toqito.state_metrics.hilbert_schmidt
====================================

.. py:module:: toqito.state_metrics.hilbert_schmidt

.. autoapi-nested-parse::

   Hilbert-Schmidt metric is a distance metric used to generate an entanglement measure.





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

.. py:function:: hilbert_schmidt(rho, sigma)

   Compute the Hilbert-Schmidt distance between two states [@WikiHilbSchOp].

   The Hilbert-Schmidt distance between density operators \(\rho\) and \(\sigma\) is defined as

   \[
       D_{\text{HS}}(\rho, \sigma) = \text{Tr}((\rho - \sigma)^2) = \left\lVert \rho - \sigma
       \right\rVert_2^2.
   \]

   .. rubric:: Examples

   One may consider taking the Hilbert-Schmidt distance between two Bell states. In `|toqito⟩`,
   one may accomplish this as

   ```python exec="1" source="above"
   import numpy as np
   from toqito.states import bell
   from toqito.state_metrics import hilbert_schmidt

   rho = bell(0) @ bell(0).conj().T
   sigma = bell(3) @ bell(3).conj().T

   print(np.around(hilbert_schmidt(rho, sigma), decimals=2))
   ```

   :raises ValueError: If matrices are not density operators.

   :param rho: An input matrix.
   :param sigma: An input matrix.

   :returns: The Hilbert-Schmidt distance between `rho` and `sigma`.