:py:mod:`matrices.gen_pauli`
============================

.. py:module:: matrices.gen_pauli

.. autoapi-nested-parse::

   Generalized Pauli matrices.



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


Functions
~~~~~~~~~

.. autoapisummary::

   matrices.gen_pauli.gen_pauli



.. py:function:: gen_pauli(k_1, k_2, dim)

   Produce generalized Pauli operator :cite:`WikiPauliGen`.

   Generates a :code:`dim`-by-:code:`dim` unitary operator. More specifically,
   it is the operator :math:`X^k_1 Z^k_2`, where :math:`X` and :math:`Z` are
   the "gen_pauli_x" and "gen_pauli_z" operators that naturally generalize the Pauli X and
   Z operators. These matrices span the entire space of
   :code:`dim`-by-:code:`dim` matrices as :code:`k_1` and :code:`k_2` range
   from 0 to :code:`dim-1`, inclusive.

   Note that the generalized Pauli operators are also known by the name of
   "discrete Weyl operators". (Lecture 6: Further Remarks On Measurements And Channels from
   :cite:`Watrous_2011_Lecture_Notes`)

   .. rubric:: Examples

   The generalized Pauli operator for :code:`k_1 = 1`, :code:`k_2 = 0`, and
   :code:`dim = 2` is given as the standard Pauli-X matrix

   .. math::
       G_{1, 0, 2} = \begin{pmatrix}
                        0 & 1 \\
                        1 & 0
                     \end{pmatrix}.

   This can be obtained in :code:`toqito` as follows.

   >>> from toqito.matrices import gen_pauli
   >>> dim = 2
   >>> k_1 = 1
   >>> k_2 = 0
   >>> gen_pauli(k_1, k_2, dim)
   array([[0.+0.j, 1.+0.j],
          [1.+0.j, 0.+0.j]])

   The generalized Pauli matrix :code:`k_1 = 1`, :code:`k_2 = 1`, and
   :code:`dim = 2` is given as the standard Pauli-Y matrix

   .. math::
       G_{1, 1, 2} = \begin{pmatrix}
                       0 & -1 \\
                       1 & 0
                     \end{pmatrix}.

   This can be obtained in :code:`toqito` as follows.

   >>> from toqito.matrices import gen_pauli
   >>> dim = 2
   >>> k_1 = 1
   >>> k_2 = 1
   >>> gen_pauli(k_1, k_2, dim)
   array([[ 0.+0.0000000e+00j, -1.+1.2246468e-16j],
          [ 1.+0.0000000e+00j,  0.+0.0000000e+00j]])

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param k_1: (a non-negative integer from 0 to :code:`dim-1` inclusive).
   :param k_2: (a non-negative integer from 0 to :code:`dim-1` inclusive).
   :param dim: (a positive integer indicating the dimension).
   :return: A generalized Pauli operator.