# This file is part of QuTiP: Quantum Toolbox in Python.
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__all__ = ['basis', 'qutrit_basis', 'coherent', 'coherent_dm', 'fock_dm',
'fock', 'thermal_dm', 'maximally_mixed_dm', 'ket2dm', 'projection',
'qstate', 'ket', 'bra', 'state_number_enumerate',
'state_number_index', 'state_index_number', 'state_number_qobj',
'phase_basis', 'zero_ket', 'spin_state', 'spin_coherent',
'bell_state', 'singlet_state', 'triplet_states', 'w_state',
'ghz_state', 'enr_state_dictionaries', 'enr_fock',
'enr_thermal_dm']
import numbers
import numpy as np
from numpy import arange, conj, prod
import scipy.sparse as sp
from qutip.qobj import Qobj
from qutip.operators import destroy, jmat
from qutip.tensor import tensor
from qutip.fastsparse import fast_csr_matrix
def _promote_to_zero_list(arg, length):
"""
Ensure `arg` is a list of length `length`. If `arg` is None it is promoted
to `[0]*length`. All other inputs are checked that they match the correct
form.
Returns
-------
list_ : list
A list of integers of length `length`.
"""
if arg is None:
arg = [0]*length
elif not isinstance(arg, list):
arg = [arg]
if not len(arg) == length:
raise ValueError("All list inputs must be the same length.")
if all(isinstance(x, numbers.Integral) for x in arg):
return arg
raise TypeError("Dimensions must be an integer or list of integers.")
[docs]def basis(dimensions, n=None, offset=None):
"""Generates the vector representation of a Fock state.
Parameters
----------
dimensions : int 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.
n : int 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.
offset : int 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.
Returns
-------
state : :class:`qutip.Qobj`
Qobj representing the requested number state ``|n>``.
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.]]
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
"""
# Promote all parameters to lists to simplify later logic.
if not isinstance(dimensions, list):
dimensions = [dimensions]
n_dimensions = len(dimensions)
ns = [m-off for m, off in zip(_promote_to_zero_list(n, n_dimensions),
_promote_to_zero_list(offset, n_dimensions))]
if any((not isinstance(x, numbers.Integral)) or x < 0 for x in dimensions):
raise ValueError("All dimensions must be >= 0.")
if not all(0 <= n < dimension for n, dimension in zip(ns, dimensions)):
raise ValueError("All basis indices must be "
"`offset <= n < dimension+offset`.")
location, size = 0, 1
for m, dimension in zip(reversed(ns), reversed(dimensions)):
location += m*size
size *= dimension
data = np.array([1], dtype=complex)
ind = np.array([0], dtype=np.int32)
ptr = np.array([0]*(location+1) + [1]*(size-location), dtype=np.int32)
return Qobj(fast_csr_matrix((data, ind, ptr), shape=(size, 1)),
dims=[dimensions, [1]*n_dimensions], isherm=False)
[docs]def qutrit_basis():
"""Basis states for a three level system (qutrit)
Returns
-------
qstates : array
Array of qutrit basis vectors
"""
return np.array([basis(3, 0), basis(3, 1), basis(3, 2)], dtype=object)
[docs]def coherent(N, alpha, offset=0, method='operator'):
"""Generates a coherent state with eigenvalue alpha.
Constructed using displacement operator on vacuum state.
Parameters
----------
N : int
Number of Fock states in Hilbert space.
alpha : float/complex
Eigenvalue of coherent state.
offset : int (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'.
method : string {'operator', 'analytic'}
Method for generating coherent state.
Returns
-------
state : qobj
Qobj quantum object for coherent state
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 ]]
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.
