as it doesn’t generate all of the SCS at theta and phi (see references).

The spin Q function is normal when integrated over the surface of the sphere

\[\frac{4 \pi}{2j + 1}\int_\phi \int_\theta Q(\theta, \phi) \sin(\theta) d\theta d\phi = 1\]

Parameters:

stateqobj: A state vector or density matrix for a spin-j quantum system.
thetaarray_like: Polar (colatitude) angle at which to calculate the Husimi-Q function.
phiarray_like: Azimuthal angle at which to calculate the Husimi-Q function.

Returns:

Q, THETA, PHI2d-array: Values representing the spin Husimi Q function at the values specified by THETA and PHI.

References

[1] Lee Loh, Y., & Kim, M. (2015). American J. of Phys., 83(1), 30–35. https://doi.org/10.1119/1.4898595

spin_wigner(rho, theta, phi)[source]

Wigner function for a spin-j system.

The spin W function is normal when integrated over the surface of the sphere

\[\sqrt{\frac{4 \pi}{2j + 1}}\int_\phi \int_\theta W(\theta,\phi) \sin(\theta) d\theta d\phi = 1\]

Parameters:

stateqobj: A state vector or density matrix for a spin-j quantum system.
thetaarray_like: Polar (colatitude) angle at which to calculate the W function.
phiarray_like: Azimuthal angle at which to calculate the W function.

Returns:

W, THETA, PHI2d-array: Values representing the spin Wigner function at the values specified by THETA and PHI.

References

[1] Agarwal, G. S. (1981). Phys. Rev. A, 24(6), 2889–2896. https://doi.org/10.1103/PhysRevA.24.2889

[2] Dowling, J. P., Agarwal, G. S., & Schleich, W. P. (1994). Phys. Rev. A, 49(5), 4101–4109. https://doi.org/10.1103/PhysRevA.49.4101

[3] Conversion between Wigner 3-j symbol and Clebsch-Gordan coefficients taken from Wikipedia (https://en.wikipedia.org/wiki/3-j_symbol)

wigner(psi, xvec, yvec, method='clenshaw', g=1.4142135623730951, sparse=False, parfor=False)[source]

Wigner function for a state vector or density matrix at points xvec + i * yvec.

Parameters:

stateqobj: A state vector or density matrix.
xvecarray_like: x-coordinates at which to calculate the Wigner function.
yvecarray_like: y-coordinates at which to calculate the Wigner function. Does not apply to the ‘fft’ method.
gfloat: Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g^2 giving the default value hbar=1.
methodstring {‘clenshaw’, ‘iterative’, ‘laguerre’, ‘fft’}: Select method ‘clenshaw’ ‘iterative’, ‘laguerre’, or ‘fft’, where ‘clenshaw’ and ‘iterative’ use an iterative method to evaluate the Wigner functions for density matrices $|m><n|$, while ‘laguerre’ uses the Laguerre polynomials in scipy for the same task. The ‘fft’ method evaluates the Fourier transform of the density matrix. The ‘iterative’ method is default, and in general recommended, but the ‘laguerre’ method is more efficient for very sparse density matrices (e.g., superpositions of Fock states in a large Hilbert space). The ‘clenshaw’ method is the preferred method for dealing with density matrices that have a large number of excitations (>~50). ‘clenshaw’ is a fast and numerically stable method.
sparsebool {False, True}: Tells the default solver whether or not to keep the input density matrix in sparse format. As the dimensions of the density matrix grow, setthing this flag can result in increased performance.
parforbool {False, True}: Flag for calculating the Laguerre polynomial based Wigner function method=’laguerre’ in parallel using the parfor function.

Returns:

Warray: Values representing the Wigner function calculated over the specified range [xvec,yvec].
yvexarray: FFT ONLY. Returns the y-coordinate values calculated via the Fourier transform.

Notes

The ‘fft’ method accepts only an xvec input for the x-coordinate. The y-coordinates are calculated internally.

