def fidelity(A, B):
"""
Calculates the fidelity (pseudo-metric) between two density matrices.
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
fid : float
Fidelity pseudo-metric between A and B.
Examples
--------
>>> x = fock_dm(5,3)
>>> y = coherent_dm(5,1)
>>> fidelity(x,y)
0.24104350624628332
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('Density matrices do not have same dimensions.')
A = A.sqrtm()
return float(np.real((A * (B * A)).sqrtm().tr()))
def fidelity(A, B):
"""
Calculates the fidelity (pseudo-metric) between two density matrices..
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
fid : float
Fidelity pseudo-metric between A and B.
Examples
--------
>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> fidelity(x,y)
0.24104350624628332
"""
if A.type != 'oper':
A = ket2dm(A)
if B.type != 'oper':
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('Density matrices do not have same dimensions.')
else:
A = A.sqrtm()
return float(np.real((A * (B * A)).sqrtm().tr()))
def hilbert_dist(A, B):
"""
Returns the Hilbert-Schmidt distance between two density matrices A & B.
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
dist : float
Hilbert-Schmidt distance between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
return (A - B).norm('fro')
def bures_angle(A, B):
"""
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
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
angle : float
Bures angle between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
return np.arccos(fidelity(A, B))
def hilbert_dist(A, B):
"""
Returns the Hilbert-Schmidt distance between two density matrices A & B.
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
dist : float
Hilbert-Schmidt distance between density matrices.
Notes
-----
See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
return ((A - B)**2).tr()
def bures_dist(A, B):
"""
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
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
dist : float
Bures distance between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
dist = np.sqrt(2.0 * (1.0 - fidelity(A, B)))
return dist
def hilbert_dist(A, B):
"""
Returns the Hilbert-Schmidt distance between two density matrices A & B.
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
dist : float
Hilbert-Schmidt distance between density matrices.
Notes
-----
See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
return ((A - B)**2).tr()
def bures_dist(A, B):
"""
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
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
dist : float
Bures distance between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
dist = np.sqrt(2.0 * (1.0 - fidelity(A, B)))
return dist
def bures_angle(A, B):
"""
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
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
angle : float
Bures angle between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
return np.arccos(fidelity(A, B))
def hilbert_dist(A, B):
"""
Returns the Hilbert-Schmidt distance between two density matrices A & B.
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
dist : float
Hilbert-Schmidt distance between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')
return (A - B).norm('fro')
def hellinger_dist(A, B, sparse=False, tol=0):
"""
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
----------
A : :class:`qutip.Qobj`
Density matrix or state vector.
B : :class:`qutip.Qobj`
Density matrix or state vector with same dimensions as A.
tol : float
Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse : {False, True}
Use sparse eigensolver.
Returns
-------
hellinger_dist : float
Quantum Hellinger distance between A and B. Ranges from 0 to sqrt(2).
Examples
--------
>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> hellinger_dist(x,y)
1.3725145002591095
"""
if A.dims != B.dims:
raise TypeError("A and B do not have same dimensions.")
if A.isket or A.isbra:
sqrtmA = ket2dm(A)
else:
sqrtmA = A.sqrtm(sparse=sparse, tol=tol)
if B.isket or B.isbra:
sqrtmB = ket2dm(B)
else:
sqrtmB = B.sqrtm(sparse=sparse, tol=tol)
product = sqrtmA * sqrtmB
eigs = sp_eigs(product.data,
isherm=product.isherm,
vecs=False,
sparse=sparse,
tol=tol)
#np.maximum() is to avoid nan appearing sometimes due to numerical
#instabilities causing np.sum(eigs) slightly (~1e-8) larger than 1
#when hellinger_dist(A, B) is called for A=B
return np.sqrt(2.0 * np.maximum(0., (1.0 - np.real(np.sum(eigs)))))
def fidelity(A, B):
"""
Calculates the fidelity (pseudo-metric) between two density matrices.
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
fid : float
Fidelity pseudo-metric between A and B.
Examples
--------
>>> x = fock_dm(5,3)
>>> y = coherent_dm(5,1)
>>> fidelity(x,y)
0.24104350624628332
"""
if A.isket or A.isbra:
# Take advantage of the fact that the density operator for A
# is a projector to avoid a sqrtm call.
sqrtmA = ket2dm(A)
# Check whether we have to turn B into a density operator, too.
if B.isket or B.isbra:
B = ket2dm(B)
else:
if B.isket or B.isbra:
# Swap the order so that we can take a more numerically
# stable square root of B.
return fidelity(B, A)
# If we made it here, both A and B are operators, so
# we have to take the sqrtm of one of them.
sqrtmA = A.sqrtm()
if sqrtmA.dims != B.dims:
raise TypeError('Density matrices do not have same dimensions.')
# We don't actually need the whole matrix here, just the trace
# of its square root, so let's just get its eigenenergies instead.
# We also truncate negative eigenvalues to avoid nan propagation;
# even for positive semidefinite matrices, small negative eigenvalues
# can be reported.
eig_vals = (sqrtmA * B * sqrtmA).eigenenergies()
return float(np.real(np.sqrt(eig_vals[eig_vals > 0]).sum()))
def fidelity(A, B):
"""
Calculates the fidelity (pseudo-metric) between two density matrices.
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"
Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns
-------
fid : float
Fidelity pseudo-metric between A and B.