"""
if method == "operator" and offset == 0:
x = basis(N, 0)
a = destroy(N)
D = (alpha * a.dag() - conj(alpha) * a).expm()
return D * x
elif method == "analytic" or offset > 0:
sqrtn = np.sqrt(np.arange(offset, offset+N, dtype=complex))
sqrtn[0] = 1 # Get rid of divide by zero warning
data = alpha/sqrtn
if offset == 0:
data[0] = np.exp(-abs(alpha)**2 / 2.0)
else:
s = np.prod(np.sqrt(np.arange(1, offset + 1))) # sqrt factorial
data[0] = np.exp(-abs(alpha)**2 / 2.0) * alpha**(offset) / s
np.cumprod(data, out=sqrtn) # Reuse sqrtn array
return Qobj(sqrtn)
else:
raise TypeError(
"The method option can only take values 'operator' or 'analytic'")
[docs]def coherent_dm(N, alpha, offset=0, method='operator'):
"""Density matrix representation of a coherent state.
Constructed via outer product of :func:`qutip.states.coherent`
Parameters
----------
N : int
Number of Fock states in Hilbert space.
alpha : float/complex
Eigenvalue for coherent state.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the state.
method : string {'operator', 'analytic'}
Method for generating coherent density matrix.
Returns
-------
dm : qobj
Density matrix representation of coherent state.
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\
]]
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.
"""
if method == "operator":
psi = coherent(N, alpha, offset=offset)
return psi * psi.dag()
elif method == "analytic":
psi = coherent(N, alpha, offset=offset, method='analytic')
return psi * psi.dag()
else:
raise TypeError(
"The method option can only take values 'operator' or 'analytic'")
[docs]def fock_dm(dimensions, n=None, offset=None):
"""Density matrix representation of a Fock state
Constructed via outer product of :func:`qutip.states.fock`.
Parameters
----------
dimensions : int 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.
n : int 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.
offset : int 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.
Returns
-------
dm : qobj
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]]
"""
psi = basis(dimensions, n, offset=offset)
return psi * psi.dag()
[docs]def fock(dimensions, n=None, offset=None):
"""Bosonic Fock (number) state.
Same as :func:`qutip.states.basis`.
Parameters
----------
dimensions : int 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.
n : int 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.
offset : int 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.
Returns
-------
Requested number state :math:`\\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]]
"""
return basis(dimensions, n, offset=offset)
[docs]def thermal_dm(N, n, method='operator'):
"""Density matrix for a thermal state of n particles
Parameters
----------
N : int
Number of basis states in Hilbert space.
n : float
Expectation value for number of particles in thermal state.
method : string {'operator', 'analytic'}
``string`` that sets the method used to generate the
thermal state probabilities
Returns
-------
dm : qobj
Thermal state density matrix.
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]]
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.
"""
if n == 0:
return fock_dm(N, 0)
else:
i = arange(N)
if method == 'operator':
beta = np.log(1.0 / n + 1.0)
diags = np.exp(-beta * i)
diags = diags / np.sum(diags)
# populates diagonal terms using truncated operator expression
rm = sp.spdiags(diags, 0, N, N, format='csr')
elif method == 'analytic':
# populates diagonal terms using analytic values
rm = sp.spdiags((1.0 + n) ** (-1.0) * (n / (1.0 + n)) ** (i),
0, N, N, format='csr')
else:
raise ValueError(
"'method' keyword argument must be 'operator' or 'analytic'")
return Qobj(rm)
[docs]def maximally_mixed_dm(N):
"""
Returns the maximally mixed density matrix for a Hilbert space of
dimension N.
Parameters
----------
N : int
Number of basis states in Hilbert space.
Returns
-------
dm : qobj
Thermal state density matrix.
"""
if (not isinstance(N, (int, np.int64))) or N <= 0:
raise ValueError("N must be integer N > 0")
dm = sp.spdiags(np.ones(N, dtype=complex)/float(N), 0, N, N, format='csr')
return Qobj(dm, isherm=True)
[docs]def ket2dm(Q):
"""Takes input ket or bra vector and returns density matrix
formed by outer product.
Parameters
----------
Q : qobj
Ket or bra type quantum object.
Returns
-------
dm : qobj
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]]
"""
if Q.type == 'ket':
out = Q * Q.dag()
elif Q.type == 'bra':
out = Q.dag() * Q
else:
raise TypeError("Input is not a ket or bra vector.")
return Qobj(out)
#
# projection operator
#
[docs]def projection(N, n, m, offset=None):
"""The projection operator that projects state :math:`|m>` on state :math:`|n>`.