References

Ulf Leonhardt, Measuring the Quantum State of Light, (Cambridge University Press, 1997)

Graphs and Visualization

Functions for visualizing results of quantum dynamics simulations, visualizations of quantum states and processes.

hinton(rho, x_basis=None, y_basis=None, color_style='scaled', label_top=True, *, cmap=None, colorbar=True, fig=None, ax=None)[source]

Draws a Hinton diagram to visualize a density matrix or superoperator.

Parameters:

rhoqobj

Input density matrix or superoperator.

Note

Hinton plots of superoperators are currently only supported for qubits.

x_basislist of strings, optional

list of x ticklabels to represent x basis of the input.

y_basislist of strings, optional

list of y ticklabels to represent y basis of the input.

color_stylestring, default=”scaled”

Determines how colors are assigned to each square:

If set to "scaled" (default), each color is chosen by passing the absolute value of the corresponding matrix element into cmap with the sign of the real part.
If set to "threshold", each square is plotted as the maximum of cmap for the positive real part and as the minimum for the negative part of the matrix element; note that this generalizes “threshold” to complex numbers.
If set to "phase", each color is chosen according to the angle of the corresponding matrix element.

label_topbool, default=True

If True, x ticklabels will be placed on top, otherwise they will appear below the plot.

cmapa matplotlib colormap instance, optional

Color map to use when plotting.

colorbarbool, default=True

Whether (True) or not (False) a colorbar should be attached.

figa matplotlib Figure instance, optional

The Figure canvas in which the plot will be drawn.

axa matplotlib axes instance, optional

The ax context in which the plot will be drawn.

Returns:

fig, outputtuple: A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.

Raises:

ValueError: Input argument is not a quantum object.

Examples

>>> import qutip
>>>
>>> dm = qutip.rand_dm(4)
>>> fig, ax = qutip.hinton(dm)
>>> fig.show()
>>>
>>> qutip.settings.colorblind_safe = True
>>> fig, ax = qutip.hinton(dm, color_style="threshold")
>>> fig.show()
>>> qutip.settings.colorblind_safe = False
>>>
>>> fig, ax = qutip.hinton(dm, color_style="phase")
>>> fig.show()

matrix_histogram(M, x_basis=None, y_basis=None, limits=None, bar_style='real', color_limits=None, color_style='real', options=None, *, cmap=None, colorbar=True, fig=None, ax=None)[source]

Draw a histogram for the matrix M, with the given x and y labels and title.

Parameters:

MMatrix of Qobj

The matrix to visualize

x_basislist of strings, optional

list of x ticklabels

y_basislist of strings, optional

list of y ticklabels

limitslist/array with two float numbers, optional

The z-axis limits [min, max]

bar_stylestring, default=”real”

If set to "real" (default), each bar is plotted as the real part of the corresponding matrix element
If set to "img", each bar is plotted as the imaginary part of the corresponding matrix element
If set to "abs", each bar is plotted as the absolute value of the corresponding matrix element
If set to "phase" (default), each bar is plotted as the angle of the corresponding matrix element

color_limitslist/array with two float numbers, optional

The limits of colorbar [min, max]

color_stylestring, default=”real”

Determines how colors are assigned to each square:

If set to "real" (default), each color is chosen according to the real part of the corresponding matrix element.
If set to "img", each color is chosen according to the imaginary part of the corresponding matrix element.
If set to "abs", each color is chosen according to the absolute value of the corresponding matrix element.
If set to "phase", each color is chosen according to the angle of the corresponding matrix element.

cmapa matplotlib colormap instance, optional

Color map to use when plotting.

colorbarbool, default=True

show colorbar

figa matplotlib Figure instance, optional

The Figure canvas in which the plot will be drawn.

axa matplotlib axes instance, optional

The axes context in which the plot will be drawn.

optionsdict, optional

A dictionary containing extra options for the plot. The names (keys) and values of the options are described below:

