:py:mod:`rand.random_density_matrix`
====================================

.. py:module:: rand.random_density_matrix

.. autoapi-nested-parse::

   Generate random density matrix.



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


Functions
~~~~~~~~~

.. autoapisummary::

   rand.random_density_matrix.random_density_matrix



.. py:function:: random_density_matrix(dim, is_real = False, k_param = None, distance_metric = 'haar')

   Generate a random density matrix.

   Generates a random :code:`dim`-by-:code:`dim` density matrix distributed according to the Hilbert-Schmidt measure.
   The matrix is of rank <= :code:`k_param` distributed according to the distribution :code:`distance_metric` If
   :code:`is_real = True`, then all of its entries will be real. The variable :code:`distance_metric` must be one of:

       - :code:`haar` (default):
           Generate a larger pure state according to the Haar measure and trace out the extra dimensions. Sometimes
           called the Hilbert-Schmidt measure when :code:`k_param = dim`.

       - :code:`bures`:
           The Bures measure.

   .. rubric:: Examples

   Using :code:`toqito`, we may generate a random complex-valued :math:`n`- dimensional density matrix. For
   :math:`d=2`, this can be accomplished as follows.

   >>> from toqito.rand import random_density_matrix
   >>> complex_dm = random_density_matrix(2)
   >>> complex_dm # doctest: +SKIP
   array([[0.83378447+0.j        , 0.25937396-0.12958422j],
          [0.25937396+0.12958422j, 0.16621553+0.j        ]])

   We can verify that this is in fact a valid density matrix using the :code:`is_denisty` function from :code:`toqito`
   as follows

   >>> from toqito.matrix_props import is_density
   >>> is_density(complex_dm)
   True

   We can also generate random density matrices that are real-valued as follows.

   >>> from toqito.rand import random_density_matrix
   >>> real_dm = random_density_matrix(2, is_real=True)
   >>> real_dm # doctest: +SKIP
   array([[0.3686455 , 0.48243625],
          [0.48243625, 0.6313545 ]])

   Again, verifying that this is a valid density matrix can be done as follows.

   >>> from toqito.matrix_props import is_density
   >>> is_density(real_dm)
   True

   By default, the random density operators are constructed using the Haar measure. We can select to generate the
   random density matrix according to the Bures metric instead as follows.

   >>> from toqito.rand import random_density_matrix
   >>> bures_mat = random_density_matrix(2, distance_metric="bures")
   >>> bures_mat # doctest: +SKIP
   array([[0.73969055+0.j        , 0.09494426-0.15279342j],
          [0.09494426+0.15279342j, 0.26030945+0.j        ]])


   As before, we can verify that this matrix generated is a valid density matrix.

   >>> from toqito.matrix_props import is_density
   >>> is_density(bures_mat)
   True


   :param dim: The number of rows (and columns) of the density matrix.
   :param is_real: Boolean denoting whether the returned matrix will have all
                   real entries or not.
   :param k_param: Default value is equal to :code:`dim`.
   :param distance_metric: The distance metric used to randomly generate the
                           density matrix. This metric is either the Haar
                           measure or the Bures measure. Default value is to
                           use the Haar measure.
   :return: A :code:`dim`-by-:code:`dim` random density matrix.