Parameters
----------
N : int
Number of basis states in Hilbert space.
n, m : float
The number states in the projection.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the projector.
Returns
-------
oper : qobj
Requested projection operator.
"""
ket1 = basis(N, n, offset=offset)
ket2 = basis(N, m, offset=offset)
return ket1 * ket2.dag()
#
# composite qubit states
#
def qstate(string):
"""Creates a tensor product for a set of qubits in either
the 'up' :math:`|0>` or 'down' :math:`|1>` state.
Parameters
----------
string : str
String containing 'u' or 'd' for each qubit (ex. 'ududd')
Returns
-------
qstate : qobj
Qobj for tensor product corresponding to input string.
Notes
-----
Look at ket and bra for more general functions
creating multiparticle states.
Examples
--------
>>> qstate('udu')
Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket
Qobj data =
[[ 0.]
[ 0.]
[ 0.]
[ 0.]
[ 0.]
[ 1.]
[ 0.]
[ 0.]]
"""
n = len(string)
if n != (string.count('u') + string.count('d')):
raise TypeError('String input to QSTATE must consist ' +
'of "u" and "d" elements only')
else:
up = basis(2, 1)
dn = basis(2, 0)
lst = []
for k in range(n):
if string[k] == 'u':
lst.append(up)
else:
lst.append(dn)
return tensor(lst)
#
# different qubit notation dictionary
#
_qubit_dict = {'g': 0, # ground state
'e': 1, # excited state
'u': 0, # spin up
'd': 1, # spin down
'H': 0, # horizontal polarization
'V': 1} # vertical polarization
def _character_to_qudit(x):
"""
Converts a character representing a one-particle state into int.
"""
if x in _qubit_dict:
return _qubit_dict[x]
else:
return int(x)
[docs]def ket(seq, dim=2):
"""
Produces a multiparticle ket state for a list or string,
where each element stands for state of the respective particle.
Parameters
----------
seq : str / 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.
dim : int (default: 2) / list of ints
Space dimension for each particle:
int if there are the same, list if they are different.
Returns
-------
ket : qobj
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.]]
"""
if isinstance(dim, int):
dim = [dim] * len(seq)
return tensor([basis(dim[i], _character_to_qudit(x))
for i, x in enumerate(seq)])
[docs]def bra(seq, dim=2):
"""
Produces a multiparticle bra state for a list or string,
where each element stands for state of the respective particle.
Parameters
----------
seq : str / 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.
dim : int (default: 2) / list of ints
Space dimension for each particle:
int if there are the same, list if they are different.
Returns
-------
bra : qobj
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.]]
"""
return ket(seq, dim=dim).dag()
#
# quantum state number helper functions
#
[docs]def state_number_enumerate(dims, excitations=None, state=None, idx=0):
"""
An iterator that enumerate all the state number arrays (quantum numbers on
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
----------
dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
state : list
Current state in the iteration. Used internally.
excitations : integer (None)
Restrict state space to states with excitation numbers below or
equal to this value.
idx : integer
Current index in the iteration. Used internally.
Returns
-------
state_number : list
Successive state number arrays that can be used in loops and other
iterations, using standard state enumeration *by definition*.
"""
if state is None:
state = np.zeros(len(dims), dtype=int)
if excitations and sum(state[0:idx]) > excitations:
pass
elif idx == len(dims):
if excitations is None:
yield np.array(state)
else:
yield tuple(state)
else:
for n in range(dims[idx]):
state[idx] = n
for s in state_number_enumerate(dims, excitations, state, idx + 1):
yield s
[docs]def state_number_index(dims, state):
"""
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
----------
dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
state : list
State number array.
Returns
-------
idx : int
The index of the state given by `state` in standard enumeration
ordering.
