def test_measure_order(self):
"""Order of measurement must not make a difference."""
b00 = bell.bell_state(0, 0)
_, b00 = ops.Measure(b00, 0, tostate=0)
_, b00 = ops.Measure(b00, 1, tostate=0)
self.assertTrue(math.isclose(b00.prob(0, 0), 1.0))
self.assertTrue(math.isclose(b00.prob(1, 1), 0.0))
b00 = bell.bell_state(0, 0)
_, b00 = ops.Measure(b00, 1, tostate=0)
_, b00 = ops.Measure(b00, 0, tostate=0)
self.assertTrue(math.isclose(b00.prob(0, 0), 1.0))
self.assertTrue(math.isclose(b00.prob(1, 1), 0.0))
def main(argv):
if len(argv) > 1:
raise app.UsageError('Too many command-line arguments.')
# Step 1: Alice and Bob share an entangled pair, and separate.
psi = bell.bell_state(0, 0)
# Step 2: Alice wants to teleport a qubit |x> to Bob,
# which is in the state:
# |x> = a|0> + b|1> (with a^2 + b^2 == 1)
a = 0.6
b = math.sqrt(1.0 - a * a)
x = state.qubit(a, b)
print('Quantum Teleportation')
print('Start with EPR Pair a={:.2f}, b={:.2f}'.format(a, b))
# Produce combined state.
alice = x * psi
# Alice lets the 1st qubit interact with the 2nd qubit, which is her
# part of the entangle state with Bob.
alice = ops.Cnot(0, 1)(alice)
# Now she applies a Hadamard to qubit 0. Bob still owns qubit 2.
alice = ops.Hadamard()(alice, idx=0)
# Alices measures and communicates the result (|00>, |01>, ...) to Bob.
alice_measures(alice, a, b, 0, 0)
alice_measures(alice, a, b, 0, 1)
alice_measures(alice, a, b, 1, 0)
alice_measures(alice, a, b, 1, 1)
def test_bell_and_pauli(self):
b00 = bell.bell_state(0, 0)
bell_xz = ops.PauliX()(b00)
bell_xz = ops.PauliZ()(bell_xz)
bell_iy = (1j * ops.PauliY())(b00)
self.assertTrue(np.allclose(bell_xz, bell_iy))
def test_not_pure(self):
"""Bell states are pure states."""
for a in [0, 1]:
for b in [0, 1]:
b = bell.bell_state(a, b)
self.assertTrue(b.density().is_pure())
self.assertTrue(
math.isclose(np.real(np.trace(b.density())),
1.0,
abs_tol=1e-6))
def test_bell(self):
"""Check successful entanglement by computing the schmidt_number."""
b00 = bell.bell_state(0, 0)
self.assertGreater(b00.schmidt_number([1]), 1.0)
self.assertTrue(
b00.is_close((state.zeros(2) + state.ones(2)) / math.sqrt(2)))
# Note the order is reversed from pictorials.
op_exp = (ops.Cnot(0, 1) @ (ops.Hadamard() * ops.Identity()))
b00_exp = op_exp(state.zeros(2))
self.assertTrue(b00.is_close(b00_exp))
b01 = bell.bell_state(0, 1)
self.assertGreater(b01.schmidt_number([1]), 1.0)
b10 = bell.bell_state(1, 0)
self.assertGreater(b10.schmidt_number([1]), 1.0)
b11 = bell.bell_state(1, 1)
self.assertGreater(b11.schmidt_number([1]), 1.0)
def test_measure(self):
b00 = bell.bell_state(0, 1)
self.assertTrue(math.isclose(b00.prob(0, 1), 0.5, abs_tol=1e-6))
self.assertTrue(math.isclose(b00.prob(1, 0), 0.5, abs_tol=1e-6))
_, b00 = ops.Measure(b00, 0, tostate=0)
self.assertTrue(math.isclose(b00.prob(0, 1), 1.0, abs_tol=1e-6))
self.assertTrue(math.isclose(b00.prob(1, 0), 0.0, abs_tol=1e-6))
