toqito.state_props.is_ppt
=========================

.. py:module:: toqito.state_props.is_ppt

.. autoapi-nested-parse::

   Checks if a quantum state violates the PPT criterion.





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

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

   Determine whether or not a matrix has positive partial transpose [@WikiPeresHorodecki].

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

   For shared systems of \(2 \otimes 2\) or \(2 \otimes 3\), 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 `True` would indicate that the state is separable and a value
   of `False` would indicate the state is entangled.

   .. rubric:: Examples

   Consider the following matrix

   \[
       X =
       \begin{pmatrix}
           1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
       \end{pmatrix}.
   \]

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

   ```python exec="1" source="above"
   from toqito.state_props import is_ppt
   import numpy as np
   mat = np.identity(9)
   print(is_ppt(mat))
   ```

   Consider the following Bell state:

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

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

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.state_props import is_ppt
   rho = bell(2) @ bell(2).conj().T
   print(is_ppt(rho))
   ```

   :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 `mat` acts.
   :param tol: Tolerance with which to check whether `mat` is PPT.

   :returns: Returns `True` if `mat` is PPT and `False` if not.