"""
return int(
sum([state[i] * prod(dims[i + 1:]) for i, d in enumerate(dims)]))
[docs]def state_index_number(dims, index):
"""
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
----------
dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
index : integer
The index of the state in standard enumeration ordering.
Returns
-------
state : list
The state number array corresponding to index `index` in standard
enumeration ordering.
"""
state = np.empty_like(dims)
D = np.concatenate([np.flipud(np.cumprod(np.flipud(dims[1:]))), [1]])
for n in range(len(dims)):
state[n] = index / D[n]
index -= state[n] * D[n]
return list(state)
[docs]def state_number_qobj(dims, state):
"""
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
----------
dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
state : list
State number array.
Returns
-------
state : :class:`qutip.Qobj.qobj`
The state as a :class:`qutip.Qobj.qobj` instance.
"""
return tensor([fock(dims[i], s) for i, s in enumerate(state)])
#
# Excitation-number restricted (enr) states
#
[docs]def enr_state_dictionaries(dims, excitations):
"""
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.
excitations : integer
The maximum numbers of dimension
Returns
-------
nstates, state2idx, idx2state: integer, dict, dict
The number of states `nstates`, a dictionary for looking up state
indices from a state tuple, and a dictionary for looking up state
state tuples from state indices.
"""
nstates = 0
state2idx = {}
idx2state = {}
for state in state_number_enumerate(dims, excitations):
state2idx[state] = nstates
idx2state[nstates] = state
nstates += 1
return nstates, state2idx, idx2state
[docs]def enr_fock(dims, excitations, state):
"""
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
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
state : list of integers
The state in the number basis representation.
Returns
-------
ket : Qobj
A Qobj instance that represent a Fock state in the exication-number-
restricted state space defined by `dims` and `exciations`.
"""
nstates, state2idx, idx2state = enr_state_dictionaries(dims, excitations)
data = sp.lil_matrix((nstates, 1), dtype=np.complex)
try:
data[state2idx[tuple(state)], 0] = 1
except:
raise ValueError("The state tuple %s is not in the restricted "
"state space" % str(tuple(state)))
return Qobj(data, dims=[dims, [1]*len(dims)])
[docs]def enr_thermal_dm(dims, excitations, n):
"""
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
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
n : integer
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.
Returns
-------
dm : Qobj
Thermal state density matrix.
"""
nstates, state2idx, idx2state = enr_state_dictionaries(dims, excitations)
if not isinstance(n, (list, np.ndarray)):
n = np.ones(len(dims)) * n
else:
n = np.asarray(n)
diags = [np.prod((n / (n + 1)) ** np.array(state))
for idx, state in idx2state.items()]
diags /= np.sum(diags)
data = sp.spdiags(diags, 0, nstates, nstates, format='csr')
return Qobj(data, dims=[dims, dims])
[docs]def phase_basis(N, m, phi0=0):
"""
Basis vector for the mth phase of the Pegg-Barnett phase operator.
Parameters
----------
N : int
Number of basis vectors in Hilbert space.
m : int
Integer corresponding to the mth discrete phase phi_m=phi0+2*pi*m/N
phi0 : float (default=0)
Reference phase angle.
Returns
-------
state : qobj
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.
"""
phim = phi0 + (2.0 * np.pi * m) / N
n = np.arange(N).reshape((N, 1))
data = 1.0 / np.sqrt(N) * np.exp(1.0j * n * phim)
return Qobj(data)
[docs]def zero_ket(N, dims=None):
"""
Creates the zero ket vector with shape Nx1 and
dimensions `dims`.
Parameters
----------
N : int
Hilbert space dimensionality
dims : list
Optional dimensions if ket corresponds to
a composite Hilbert space.
Returns
-------
zero_ket : qobj
Zero ket on given Hilbert space.
"""
return Qobj(sp.csr_matrix((N, 1), dtype=complex), dims=dims)
[docs]def spin_state(j, m, type='ket'):
"""Generates the spin state |j, m>, i.e. the eigenstate
of the spin-j Sz operator with eigenvalue m.
