def test_partial_transpose_bipartite():
"""partial transpose of bipartite systems"""
rho = Qobj(np.arange(16).reshape(4, 4), dims=[[2, 2], [2, 2]])
# no transpose
rho_pt = partial_transpose(rho, [0, 0])
assert_(np.abs(np.max(rho_pt.full() - rho.full())) < 1e-12)
# partial transpose subsystem 1
rho_pt = partial_transpose(rho, [1, 0])
rho_pt_expected = np.array([[0, 1, 8, 9],
[4, 5, 12, 13],
[2, 3, 10, 11],
[6, 7, 14, 15]])
assert_(np.abs(np.max(rho_pt.full() - rho_pt_expected)) < 1e-12)
# partial transpose subsystem 2
rho_pt = partial_transpose(rho, [0, 1])
rho_pt_expected = np.array([[0, 4, 2, 6],
[1, 5, 3, 7],
[8, 12, 10, 14],
[9, 13, 11, 15]])
assert_(np.abs(np.max(rho_pt.full() - rho_pt_expected)) < 1e-12)
# full transpose
rho_pt = partial_transpose(rho, [1, 1])
assert_(np.abs(np.max(rho_pt.full() - rho.trans().full())) < 1e-12)
def test_partial_transpose_comparison():
"""partial transpose: comparing sparse and dense implementations"""
N = 10
rho = tensor(rand_dm(N, density=0.5), rand_dm(N, density=0.5))
# partial transpose of system 1
rho_pt1 = partial_transpose(rho, [1, 0], method="dense")
rho_pt2 = partial_transpose(rho, [1, 0], method="sparse")
np.abs(np.max(rho_pt1.full() - rho_pt1.full())) < 1e-12
# partial transpose of system 2
rho_pt1 = partial_transpose(rho, [0, 1], method="dense")
rho_pt2 = partial_transpose(rho, [0, 1], method="sparse")
np.abs(np.max(rho_pt1.full() - rho_pt2.full())) < 1e-12
def test_partial_transpose_randomized():
"""partial transpose: randomized tests on tripartite system"""
rho = tensor(rand_dm(2, density=1),
rand_dm(2, density=1),
rand_dm(2, density=1))
mask = np.random.randint(2, size=3)
rho_pt_ref = _partial_transpose_reference(rho, mask)
rho_pt1 = partial_transpose(rho, mask, method="dense")
np.abs(np.max(rho_pt1.full() - rho_pt_ref.full())) < 1e-12
rho_pt2 = partial_transpose(rho, mask, method="sparse")
np.abs(np.max(rho_pt2.full() - rho_pt_ref.full())) < 1e-12
def peres_horodecki_bipartite(rho, mask=[0,1]):
""" Tests the given bipartite state for Peres-Horodecki criterion
Parameters
----------
rho: qobj/array-like
Density operator for the state
mask: list of int, length 2
mask used for partial transpose
Returns
-------
isentangled: bool
True for entangled, False for disentangled
"""
if rho.type != 'oper':
raise TypeError("Input must be a density matrix")
rhopt = partial_transpose(rho,mask)
rhopt_eigs = rhopt.eigenenergies()
if min(rhopt_eigs) < 0:
isentangled = True
else:
isentangled = False
return isentangled
def log_neg(rho,mask=[1,0]):
""" Calculate the logarithmic negativity for a density matrix
Parameters:
-----------
rho : qobj/array-like
Input density matrix
Returns:
--------
logneg: Logarithmic Negativity
"""
if rho.type != 'oper':
raise TypeError("Input must be a density matrix")
rhopt = partial_transpose(rho,mask)
logneg = log2( rhopt.norm() )
return logneg
def negativity(rho,mask=[1,0]):
""" Calculate the negativity for a density matrix
Parameters:
-----------
rho : qobj/array-like
Input density matrix
Returns:
--------
neg : Negativity
"""
if rho.type != 'oper':
raise TypeError("Input must be a density matrix")
rhopt = partial_transpose(rho,mask)
neg = ( rhopt.norm() - 1 ) / 2
return neg
def test_partial_transpose_randomized():
"""partial transpose: randomized tests on tripartite system"""
rho = tensor(rand_dm(2, density=1), rand_dm(2, density=1),
rand_dm(2, density=1))
mask = np.random.randint(2, size=3)
rho_pt_ref = _partial_transpose_reference(rho, mask)
rho_pt1 = partial_transpose(rho, mask, method="dense")
np.abs(np.max(rho_pt1.full() - rho_pt_ref.full())) < 1e-12
rho_pt2 = partial_transpose(rho, mask, method="sparse")
np.abs(np.max(rho_pt2.full() - rho_pt_ref.full())) < 1e-12
def test_partial_transpose_comparison():