Examples
--------
>>> x = fock_dm(5,3)
>>> y = coherent_dm(5,1)
>>> fidelity(x,y)
0.24104350624628332
"""
if A.isket or A.isbra:
# Take advantage of the fact that the density operator for A
# is a projector to avoid a sqrtm call.
sqrtmA = ket2dm(A)
# Check whether we have to turn B into a density operator, too.
if B.isket or B.isbra:
B = ket2dm(B)
else:
if B.isket or B.isbra:
# Swap the order so that we can take a more numerically
# stable square root of B.
return fidelity(B, A)
# If we made it here, both A and B are operators, so
# we have to take the sqrtm of one of them.
sqrtmA = A.sqrtm()
if sqrtmA.dims != B.dims:
raise TypeError('Density matrices do not have same dimensions.')
# We don't actually need the whole matrix here, just the trace
# of its square root, so let's just get its eigenenergies instead.
# We also truncate negative eigenvalues to avoid nan propagation;
# even for positive semidefinite matrices, small negative eigenvalues
# can be reported.
eig_vals = (sqrtmA * B * sqrtmA).eigenenergies()
return float(np.real(np.sqrt(eig_vals[eig_vals > 0]).sum()))
def test_QobjPurity():
"Tests the purity method of `Qobj`"
psi = basis(2, 1)
# check purity of pure ket state
assert_almost_equal(psi.purity(), 1)
# check purity of pure ket state (superposition)
psi2 = basis(2, 0)
psi_tot = (psi+psi2).unit()
assert_almost_equal(psi_tot.purity(), 1)
# check purity of density matrix of pure state
assert_almost_equal(ket2dm(psi_tot).purity(), 1)
# check purity of maximally mixed density matrix
rho_mixed = (ket2dm(psi)+ket2dm(psi2)).unit()
assert_almost_equal(rho_mixed.purity(), 0.5)
def _correlation_me_4op_2t(H, rho0, tlist, taulist, c_ops,
a_op, b_op, c_op, d_op, reverse=False,
args=None, options=Odeoptions()):
"""
Calculate the four-operator two-time correlation function on the form
<A(t)B(t+tau)C(t+tau)D(t)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t = mesolve(
H, rho0, tlist, c_ops, [], args=args, options=options).states
for t_idx, rho in enumerate(rho_t):
C_mat[t_idx, :] = mesolve(H, d_op * rho * a_op, taulist,
c_ops, [b_op * c_op],
args=args, options=options).expect[0]
return C_mat
def test_call():
"""
Test Qobj: Call
"""
# Make test objects.
psi = rand_ket(3)
rho = rand_dm_ginibre(3)
U = rand_unitary(3)
S = rand_super_bcsz(3)
# Case 0: oper(ket).
assert U(psi) == U * psi
# Case 1: oper(oper). Should raise TypeError.
with expect_exception(TypeError):
U(rho)
# Case 2: super(ket).
assert S(psi) == vector_to_operator(S * operator_to_vector(ket2dm(psi)))
# Case 3: super(oper).
assert S(rho) == vector_to_operator(S * operator_to_vector(rho))
# Case 4: super(super). Should raise TypeError.
with expect_exception(TypeError):
S(S)
def _correlation_es_2op_1t(H,
rho0,
tlist,
c_ops,
a_op,
b_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation_ss` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
solC_tau = ode2es(L, rho0 * a_op)
return esval(expect(b_op, solC_tau), tlist)
else:
# default: <A(t+tau)B(t)>
solC_tau = ode2es(L, b_op * rho0)
return esval(expect(a_op, solC_tau), tlist)
def _correlation_me_4op_1t(H,
rho0,
tlist,
c_ops,
a_op,
b_op,
c_op,
d_op,
args=None,
options=Odeoptions()):
"""
Calculate the four-operator two-time correlation function on the form
<A(0)B(tau)C(tau)D(0)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
return mesolve(H,
d_op * rho0 * a_op,
tlist,
c_ops, [b_op * c_op],
args=args,
options=options).expect[0]
def _subsystem_apply_reference(state, channel, mask):
if isket(state):
state = ket2dm(state)
if isoper(channel):
full_oper = tensor([
channel if mask[j] else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))
])
return full_oper * state * full_oper.dag()
else:
# Go to Choi, then Kraus
# chan_mat = array(channel.data.todense())
choi_matrix = super_to_choi(channel)
vals, vecs = eig(choi_matrix.full())
vecs = list(map(array, zip(*vecs)))
kraus_list = [
sqrt(vals[j]) * vec2mat(vecs[j]) for j in range(len(vals))
]
# Kraus operators to be padded with identities:
k_qubit_kraus_list = product(kraus_list, repeat=sum(mask))
rho_out = Qobj(inpt=zeros(state.shape), dims=state.dims)
for operator_iter in k_qubit_kraus_list:
operator_iter = iter(operator_iter)
op_iter_list = [
next(operator_iter) if mask[j] else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))
]
full_oper = tensor(list(map(Qobj, op_iter_list)))
rho_out = rho_out + full_oper * state * full_oper.dag()
return Qobj(rho_out)
def _correlation_es_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using an exponential series solver.
