def testSingletBetweenAlice1AndBobAndLocalOtherwiseIsNotInPolytope(self):
scenario = SequentialBellScenario([[2, 2], [2, 2]], [2, 2])
alice1blochVectors = [[1, 0, 0], [0, 1, 0]]
alice1Observables = list(
map(lambda bloch: createQubitObservable(bloch),
alice1blochVectors))
alice1Krauss = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), alice1Observables))
bobUnBlochVectors = [[-1, -1, 0], [-1, 1, 0]]
bobObservables = list(
map(lambda bloch: createQubitObservable(bloch), bobUnBlochVectors))
bobEffects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), bobObservables))
psi = createMaxEntState(2)
expectedCorrelations = {((x1, x2), y, (a1, a2), b): int(
(a2 == 0)) * ((qt.tensor(alice1Krauss[x1][a1], bobEffects[y][b]) *
psi * psi.dag()).tr())
for ((x1, x2), y, (a1, a2),
b) in scenario.getTuplesOfEvents()}
expectedBehaviour = Behaviour(scenario, expectedCorrelations)
poly = SequentialBellPolytope(scenario)
self.assertFalse(poly.contains(expectedBehaviour.getProbabilityList()))
def testRandProjMeasBetweenAlice1andBob(self):
scenario = SequentialBellScenario([[2, 2], [2, 2]], [2, 2])
alice1blochVectors = [[1, 0, 0], [0, 1, 0]]
alice1Observables = list(
map(lambda bloch: createQubitObservable(bloch),
alice1blochVectors))
alice1Krauss = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), alice1Observables))
phases = [
np.random.uniform(-np.pi / 2, np.pi / 2),
np.random.uniform(-np.pi / 2, np.pi / 2)
]
alice2blochVectors = [[np.sin(theta), 0,
np.cos(theta)] for theta in phases]
alice2Observables = list(
map(lambda bloch: createQubitObservable(bloch),
alice2blochVectors))
alice2Effects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), alice2Observables))
phasesBob = [
np.random.uniform(-np.pi / 2, np.pi / 2),
np.random.uniform(-np.pi / 2, np.pi / 2)
]
bobVectors = [[np.sin(theta), 0, np.cos(theta)] for theta in phasesBob]
bobObservables = list(
map(lambda bloch: createQubitObservable(bloch), bobVectors))
bobEffects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), bobObservables))
aux = alice1Krauss
alice1Krauss = alice2Effects
alice2Effects = aux
psi = createMaxEntState(2)
rho = psi * psi.dag()
expectedCorrelations = {}
for x1 in range(2):
for a1 in range(2):
postMeasrmntState = qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2)) * rho * (qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2))).dag()
for x2, y, a2, b in product(range(2), repeat=4):
expectedCorrelations[(x1, x2), y, (a1, a2), b] = (
qt.tensor(alice2Effects[x2][a2], bobEffects[y][b]) *
postMeasrmntState).tr().real
expectedBehaviour = Behaviour(scenario, expectedCorrelations)
poly = BellPolytopeWithOneWayCommunication(
SequentialBellPolytope(scenario))
self.assertTrue(poly.contains(expectedBehaviour.getProbabilityList()),
'phases:' + str(phases[0]) + ', ' + str(phases[1]))
def testBuildBehaviourFromDictionaryAndConvertToList(self):
pr = {(x, y, a, b): 1 / 2 * int(x * y == (a + b) % 2)
for x, y, a, b in product(range(2), repeat=4)}
behaviour = Behaviour(BellScenario([2, 2], [2, 2]), pr)
probabilities = behaviour.getProbabilityList()
self.assertEqual(probabilities, [
1 / 2, 0, 0, 1 / 2, 1 / 2, 0, 0, 1 / 2, 1 / 2, 0, 0, 1 / 2, 0,
1 / 2, 1 / 2, 0
])
def testPRCorrBetweenAlice1andBobAndLocalOtherwiseIsNotInPolytope(self):
scenario = SequentialBellScenario([[2, 2], [2, 2]], [2, 2])
expectedCorrelations = {
((x1, x2), y, (a1, a2), b):
1 / 2 * int((a2 == 0) & (x1 * y == (a1 + b) % 2))
for ((x1, x2), y, (a1, a2), b) in scenario.getTuplesOfEvents()
}
expectedBehaviour = Behaviour(scenario, expectedCorrelations)
poly = SequentialBellPolytope(scenario)
self.assertFalse(poly.contains(expectedBehaviour.getProbabilityList()))
def testCHSHBetweenAlice1AndBobAndIdentityInAlice2IsInPolytope(self):
scenario = SequentialBellScenario([[2, 2], [2, 2]], [2, 2])
alice1blochVectors = [[1, 0, 0], [0, 1, 0]]
alice1Observables = list(
map(lambda bloch: createQubitObservable(bloch),
alice1blochVectors))
alice1Krauss = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), alice1Observables))
alice2Effects = [[qt.qeye(2), 0 * qt.qeye(2)],
[qt.qeye(2), 0 * qt.qeye(2)]]
