def test_ppt_distinguishability_yyd_density_matrices():
"""
PPT distinguishing the YYD states from [1] should yield `7/8 ~ 0.875`
Feeding the input to the function as density matrices.
References:
[1]: Yu, Nengkun, Runyao Duan, and Mingsheng Ying.
"Four locally indistinguishable ququad-ququad orthogonal
maximally entangled states."
Physical review letters 109.2 (2012): 020506.
https://arxiv.org/abs/1107.3224
"""
psi_0 = bell(0)
psi_1 = bell(2)
psi_2 = bell(3)
psi_3 = bell(1)
x_1 = np.kron(psi_0, psi_0)
x_2 = np.kron(psi_1, psi_3)
x_3 = np.kron(psi_2, psi_3)
x_4 = np.kron(psi_3, psi_3)
rho_1 = x_1 * x_1.conj().T
rho_2 = x_2 * x_2.conj().T
rho_3 = x_3 * x_3.conj().T
rho_4 = x_4 * x_4.conj().T
states = [rho_1, rho_2, rho_3, rho_4]
probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
# Min-error tests:
primal_res = ppt_distinguishability(states,
probs=probs,
dist_method="min-error",
strategy=True)
dual_res = ppt_distinguishability(states,
probs=probs,
dist_method="min-error",
strategy=False)
np.testing.assert_equal(np.isclose(primal_res, 7 / 8, atol=0.001), True)
np.testing.assert_equal(np.isclose(dual_res, 7 / 8, atol=0.001), True)
primal_res = ppt_distinguishability(states,
probs=probs,
dist_method="unambiguous",
strategy=True)
dual_res = ppt_distinguishability(states,
probs=probs,
dist_method="unambiguous",
strategy=False)
np.testing.assert_equal(np.isclose(primal_res, 3 / 4, atol=0.001), True)
np.testing.assert_equal(np.isclose(dual_res, 3 / 4, atol=0.001), True)
def test_ppt_distinguishability_yyd_states_no_probs():
"""
PPT distinguishing the YYD states from [1] should yield 7/8 ~ 0.875
If no probability vector is explicitly given, assume uniform
probabilities are given.
References:
[1]: Yu, Nengkun, Runyao Duan, and Mingsheng Ying.
"Four locally indistinguishable ququad-ququad orthogonal
maximally entangled states."
Physical review letters 109.2 (2012): 020506.
https://arxiv.org/abs/1107.3224
"""
psi_0 = bell(0)
psi_1 = bell(2)
psi_2 = bell(3)
psi_3 = bell(1)
x_1 = np.kron(psi_0, psi_0)
x_2 = np.kron(psi_1, psi_3)
x_3 = np.kron(psi_2, psi_3)
x_4 = np.kron(psi_3, psi_3)
rho_1 = x_1 * x_1.conj().T
rho_2 = x_2 * x_2.conj().T
rho_3 = x_3 * x_3.conj().T
rho_4 = x_4 * x_4.conj().T
states = [rho_1, rho_2, rho_3, rho_4]
primal_res = ppt_distinguishability(states,
probs=None,
dist_method="min-error",
strategy=True)
dual_res = ppt_distinguishability(states,
probs=None,
dist_method="min-error",
strategy=False)
np.testing.assert_equal(np.isclose(primal_res, 7 / 8, atol=0.001), True)
np.testing.assert_equal(np.isclose(dual_res, 7 / 8, atol=0.001), True)
primal_res = ppt_distinguishability(states,
probs=None,
dist_method="unambiguous",
strategy=True)
dual_res = ppt_distinguishability(states,
probs=None,
dist_method="unambiguous",
strategy=False)
np.testing.assert_equal(np.isclose(primal_res, 3 / 4, atol=0.001), True)
np.testing.assert_equal(np.isclose(dual_res, 3 / 4, atol=0.001), True)
def test_ppt_distinguishability_yyd_vectors():
"""
PPT distinguishing the YYD states from [1] should yield `7/8 ~ 0.875`
Feeding the input to the function as state vectors.
References:
[1]: Yu, Nengkun, Runyao Duan, and Mingsheng Ying.