"""
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
rho0 = steadystate(H, c_ops)
tlist = [0]
elif isket(state0):
rho0 = ket2dm(state0)
else:
rho0 = state0
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, c_op * rho_t * a_op)
corr_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
return corr_mat
def update(self, rho):
"""
Calculate the probability function for the given state of an harmonic
oscillator (as density matrix)
"""
if isket(rho):
rho = ket2dm(rho)
self.data = np.zeros(len(self.xvecs[0]), dtype=complex)
M, N = rho.shape
for m in range(M):
k_m = pow(self.omega / pi, 0.25) / \
sqrt(2 ** m * factorial(m)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(m), self.xvecs[0])
for n in range(N):
k_n = pow(self.omega / pi, 0.25) / \
sqrt(2 ** n * factorial(n)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(n), self.xvecs[0])
self.data += np.conjugate(k_n) * k_m * rho.data[m, n]
def _correlation_me_2op_2t(H, rho0, tlist, taulist, c_ops, a_op, b_op,
reverse=False, args=None, options=Odeoptions()):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t_list = mesolve(
H, rho0, tlist, c_ops, [], args=args, options=options).states
if reverse:
# <A(t)B(t+tau)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H, rho_t * a_op, taulist,
c_ops, [b_op], args=args,
options=options).expect[0]
else:
# <A(t+tau)B(t)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H, b_op * rho_t, taulist,
c_ops, [a_op], args=args,
options=options).expect[0]
return C_mat
def _correlation_es_2op_1t(H, rho0, tlist, c_ops, a_op, b_op, reverse=False,
args=None, options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation_ss` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
solC_tau = ode2es(L, rho0 * a_op)
return esval(expect(b_op, solC_tau), tlist)
else:
# default: <A(t+tau)B(t)>
solC_tau = ode2es(L, b_op * rho0)
return esval(expect(a_op, solC_tau), tlist)
def test_composite_vec():
"""
Composite: Tests compositing states and density operators.
"""
k1 = rand_ket(5)
k2 = rand_ket(7)
r1 = operator_to_vector(ket2dm(k1))
r2 = operator_to_vector(ket2dm(k2))
r3 = operator_to_vector(rand_dm(3))
r4 = operator_to_vector(rand_dm(4))
assert_(composite(k1, k2) == tensor(k1, k2))
assert_(composite(r3, r4) == super_tensor(r3, r4))
assert_(composite(k1, r4) == super_tensor(r1, r4))
assert_(composite(r3, k2) == super_tensor(r3, r2))
def _subsystem_apply_reference(state, channel, mask):
if isket(state):
state = ket2dm(state)
if isoper(channel):
full_oper = tensor([channel if mask[j]
else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))])
return full_oper * state * full_oper.dag()
else:
# Go to Choi, then Kraus
# chan_mat = array(channel.data.todense())
choi_matrix = super_to_choi(channel)
vals, vecs = eig(choi_matrix.full())
vecs = list(map(array, zip(*vecs)))
kraus_list = [sqrt(vals[j]) * vec2mat(vecs[j])
for j in range(len(vals))]
# Kraus operators to be padded with identities:
k_qubit_kraus_list = product(kraus_list, repeat=sum(mask))
rho_out = Qobj(inpt=zeros(state.shape), dims=state.dims)
for operator_iter in k_qubit_kraus_list:
operator_iter = iter(operator_iter)
op_iter_list = [next(operator_iter).conj().T if mask[j]
else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))]
full_oper = tensor(list(map(Qobj, op_iter_list)))
rho_out = rho_out + full_oper * state * full_oper.dag()
return Qobj(rho_out)
def update(self, rho):
"""
Calculate the probability function for the given state of an harmonic
oscillator (as density matrix)
"""
if isket(rho):
rho = ket2dm(rho)
self.data = np.zeros(len(self.xvecs[0]), dtype=complex)
M, N = rho.shape
for m in range(M):
k_m = pow(self.omega / pi, 0.25) / \
sqrt(2 ** m * factorial(m)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(m), self.xvecs[0])
for n in range(N):
k_n = pow(self.omega / pi, 0.25) / \
sqrt(2 ** n * factorial(n)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(n), self.xvecs[0])
self.data += np.conjugate(k_n) * k_m * rho.data[m, n]
def test_call():
"""
Test Qobj: Call
"""
# Make test objects.
psi = rand_ket(3)
rho = rand_dm_ginibre(3)
U = rand_unitary(3)
S = rand_super_bcsz(3)
# Case 0: oper(ket).
assert U(psi) == U * psi
# Case 1: oper(oper). Should raise TypeError.
with expect_exception(TypeError):
U(rho)
# Case 2: super(ket).
assert S(psi) == vector_to_operator(S * operator_to_vector(ket2dm(psi)))
# Case 3: super(oper).
assert S(rho) == vector_to_operator(S * operator_to_vector(rho))
# Case 4: super(super). Should raise TypeError.
with expect_exception(TypeError):
S(S)
def plot_fock_distribution(rho,
offset=0,
fig=None,
ax=None,
figsize=(8, 6),
title=None,
unit_y_range=True):
"""
Plot the Fock distribution for a density matrix (or ket) that describes
an oscillator mode.
