state_props.concurrence

Concurrence property.

Module Contents

Functions

concurrence(rho)

Calculate the concurrence of a bipartite state [1].

state_props.concurrence.concurrence(rho)

Calculate the concurrence of a bipartite state [1].

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

\[\rho\tilde{\rho} = \rho \sigma_y \otimes \sigma_y \rho^* \sigma_y \otimes \sigma_y.\]

Concurrence can serve as a measure of entanglement.

Examples

Consider the following Bell state:

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

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.

>>> 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

\[v = |0\rangle \otimes |1 \rangle.\]

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

[1] (1,2)

Wikipedia. Concurrence (quantum computing). https://en.wikipedia.org/wiki/Concurrence_(quantum_computing).

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