"""partial transpose: comparing sparse and dense implementations"""
N = 10
rho = tensor(rand_dm(N, density=0.5), rand_dm(N, density=0.5))
# partial transpose of system 1
rho_pt1 = partial_transpose(rho, [1, 0], method="dense")
rho_pt2 = partial_transpose(rho, [1, 0], method="sparse")
np.abs(np.max(rho_pt1.full() - rho_pt1.full())) < 1e-12
# partial transpose of system 2
rho_pt1 = partial_transpose(rho, [0, 1], method="dense")
rho_pt2 = partial_transpose(rho, [0, 1], method="sparse")
np.abs(np.max(rho_pt1.full() - rho_pt2.full())) < 1e-12
def peres_horodecki_bipartite(rho, mask=[0, 1]):
""" Tests the given bipartite state for Peres-Horodecki criterion
Parameters
----------
rho: qobj/array-like
Density operator for the state
mask: list of int, length 2
mask used for partial transpose
Returns
-------
isentangled: bool
True for entangled, False for disentangled
"""
if rho.type != 'oper':
raise TypeError("Input must be a density matrix")
rhopt = partial_transpose(rho, mask)
rhopt_eigs = rhopt.eigenenergies()
if min(rhopt_eigs) < 0:
isentangled = True
else:
isentangled = False
return isentangled
def least_ent_eig(r, theta, a_z):
rho=(1/4)*(qt.tensor(qt.qeye(2), qt.qeye(2)) + qt.tensor(a_z*qt.sigmaz(), qt.qeye(2))\
- r*np.sin(theta)/np.sqrt(2)*qt.tensor(qt.sigmax(),qt.sigmax())\
- r*np.sin(theta)/np.sqrt(2)*qt.tensor(qt.sigmay(),qt.sigmay())\
- r*np.cos(theta)*qt.tensor(qt.sigmaz(),qt.sigmaz()))
rho_pt = qt.partial_transpose(rho, [0, 1])
eigs = np.linalg.eigvalsh(rho_pt.full())
return min(eigs)
def test_partial_transpose_bipartite():
"""partial transpose of bipartite systems"""
rho = Qobj(np.arange(16).reshape(4, 4), dims=[[2, 2], [2, 2]])
# no transpose
rho_pt = partial_transpose(rho, [0, 0])
assert_(np.abs(np.max(rho_pt.full() - rho.full())) < 1e-12)
# partial transpose subsystem 1
rho_pt = partial_transpose(rho, [1, 0])
rho_pt_expected = np.array([[0, 1, 8, 9], [4, 5, 12, 13], [2, 3, 10, 11],
[6, 7, 14, 15]])
assert_(np.abs(np.max(rho_pt.full() - rho_pt_expected)) < 1e-12)
# partial transpose subsystem 2
rho_pt = partial_transpose(rho, [0, 1])
rho_pt_expected = np.array([[0, 4, 2, 6], [1, 5, 3, 7], [8, 12, 10, 14],
[9, 13, 11, 15]])
assert_(np.abs(np.max(rho_pt.full() - rho_pt_expected)) < 1e-12)
# full transpose
rho_pt = partial_transpose(rho, [1, 1])
assert_(np.abs(np.max(rho_pt.full() - rho.trans().full())) < 1e-12)
def negativity(rho, mask=[1, 0]):
""" Calculate the negativity for a density matrix
Parameters:
-----------
rho : qobj/array-like
Input density matrix
Returns:
--------
neg : Negativity
"""
if rho.type != 'oper':
raise TypeError("Input must be a density matrix")
rhopt = partial_transpose(rho, mask)
neg = (rhopt.norm() - 1) / 2
return neg
def create_dataset(n_samples):
"""Create dataset.
Parameters:
n_samples(int): Number of samples
Output:
_states(list): States associated with each set of measurements.
_measurements(list): Measurements.
_labels(list): Label associated with each set of measurements.
"""
_states = []
_labels = []
_measurements = []
#Basis Measured
name_basis = ['I', 'X', 'Y', 'Z']
basis = [qutip.identity(2), qutip.sigmax(),qutip.sigmay(),qutip.sigmaz()]
for _ in range(n_samples):
density = qutip.rand_dm(4, density=0.75, dims=[[2,2],[2,2]])
#Partial Transpose
density_partial_T = qutip.partial_transpose(density, [0,1])
#Labels: 1 if entangled 0 if separable (PPT Criterion)
if (density_partial_T.eigenenergies() < 0).any():
_labels.append(1)
else:
_labels.append(-1)
_states.append(density)
val_measurements = measurement(density_matrix=density,
base=basis,
name_base=name_basis)
_measurements.append(val_measurements)
return _states, _measurements, _labels
def generate_separable(n_samples):
"""Create n_samples separable states.