‘zticks’list of numbers, optional: A list of z-axis tick locations.
‘bars_spacing’float, default=0.1: spacing between bars.
‘bars_alpha’float, default=1.: transparency of bars, should be in range 0 - 1
‘bars_lw’float, default=0.5: linewidth of bars’ edges.
‘bars_edgecolor’color, default=’k’: The colors of the bars’ edges. Examples: ‘k’, (0.1, 0.2, 0.5) or ‘#0f0f0f80’.
‘shade’bool, default=True: Whether to shade the dark sides of the bars (True) or not (False). The shading is relative to plot’s source of light.
‘azim’float, default=-35: The azimuthal viewing angle.
‘elev’float, default=35: The elevation viewing angle.
‘stick’bool, default=False: Changes xlim and ylim in such a way that bars next to XZ and YZ planes will stick to those planes. This option has no effect if ax is passed as a parameter.
‘cbar_pad’float, default=0.04: The fraction of the original axes between the colorbar and the new image axes. (i.e. the padding between the 3D figure and the colorbar).
‘cbar_to_z’bool, default=False: Whether to set the color of maximum and minimum z-values to the maximum and minimum colors in the colorbar (True) or not (False).
‘threshold’: float, optional: Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.

Returns:

fig, outputtuple: A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.

Raises:

ValueError: Input argument is not valid.

plot_energy_levels(H_list, h_labels=None, energy_levels=None, N=0, *, fig=None, ax=None)[source]

Plot the energy level diagrams for a list of Hamiltonians. Include up to N energy levels. For each element in H_list, the energy levels diagram for the cummulative Hamiltonian sum(H_list[0:n]) is plotted, where n is the index of an element in H_list.

Parameters:

H_listList of Qobj: A list of Hamiltonians.
h_lablesList of string, optional: A list of xticklabels for each Hamiltonian
energy_levelsList of string, optional: A list of yticklabels to the left of energy levels of the initial Hamiltonian.
Nint, default=0: The number of energy levels to plot
figa matplotlib Figure instance, optional: The Figure canvas in which the plot will be drawn.
axa matplotlib axes instance, optional: The axes context in which the plot will be drawn.

Returns:

fig, axtuple: A tuple of the matplotlib figure and axes instances used to produce the figure.

Raises:

ValueError: Input argument is not valid.

plot_expectation_values(results, ylabels=None, *, fig=None, axes=None)[source]

Visualize the results (expectation values) for an evolution solver. results is assumed to be an instance of Result, or a list of Result instances. :param results: List of results objects returned by any of the QuTiP evolution solvers. :type results: (list of) qutip.solver.Result :param ylabels: The y-axis labels. List should be of the same length as results. :type ylabels: list of strings, optional :param fig: The Figure canvas in which the plot will be drawn. :type fig: a matplotlib Figure instance, optional :param axes: The axes context in which the plot will be drawn. :type axes: (list of) axes instances, optional

Returns:

fig, axestuple: A tuple of the matplotlib figure and array of axes instances used to produce the figure.

plot_fock_distribution(rho, fock_numbers=None, color='green', unit_y_range=True, *, fig=None, ax=None)[source]

Plot the Fock distribution for a density matrix (or ket) that describes an oscillator mode.

Parameters:

rhoqutip.Qobj: The density matrix (or ket) of the state to visualize.
fock_numberslist of strings, optional: list of x ticklabels to represent fock numbers
colorcolor or list of colors, default=”green”: The colors of the bar faces.
unit_y_rangebool, default=True: Set y-axis limits [0, 1] or not
figa matplotlib Figure instance, optional: The Figure canvas in which the plot will be drawn.
axa matplotlib axes instance, optional: The axes context in which the plot will be drawn.

Returns:

fig, outputtuple: A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.

plot_qubism(ket, theme='light', how='pairs', grid_iteration=1, legend_iteration=0, *, fig=None, ax=None)[source]

Qubism plot for pure states of many qudits. Works best for spin chains, especially with even number of particles of the same dimension. Allows to see entanglement between first 2k particles and the rest.