Parameters
----------
rho : :class:`qutip.qobj.Qobj`
The density matrix (or ket) of the state to visualize.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
title : string
An optional title for the figure.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
if isket(rho):
rho = ket2dm(rho)
N = rho.shape[0]
ax.bar(np.arange(offset, offset + N) - .4,
np.real(rho.diag()),
color="green",
alpha=0.6,
width=0.8)
if unit_y_range:
ax.set_ylim(0, 1)
ax.set_xlim(-.5 + offset, N + offset)
ax.set_xlabel('Fock number', fontsize=12)
ax.set_ylabel('Occupation probability', fontsize=12)
if title:
ax.set_title(title)
return fig, ax
def test_composite_vec():
"""
Composite: Tests compositing states and density operators.
"""
k1 = rand_ket(5)
k2 = rand_ket(7)
r1 = operator_to_vector(ket2dm(k1))
r2 = operator_to_vector(ket2dm(k2))
r3 = operator_to_vector(rand_dm(3))
r4 = operator_to_vector(rand_dm(4))
assert_(composite(k1, k2) == tensor(k1, k2))
assert_(composite(r3, r4) == super_tensor(r3, r4))
assert_(composite(k1, r4) == super_tensor(r1, r4))
assert_(composite(r3, k2) == super_tensor(r3, r2))
def _mesolve_func_td(H_func, rho0, tlist, c_op_list, expt_ops, H_args, opt):
"""!
Evolve the density matrix using an ODE solver with time dependent
Hamiltonian.
"""
n_op = len(c_op_list)
#
# check initial state
#
if isket(rho0):
# if initial state is a ket and no collapse operator where given,
# fallback on the unitary schrodinger equation solver
if n_op == 0:
return _wfsolve_list_td(H_func, rho0, tlist, expt_ops, H_args, opt)
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
L = 0
for m in range(0, n_op):
cdc = c_op_list[m].dag() * c_op_list[m]
if L == 0:
L = spre(c_op_list[m])*spost(c_op_list[m].dag())-0.5*spre(cdc)-0.5*spost(cdc)
else:
L += spre(c_op_list[m])*spost(c_op_list[m].dag())-0.5*spre(cdc)-0.5*spost(cdc)
if n_op > 0:
L_func_and_args = [H_func, L.data]
else:
n,m = rho0.shape
L_func_and_args = [H_func, sp.lil_matrix((n**2,m**2)).tocsr()]
for arg in H_args:
if isinstance(arg,Qobj):
L_func_and_args.append((-1j*(spre(arg) - spost(arg))).data)
else:
L_func_and_args.append(arg)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full())
r = scipy.integrate.ode(_ode_rho_func_td)
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, nsteps=opt.nsteps,
first_step=opt.first_step, min_step=opt.min_step,
max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
r.set_f_params(L_func_and_args)
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, expt_ops, opt, vec2mat)
def _mesolve_const(H, rho0, tlist, c_op_list, e_ops, args, opt,
progress_bar):
"""
Evolve the density matrix using an ODE solver, for constant hamiltonian
and collapse operators.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
# if initial state is a ket and no collapse operator where given,
# fall back on the unitary schrodinger equation solver
if len(c_op_list) == 0 and isoper(H):
return _sesolve_const(H, rho0, tlist, e_ops, args, opt,
progress_bar)
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
if opt.tidy:
H = H.tidyup(opt.atol)
L = liouvillian(H, c_op_list)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel('F')
if issuper(rho0):
r = scipy.integrate.ode(_ode_super_func)
r.set_f_params(L.data)
else:
if opt.use_openmp and L.data.nnz >= qset.openmp_thresh:
r = scipy.integrate.ode(cy_ode_rhs_openmp)
r.set_f_params(L.data.data, L.data.indices, L.data.indptr,
opt.openmp_threads)
else:
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(L.data.data, L.data.indices, L.data.indptr)
# r = scipy.integrate.ode(_ode_rho_test)
# r.set_f_params(L.data)
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, nsteps=opt.nsteps,
first_step=opt.first_step, min_step=opt.min_step,
max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
def _mesolve_const(H, rho0, tlist, c_op_list, e_ops, args, opt,
progress_bar):
"""
Evolve the density matrix using an ODE solver, for constant hamiltonian
and collapse operators.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
# if initial state is a ket and no collapse operator where given,
# fall back on the unitary schrodinger equation solver
if len(c_op_list) == 0 and isoper(H):
return _sesolve_const(H, rho0, tlist, e_ops, args, opt,
progress_bar)
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
if opt.tidy:
H = H.tidyup(opt.atol)
L = liouvillian(H, c_op_list)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel('F')
if issuper(rho0):
r = scipy.integrate.ode(_ode_super_func)
r.set_f_params(L.data)
else:
if opt.use_openmp and L.data.nnz >= qset.openmp_thresh:
r = scipy.integrate.ode(cy_ode_rhs_openmp)
r.set_f_params(L.data.data, L.data.indices, L.data.indptr,
opt.openmp_threads)
else:
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(L.data.data, L.data.indices, L.data.indptr)
# r = scipy.integrate.ode(_ode_rho_test)
# r.set_f_params(L.data)
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, nsteps=opt.nsteps,
first_step=opt.first_step, min_step=opt.min_step,
max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
def steadystate_nonlinear(L_func,
rho0,
args={},
maxiter=10,
random_initial_state=False,
tol=1e-6,
itertol=1e-5,
use_umfpack=True,
verbose=False):
"""
Steady state for the evolution subject to the nonlinear Liouvillian
(which depends on the density matrix).
.. note:: Experimental. Not at all certain that the inverse power method
works for state-dependent Liouvillian operators.
