def test_fidelity_non_identical_states_2(self):
"""Test the fidelity between two non-identical states."""
e_0, e_1 = ket(2, 0), ket(2, 1)
rho = 3 / 4 * e_0 * e_0.conj().T + 1 / 4 * e_1 * e_1.conj().T
sigma = 1 / 8 * e_0 * e_0.conj().T + 7 / 8 * e_1 * e_1.conj().T
self.assertEqual(np.isclose(fidelity(rho, sigma), 0.774, rtol=1e-03),
True)
def test_domino_2(self):
"""Domino with index = 2."""
e_0, e_1 = ket(3, 0), ket(3, 1)
expected_res = np.kron(e_0, 1 / np.sqrt(2) * (e_0 - e_1))
res = domino(2)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_domino_8(self):
"""Domino with index = 8."""
e_0, e_1, e_2 = ket(3, 0), ket(3, 1), ket(3, 2)
expected_res = np.kron(1 / np.sqrt(2) * (e_0 - e_1), e_2)
res = domino(8)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_concurrence_separable(self):
"""The concurrence of a product state is zero."""
e_0, e_1 = ket(2, 0), ket(2, 1)
v_vec = np.kron(e_0, e_1)
sigma = v_vec * v_vec.conj().T
res = concurrence(sigma)
self.assertEqual(np.isclose(res, 0), True)
def test_state_distinguishability_three_state_vec(self):
"""State distinguishability for two state vectors."""
e_0, e_1 = ket(2, 0), ket(2, 1)
states = [e_0, e_1]
probs = [1 / 2, 1 / 2]
res = state_distinguishability(states, probs)
self.assertEqual(np.isclose(res, 1 / 2), True)
def test_sub_fidelity_lower_bound_2(self):
"""Test sub_fidelity is lower bound on fidelity for rho and pi."""
e_0, e_1 = ket(2, 0), ket(2, 1)
rho = 3 / 4 * e_0 * e_0.conj().T + 1 / 4 * e_1 * e_1.conj().T
sigma = 1 / 8 * e_0 * e_0.conj().T + 7 / 8 * e_1 * e_1.conj().T
res = sub_fidelity(rho, sigma)
self.assertLessEqual(res, fidelity(rho, sigma))
def test_max_ent_2(self):
"""Generate maximally entangled state: `1/sqrt(2) * (|00> + |11>)`."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = 1 / np.sqrt(2) * (np.kron(e_0, e_0) + np.kron(e_1, e_1))
res = max_entangled(2)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_max_ent_2_0_0(self):
"""Generate maximally entangled state: `|00> + |11>`."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = 1 * (np.kron(e_0, e_0) + np.kron(e_1, e_1))
res = max_entangled(2, False, False)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_domino_4(self):
"""Domino with index = 4."""
e_1, e_2 = ket(3, 1), ket(3, 2)
expected_res = np.kron(e_2, 1 / np.sqrt(2) * (e_1 - e_2))
res = domino(4)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_is_not_mub_dim_2(self):
"""Return False for non-MUB of dimension 2."""
e_0, e_1 = ket(2, 0), ket(2, 1)
mub_1 = [e_0, e_1]
mub_2 = [1 / np.sqrt(2) * (e_0 + e_1), e_1]
mub_3 = [1 / np.sqrt(2) * (e_0 + 1j * e_1), e_0]
mubs = [mub_1, mub_2, mub_3]
self.assertEqual(is_mub(mubs), False)
def test_schmidt_rank_singlet_state(self):
"""
Computing the Schmidt rank of the entangled singlet state should yield
a value greater than 1.
"""
e_0, e_1 = ket(2, 0), ket(2, 1)
rho = 1 / np.sqrt(2) * (np.kron(e_0, e_1) - np.kron(e_1, e_0))
rho = rho.conj().T * rho
self.assertEqual(schmidt_rank(rho) > 1, True)
def test_tensor_list_3(self):
"""Test tensor list with three items."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = np.kron(np.kron(e_0, e_1), e_0)
res = tensor([e_0, e_1, e_0])
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_is_pure_list(self):
"""Check that list of pure states returns True."""
