:py:mod:`matrices.shift`
========================

.. py:module:: matrices.shift

.. autoapi-nested-parse::

   Shift matrix.



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


Functions
~~~~~~~~~

.. autoapisummary::

   matrices.shift.shift



.. py:function:: shift(dim)

   Produce a :code:`dim`-by-:code:`dim` shift matrix :cite:`WikiPauliGen`.

   Returns the shift matrix of dimension :code:`dim` described in :cite:`WikiPauliGen`.
   The shift matrix generates the following :code:`dim`-by-:code:`dim` matrix:

   .. math::
       \Sigma_{1, d} = \begin{pmatrix}
                       0 & 0 & 0 & \ldots & 0 & 1 \\
                       1 & 0 & 0 & \ldots & 0 & 0 \\
                       0 & 1 & 0 & \ldots & 0 & 0 \\
                       0 & 0 & 1 & \ldots & 0 & 0 \\
                       \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
                       0 & 0 & 0 & \ldots & 1 & 0
                   \end{pmatrix}

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

   .. rubric:: Examples

   The shift matrix generated from :math:`d = 3` yields the following matrix:

   .. math::
       \Sigma_{1, 3} =
       \begin{pmatrix}
           0 & 0 & 1 \\
           1 & 0 & 0 \\
           0 & 1 & 0
       \end{pmatrix}

   >>> from toqito.matrices import shift
   >>> shift(3)
   [[0., 0., 1.],
    [1., 0., 0.],
    [0., 1., 0.]]

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param dim: Dimension of the matrix.
   :return: :code:`dim`-by-:code:`dim` shift matrix.