Note

colorblind_safe does not apply because of its unique colormap

Parameters:

ketQobj

Pure state for plotting.

theme‘light’ or ‘dark’, default=’light’

Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.

how‘pairs’, ‘pairs_skewed’ or ‘before_after’, default=’pairs’

Type of Qubism plotting. Options:

‘pairs’ - typical coordinates,
‘pairs_skewed’ - for ferromagnetic/antriferromagnetic plots,
‘before_after’ - related to Schmidt plot (see also: plot_schmidt).

grid_iterationint, default=1

Helper lines to be drawn on plot. Show tiles for 2*grid_iteration particles vs all others.

legend_iterationint or ‘grid_iteration’ or ‘all’, default=0

Show labels for first 2*legend_iteration particles. Option ‘grid_iteration’ sets the same number of particles as for grid_iteration. Option ‘all’ makes label for all particles. Typically it should be 0, 1, 2 or perhaps 3.

figa matplotlib figure instance, optional

The figure canvas on which the plot will be drawn.

axa matplotlib axis instance, optional

The axis context in which the plot will be drawn.

Returns:

fig, outputtuple: A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.

Notes

Quantum Process Tomography

qpt(U, op_basis_list)[source]

Calculate the quantum process tomography chi matrix for a given (possibly nonunitary) transformation matrix U, which transforms a density matrix in vector form according to:

vec(rho) = U * vec(rho0)

or

rho = unstack_columns(U * stack_columns(rho0))

U can be calculated for an open quantum system using the QuTiP propagator function.

Parameters:

UQobj: Transformation operator. Can be calculated using QuTiP propagator function.
op_basis_listlist: A list of Qobj’s representing the basis states.

Returns:

chiarray: QPT chi matrix

qpt_plot(chi, lbls_list, title=None, fig=None, axes=None)[source]

Visualize the quantum process tomography chi matrix. Plot the real and imaginary parts separately.

Parameters:

chiarray: Input QPT chi matrix.
lbls_listlist: List of labels for QPT plot axes.
titlestring: Plot title.
figfigure instance: User defined figure instance used for generating QPT plot.
axeslist of figure axis instance: User defined figure axis instance (list of two axes) used for generating QPT plot.

Returns:

fig, axtuple: A tuple of the matplotlib figure and axes instances used to produce the figure.

qpt_plot_combined(chi, lbls_list, title=None, fig=None, ax=None, figsize=(8, 6), threshold=None)[source]

Visualize the quantum process tomography chi matrix. Plot bars with height and color corresponding to the absolute value and phase, respectively.

Parameters:

chiarray: Input QPT chi matrix.
lbls_listlist: List of labels for QPT plot axes.
titlestring: Plot title.
figfigure instance: User defined figure instance used for generating QPT plot.
axfigure axis instance: User defined figure axis instance used for generating QPT plot (alternative to the fig argument).
threshold: float (None): Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.

Returns:

fig, axtuple: A tuple of the matplotlib figure and axes instances used to produce the figure.

Non-Markovian Solvers

This module contains an implementation of the non-Markovian transfer tensor method (TTM), introduced in [1].

[1] Javier Cerrillo and Jianshu Cao, Phys. Rev. Lett 112, 110401 (2014)

ttmsolve(dynmaps, state0, times, e_ops=[], num_learning=0, options=None)[source]

Expand time-evolution using the Transfer Tensor Method [1], based on a set of precomputed dynamical maps.

Parameters:

dynmapslist of qutip.Qobj, callable

List of precomputed dynamical maps (superoperators) for the first times of times or a callback function that returns the superoperator at a given time.

state0qutip.Qobj

Initial density matrix or state vector (ket).

timesarray_like

List of times $t_n$ at which to compute results. Must be uniformily spaced.

e_opsqutip.qobj, callable, or list.

Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See expect for more detail of operator expectation.

num_learningint

Number of times used to construct the dynmaps operators when dynmaps is a callable.

optionsdictionary

Dictionary of options for the solver.

store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool Normalize output state to hide ODE numerical errors.
threshold : float Threshold for halting. Halts if $||T_{n}-T_{n-1}||$ is below treshold.

Returns:

output: qutip.solver.Result: An instance of the class qutip.solver.Result.

[1]

Javier Cerrillo and Jianshu Cao, Phys. Rev. Lett 112, 110401 (2014) ..

Utility Functions

This module contains utility functions that are commonly needed in other qutip modules.

clebsch(j1, j2, j3, m1, m2, m3)[source]

Calculates the Clebsch-Gordon coefficient for coupling (j1,m1) and (j2,m2) to give (j3,m3).

