def test_schmidt_decomp_dim_list_pure_state():
"""Schmidt decomposition of a pure state with a dimension list."""
pure_vec = -1 / np.sqrt(2) * np.array([[1], [0], [1], [0]])
# Test when dimension default and k_param is default (0):
singular_vals, vt_mat, u_mat = schmidt_decomposition(pure_vec)
expected_singular_vals = np.array([[1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = 1 / np.sqrt(2) * np.array([[-1], [-1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = np.array([[1], [0]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
# Test when dimension [2, 2] and k_param is 1:
singular_vals, vt_mat, u_mat = schmidt_decomposition(pure_vec, [2, 2], 1)
expected_singular_vals = np.array([[1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = 1 / np.sqrt(2) * np.array([[-1], [-1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = np.array([[1], [0]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
# Test when dimension [2, 2] and k_param is 2:
singular_vals, vt_mat, u_mat = schmidt_decomposition(pure_vec, [2, 2], 2)
expected_singular_vals = np.array([[1], [0]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = 1 / np.sqrt(2) * np.array([[-1, -1], [-1, 1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = np.identity(2)
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
def test_schmidt_decomp_two_qubit_3():
"""
Schmidt decomposition of two-qubit state.
The Schmidt decomposition of 1/2* (|00> + |11>) has Schmidt coefficients
equal to 1/2[1, 1]
"""
e_0, e_1 = basis(2, 0), basis(2, 1)
phi = 1 / 2 * (np.kron(e_0, e_0) + np.kron(e_1, e_1))
singular_vals, vt_mat, u_mat = schmidt_decomposition(phi)
expected_singular_vals = 1 / 2 * np.array([[1], [1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = np.array([[1, 0], [0, 1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = np.array([[1, 0], [0, 1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
s_decomp = (
singular_vals[0] * np.atleast_2d(np.kron(vt_mat[:, 0], u_mat[:, 0])).T
+
singular_vals[1] * np.atleast_2d(np.kron(vt_mat[:, 1], u_mat[:, 1])).T)
np.testing.assert_equal(np.isclose(np.linalg.norm(phi - s_decomp), 0),
True)
def test_schmidt_decomp_two_qubit_2():
"""
Schmidt decomposition of two-qubit state.
The Schmidt decomposition of | phi > = 1/2(|00> + |01> + |10> - |11>) is
the state 1/sqrt(2) * (|0>|+> + |1>|->).
"""
e_0, e_1 = basis(2, 0), basis(2, 1)
phi = 1 / 2 * (np.kron(e_0, e_0) + np.kron(e_0, e_1) + np.kron(e_1, e_0) -
np.kron(e_1, e_1))
singular_vals, vt_mat, u_mat = schmidt_decomposition(phi)
expected_singular_vals = 1 / np.sqrt(2) * np.array([[1], [1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = np.array([[-1, 0], [0, -1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = 1 / np.sqrt(2) * np.array([[-1, -1], [-1, 1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
s_decomp = (
singular_vals[0] * np.atleast_2d(np.kron(vt_mat[:, 0], u_mat[:, 0])).T
+
singular_vals[1] * np.atleast_2d(np.kron(vt_mat[:, 1], u_mat[:, 1])).T)
np.testing.assert_equal(np.isclose(np.linalg.norm(phi - s_decomp), 0),
True)
def _is_product(rho: np.ndarray,
dim: Union[int, List[int]] = None) -> [int, bool]:
"""
Determine if input is a product state recursive helper.
:param rho: The vector or matrix to check.
:param dim: The dimension of the input.
:return: :code:`True` if :code:`rho` is a product vector and :code:`False` otherwise.
"""
# If the input is provided as a matrix, compute the operator Schmidt rank.
if len(rho.shape) == 2:
if rho.shape[0] != 1 and rho.shape[1] != 1:
return _operator_is_product(rho, dim)
if dim is None:
dim = np.round(np.sqrt(len(rho)))
if isinstance(dim, list):
dim = np.array(dim)
# Allow the user to enter a single number for dim.
if isinstance(dim, float):
num_sys = 1
else:
num_sys = len(dim)
if num_sys == 1:
dim = np.array([dim, len(rho) // dim])
dim[1] = np.round(dim[1])
num_sys = 2
dec = None
# If there are only two subsystems, just use the Schmidt decomposition.
if num_sys == 2:
singular_vals, u_mat, vt_mat = schmidt_decomposition(rho, dim, 2)
ipv = singular_vals[1] <= np.prod(dim) * np.spacing(singular_vals[0])
# Provide this even if not requested, since it is needed if this
# function was called as part of its recursive algorithm (see below)
if ipv:
u_mat = u_mat * np.sqrt(singular_vals[0])
vt_mat = vt_mat * np.sqrt(singular_vals[0])
dec = [u_mat[:, 0], vt_mat[:, 0]]
else:
new_dim = [dim[0] * dim[1]]
new_dim.extend(dim[2:])
ipv, dec = _is_product(rho, new_dim)
if ipv:
ipv, tdec = _is_product(dec[0], [dim[0], dim[1]])
if ipv:
dec = [tdec, dec[1:]]
return ipv, dec
def test_schmidt_decomp_max_ent():
"""Schmidt decomposition of the 3-D maximally entangled state."""
