toqito.state_props.entanglement_of_formation
============================================

.. py:module:: toqito.state_props.entanglement_of_formation

.. autoapi-nested-parse::

   Computes the entanglement of formation of a bipartite quantum state.





Module Contents
---------------

.. py:function:: entanglement_of_formation(rho, dim = None)

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

   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.

   .. rubric:: Examples

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

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

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

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

   ```python exec="1" source="above"
   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
   print(entanglement_of_formation(rho))
   ```

   :raises ValueError: If matrices have improper dimension.

   :param rho: A matrix or vector.
   :param dim: The default has both subsystems of equal dimension.

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