:py:mod:`state_props.is_ppt`
============================

.. py:module:: state_props.is_ppt

.. autoapi-nested-parse::

   Check if violates the PPT criterion.



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


Functions
~~~~~~~~~

.. autoapisummary::

   state_props.is_ppt.is_ppt



.. py:function:: is_ppt(mat, sys = 2, dim = None, tol = None)

   Determine whether or not a matrix has positive partial transpose :cite:`WikiPeresHorodecki`.

   Yields either :code:`True` or :code:`False`, indicating that :code:`mat` does or does not have
   positive partial transpose (within numerical error). The variable :code:`mat` is assumed to act
   on bipartite space.

   For shared systems of :math:`2 \otimes 2` or :math:`2 \otimes 3`, the PPT criterion serves as a
   method to determine whether a given state is entangled or separable. Therefore, for systems of
   this size, the return value :code:`True` would indicate that the state is separable and a value
   of :code:`False` would indicate the state is entangled.

   .. rubric:: Examples

   Consider the following matrix

   .. math::
       X =
       \begin{pmatrix}
           1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
       \end{pmatrix}.

   This matrix trivially satisfies the PPT criterion as can be seen using the
   :code:`toqito` package.

   >>> from toqito.state_props import is_ppt
   >>> import numpy as np
   >>> mat = np.identity(9)
   >>> is_ppt(mat)
   True

   Consider the following Bell state:

   .. math::
       u = \frac{1}{\sqrt{2}}\left( |01 \rangle + |10 \rangle \right).

   For the density matrix :math:`\rho = u u^*`, as this is an entangled state
   of dimension :math:`2`, it will violate the PPT criterion, which can be seen
   using the :code:`toqito` package.

   >>> from toqito.states import bell
   >>> from toqito.state_props import is_ppt
   >>> rho = bell(2) * bell(2).conj().T
   >>> is_ppt(rho)
   False

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param mat: A square matrix.
   :param sys: Scalar or vector indicating which subsystems the transpose
               should be applied on.
   :param dim: The dimension is a vector containing the dimensions of the
               subsystems on which :code:`mat` acts.
   :param tol: Tolerance with which to check whether `mat` is PPT.
   :return: Returns :code:`True` if :code:`mat` is PPT and :code:`False` if
            not.