state_metrics.trace_distance
============================

.. py:module:: state_metrics.trace_distance

.. autoapi-nested-parse::

   Trace distance metric gives a measure of distinguishability between two quantum states.

   The trace distance is calculated via density matrices.



Functions
---------

.. autoapisummary::

   state_metrics.trace_distance.trace_distance


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

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

   Compute the trace distance between density operators `rho` and `sigma`.

   The trace distance between :math:`\rho` and :math:`\sigma` is defined as

   .. math::
       \delta(\rho, \sigma) = \frac{1}{2} \left( \text{Tr}(\left| \rho - \sigma
        \right| \right).

   More information on the trace distance can be found in :cite:`Quantiki_TrDist`.

   .. rubric:: Examples

   Consider the following Bell state

   .. math::
       u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.

   The corresponding density matrix of :math:`u` may be calculated by:

   .. math::
       \rho = u u^* = \begin{pmatrix}
                        1 & 0 & 0 & 1 \\
                        0 & 0 & 0 & 0 \\
                        0 & 0 & 0 & 0 \\
                        1 & 0 & 0 & 1
                      \end{pmatrix} \in \text{D}(\mathcal{X}).

   The trace distance between :math:`\rho` and another state :math:`\sigma` is equal to :math:`0` if any only if
   :math:`\rho = \sigma`. We can check this using the :code:`|toqito⟩` package.

   .. jupyter-execute::

    from toqito.states import bell
    from toqito.state_metrics import trace_distance

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

    trace_distance(rho, sigma)

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: If matrices are not of density operators.
   :param rho: An input matrix.
   :param sigma: An input matrix.
   :return: The trace distance between :code:`rho` and :code:`sigma`.