def test_pauli_str_list():
"""Test with list of Paulis of str."""
expected_res = np.array([[0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0]])
res = pauli(["x", "x"])
bool_mat = np.isclose(res, expected_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_pauli_3():
"""Pauli-Z operator with argument 3."""
expected_res = np.array([[1, 0], [0, -1]])
res = pauli(3)
bool_mat = np.isclose(res, expected_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_pauli_2():
"""Pauli-Y operator with argument 2."""
expected_res = np.array([[0, -1j], [1j, 0]])
res = pauli(2)
bool_mat = np.isclose(res, expected_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_pauli_1():
"""Pauli-X operator with argument 1."""
expected_res = np.array([[0, 1], [1, 0]])
res = pauli(1)
bool_mat = np.isclose(res, expected_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_pauli_str_sparse():
"""Pauli-I operator with argument "I"."""
res = pauli("I", True)
np.testing.assert_equal(issparse(res), True)
def concurrence(rho: np.ndarray) -> float:
r"""
Calculate the concurrence of a bipartite state [WikCon]_.
The concurrence of a bipartite state :math:`\rho` is defined as
.. math::
\max(0, \lambda_1 - \lambda_2 - \lambda_3 - \lambda_4),
where :math:`\lambda_1, \ldots, \lambda_4` are the square roots of the
eigenvalues in decreasing order of the matrix
.. math::
\rho\tilde{\rho} = \rho \sigma_y \otimes \sigma_y \rho^* \sigma_y \otimes \sigma_y.
Concurrence can serve as a measure of entanglement.
Examples
==========
Consider the following Bell state:
.. math::
u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right).
The concurrence of the density matrix :math:`\rho = u u^*` defined by the vector :math:`u` is
given as
.. math::
\mathcal{C}(\rho) \approx 1.
The following example calculates this quantity using the :code:`toqito` package.
>>> import numpy as np
>>> from toqito.states import basis
>>> from toqito.state_props import concurrence
>>> e_0, e_1 = basis(2, 0), basis(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
>>> concurrence(rho)
0.9999999999999998
Consider the concurrence of the following product state
.. math::
v = |0\rangle \otimes |1 \rangle.
As this state has no entanglement, the concurrence is zero.
>>> import numpy as np
>>> from toqito.states import basis
>>> from toqito.state_props import concurrence
>>> e_0, e_1 = basis(2, 0), basis(2, 1)
>>> v_vec = np.kron(e_0, e_1)
>>> sigma = v_vec * v_vec.conj().T
>>> concurrence(sigma)
0
References
==========
.. [WikCon] Wikipedia page for concurrence (quantum computing)
https://en.wikipedia.org/wiki/Concurrence_(quantum_computing)
:param rho: The bipartite system specified as a matrix.
:return: The concurrence of the bipartite state :math:`\rho`.
"""
if rho.shape != (4, 4):
raise ValueError(
"InvalidDim: Concurrence is only defined for bipartite systems.")
sigma_y = pauli("Y", False)
sigma_y_y = np.kron(sigma_y, sigma_y)
rho_tilde = sigma_y_y @ rho.conj() @ sigma_y_y
eig_vals = np.linalg.eigvals(rho @ rho_tilde)
eig_vals = np.sort(np.abs(np.sqrt(eig_vals)))[::-1]
return max(0, eig_vals[0] - eig_vals[1] - eig_vals[2] - eig_vals[3])