e_0, e_1, e_2 = ket(3, 0), ket(3, 1), ket(3, 2)
e0_dm = e_0 * e_0.conj().T
e1_dm = e_1 * e_1.conj().T
e2_dm = e_2 * e_2.conj().T
self.assertEqual(is_pure([e0_dm, e1_dm, e2_dm]), True)
def test_bell_3(self):
"""Generates the Bell state: `1/sqrt(2) * (|01> - |10>)`."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = 1 / np.sqrt(2) * (np.kron(e_0, e_1) - np.kron(e_1, e_0))
res = bell(3)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_ghz_2_3(self):
"""Produces the 3-qubit GHZ state: `1/sqrt(2) * (|000> + |111>)`."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = 1 / np.sqrt(2) * (tensor(e_0, e_0, e_0) + tensor(e_1, e_1, e_1))
res = ghz(2, 3).toarray()
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_concurrence_entangled(self):
"""The concurrence on maximally entangled Bell state."""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_00, e_11 = np.kron(e_0, e_0), np.kron(e_1, e_1)
u_vec = 1 / np.sqrt(2) * (e_00 + e_11)
rho = u_vec * u_vec.conj().T
res = concurrence(rho)
self.assertEqual(np.isclose(res, 1), True)
def test_state_distinguishability_two_states(self):
"""State distinguishability for two state density matrices."""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_00 = e_0 * e_0.conj().T
e_11 = e_1 * e_1.conj().T
states = [e_00, e_11]
probs = [1 / 2, 1 / 2]
res = state_distinguishability(states, probs)
self.assertEqual(np.isclose(res, 1 / 2), True)
def bell(idx: int) -> np.ndarray:
r"""
Produce a Bell state [WIKBELL]_.
Returns one of the following four Bell states depending on the value
of `idx`:
.. math::
\begin{equation}
\begin{aligned}
\frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) &
\qquad &
\frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right) \\
\frac{1}{\sqrt{2}} \left( |01 \rangle + |10 \rangle \right) &
\qquad &
\frac{1}{\sqrt{2}} \left( |01 \rangle - |10 \rangle \right)
\end{aligned}
\end{equation}
Examples
==========
When `idx = 0`, this produces the following Bell state
.. math::
\frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right).
Using `toqito`, we can see that this yields the proper state.
>>> from toqito.states.states.bell import bell
>>> import numpy as np
>>> bell(0)
array([[0.70710678],
[0. ],
[0. ],
[0.70710678]])
References
==========
.. [WIKBELL] Wikipedia: Bell state
https://en.wikipedia.org/wiki/Bell_state
:param idx: A parameter in [0, 1, 2, 3]
"""
e_0, e_1 = ket(2, 0), ket(2, 1)
if idx == 0:
return 1 / np.sqrt(2) * (np.kron(e_0, e_0) + np.kron(e_1, e_1))
if idx == 1:
return 1 / np.sqrt(2) * (np.kron(e_0, e_0) - np.kron(e_1, e_1))
if idx == 2:
return 1 / np.sqrt(2) * (np.kron(e_0, e_1) + np.kron(e_1, e_0))
if idx == 3:
return 1 / np.sqrt(2) * (np.kron(e_0, e_1) - np.kron(e_1, e_0))
raise ValueError("Invalid integer value for Bell state.")
def test_is_mub_dim_2(self):
"""Return True for MUB of dimension 2."""
e_0, e_1 = ket(2, 0), ket(2, 1)
mub_1 = [e_0, e_1]
mub_2 = [1 / np.sqrt(2) * (e_0 + e_1), 1 / np.sqrt(2) * (e_0 - e_1)]
mub_3 = [
1 / np.sqrt(2) * (e_0 + 1j * e_1),
1 / np.sqrt(2) * (e_0 - 1j * e_1)
]
mubs = [mub_1, mub_2, mub_3]
self.assertEqual(is_mub(mubs), True)
def test_tile_3(self):
r"""Generate the Tile state for index = 3:
.. math::
|\psi_3\rangle = \frac{1}{\sqrt{2}}
\left(|1\rangle - |2\rangle \right) |0\rangle
"""
e_0, e_1, e_2 = ket(3, 0), ket(3, 1), ket(3, 2)
expected_res = 1 / np.sqrt(2) * (e_1 - e_2) * e_0
res = tile(3)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_tile_0(self):
r"""Generate the Tile state for index = 0:
.. math::
|\psi_0 \rangle = \frac{1}{\sqrt{2}} |0 \rangle
\left(|0\rangle - |1\rangle \right).