# This state can't be measured, all zeros.
_, b00 = ops.Measure(b00, 1, tostate=1)
self.assertTrue(math.isclose(b00.prob(1, 0), 0.0, abs_tol=1e-6))
b00 = bell.bell_state(0, 1)
self.assertTrue(math.isclose(b00.prob(0, 1), 0.5, abs_tol=1e-6))
self.assertTrue(math.isclose(b00.prob(1, 0), 0.5, abs_tol=1e-6))
_, b00 = ops.Measure(b00, 0, tostate=1)
self.assertTrue(math.isclose(b00.prob(0, 1), 0.0, abs_tol=1e-6))
self.assertTrue(math.isclose(b00.prob(1, 0), 1.0, abs_tol=1e-6))
# This state can't be measured, all zeros.
p, _ = ops.Measure(b00, 1, tostate=1, collapse=False)
self.assertEqual(p, 0.0)
def main(argv):
if len(argv) > 1:
raise app.UsageError('Too many command-line arguments.')
# Step 1: Alice and Bob share an entangled pair, and separate.
psi = bell.bell_state(0, 0)
# Alices manipulates her qubit and sends her 1 qubit back to Bob,
# who measures. In the Hadamard basis he would get b00, b01, etc.
# but we're measuring in the computational basis by reverse
# applying Hadamard and Cnot.
for bit0 in range(2):
for bit1 in range(2):
psi_alice = alice_manipulates(psi, bit0, bit1)
bob_measures(psi_alice, bit0, bit1)
def test_partial(self):
"""Test partial trace."""
psi = bell.bell_state(0, 0)
reduced = ops.TraceOut(psi.density(), [0])
self.assertTrue(
math.isclose(np.real(np.trace(reduced)), 1.0, abs_tol=1e-6))
self.assertTrue(math.isclose(np.real(reduced[0, 0]), 0.5,
abs_tol=1e-6))
self.assertTrue(math.isclose(np.real(reduced[1, 1]), 0.5,
abs_tol=1e-6))
q0 = state.qubit(alpha=0.5)
q1 = state.qubit(alpha=0.8660254)
psi = q0 * q1
reduced = ops.TraceOut(psi.density(), [0])
self.assertTrue(math.isclose(np.real(np.trace(reduced)), 1.0))
self.assertTrue(
math.isclose(np.real(reduced[0, 0]), 0.75, abs_tol=1e-6))
self.assertTrue(
math.isclose(np.real(reduced[1, 1]), 0.25, abs_tol=1e-6))
reduced = ops.TraceOut(psi.density(), [1])
self.assertTrue(math.isclose(np.real(np.trace(reduced)), 1.0))
self.assertTrue(
math.isclose(np.real(reduced[0, 0]), 0.25, abs_tol=1e-6))
self.assertTrue(
math.isclose(np.real(reduced[1, 1]), 0.75, abs_tol=1e-6))
psi = q1 * q0 * state.qubit(alpha=(0.8660254))
reduced = ops.TraceOut(psi.density(), [2])
reduced = ops.TraceOut(reduced, [0])
self.assertTrue(math.isclose(np.real(np.trace(reduced)), 1.0))
self.assertTrue(math.isclose(np.real(reduced[0, 0]), 0.25))
self.assertTrue(math.isclose(np.real(reduced[1, 1]), 0.75))
psi = q1 * q0 * state.qubit(alpha=(0.8660254))
reduced = ops.TraceOut(psi.density(), [0, 2])
self.assertTrue(math.isclose(np.real(np.trace(reduced)), 1.0))
self.assertTrue(math.isclose(np.real(reduced[0, 0]), 0.25))
self.assertTrue(math.isclose(np.real(reduced[1, 1]), 0.75))
