state_opt.ppt_distinguishability
================================

.. py:module:: state_opt.ppt_distinguishability

.. autoapi-nested-parse::

   Calculates the probability of PPT state distinguishability when done optimally.



Functions
---------

.. autoapisummary::

   state_opt.ppt_distinguishability.ppt_distinguishability
   state_opt.ppt_distinguishability._min_error_primal
   state_opt.ppt_distinguishability._min_error_dual


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

.. py:function:: ppt_distinguishability(vectors, subsystems, dimensions, probs = None, strategy = 'min_error', solver = 'cvxopt', primal_dual = 'dual')

   Compute probability of optimally distinguishing a state via PPT measurements :cite:`Cosentino_2013_PPT`.

   Implements the semidefinite program (SDP) whose optimal value is equal to the maximum
   probability of perfectly distinguishing orthogonal maximally entangled states using any PPT
   measurement; a measurement whose operators are positive under partial transpose. This SDP was
   explicitly provided in :cite:`Cosentino_2013_PPT`.

   One can specify the distinguishability method using the :code:`dist_method` argument.

   For :code:`dist_method = "min_error"`, this is the default method that yields the probability of
   distinguishing quantum states via PPT measurements that minimize the probability of error.

   For :code:`dist_method = "unambig"`, Alice and Bob never provide an incorrect answer,
   although it is possible that their answer is inconclusive.

   For more info, go to the tutorial in the documentation :ref:`ref-label-state-dist-ppt`.

   .. rubric:: Examples

   Consider the following Bell states:

   .. math::
       \begin{equation}
           \begin{aligned}
           |\psi_0 \rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}, &\quad
           |\psi_1 \rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}}, \\
           |\psi_2 \rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}, &\quad
           |\psi_3 \rangle = \frac{|00\rangle - |11\rangle}{\sqrt{2}}.
           \end{aligned}
       \end{equation}

   It was illustrated in :cite:`Yu_2012_Four` that for the following set of states

   .. math::
       \begin{equation}
           \begin{aligned}
           \rho_1^{(2)} &= |\psi_0 \rangle | \psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \quad
           \rho_2^{(2)} &= |\psi_1 \rangle | \psi_3 \rangle \langle \psi_1 | \langle \psi_3 |, \\
           \rho_3^{(2)} &= |\psi_2 \rangle | \psi_3 \rangle \langle \psi_2 | \langle \psi_3 |, \quad
           \rho_4^{(2)} &= |\psi_3 \rangle | \psi_3 \rangle \langle \psi_3 | \langle \psi_3 |, \\
           \end{aligned}
       \end{equation}

   that the optimal probability of distinguishing via a PPT measurement should yield
   :math:`7/8 \approx 0.875` as was proved in :cite:`Yu_2012_Four`.

   .. jupyter-execute::

    import numpy as np
    from toqito.states import bell
    from toqito.state_opt import ppt_distinguishability

    # Bell vectors:
    psi_0 = bell(0)
    psi_1 = bell(2)
    psi_2 = bell(3)
    psi_3 = bell(1)

    # YDY vectors from :cite:`Yu_2012_Four`.
    x_1 = np.kron(psi_0, psi_0)
    x_2 = np.kron(psi_1, psi_3)
    x_3 = np.kron(psi_2, psi_3)
    x_4 = np.kron(psi_3, psi_3)

    # YDY density matrices.
    rho_1 = x_1 @ x_1.conj().T
    rho_2 = x_2 @ x_2.conj().T
    rho_3 = x_3 @ x_3.conj().T
    rho_4 = x_4 @ x_4.conj().T

    states = [rho_1, rho_2, rho_3, rho_4]
    probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]

    opt_val, _ = ppt_distinguishability(vectors=states, probs=probs, dimensions=[2, 2, 2, 2], subsystems=[0, 2])

    print(f"Optimal value: {opt_val:.3f}")

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param vectors: A list of states provided as either matrices or vectors.
   :param probs: Respective list of probabilities each state is selected.
   :param subsystems: A list of integers that correspond to the complex Euclidean space dimensions.
   :param dimensions: A list of integers that correspond to the dimensions of the subsystems.
   :param strategy: The method of distinguishing states.
   :param solver: The SDP solver to use.
   :param primal_dual: Option for the optimization problem.
   :return: The optimal probability with which the states can be distinguished
            via PPT measurements.



.. py:function:: _min_error_primal(vectors, subsystems, dimensions, probs, solver = 'cvxopt', strategy = 'min_error')

   Primal problem for the SDP with PPT constraints.


.. py:function:: _min_error_dual(vectors, subsystems, dimensions, probs, solver = 'cvxopt', strategy = 'min_error')

   Semidefinite program with PPT constraints (dual problem).