state_props.entanglement_of_formation

Computes the entanglement of formation of a bipartite quantum state.

Functions

entanglement_of_formation(rho[, dim])

Compute entanglement-of-formation of a bipartite quantum state [1].

Module Contents

state_props.entanglement_of_formation.entanglement_of_formation(rho, dim=None)

Compute entanglement-of-formation of a bipartite quantum state [1].

Entanglement-of-formation is the entropy of formation of the bipartite quantum state rho. Note that this function currently only supports rho being a pure state or a 2-qubit state: it is not known how to compute the entanglement-of-formation of higher-dimensional mixed states.

This function was adapted from QETLAB.

Examples

Compute the entanglement-of-formation of a Bell state.

Let \(u = \frac{1}{\sqrt{2}} \left(|00\rangle + |11\rangle \right)\) and let

\[\begin{split}\rho = uu^* = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix}.\end{split}\]

The entanglement-of-formation of \(\rho\) is equal to 1.

>>> import numpy as np
>>> from toqito.state_props import entanglement_of_formation
>>> from toqito.states import bell
>>>
>>> u_vec = bell(0)
>>> rho = u_vec @ u_vec.conj().T
>>> np.around(entanglement_of_formation(rho), decimals=3)
np.float64(1.0)

References

[1] (1,2)

Quantiki. Entanglement of formation. URL: https://www.quantiki.org/wiki/entanglement-formation.

Raises:

ValueError – If matrices have improper dimension.

Parameters:

• rho (numpy.ndarray) – A matrix or vector.

• dim (list[int] | int) – The default has both subsystems of equal dimension.

Returns:

A value between 0 and 1 that corresponds to the entanglement-of-formation of rho.

Return type:

float