bobUnBlochVectors = [[-1, -1, 0], [-1, 1, 0]]
bobObservables = list(
map(lambda bloch: createQubitObservable(bloch), bobUnBlochVectors))
bobEffects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), bobObservables))
psi = createMaxEntState(2)
rho = psi * psi.dag()
expectedCorrelations = {}
for x1 in range(2):
for a1 in range(2):
postMeasrmntState = qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2)) * rho * (qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2))).dag()
for x2, y, a2, b in product(range(2), repeat=4):
expectedCorrelations[(x1, x2), y, (a1, a2), b] = (
qt.tensor(alice2Effects[x2][a2], bobEffects[y][b]) *
postMeasrmntState).tr().real
expectedBehaviour = Behaviour(scenario, expectedCorrelations)
poly = BellPolytopeWithOneWayCommunication(
SequentialBellPolytope(scenario))
self.assertTrue(poly.contains(expectedBehaviour.getProbabilityList()))
bobObservables = list(
map(lambda bloch: createQubitObservable(bloch), bobUnBlochVectors))
bobEffects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), bobObservables))
psi = createMaxEntState(2)
rho = psi * psi.dag()
expectedCorrelations = {}
for x1, x2, y in product(range(len(alice1outputs)),
range(len(alice2outputs)),
range(len(bobOutputs))):
for a1, a2, b in product(range(alice1outputs[x1]),
range(alice2outputs[x2]),
range(bobOutputs[y])):
postMeasrmntState = qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2)) * rho * (qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2))).dag()
expectedCorrelations[(x1, x2), y, (a1, a2), b] = (
qt.tensor(alice2Effects[x2][a2], bobEffects[y][b]) *
postMeasrmntState).tr().real
expectedBehaviour = Behaviour(scenario, expectedCorrelations)
polytope = BellPolytopeWithOneWayCommunication(
SequentialBellPolytope(scenario))
print(polytope.contains(expectedBehaviour.getProbabilityList()))
def testMeasurementsOverSeparableStateAreLocal(self):
alice1outputs = [2, 2]
alice2outputs = [2, 3]
bobOutputs = [2, 2]
scenario = SequentialBellScenario([alice1outputs, alice2outputs],
bobOutputs)
epsilon = 0
plus = 1 / np.sqrt(2) * (qt.basis(2, 0) + qt.basis(2, 1))
minus = 1 / np.sqrt(2) * (qt.basis(2, 0) - qt.basis(2, 1))
Kplus = np.cos(epsilon) * plus * plus.dag() + np.sin(
epsilon) * minus * minus.dag()
Kminus = -np.cos(epsilon) * minus * minus.dag() + np.sin(
epsilon) * plus * plus.dag()
alice1Krauss = [
projectorsForQubitObservable(createQubitObservable([0, 0, 1])),
[Kplus, Kminus]
]
alice2blochVectors = [[0, 0, 1]]
alice2Observables = list(
map(lambda bloch: createQubitObservable(bloch),
alice2blochVectors))
alice2Effects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), alice2Observables))
trineAlice2 = [[0, 0, 1],
[np.sin(2 * np.pi / 3), 0,
np.cos(2 * np.pi / 3)],
[np.sin(4 * np.pi / 3), 0,
np.cos(4 * np.pi / 3)]]
paulies = [qt.sigmax(), qt.sigmay(), qt.sigmaz()]
alice2Effects.append(
list(
map(
lambda bloch: effectForQubitPovm(
1 / 3, sum([paulies[i] * bloch[i] for i in range(3)])),
trineAlice2)))
mu = np.arctan(np.sin(2 * epsilon))
bobUnBlochVectors = [[np.sin(mu), 0, np.cos(mu)],
[-np.sin(mu), 0, np.cos(mu)]]
bobObservables = list(
map(lambda bloch: createQubitObservable(bloch), bobUnBlochVectors))
bobEffects = list(
map(
lambda qubitObservable: projectorsForQubitObservable(
qubitObservable), bobObservables))
psi = qt.tensor(qt.basis(2, 0), qt.basis(2, 0))
rho = psi * psi.dag()
expectedCorrelations = {}
for x1, x2, y in product(range(len(alice1outputs)),
range(len(alice2outputs)),
range(len(bobOutputs))):
for a1, a2, b in product(range(alice1outputs[x1]),
range(alice2outputs[x2]),
range(bobOutputs[y])):
postMeasrmntState = qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2)) * rho * (qt.tensor(
alice1Krauss[x1][a1], qt.qeye(2))).dag()
expectedCorrelations[(x1, x2), y, (a1, a2), b] = (
qt.tensor(alice2Effects[x2][a2], bobEffects[y][b]) *
postMeasrmntState).tr().real
expectedBehaviour = Behaviour(scenario, expectedCorrelations)
polytope = BellPolytopeWithOneWayCommunication(
SequentialBellPolytope(scenario))
self.assertTrue(
polytope.contains(expectedBehaviour.getProbabilityList()))
