toqito.state_props.concurrence

Concurrence property calculates the concurrence of a bipartite state.

The concurrence property is an entanglement measure defined for the product states of two qubits.

Module Contents

toqito.state_props.concurrence.concurrence(rho)

Calculate the concurrence of a bipartite state [@WikiConcurrence].

The concurrence of a bipartite state (rho) is defined as

[

max(0, lambda_1 - lambda_2 - lambda_3 - lambda_4),

]

where (lambda_1, ldots, lambda_4) are the square roots of the eigenvalues in decreasing order of the matrix

[

rhotilde{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:

[

u = frac{1}{sqrt{2}} left( |00 rangle + |11 rangle right).

]

The concurrence of the density matrix (rho = u u^*) defined by the vector (u) is given as

[

mathcal{C}(rho) approx 1.

]

The following example calculates this quantity using the |toqito⟩ package.

`python exec="1" source="above" import numpy as np from toqito.matrices import standard_basis from toqito.state_props import concurrence e_0, e_1 = standard_basis(2) 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 print(concurrence(rho)) `

Consider the concurrence of the following product state

[

v = |0rangle otimes |1 rangle.

]

As this state has no entanglement, the concurrence is zero.

`python exec="1" source="above" 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 print(concurrence(sigma)) `

Raises:

ValueError – If system is not bipartite.

Parameters:

rho (numpy.ndarray) – The bipartite system specified as a matrix.

Returns:

The concurrence of the bipartite state (rho).

Return type:

float