:py:mod:`perms.antisymmetric_projection`
========================================

.. py:module:: perms.antisymmetric_projection

.. autoapi-nested-parse::

   Antisymmetric projection operator.



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


Functions
~~~~~~~~~

.. autoapisummary::

   perms.antisymmetric_projection.antisymmetric_projection



.. py:function:: antisymmetric_projection(dim, p_param = 2, partial = False)

   Produce the projection onto the antisymmetric subspace :cite:`WikiAsymmOp`.

   Produces the orthogonal projection onto the anti-symmetric subspace of :code:`p_param` copies of
   :code:`dim`-dimensional space. If :code:`partial = True`, then the antisymmetric projection (PA) isn't the
   orthogonal projection itself, but rather a matrix whose columns form an orthonormal basis for the symmetric subspace
   (and hence the PA * PA' is the orthogonal projection onto the symmetric subspace.)

   .. rubric:: Examples

   The :math:`2`-dimensional antisymmetric projection with :math:`p=1` is given as
   :math:`2`-by-:math:`2` identity matrix

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

   Using :code:`toqito`, we can see this gives the proper result.

   >>> from toqito.perms import antisymmetric_projection
   >>> antisymmetric_projection(2, 1)
   array([[1., 0.],
          [0., 1.]])

   When the :math:`p` value is greater than the dimension of the antisymmetric projection, this just gives the matrix
   consisting of all zero entries. For instance, when :math:`d = 2` and :math:`p = 3` we have that

   .. math::
       A_{2, 3} =
       \begin{pmatrix}
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
       \end{pmatrix}.

   Using :code:`toqito` we can see this gives the proper result.

   >>> from toqito.perms import antisymmetric_projection
   >>> antisymmetric_projection(2, 3)
   array([[0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0., 0.]])

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param dim: The dimension of the local systems.
   :param p_param: Default value of 2.
   :param partial: Default value of 0.
   :return: Projection onto the antisymmetric subspace.