Parameters:
n_samples(int): Number of samples
Output:
states_train(list): States associated with each set of measurements.
measurements_train(list): measurements
"""
_states = []
_measurements = []
#Basis Measured
name_basis = ['I', 'X', 'Y', 'Z']
basis = [qutip.identity(2), qutip.sigmax(),qutip.sigmay(),qutip.sigmaz()]
counter = 0
while not counter == n_samples:
density = qutip.rand_dm(4, density=0.75, dims=[[2,2],[2,2]])
#Partial Transpose
density_partial_T = qutip.partial_transpose(density, [0,1])
#Labels: 1 if entangled 0 if separable (PPT Criterion)
if (density_partial_T.eigenenergies() < 0).any():
pass
else:
_states.append(density)
val_measurements = measurement(density_matrix=density,
base=basis,
name_base=name_basis)
_measurements.append(val_measurements)
counter += 1
return _states, _measurements
def entangled(state):
if state.shape == (4, 4):
return (np.any(q.partial_transpose(state, [1, 0]).eigenenergies() < 0))
else:
raise TypeError(
'The Peres-Horodecki criterion only works for pairs of qubits.')
def simulate_process_prep_measure(E_AB, prep_meas_settings):
"""
Simulate measurements for process tomography using the prepare-and-measure
scheme, given a "true" quantum process `E_AB`, a list of input states and
measurement settings specified by `prep_meas_settings`.
The process `E_AB` should be a :py:class:`qutip.Qobj` containing the Choi
matrix of the channel being applied, as a :py:class:`qutip.Qobj` matrix (NOT
as a superoperator). It should not be normalized, i.e., we expect
:math:`\mathrm{tr}_B(E_{AB}) = \mathbb{I}_A`.
The argument `prep_meas_settings` must be a list of tuples `(sigma_in,
Mk_out, num_repeats)`, where `sigma_in` is an input state specified as a
:py:class:`qutip.Qobj` object, where `Mk_out` is a POVM specified as a list
of POVM effects (each POVM effect is a positive semidefinite matrix of norm
≤ 1, and is specified as a :py:class:`qutip.Qobj` matrix), and finally
`num_repeats` is the number of times to repeat this setting.
Returns: an object `d` with properties `d.Emn_ch`, `d.Nm`, representing the
arguments suitable for :py:meth:`QPtomographer.channelspace.run()` and the
simulated frequency counts.
"""
Emn_ch = []
Nm = []
dimA = E_AB.dims[0][0]
dimB = E_AB.dims[0][1]
dimAB = dimA * dimB
for (sigma_in, Mk_out, num_repeats) in prep_meas_settings:
# calculate the output state after sending sigma_in through the process
rho_out = (E_AB * qutip.tensor(qutip.partial_transpose(sigma_in, [1]),
qutip.qeye(dimB))).ptrace(1)
# output POVM
num_out_effects = len(Mk_out)
# get some random numbers for this measurement setting
x = np.random.rand(num_repeats)
proboffset = 0
# We sample the measurement outcomes as follows: we split the interval
# [0,1] in number sections equal to the number of possible outcomes,
# each of length = probability of that outcome. Then, for each possible
# outcome, we count the number of random numbers in `x` that fall into
# the corresponding section.
for Mk in Mk_out:
p = qutip.expect(Mk, rho_out) # = trace(Mk * rho_out)
Emn_ch.append(
qutip.tensor(qutip.partial_transpose(sigma_in, [1]),
Mk).data.toarray().astype(dtype=complex))
Nm.append(
np.count_nonzero((proboffset <= x) & (x < proboffset + p)))
proboffset += p
# sanity check
assert np.abs(proboffset - 1.0) < 1e-6
d = _Store()
d.Emn_ch = Emn_ch
d.Nm = np.array(Nm)
return d
# Set initial state
sys = cv.System(N, 2)
r = np.arcsinh(np.sqrt(mu))
#sys.apply_SMS(r)
sys.apply_TMS(r)
# Make state no Gaussian
k = 0.4
sys.apply_photon_subtraction(k, 1)
psi0 = sys.state
if psi0.isket:
rho0 = psi0 * psi0.dag()
# State with partial transpose
rho = qt.partial_transpose(rho0, [1, 0])
# Method 1 - Fastest
norm = rho.norm()
log_neg = np.log2(norm)
# Method 2
rho2 = rho * rho.dag()
norm2 = (rho2.sqrtm()).tr()
log_neg2 = np.log2(norm2)
# Method 3 - for Gaussian states - best approx for small N?
sys.set_state(rho)
sys.set_quadratures_basis()
cm = sys.get_full_CM()
print(cm)