toqito.perms.antisymmetric_projection
=====================================

.. py:module:: toqito.perms.antisymmetric_projection

.. autoapi-nested-parse::

   Antisymmetric projection operator produces an orthogonal projection onto an anti-symmetric subspace.





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

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

   Produce the projection onto the antisymmetric subspace [@WikiAsymmOp].

   Produces the orthogonal projection onto the anti-symmetric subspace of `p_param` copies of
   `dim`-dimensional space. If `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 \(2\)-dimensional antisymmetric projection with \(p=1\) is given as
   \(2\)-by-\(2\) identity matrix

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

   Using `|toqito⟩`, we can see this gives the proper result.

   ```python exec="1" source="above"
   from toqito.perms import antisymmetric_projection

   print(antisymmetric_projection(2, 1))
   ```

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

   \[
       A_{2, 3} =
       \begin{pmatrix}
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
       \end{pmatrix}.
   \]

   Using `|toqito⟩` we can see this gives the proper result.

   ```python exec="1" source="above"
   from toqito.perms import antisymmetric_projection

   print(antisymmetric_projection(2, 3))
   ```

   :param dim: The dimension of the local systems.
   :param p_param: Default value of 2.
   :param partial: Default value of 0.

   :returns: Projection onto the antisymmetric subspace.