Parameters:

j1float: Total angular momentum 1.
j2float: Total angular momentum 2.
j3float: Total angular momentum 3.
m1float: z-component of angular momentum 1.
m2float: z-component of angular momentum 2.
m3float: z-component of angular momentum 3.

Returns:

cg_coefffloat: Requested Clebsch-Gordan coefficient.

convert_unit(value, orig='meV', to='GHz')[source]

Convert an energy from unit orig to unit to.

Parameters:

valuefloat / array: The energy in the old unit.
origstring: The name of the original unit (“J”, “eV”, “meV”, “GHz”, “mK”)
tostring: The name of the new unit (“J”, “eV”, “meV”, “GHz”, “mK”)

Returns:

value_new_unitfloat / array: The energy in the new unit.

n_thermal(w, w_th)[source]

Return the number of photons in thermal equilibrium for an harmonic oscillator mode with frequency ‘w’, at the temperature described by ‘w_th’ where $\omega_{\rm th} = k_BT/\hbar$.

Parameters:

wfloat or array: Frequency of the oscillator.
w_thfloat: The temperature in units of frequency (or the same units as w).

Returns:

n_avgfloat or array: Return the number of average photons in thermal equilibrium for a an oscillator with the given frequency and temperature.

File I/O Functions

file_data_read(filename, sep=None)[source]

Retrieves an array of data from the requested file.

Parameters:

filenamestr or pathlib.Path: Name of file containing reqested data.
sepstr: Seperator used to store data.

Returns:

dataarray_like: Data from selected file.

file_data_store(filename, data, numtype='complex', numformat='decimal', sep=',')[source]

Stores a matrix of data to a file to be read by an external program.

Parameters:

filenamestr or pathlib.Path: Name of data file to be stored, including extension.
data: array_like: Data to be written to file.
numtypestr {‘complex, ‘real’}: Type of numerical data.
numformatstr {‘decimal’,’exp’}: Format for written data.
sepstr: Single-character field seperator. Usually a tab, space, comma, or semicolon.

qload(name)[source]

Loads data file from file named ‘filename.qu’ in current directory.

Parameters:

namestr or pathlib.Path: Name of data file to be loaded.

Returns:

qobjectinstance / array_like: Object retrieved from requested file.

qsave(data, name='qutip_data')[source]

Saves given data to file named ‘filename.qu’ in current directory.

Parameters:

datainstance/array_like: Input Python object to be stored.
filenamestr or pathlib.Path: Name of output data file.

Parallelization

This module provides functions for parallel execution of loops and function mappings, using the builtin Python module multiprocessing or the loky parallel execution library.

parallel_map(task, values, task_args=None, task_kwargs=None, reduce_func=None, map_kw=None, progress_bar=None, progress_bar_kwargs={})[source]

Parallel execution of a mapping of values to the function task. This is functionally equivalent to:

result = [task(value, *task_args, **task_kwargs) for value in values]

Parameters:

taska Python function: The function that is to be called for each value in task_vec.
valuesarray / list: The list or array of values for which the task function is to be evaluated.
task_argslist / dictionary: The optional additional argument to the task function.
task_kwargslist / dictionary: The optional additional keyword argument to the task function.
reduce_funcfunc (optional): If provided, it will be called with the output of each tasks instead of storing a them in a list. Note that the order in which results are passed to reduce_func is not defined. It should return None or a number. When returning a number, it represent the estimation of the number of task left. On a return <= 0, the map will end early.
progress_barstring: Progress bar options’s string for showing progress.
progress_bar_kwargsdict: Options for the progress bar.
map_kw: dict (optional): Dictionary containing entry for: - timeout: float, Maximum time (sec) for the whole map. - job_timeout: float, Maximum time (sec) for each job in the map. - num_cpus: int, Number of job to run at once. - fail_fast: bool, Raise an error at the first.