"""
use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
if random_initial_state:
rhoss = rand_dm(rho0.shape[0], 1.0, dims=rho0.dims)
elif isket(rho0):
rhoss = ket2dm(rho0)
else:
rhoss = Qobj(rho0)
v = mat2vec(rhoss.full())
n = prod(rhoss.shape)
tr_vec = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
tr_vec = tr_vec.reshape((1, n))
it = 0
while it < maxiter:
L = L_func(rhoss, args)
L = L.data.tocsc() - (tol**2) * sp.eye(n, n, format='csc')
L.sort_indices()
v = spsolve(L, v, use_umfpack=use_umfpack)
v = v / la.norm(v, np.inf)
data = v / sum(tr_vec.dot(v))
data = reshape(data, (rhoss.shape[0], rhoss.shape[1])).T
rhoss.data = sp.csr_matrix(data)
it += 1
if la.norm(L * v, np.inf) <= tol:
break
if it >= maxiter:
raise ValueError('Failed to find steady state after ' + str(maxiter) +
' iterations')
rhoss = 0.5 * (rhoss + rhoss.dag())
return rhoss.tidyup() if qset.auto_tidyup else rhoss
def plot_wigner_fock_distribution(rho, fig=None, axes=None, figsize=(8, 4),
cmap=None, alpha_max=7.5, colorbar=False,
method='iterative'):
"""
Plot the Fock distribution and the Wigner function for a density matrix
(or ket) that describes an oscillator mode.
Parameters
----------
rho : :class:`qutip.qobj.Qobj`
The density matrix (or ket) of the state to visualize.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
axes : a list of two matplotlib axes instances
The axes context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
cmap : a matplotlib cmap instance
The colormap.
alpha_max : float
The span of the x and y coordinates (both [-alpha_max, alpha_max]).
colorbar : bool
Whether (True) or not (False) a colorbar should be attached to the
Wigner function graph.
method : string {'iterative', 'laguerre', 'fft'}
The method used for calculating the wigner function. See the
documentation for qutip.wigner for details.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig and not axes:
fig, axes = plt.subplots(1, 2, figsize=figsize)
if isket(rho):
rho = ket2dm(rho)
plot_fock_distribution(rho, fig=fig, ax=axes[0])
plot_wigner(rho, fig=fig, ax=axes[1], figsize=figsize, cmap=cmap,
alpha_max=alpha_max, colorbar=colorbar, method=method)
return fig, axes
def tracedist(A, B, sparse=False, tol=0):
"""
Calculates the trace distance between two density matrices..
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"
Parameters
----------!=
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
tol : float
Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse : {False, True}
Use sparse eigensolver.
Returns
-------
tracedist : float
Trace distance between A and B.
Examples
--------
>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> tracedist(x,y)
0.9705143161472971
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError("A and B do not have same dimensions.")
diff = A - B
diff = diff.dag() * diff
vals = sp_eigs(diff.data, diff.isherm, vecs=False, sparse=sparse, tol=tol)
return float(np.real(0.5 * np.sum(np.sqrt(np.abs(vals)))))
def tracedist(A, B, sparse=False, tol=0):
"""
Calculates the trace distance between two density matrices..
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"
Parameters
----------!=
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
tol : float
Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse : {False, True}
Use sparse eigensolver.
Returns
-------
tracedist : float
Trace distance between A and B.
Examples
--------
>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> tracedist(x,y)
0.9705143161472971
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)
if A.dims != B.dims:
raise TypeError("A and B do not have same dimensions.")
diff = A - B
diff = diff.dag() * diff
vals = sp_eigs(diff.data, diff.isherm, vecs=False, sparse=sparse, tol=tol)
return float(np.real(0.5 * np.sum(np.sqrt(np.abs(vals)))))
def plot_fock_distribution(rho, offset=0, fig=None, ax=None,
figsize=(8, 6), title=None, unit_y_range=True):
"""
Plot the Fock distribution for a density matrix (or ket) that describes
an oscillator mode.
Parameters
----------
rho : :class:`qutip.qobj.Qobj`
The density matrix (or ket) of the state to visualize.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
title : string
An optional title for the figure.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
if isket(rho):
rho = ket2dm(rho)
N = rho.shape[0]
ax.bar(np.arange(offset, offset + N) - .4, np.real(rho.diag()),
color="green", alpha=0.6, width=0.8)
if unit_y_range:
ax.set_ylim(0, 1)
ax.set_xlim(-.5 + offset, N + offset)
ax.set_xlabel('Fock number', fontsize=12)
ax.set_ylabel('Occupation probability', fontsize=12)
if title:
ax.set_title(title)
return fig, ax
def _correlation_me_2t(H,
state0,
tlist,
taulist,
c_ops,
a_op,
b_op,
c_op,
args=None,
options=Options()):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using a master equation solver.
"""
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
rho0 = steadystate(H, c_ops)
tlist = [0]
elif isket(state0):
rho0 = ket2dm(state0)
else:
rho0 = state0
if debug:
print(inspect.stack()[0][3])
rho_t = mesolve(H, rho0, tlist, c_ops, [], args=args,
options=options).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
H_shifted, _args = _transform_H_t_shift(H, args)
for t_idx, rho in enumerate(rho_t):
if not isinstance(H, Qobj):
_args["_t0"] = tlist[t_idx]
corr_mat[t_idx, :] = mesolve(H_shifted,
c_op * rho * a_op,
taulist,
c_ops, [b_op],
args=_args,
options=options).expect[0]
if t_idx == 1:
options.rhs_reuse = True
return corr_mat
def spin_q_function(rho, theta, phi):
"""Husimi Q-function for spins.