"Four locally indistinguishable ququad-ququad orthogonal
maximally entangled states."
Physical review letters 109.2 (2012): 020506.
https://arxiv.org/abs/1107.3224
"""
psi_0 = bell(0)
psi_1 = bell(2)
psi_2 = bell(3)
psi_3 = bell(1)
x_1 = np.kron(psi_0, psi_0)
x_2 = np.kron(psi_1, psi_3)
x_3 = np.kron(psi_2, psi_3)
x_4 = np.kron(psi_3, psi_3)
states = [x_1, x_2, x_3, x_4]
probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
res = ppt_distinguishability(states, probs)
np.testing.assert_equal(np.isclose(res, 7 / 8), True)
def test_ppt_distinguishability_four_bell_states():
r"""
PPT distinguishing the four Bell states.
There exists a closed form formula for the probability with which one
is able to distinguish one of the four Bell states given with equal
probability when Alice and Bob have access to a resource state [1].
The resource state is defined by
..math::
The closed form probability with which Alice and Bob can distinguish via
PPT measurements is given as follows
.. math::
\frac{1}{2} \left(1 + \sqrt{1 - \epsilon^2} \right).
This formula happens to be equal to LOCC and SEP as well for this case.
Refer to Theorem 5 in [1] for more details.
References:
[1]: Bandyopadhyay, Somshubhro, et al.
"Limitations on separable measurements by convex optimization."
IEEE Transactions on Information Theory 61.6 (2015): 3593-3604.
https://arxiv.org/abs/1408.6981
"""
rho_1 = bell(0) * bell(0).conj().T
rho_2 = bell(1) * bell(1).conj().T
rho_3 = bell(2) * bell(2).conj().T
rho_4 = bell(3) * bell(3).conj().T
e_0, e_1 = basis(2, 0), basis(2, 1)
e_00 = np.kron(e_0, e_0)
e_11 = np.kron(e_1, e_1)
eps = 0.5
resource_state = np.sqrt((1 + eps) / 2) * e_00 + np.sqrt(
(1 - eps) / 2) * e_11
resource_state = resource_state * resource_state.conj().T
states = [
np.kron(rho_1, resource_state),
np.kron(rho_2, resource_state),
np.kron(rho_3, resource_state),
np.kron(rho_4, resource_state),
]
probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
res = ppt_distinguishability(states, probs)
exp_res = 1 / 2 * (1 + np.sqrt(1 - eps**2))
np.testing.assert_equal(np.isclose(res, exp_res), True)
def test_ppt_distinguishability_werner_hiding_pairs():
r"""
One quantum data hiding scheme involves the Werner hiding pair.
A Werner hiding pair is defined by
.. math::
\begin{equation}
\sigma_0^{(n)} = \frac{\mathbb{I} \otimes \mathbb{I} + W_n}{n(n+1)}
\quad \text{and} \quad
\sigma_1^{(n)} = \frac{\mathbb{I} \otimes \mathbb{I} - W_n}{n(n-1)}
\end{equation}
The optimal probability to distinguish the Werner hiding pair is known
to be upper bounded by the following equation
.. math::
\begin{equation}
\frac{1}{2} + \frac{1}{n+1}
\end{equation}
References:
[1]: Terhal, Barbara M., David P. DiVincenzo, and Debbie W. Leung.
"Hiding bits in Bell states."
Physical review letters 86.25 (2001): 5807.
https://arxiv.org/abs/quant-ph/0011042
[2]: Cosentino, Alessandro
"Quantum state local distinguishability via convex optimization".
University of Waterloo, Thesis
https://uwspace.uwaterloo.ca/handle/10012/9572
"""
dim = 2
sigma_0 = (np.kron(np.identity(dim), np.identity(dim)) +
swap_operator(dim)) / (dim * (dim + 1))
sigma_1 = (np.kron(np.identity(dim), np.identity(dim)) -
swap_operator(dim)) / (dim * (dim - 1))
states = [sigma_0, sigma_1]
expected_val = 1 / 2 + 1 / (dim + 1)
res = ppt_distinguishability(states)
np.testing.assert_equal(np.isclose(res, expected_val), True)