Returns:

resultlist: The result list contains the value of task(value, *task_args, **task_kwargs) for each value in values. If a reduce_func is provided, and empty list will be returned.

serial_map(task, values, task_args=None, task_kwargs=None, reduce_func=None, map_kw=None, progress_bar=None, progress_bar_kwargs={})[source]

Serial mapping function with the same call signature as parallel_map, for easy switching between serial and parallel execution. This is functionally equivalent to:

result = [task(value, *task_args, **task_kwargs) for value in values]

This function work as a drop-in replacement of qutip.parallel_map.

Parameters:

taska Python function: The function that is to be called for each value in task_vec.
valuesarray / list: The list or array of values for which the task function is to be evaluated.
task_argslist / dictionary: The optional additional argument to the task function.
task_kwargslist / dictionary: The optional additional keyword argument to the task function.
reduce_funcfunc (optional): If provided, it will be called with the output of each tasks instead of storing a them in a list. It should return None or a number. When returning a number, it represent the estimation of the number of task left. On a return <= 0, the map will end early.
progress_barstring: Progress bar options’s string for showing progress.
progress_bar_kwargsdict: Options for the progress bar.
map_kw: dict (optional): Dictionary containing: - timeout: float, Maximum time (sec) for the whole map. - fail_fast: bool, Raise an error at the first.

Returns:

resultlist: The result list contains the value of task(value, *task_args, **task_kwargs) for each value in values. If a reduce_func is provided, and empty list will be returned.

Semidefinite Programming

IPython Notebook Tools

This module contains utility functions for using QuTiP with IPython notebooks.

parallel_map(task, values, task_args=None, task_kwargs=None, client=None, view=None, progress_bar=None, show_scheduling=False, **kwargs)[source]

Call the function task for each value in values using a cluster of IPython engines. The function task should have the signature task(value, *args, **kwargs).

The client and view are the IPython.parallel client and load-balanced view that will be used in the parfor execution. If these are None, new instances will be created.

Parameters:

task: a Python function: The function that is to be called for each value in task_vec.
values: array / list: The list or array of values for which the task function is to be evaluated.
task_args: list / dictionary: The optional additional argument to the task function.
task_kwargs: list / dictionary: The optional additional keyword argument to the task function.
client: IPython.parallel.Client: The IPython.parallel Client instance that will be used in the parfor execution.
view: a IPython.parallel.Client view: The view that is to be used in scheduling the tasks on the IPython cluster. Preferably a load-balanced view, which is obtained from the IPython.parallel.Client instance client by calling, view = client.load_balanced_view().
show_scheduling: bool {False, True}, default False: Display a graph showing how the tasks (the evaluation of task for for the value in task_vec1) was scheduled on the IPython engine cluster.
show_progressbar: bool {False, True}, default False: Display a HTML-based progress bar during the execution of the parfor loop.

Returns:

resultlist: The result list contains the value of task(value, task_args, task_kwargs) for each value in values.

version_table(verbose=False)[source]

Print an HTML-formatted table with version numbers for QuTiP and its dependencies. Use it in a IPython notebook to show which versions of different packages that were used to run the notebook. This should make it possible to reproduce the environment and the calculation later on.

Returns:

version_table: string: Return an HTML-formatted string containing version information for QuTiP dependencies.

Miscellaneous

about()[source]: About box for QuTiP. Gives version numbers for QuTiP, NumPy, SciPy, Cython, and MatPlotLib.

simdiag(ops, evals: bool = True, *, tol: float = 1e-14, safe_mode: bool = True)[source]

Simultaneous diagonalization of commuting Hermitian matrices.

Parameters:

opslist/array: list or array of qobjs representing commuting Hermitian operators.
evalsbool [True]: Whether to return the eigenvalues for each ops and eigenvectors or just the eigenvectors.
tolfloat [1e-14]: Tolerance for detecting degenerate eigenstates.
safe_modebool [True]: Whether to check that all ops are Hermitian and commuting. If set to False and operators are not commuting, the eigenvectors returned will often be eigenvectors of only the first operator.

Returns:

eigstuple: Tuple of arrays representing eigvecs and eigvals of quantum objects corresponding to simultaneous eigenvectors and eigenvalues for each operator.