Parameters
----------
state : qobj
A state vector or density matrix for a spin-j quantum system.
theta : array_like
theta-coordinates at which to calculate the Q function.
phi : array_like
phi-coordinates at which to calculate the Q function.
Returns
-------
Q, THETA, PHI : 2d-array
Values representing the spin Q function at the values specified
by THETA and PHI.
"""
if rho.type == 'bra':
rho = rho.dag()
if rho.type == 'ket':
rho = ket2dm(rho)
J = rho.shape[0]
j = (J - 1) / 2
THETA, PHI = meshgrid(theta, phi)
Q = np.zeros_like(THETA, dtype=complex)
for m1 in arange(-j, j+1):
Q += binom(2*j, j+m1) * cos(THETA/2) ** (2*(j-m1)) * sin(THETA/2) ** (2*(j+m1)) * \
rho.data[int(j-m1), int(j-m1)]
for m2 in arange(m1+1, j+1):
Q += (sqrt(binom(2*j, j+m1)) * sqrt(binom(2*j, j+m2)) *
cos(THETA/2) ** (2*j-m1-m2) * sin(THETA/2) ** (2*j+m1+m2)) * \
(exp(1j * (m2-m1) * PHI) * rho.data[int(j-m1), int(j-m2)] +
exp(1j * (m1-m2) * PHI) * rho.data[int(j-m2), int(j-m1)])
return Q.real, THETA, PHI
def steady_nonlinear(L_func, rho0, args={}, maxiter=10,
random_initial_state=False,
tol=1e-6, itertol=1e-5, use_umfpack=True):
"""
Steady state for the evolution subject to the nonlinear Liouvillian
(which depends on the density matrix).
.. note:: Experimental. Not at all certain that the inverse power method
works for state-dependent liouvillian operators.
"""
use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
if random_initial_state:
rhoss = rand_dm(rho0.shape[0], 1.0, dims=rho0.dims)
elif isket(rho0):
rhoss = ket2dm(rho0)
else:
rhoss = Qobj(rho0)
v = mat2vec(rhoss.full())
n = prod(rhoss.shape)
tr_vec = sp.eye(rhoss.shape[0], rhoss.shape[0], format='lil')
tr_vec = tr_vec.reshape((1, n)).tocsr()
it = 0
while it < maxiter:
L = L_func(rhoss, args)
L = L.data.tocsc() - (tol ** 2) * sp.eye(n, n, format='csc')
L.sort_indices()
v = spsolve(L, v, permc_spec="MMD_AT_PLUS_A", use_umfpack=use_umfpack)
v = v / la.norm(v, np.inf)
data = v / sum(tr_vec.dot(v))
data = reshape(data, (rhoss.shape[0], rhoss.shape[1])).T
rhoss.data = sp.csr_matrix(data)
it += 1
if la.norm(L * v, np.inf) <= tol:
break
if it >= maxiter:
raise ValueError('Failed to find steady state after ' +
str(maxiter) + ' iterations')
#rhoss.data = 0.5 * (data + data.conj().T)
return rhoss.tidyup() if qset.auto_tidyup else rhoss
def _correlation_me_2op_2t(H,
rho0,
tlist,
taulist,
c_ops,
a_op,
b_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t_list = mesolve(H, rho0, tlist, c_ops, [], args=args,
options=options).states
if reverse:
# <A(t)B(t+tau)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H,
rho_t * a_op,
taulist,
c_ops, [b_op],
args=args,
options=options).expect[0]
else:
# <A(t+tau)B(t)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H,
b_op * rho_t,
taulist,
c_ops, [a_op],
args=args,
options=options).expect[0]
return C_mat
def _correlation_me_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
args={}, options=Options()):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using a master equation solver.
"""
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
rho0 = steadystate(H, c_ops)
tlist = [0]
elif isket(state0):
rho0 = ket2dm(state0)
else:
rho0 = state0
if debug:
print(inspect.stack()[0][3])
rho_t = mesolve(H, rho0, tlist, c_ops, [],
args=args, options=options).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
H_shifted, c_ops_shifted, _args = _transform_L_t_shift(H, c_ops, args)
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
for t_idx, rho in enumerate(rho_t):
if not isinstance(H, Qobj):
_args["_t0"] = tlist[t_idx]
corr_mat[t_idx, :] = mesolve(
H_shifted, c_op * rho * a_op, taulist, c_ops_shifted,
[b_op], args=_args, options=options
).expect[0]
if t_idx == 1:
options.rhs_reuse = True
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
return corr_mat
def mesolve_const_checkpoint(H, rho0, tlist, c_op_list, e_ops, args, opt,
progress_bar, save, subdir):
"""
Evolve the density matrix using an ODE solver, for constant hamiltonian
and collapse operators.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
if opt.tidy:
H = H.tidyup(opt.atol)
L = liouvillian(H, c_op_list)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel()
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(L.data.data, L.data.indices, L.data.indptr)
r.set_integrator('zvode',
method=opt.method,
order=opt.order,
atol=opt.atol,
rtol=opt.rtol,
nsteps=opt.nsteps,
first_step=opt.first_step,
min_step=opt.min_step,
max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# call generic ODE code
#
return generic_ode_solve_checkpoint(r, rho0, tlist, e_ops, opt,
progress_bar, save, subdir)
def _correlation_me_4op_1t(H, rho0, tlist, c_ops, a_op, b_op, c_op, d_op, args=None, options=Options()):
"""
Calculate the four-operator two-time correlation function on the form
<A(0)B(tau)C(tau)D(0)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
return mesolve(H, d_op * rho0 * a_op, tlist, c_ops, [b_op * c_op], args=args, options=options).expect[0]
def _correlation_es_2op_2t(H,
rho0,
tlist,
taulist,
c_ops,
a_op,
b_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, rho_t * a_op)
C_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
else:
# default: <A(t+tau)B(t)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, b_op * rho_t)
C_mat[t_idx, :] = esval(expect(a_op, solES_tau), taulist)
return C_mat
def spin_wigner(rho, theta, phi):
"""Wigner function for spins.
