Source code for qutip.states

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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) # doctest: +SKIP 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]) # doctest: +SKIP 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 """ out = np.empty((3,), dtype=object) out[:] = [basis(3, 0), basis(3, 1), basis(3, 2)] return out
[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) # doctest: +SKIP 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 ValueError( "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) # doctest: +SKIP 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. """ psi = coherent(N, alpha, offset=offset, method=method) return psi * psi.dag()
[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) # doctest: +SKIP 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) # doctest: +SKIP 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) # doctest: +SKIP 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') # doctest: +SKIP 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("The method option can only take " "values '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) # doctest: +SKIP 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): r""" The projection operator that projects state :math:`\lvert m\rangle` on state :math:`\lvert n\rangle`. 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): r"""Creates a tensor product for a set of qubits in either the 'up' :math:`\lvert0\rangle` or 'down' :math:`\lvert1\rangle` 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') # doctest: +SKIP 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") # doctest: +SKIP Quantum object: dims = [[2, 2], [1, 1]], shape = [4, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 1.] [ 0.]] >>> ket("Hue") # doctest: +SKIP 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) # doctest: +SKIP 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]) # doctest: +SKIP 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") # doctest: +SKIP Quantum object: dims = [[1, 1], [2, 2]], shape = [1, 4], type = bra Qobj data = [[ 0. 0. 1. 0.]] >>> bra("Hue") # doctest: +SKIP 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) # doctest: +SKIP 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]) # doctest: +SKIP 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, nexc=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]): # doctest: +SKIP >>> print(state) # doctest: +SKIP [ 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. nexc : integer Number of excitations in modes [0..idx-1]. 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 idx == len(dims): if excitations is None: yield np.array(state) else: yield tuple(state) else: if excitations is None: nlim = dims[idx] else: # modes [0..idx-1] have nexc excitations, # so mode idx can have at most excitations-nexc excitations nlim = min(dims[idx], 1 + excitations - nexc) for n in range(nlim): state[idx] = n for s in state_number_enumerate(dims, excitations, state, idx + 1, nexc + n): 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]) # doctest: +SKIP 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.complex128) 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'): r"""Generates the spin state :math:`\lvert j, m\rangle`, 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'): r"""Generate the coherent spin state :math:`\lvert \theta, \phi\rangle`. 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'): r""" Returns the selected Bell state: .. math:: \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} 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(): r""" Returns the two particle singlet-state: .. math:: \lvert S\rangle = \frac1{\sqrt2}(\lvert01\rangle-\lvert10\rangle) that is identical to the fourth bell state. Returns ------- Bell_state : qobj :math:`\lvert B_{11}\rangle` Bell state """ return bell_state('11')
[docs]def triplet_states(): r""" Returns a list of the two particle triplet-states: .. math:: \lvert T_1\rangle = \lvert11\rangle \lvert T_2\rangle = \frac1{\sqrt2}(\lvert01\rangle + \lvert10\rangle) \lvert T_3\rangle = \lvert00\rangle 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
[docs]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)