toqito.state_props.has_symmetric_extension
==========================================

.. py:module:: toqito.state_props.has_symmetric_extension

.. autoapi-nested-parse::

   Determine whether there exists a symmetric extension for a given quantum state.





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

.. py:function:: has_symmetric_extension(rho, level = 2, dim = None, ppt = True, tol = 0.0001)

   Determine whether there exists a symmetric extension for a given quantum state.

   For more information, see [@Doherty_2002_Distinguishing].

   Determining whether an operator possesses a symmetric extension at some level `level`
   can be used as a check to determine if the operator is entangled or not.

   This function was adapted from QETLAB.

   .. rubric:: Examples

   2-qubit symmetric extension:

   In [@Chen_2014_Symmetric], it was shown that a 2-qubit state \(\rho_{AB}\) has a
   symmetric extension if and only if

   \[
       \text{Tr}(\rho_B^2) \geq \text{Tr}(\rho_{AB}^2) - 4 \sqrt{\text{det}(\rho_{AB})}.
   \]

   This closed-form equation is much quicker to check than running the semidefinite program.

   ```python exec="1" source="above" session="has_symmetric_example"
   import numpy as np
   from toqito.state_props import has_symmetric_extension
   from toqito.matrix_ops import partial_trace
   rho = np.array([[1, 0, 0, -1], [0, 1, 1/2, 0], [0, 1/2, 1, 0], [-1, 0, 0, 1]])
   # Show the closed-form equation holds
   print(
   np.trace(np.linalg.matrix_power(partial_trace(rho, 1), 2))
   >= np.trace(rho**2) - 4 * np.sqrt(np.linalg.det(rho)))
   ```

   ```python exec="1" source="above" session="has_symmetric_example"
   # Now show that the `has_symmetric_extension` function recognizes this case.
   print(has_symmetric_extension(rho))
   ```

   Higher qubit systems:

   Consider a density operator corresponding to one of the Bell states.

   \[
       \rho = \frac{1}{2} \begin{pmatrix}
                           1 & 0 & 0 & 1 \\
                           0 & 0 & 0 & 0 \\
                           0 & 0 & 0 & 0 \\
                           1 & 0 & 0 & 1
                          \end{pmatrix}
   \]

   To make this state over more than just two qubits, let's construct the following state

   \[
       \sigma = \rho \otimes \rho.
   \]

   As the state \(\sigma\) is entangled, there should not exist a symmetric extension at some
   level. We see this being the case for a relatively low level of the hierarchy.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.states import bell
   from toqito.state_props import has_symmetric_extension
   rho = bell(0) @ bell(0).conj().T
   sigma = np.kron(rho, rho)
   print(has_symmetric_extension(sigma))
   ```

   :raises ValueError: If dimension does not evenly divide matrix length.

   :param rho: A matrix or vector.
   :param level: Level of the hierarchy to compute.
   :param dim: The default has both subsystems of equal dimension.
   :param ppt: If `True`, this enforces that the symmetric extension must be PPT.
   :param tol: Tolerance when determining whether a symmetric extension exists.

   :returns: `True` if `mat` has a symmetric extension; `False` otherwise.