Parameters
----------
state : qobj
A state vector or density matrix for a spin-j quantum system.
theta : array_like
theta-coordinates at which to calculate the Q function.
phi : array_like
phi-coordinates at which to calculate the Q function.
Returns
-------
W, THETA, PHI : 2d-array
Values representing the spin Wigner function at the values specified
by THETA and PHI.
.. note::
Experimental.
"""
if rho.type == 'bra':
rho = rho.dag()
if rho.type == 'ket':
rho = ket2dm(rho)
J = rho.shape[0]
j = (J - 1) / 2
THETA, PHI = meshgrid(theta, phi)
W = np.zeros_like(THETA, dtype=complex)
for k in range(int(2 * j)+1):
for q in range(-k, k+1):
W += _rho_kq(rho, j, k, q) * sph_harm(q, k, PHI, THETA)
return W, THETA, PHI
def concurrence(rho):
"""
Calculate the concurrence entanglement measure for a two-qubit state.
Parameters
----------
state : qobj
Ket, bra, or density matrix for a two-qubit state.
Returns
-------
concur : float
Concurrence
References
----------
.. [1] http://en.wikipedia.org/wiki/Concurrence_(quantum_computing)
"""
if rho.isket and rho.dims != [[2, 2], [1, 1]]:
raise Exception("Ket must be tensor product of two qubits.")
elif rho.isbra and rho.dims != [[1, 1], [2, 2]]:
raise Exception("Bra must be tensor product of two qubits.")
elif rho.isoper and rho.dims != [[2, 2], [2, 2]]:
raise Exception("Density matrix must be tensor product of two qubits.")
if rho.isket or rho.isbra:
rho = ket2dm(rho)
sysy = tensor(sigmay(), sigmay())
rho_tilde = (rho * sysy) * (rho.conj() * sysy)
evals = rho_tilde.eigenenergies()
# abs to avoid problems with sqrt for very small negative numbers
evals = abs(sort(real(evals)))
lsum = sqrt(evals[3]) - sqrt(evals[2]) - sqrt(evals[1]) - sqrt(evals[0])
return max(0, lsum)
def testExpandGate2toN(self):
"""
gates: expand 2 to N (using cnot, iswap, sqrtswap)
"""
a, b = np.random.rand(), np.random.rand()
k1 = (a * basis(2, 0) + b * basis(2, 1)).unit()
c, d = np.random.rand(), np.random.rand()
k2 = (c * basis(2, 0) + d * basis(2, 1)).unit()
psi_ref_in = tensor(k1, k2)
N = 6
psi_rand_list = [rand_ket(2) for k in range(N)]
for g in [cnot, iswap, sqrtswap]:
psi_ref_out = g() * psi_ref_in
rho_ref_out = ket2dm(psi_ref_out)
for m in range(N):
for n in set(range(N)) - {m}:
psi_list = [psi_rand_list[k] for k in range(N)]
psi_list[m] = k1
psi_list[n] = k2
psi_in = tensor(psi_list)
if g == cnot:
G = g(N, m, n)
else:
G = g(N, [m, n])
psi_out = G * psi_in
o1 = psi_out.overlap(psi_in)
o2 = psi_ref_out.overlap(psi_ref_in)
assert_(abs(o1 - o2) < 1e-12)
p = [0, 1] if m < n else [1, 0]
rho_out = psi_out.ptrace([m, n]).permute(p)
assert_((rho_ref_out - rho_out).norm() < 1e-12)
def spin_wigner(rho, theta, phi):
"""Wigner function for a spin-j system on the 2-sphere of radius j
(for j = 1/2 this is the Bloch sphere).
Parameters
----------
state : qobj
A state vector or density matrix for a spin-j quantum system.
theta : array_like
Polar angle at which to calculate the W function.
phi : array_like
Azimuthal angle at which to calculate the W function.
Returns
-------
W, THETA, PHI : 2d-array
Values representing the spin Wigner function at the values specified
by THETA and PHI.
Notes
-----
Experimental.
