toqito.matrices.gen_pauli
=========================

.. py:module:: toqito.matrices.gen_pauli

.. autoapi-nested-parse::

   Produces the generalized Pauli operator matrices.





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

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

   Produce generalized Pauli operator [@WikiPauliGen].

   Generates a `dim`-by-`dim` unitary operator. More specifically,
   it is the operator \(X^k_1 Z^k_2\), where \(X\) and \(Z\) 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
   `dim`-by-`dim` matrices as `k_1` and `k_2` range
   from 0 to `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
   [@Watrous_2011_Lecture_Notes])

   .. rubric:: Examples

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

   \[
       G_{1, 0, 2} = \begin{pmatrix}
                        0 & 1 \\
                        1 & 0
                     \end{pmatrix}.
   \]

   This can be obtained in `|toqito⟩` as follows.

   ```python exec="1" source="above"
   from toqito.matrices import gen_pauli

   print(gen_pauli(k_1=1, k_2=0, dim=2))
   ```


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

   \[
       G_{1, 1, 2} = \begin{pmatrix}
                       0 & -1 \\
                       1 & 0
                     \end{pmatrix}.
   \]

   This can be obtained in `|toqito⟩` as follows.

   ```python exec="1" source="above"
   from toqito.matrices import gen_pauli

   print(gen_pauli(k_1=1, k_2=1, dim=2))
   ```

   :param k_1: (a non-negative integer from 0 to `dim-1` inclusive).
   :param k_2: (a non-negative integer from 0 to `dim-1` inclusive).
   :param dim: (a positive integer indicating the dimension).

   :returns: A generalized Pauli operator.