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__all__ = ['liouvillian', 'liouvillian_ref', 'lindblad_dissipator',
'operator_to_vector', 'vector_to_operator', 'mat2vec', 'vec2mat',
'vec2mat_index', 'mat2vec_index', 'spost', 'spre', 'sprepost']
import scipy.sparse as sp
import numpy as np
from qutip.qobj import Qobj
from qutip.sparse import sp_reshape
[docs]def liouvillian(H, c_ops=[], data_only=False, chi=None):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators. Like liouvillian, but with an
experimental implementation which avoids creating extra Qobj instances,
which can be advantageous for large systems.
Parameters
----------
H : qobj
System Hamiltonian.
c_ops : array_like
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : qobj
Liouvillian superoperator.
"""
if chi and len(chi) != len(c_ops):
raise ValueError('chi must be a list with same length as c_ops')
if H is not None:
if H.isoper:
op_dims = H.dims
op_shape = H.shape
elif H.issuper:
op_dims = H.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for Hamiltonian.")
else:
# no hamiltonian given, pick system size from a collapse operator
if isinstance(c_ops, list) and len(c_ops) > 0:
c = c_ops[0]
if c.isoper:
op_dims = c.dims
op_shape = c.shape
elif c.issuper:
op_dims = c.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for collapse operator.")
else:
raise TypeError("Either H or c_ops must be given.")
sop_dims = [[op_dims[0], op_dims[0]], [op_dims[1], op_dims[1]]]
sop_shape = [np.prod(op_dims), np.prod(op_dims)]
spI = sp.identity(op_shape[0], format='csr')
if H:
if H.isoper:
data = -1j * (sp.kron(spI, H.data, format='csr')
- sp.kron(H.data.T, spI, format='csr'))
else:
data = H.data
else:
data = sp.csr_matrix((sop_shape[0], sop_shape[1]), dtype=complex)
for idx, c_op in enumerate(c_ops):
if c_op.issuper:
data = data + c_op.data
else:
cd = c_op.data.T.conj()
c = c_op.data
if chi:
data = data + np.exp(1j * chi[idx]) * sp.kron(cd.T, c,
format='csr')
else:
data = data + sp.kron(cd.T, c, format='csr')
cdc = cd * c
data = data - 0.5 * sp.kron(spI, cdc, format='csr')
data = data - 0.5 * sp.kron(cdc.T, spI, format='csr')
if data_only:
return data
else:
L = Qobj()
L.dims = sop_dims
L.data = data
L.isherm = False
L.superrep = 'super'
return L
def liouvillian_ref(H, c_ops=[]):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators.
Parameters
----------
H : qobj
System Hamiltonian.
c_ops : array_like
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : qobj
Liouvillian superoperator.
"""
L = -1.0j * (spre(H) - spost(H)) if H else 0
for c in c_ops:
if c.issuper:
L += c
else:
cdc = c.dag() * c
L += spre(c) * spost(c.dag()) - 0.5 * spre(cdc) - 0.5 * spost(cdc)
return L
[docs]def lindblad_dissipator(a, b=None, data_only=False):
"""
Lindblad dissipator (generalized) for a single pair of collapse operators
(a, b), or for a single collapse operator (a) when b is not specified:
.. math::
\\mathcal{D}[a,b]\\rho = a \\rho b^\\dagger -
\\frac{1}{2}a^\\dagger b\\rho - \\frac{1}{2}\\rho a^\\dagger b
Parameters
----------
a : qobj
Left part of collapse operator.
b : qobj (optional)
Right part of collapse operator. If not specified, b defaults to a.
Returns
-------
D : qobj
Lindblad dissipator superoperator.
"""
if b is None:
b = a
ad_b = a.dag() * b
D = spre(a) * spost(b.dag()) - 0.5 * spre(ad_b) - 0.5 * spost(ad_b)
return D.data if data_only else D
[docs]def operator_to_vector(op):
"""
Create a vector representation of a quantum operator given
the matrix representation.
"""
q = Qobj()
q.dims = [op.dims, [1]]
q.data = sp_reshape(op.data.T, (np.prod(op.shape), 1))
return q
[docs]def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in
vector form.
"""
q = Qobj()
q.dims = op.dims[0]
n = int(np.sqrt(op.shape[0]))
q.data = sp_reshape(op.data.T, (n, n)).T
return q
def mat2vec(mat):
"""
Private function reshaping matrix to vector.
"""
return mat.T.reshape(np.prod(np.shape(mat)), 1)
def vec2mat(vec):
"""
Private function reshaping vector to matrix.
"""
n = int(np.sqrt(len(vec)))
return vec.reshape((n, n)).T
def vec2mat_index(N, I):
"""
Convert a vector index to a matrix index pair that is compatible with the
vector to matrix rearrangement done by the vec2mat function.
"""
j = int(I / N)
i = I - N * j
return i, j
def mat2vec_index(N, i, j):
"""
Convert a matrix index pair to a vector index that is compatible with the
matrix to vector rearrangement done by the mat2vec function.
"""
return i + N * j
[docs]def spost(A):
"""Superoperator formed from post-multiplication by operator A
Parameters
----------
A : qobj
Quantum operator for post multiplication.
Returns
-------
super : qobj
Superoperator formed from input qauntum object.
"""
if not isinstance(A, Qobj):
raise TypeError('Input is not a quantum object')
if not A.isoper:
raise TypeError('Input is not a quantum operator')
S = Qobj(isherm=A.isherm, superrep='super')
S.dims = [[A.dims[0], A.dims[1]], [A.dims[0], A.dims[1]]]
S.data = sp.kron(A.data.T, sp.identity(np.prod(A.shape[0])), format='csr')
return S
[docs]def spre(A):
"""Superoperator formed from pre-multiplication by operator A.
Parameters
----------
A : qobj
Quantum operator for pre-multiplication.
Returns
--------
super :qobj
Superoperator formed from input quantum object.
"""
if not isinstance(A, Qobj):
raise TypeError('Input is not a quantum object')
if not A.isoper:
raise TypeError('Input is not a quantum operator')
S = Qobj(isherm=A.isherm, superrep='super')
S.dims = [[A.dims[0], A.dims[1]], [A.dims[0], A.dims[1]]]
S.data = sp.kron(sp.identity(np.prod(A.shape[1])), A.data, format='csr')
return S
def _drop_projected_dims(dims):
"""
Eliminate subsystems that has been collapsed to only one state due to
a projection.
"""
return [d for d in dims if d != 1]
[docs]def sprepost(A, B):
"""Superoperator formed from pre-multiplication by operator A and post-
multiplication of operator B.
Parameters
----------
A : Qobj
Quantum operator for pre-multiplication.
B : Qobj
Quantum operator for post-multiplication.
Returns
--------
super : Qobj
Superoperator formed from input quantum objects.
"""
dims = [[_drop_projected_dims(A.dims[0]), _drop_projected_dims(B.dims[1])],
[_drop_projected_dims(A.dims[1]), _drop_projected_dims(B.dims[0])]]
data = sp.kron(B.data.T, A.data, format='csr')
return Qobj(data, dims=dims, superrep='super')