"""
if rho.type == 'bra':
rho = rho.dag()
if rho.type == 'ket':
rho = ket2dm(rho)
J = rho.shape[0]
j = (J - 1) / 2
THETA, PHI = meshgrid(theta, phi)
W = np.zeros_like(THETA, dtype=complex)
for k in range(int(2 * j)+1):
for q in arange(-k, k+1):
# sph_harm takes azimuthal angle then polar angle as arguments
W += _rho_kq(rho, j, k, q) * sph_harm(q, k, PHI, THETA)
return W, THETA, PHI
def testExpandGate2toN(self):
"""
gates: expand 2 to N (using cnot, iswap, sqrtswap)
"""
a, b = np.random.rand(), np.random.rand()
k1 = (a * basis(2, 0) + b * basis(2, 1)).unit()
c, d = np.random.rand(), np.random.rand()
k2 = (c * basis(2, 0) + d * basis(2, 1)).unit()
psi_ref_in = tensor(k1, k2)
N = 6
psi_rand_list = [rand_ket(2) for k in range(N)]
for g in [cnot, iswap, sqrtswap]:
psi_ref_out = g() * psi_ref_in
rho_ref_out = ket2dm(psi_ref_out)
for m in range(N):
for n in set(range(N)) - {m}:
psi_list = [psi_rand_list[k] for k in range(N)]
psi_list[m] = k1
psi_list[n] = k2
psi_in = tensor(psi_list)
if g == cnot:
G = g(N, m, n)
else:
G = g(N, [m, n])
psi_out = G * psi_in
o1 = psi_out.overlap(psi_in)
o2 = psi_ref_out.overlap(psi_ref_in)
assert_(abs(o1 - o2) < 1e-12)
p = [0, 1] if m < n else [1, 0]
rho_out = psi_out.ptrace([m, n]).permute(p)
assert_((rho_ref_out - rho_out).norm() < 1e-12)
def concurrence(rho):
"""
Calculate the concurrence entanglement measure for a two-qubit state.
Parameters
----------
state : qobj
Ket, bra, or density matrix for a two-qubit state.
Returns
-------
concur : float
Concurrence
References
----------
.. [1] http://en.wikipedia.org/wiki/Concurrence_(quantum_computing)
"""
if rho.isket and rho.dims != [[2, 2], [1, 1]]:
raise Exception("Ket must be tensor product of two qubits.")
elif rho.isbra and rho.dims != [[1, 1], [2, 2]]:
raise Exception("Bra must be tensor product of two qubits.")
elif rho.isoper and rho.dims != [[2, 2], [2, 2]]:
raise Exception("Density matrix must be tensor product of two qubits.")
if rho.isket or rho.isbra:
rho = ket2dm(rho)
sysy = tensor(sigmay(), sigmay())
rho_tilde = (rho * sysy) * (rho.conj() * sysy)
evals = rho_tilde.eigenenergies()
# abs to avoid problems with sqrt for very small negative numbers
evals = abs(sort(real(evals)))
lsum = sqrt(evals[3]) - sqrt(evals[2]) - sqrt(evals[1]) - sqrt(evals[0])
return max(0, lsum)
def _correlation_me_2op_1t(H, rho0, tlist, c_ops, a_op, b_op, reverse=False, args=None, options=Options()):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation_ss` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
if reverse:
# <A(t)B(t+tau)>
return mesolve(H, rho0 * a_op, tlist, c_ops, [b_op], args=args, options=options).expect[0]
else:
# <A(t+tau)B(t)>
return mesolve(H, b_op * rho0, tlist, c_ops, [a_op], args=args, options=options).expect[0]
def _correlation_me_4op_2t(H,
rho0,
tlist,
taulist,
c_ops,
a_op,
b_op,
c_op,
d_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Calculate the four-operator two-time correlation function on the form
<A(t)B(t+tau)C(t+tau)D(t)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t = mesolve(H, rho0, tlist, c_ops, [], args=args,
options=options).states
for t_idx, rho in enumerate(rho_t):
C_mat[t_idx, :] = mesolve(H,
d_op * rho * a_op,
taulist,
c_ops, [b_op * c_op],
args=args,
options=options).expect[0]
return C_mat
def entropy_vn(rho,base=e,sparse=False):
"""
Von-Neumann entropy of density matrix
Parameters
----------
rho : qobj
Density matrix.
base : {e,2}
Base of logarithm.
Other Parameters
----------------
sparse : {False,True}
Use sparse eigensolver.
Returns
-------
entropy : float
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
"""
if rho.type=='ket' or rho.type=='bra':
rho=ket2dm(rho)
vals=sp_eigs(rho,vecs=False,sparse=sparse)
nzvals=vals[vals!=0]
if base==2:
logvals=log2(nzvals)
elif base==e:
logvals=log(nzvals)
else:
raise ValueError("Base must be 2 or e.")
return float(real(-sum(nzvals*logvals)))
def _correlation_es_2op_2t(H, rho0, tlist, taulist, c_ops, a_op, b_op,
reverse=False, args=None, options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, rho_t * a_op)
C_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
else:
# default: <A(t+tau)B(t)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, b_op * rho_t)
C_mat[t_idx, :] = esval(expect(a_op, solES_tau), taulist)
return C_mat
def entropy_linear(rho):
"""
Linear entropy of a density matrix.
Parameters
----------
rho : qobj
sensity matrix or ket/bra vector.
Returns
-------
entropy : float
Linear entropy of rho.
Examples
--------
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_linear(rho)
0.5
"""
if rho.type == 'ket' or rho.type == 'bra':
rho = ket2dm(rho)
return float(real(1.0 - (rho ** 2).tr()))