singular_vals, u_mat, vt_mat = schmidt_decomposition(max_entangled(3))
expected_u_mat = np.identity(3)
expected_vt_mat = np.identity(3)
expected_singular_vals = 1 / np.sqrt(3) * np.array([[1], [1], [1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
def test_schmidt_decomp_standard_basis():
"""Test on standard basis vectors."""
e_1 = basis(2, 1)
singular_vals, vt_mat, u_mat = schmidt_decomposition(np.kron(e_1, e_1))
expected_singular_vals = np.array([[1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = np.array([[0], [1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = np.array([[0], [1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
def test_schmidt_decomp_dim_list():
"""Schmidt decomposition with list specifying dimension."""
singular_vals, u_mat, vt_mat = schmidt_decomposition(max_entangled(3),
dim=[3, 3])
expected_u_mat = np.identity(3)
expected_vt_mat = np.identity(3)
expected_singular_vals = 1 / np.sqrt(3) * np.array([[1], [1], [1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
def test_schmidt_decomp_example():
"""Test for example Schmidt decomposition."""
e_0, e_1 = basis(2, 0), basis(2, 1)
phi = ((1 + np.sqrt(6)) / (2 * np.sqrt(6)) * np.kron(e_0, e_0) +
(1 - np.sqrt(6)) / (2 * np.sqrt(6)) * np.kron(e_0, e_1) +
(np.sqrt(2) - np.sqrt(3)) / (2 * np.sqrt(6)) * np.kron(e_1, e_0) +
(np.sqrt(2) + np.sqrt(3)) / (2 * np.sqrt(6)) * np.kron(e_1, e_1))
singular_vals, vt_mat, u_mat = schmidt_decomposition(phi)
expected_singular_vals = np.array([[np.sqrt(3 / 4)], [np.sqrt(1 / 4)]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = np.array([[-0.81649658, 0.57735027],
[0.57735027, 0.81649658]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = 1 / np.sqrt(2) * np.array([[-1, 1], [1, 1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
def test_schmidt_decomp_two_qubit_4():
"""
Schmidt decomposition of two-qubit state.
The Schmidt decomposition of 1/2 * (|00> - |01> + |10> + |11>) has Schmidt coefficients
equal to [1, 1]
"""
e_0, e_1 = basis(2, 0), basis(2, 1)
phi = 1 / 2 * (np.kron(e_0, e_0) - np.kron(e_0, e_1) + np.kron(e_1, e_0) +
np.kron(e_1, e_1))
singular_vals, vt_mat, u_mat = schmidt_decomposition(phi)
expected_singular_vals = 1 / np.sqrt(2) * np.array([[1], [1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = np.array([[-1, 0], [0, 1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = 1 / np.sqrt(2) * np.array([[-1, 1], [1, 1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)
def test_schmidt_decomp_two_qubit_1():
"""
Schmidt decomposition of two-qubit state.
The Schmidt decomposition of | phi > = 1/2(|00> + |01> + |10> + |11>) is
the state |+>|+> where |+> = 1/sqrt(2) * (|0> + |1>).
"""
e_0, e_1 = basis(2, 0), basis(2, 1)
phi = 1 / 2 * (np.kron(e_0, e_0) + np.kron(e_0, e_1) + np.kron(e_1, e_0) +
np.kron(e_1, e_1))
singular_vals, vt_mat, u_mat = schmidt_decomposition(phi)
expected_singular_vals = np.array([[1]])
bool_mat = np.isclose(expected_singular_vals, singular_vals)
np.testing.assert_equal(np.all(bool_mat), True)
expected_vt_mat = 1 / np.sqrt(2) * np.array([[-1], [-1]])
bool_mat = np.isclose(expected_vt_mat, vt_mat)
np.testing.assert_equal(np.all(bool_mat), True)
expected_u_mat = 1 / np.sqrt(2) * np.array([[-1], [-1]])
bool_mat = np.isclose(expected_u_mat, u_mat)
np.testing.assert_equal(np.all(bool_mat), True)