toqito.matrices.gen_pauli_z
===========================

.. py:module:: toqito.matrices.gen_pauli_z

.. autoapi-nested-parse::

   Produces a generalized Pauli-Z operator matrix.





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

.. py:function:: gen_pauli_z(dim)

   Produce gen_pauli_z matrix [@WikiClock].

   Returns the gen_pauli_z matrix of dimension `dim` described in [@WikiClock].
   The gen_pauli_z matrix generates the following `dim`-by-`dim` matrix

   \[
       \Sigma_{1, d} = \begin{pmatrix}
                       1 & 0 & 0 & \ldots & 0 \\
                       0 & \omega & 0 & \ldots & 0 \\
                       0 & 0 & \omega^2 & \ldots & 0 \\
                       \vdots & \vdots & \vdots & \ddots & \vdots \\
                       0 & 0 & 0 & \ldots & \omega^{d-1}
                  \end{pmatrix}
   \]

   where \(\omega\) is the n-th primitive root of unity.

   The gen_pauli_z matrix is primarily used in the construction of the generalized
   Pauli operators.

   .. rubric:: Examples

   The gen_pauli_z matrix generated from \(d = 3\) yields the following matrix:

   \[
       \Sigma_{1, 3} = \begin{pmatrix}
           1 & 0 & 0 \\
           0 & \omega & 0 \\
           0 & 0 & \omega^2
       \end{pmatrix}
   \]

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

   print(gen_pauli_z(3))
   ```

   :param dim: Dimension of the matrix.

   :returns: `dim`-by-`dim` gen_pauli_z matrix.