def test_monotonicity(self, dimension):
"""
Check monotonicity w.r.t. tensor product, see. Eq. (45) in
arXiv:1611.03449v2:
hellinger_dist(rhoA & rhoB, sigmaA & sigmaB)
>= hellinger_dist(rhoA, sigmaA)
with equality iff sigmaB = rhoB where '&' is the tensor product.
"""
tol = 1e-5
rhoA, rhoB, sigmaA, sigmaB = [rand_dm(dimension) for _ in [None] * 4]
rho = tensor(rhoA, rhoB)
rho_sim = tensor(rhoA, sigmaB)
sigma = tensor(sigmaA, sigmaB)
dist = hellinger_dist(rhoA, sigmaA)
assert hellinger_dist(rho, sigma) + tol > dist
assert hellinger_dist(rho_sim, sigma) == pytest.approx(dist, abs=tol)
def test_orthogonal(self, left_dm, right_dm, dimension):
left = basis(dimension, 0)
right = basis(dimension, dimension // 2)
if left_dm:
left = left.proj()
if right_dm:
right = right.proj()
expected = np.sqrt(2)
assert hellinger_dist(left, right) == pytest.approx(expected, abs=1e-6)
def test_inequality_hellinger_dist_to_bures_dist(self, left, right):
tol = 1e-7
hellinger = hellinger_dist(left, right)
bures = bures_dist(left, right)
assert bures <= hellinger + tol
def test_known_cases_pure_states(self, dimension):
left = rand_ket(dimension)
right = rand_ket(dimension)
expected = np.sqrt(2 * (1 - np.abs(left.overlap(right))**2))
assert hellinger_dist(left, right) == pytest.approx(expected, abs=1e-7)
def test_state_with_itself(self, state):
assert hellinger_dist(state, state) == pytest.approx(0, abs=1e-6)