"""
e_0, e_1 = ket(3, 0), ket(3, 1)
expected_res = 1 / np.sqrt(2) * e_0 * (e_0 - e_1)
res = tile(0)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_w_state_3(self):
"""The 3-qubit W-state."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = (
1
/ np.sqrt(3)
* (tensor(e_1, e_0, e_0) + tensor(e_0, e_1, e_0) + tensor(e_0, e_0, e_1))
)
res = w_state(3)
bool_mat = np.isclose(res, expected_res, atol=0.2)
self.assertEqual(np.all(bool_mat), True)
def test_tile_4(self):
r"""Generate the Tile state for index = 4:
.. math::
|\psi_4\rangle = \frac{1}{3}
\left(|0\rangle + |1\rangle + |2\rangle)\right)
\left(|0\rangle + |1\rangle + |2\rangle.
"""
e_0, e_1, e_2 = ket(3, 0), ket(3, 1), ket(3, 2)
expected_res = 1 / 3 * (e_0 + e_1 + e_2) * (e_0 + e_1 + e_2)
res = tile(4)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_schmidt_rank_separable_state(self):
"""
Computing the Schmidt rank of a separable state should yield a value
equal to 1.
"""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_00 = np.kron(e_0, e_0)
e_01 = np.kron(e_0, e_1)
e_10 = np.kron(e_1, e_0)
e_11 = np.kron(e_1, e_1)
rho = 1 / 2 * (e_00 - e_01 - e_10 + e_11)
rho = rho.conj().T * rho
self.assertEqual(schmidt_rank(rho) == 1, True)
def test_trace_distance_same_state(self):
r"""Test that: :math:`T(\rho, \sigma) = 0` iff `\rho = \sigma`."""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_00 = np.kron(e_0, e_0)
e_11 = np.kron(e_1, e_1)
u_vec = 1 / np.sqrt(2) * (e_00 + e_11)
rho = u_vec * u_vec.conj().T
sigma = rho
res = trace_distance(rho, sigma)
self.assertEqual(np.isclose(res, 0), True)
def test_trace_norm(self):
"""Test trace norm."""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_00 = np.kron(e_0, e_0)
e_11 = np.kron(e_1, e_1)
u_vec = 1 / np.sqrt(2) * (e_00 + e_11)
rho = u_vec * u_vec.conj().T
res = trace_norm(rho)
_, singular_vals, _ = np.linalg.svd(rho)
expected_res = float(np.sum(singular_vals))
self.assertEqual(np.isclose(res, expected_res), True)
def test_counterfeit_attack_wiesner_money(self):
"""Probability of counterfeit attack on Wiesner's quantum money."""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_p = (e_0 + e_1) / np.sqrt(2)
e_m = (e_0 - e_1) / np.sqrt(2)
e_000 = tensor(e_0, e_0, e_0)
e_111 = tensor(e_1, e_1, e_1)
e_ppp = tensor(e_p, e_p, e_p)
e_mmm = tensor(e_m, e_m, e_m)
q_a = (1 / 4 * (e_000 * e_000.conj().T + e_111 * e_111.conj().T +
e_ppp * e_ppp.conj().T + e_mmm * e_mmm.conj().T))
res = counterfeit_attack(q_a)
self.assertEqual(np.isclose(res, 3 / 4), True)
def test_weak_coin_flipping(self):
"""
Test for maximally entangled state.
Refer to the appendix of https://arxiv.org/abs/1703.03887
"""
e_0, e_1 = ket(2, 0), ket(2, 1)
e_m = (e_0 - e_1) / np.sqrt(2)
rho = np.kron(e_1 * e_1.conj().T, e_0 * e_0.conj().T) + np.kron(
e_m * e_m.conj().T, e_1 * e_1.conj().T
)
self.assertEqual(
np.isclose(weak_coin_flipping(rho), np.cos(np.pi / 8) ** 2), True
)
def test_tensor_n_0(self):
"""Test tensor n=0 times."""
e_0 = ket(2, 0)
expected_res = None
res = tensor(e_0, 0)
self.assertEqual(res, expected_res)
def test_tensor_n_3(self):
"""Test tensor n=3 times."""
e_0 = ket(2, 0)
expected_res = np.kron(np.kron(e_0, e_0), e_0)
res = tensor(e_0, 3)
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