Parameters
----------
j : float
The spin of the state ().
m : int
Eigenvalue of the spin-j Sz operator.
type : string {'ket', 'bra', 'dm'}
Type of state to generate.
Returns
-------
state : qobj
Qobj quantum object for spin state
"""
J = 2 * j + 1
if type == 'ket':
return basis(int(J), int(j - m))
elif type == 'bra':
return basis(int(J), int(j - m)).dag()
elif type == 'dm':
return fock_dm(int(J), int(j - m))
else:
raise ValueError("invalid value keyword argument 'type'")
[docs]def spin_coherent(j, theta, phi, type='ket'):
"""Generate the coherent spin state |theta, phi>.
Parameters
----------
j : float
The spin of the state.
theta : float
Angle from z axis.
phi : float
Angle from x axis.
type : string {'ket', 'bra', 'dm'}
Type of state to generate.
Returns
-------
state : qobj
Qobj quantum object for spin coherent state
"""
Sp = jmat(j, '+')
Sm = jmat(j, '-')
psi = (0.5 * theta * np.exp(1j * phi) * Sm -
0.5 * theta * np.exp(-1j * phi) * Sp).expm() * spin_state(j, j)
if type == 'ket':
return psi
elif type == 'bra':
return psi.dag()
elif type == 'dm':
return ket2dm(psi)
else:
raise ValueError("invalid value keyword argument 'type'")
[docs]def bell_state(state='00'):
"""
Returns the Bell state:
|B00> = 1 / sqrt(2)*[|0>|0>+|1>|1>]
|B01> = 1 / sqrt(2)*[|0>|0>-|1>|1>]
|B10> = 1 / sqrt(2)*[|0>|1>+|1>|0>]
|B11> = 1 / sqrt(2)*[|0>|1>-|1>|0>]
Returns
-------
Bell_state : qobj
Bell state
"""
if state == '00':
Bell_state = tensor(
basis(2), basis(2))+tensor(basis(2, 1), basis(2, 1))
elif state == '01':
Bell_state = tensor(
basis(2), basis(2))-tensor(basis(2, 1), basis(2, 1))
elif state == '10':
Bell_state = tensor(
basis(2), basis(2, 1))+tensor(basis(2, 1), basis(2))
elif state == '11':
Bell_state = tensor(
basis(2), basis(2, 1))-tensor(basis(2, 1), basis(2))
return Bell_state.unit()
[docs]def singlet_state():
"""
Returns the two particle singlet-state:
|S>=1/sqrt(2)*[|0>|1>-|1>|0>]
that is identical to the fourth bell state.
Returns
-------
Bell_state : qobj
|B11> Bell state
"""
return bell_state('11')
def triplet_states():
"""
Returns the two particle triplet-states:
|T>= |1>|1>
= 1 / sqrt(2)*[|0>|1>-|1>|0>]
= |0>|0>
that is identical to the fourth bell state.
Returns
-------
trip_states : list
2 particle triplet states
"""
trip_states = []
trip_states.append(tensor(basis(2, 1), basis(2, 1)))
trip_states.append(
(tensor(basis(2), basis(2, 1)) + tensor(basis(2, 1), basis(2))).unit()
)
trip_states.append(tensor(basis(2), basis(2)))
return trip_states
def w_state(N=3):
"""
Returns the N-qubit W-state.
Parameters
----------
N : int (default=3)
Number of qubits in state
Returns
-------
W : qobj
N-qubit W-state
"""
inds = np.zeros(N, dtype=int)
inds[0] = 1
state = tensor([basis(2, x) for x in inds])
for kk in range(1, N):
perm_inds = np.roll(inds, kk)
state += tensor([basis(2, x) for x in perm_inds])
return state.unit()
[docs]def ghz_state(N=3):
"""
Returns the N-qubit GHZ-state.
Parameters
----------
N : int (default=3)
Number of qubits in state
Returns
-------
G : qobj
N-qubit GHZ-state
"""
state = (tensor([basis(2) for k in range(N)]) +
tensor([basis(2, 1) for k in range(N)]))
return state/np.sqrt(2)