toqito.state_props.concurrence
==============================

.. py:module:: toqito.state_props.concurrence

.. autoapi-nested-parse::

   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
---------------

.. py:function:: 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

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

   Concurrence can serve as a measure of entanglement.

   .. rubric:: 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

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

   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.

   :param rho: The bipartite system specified as a matrix.

   :returns: The concurrence of the bipartite state \(\rho\).