:py:mod:`state_metrics.fidelity_of_separability`
================================================

.. py:module:: state_metrics.fidelity_of_separability

.. autoapi-nested-parse::

   Add function for fidelity of separability as defined in :cite:`Philip_2023_Schrodinger`.

   The constraints for this function are positive partial transpose (PPT) & k-extendible states.



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


Functions
~~~~~~~~~

.. autoapisummary::

   state_metrics.fidelity_of_separability.fidelity_of_separability



.. py:function:: fidelity_of_separability(input_state_rho, input_state_rho_dims, k = 1, verbosity_option = 0, solver_option = 'cvxopt')

   Define the first benchmark introduced in Appendix H of :cite:`Philip_2023_Schrodinger`.

   If you would like to instead use the benchmark introduced in Appendix I, go to
   :obj:`toqito.channel_metrics.fidelity_of_separability`.

   In :cite:`Philip_2023_Schrodinger` a variational quantum algorithm (VQA) is introduced to test
   the separability of a general bipartite state. The algorithm utilizes
   quantum steering between two separated systems such that the separability
   of the state is quantified.

   Due to the limitations of currently available quantum computers, two
   optimization semidefinite programs (SDP) benchmarks were introduced to
   maximize the fidelity of separability subject to some state constraints
   (Positive Partial Transpose (PPT), symmetric extensions (k-extendibility
   ) :cite:`Hayden_2013_TwoMessage` ) This function approximites the fidelity of separability by
   maximizing over PPT states & k-extendible states i.e. an optimization
   problem over states :cite:`Watrous_2018_TQI`.

   The following expression (Equation (H2) from :cite:`Philip_2023_Schrodinger` ) defines the
   constraints for approxiamting

   :math:`\sqrt{\widetilde{F}_s^1}(\rho_{AB}) {:}=`

   .. math::

       \begin{multline}
       \max_{\substack{X_{AB} \in\mathcal{L}(\mathcal{H}_{AB}),\\\sigma_{AB^{k}}\geq0}}
       \left\{\begin{array}
               [c]{c}
               \operatorname{Re}[\operatorname{Tr}[X_{AB}]]:\\%
               \begin{bmatrix}
               \rho_{AB} & X_{AB}\\
               X_{AB}^{\dagger} & \sigma_{AB_{1}}%
               \end{bmatrix}
               \geq0,\\
               \operatorname{Tr}[\sigma_{AB^{k}}]=1,\\
               \sigma_{AB^{k}}=\mathcal{P}_{B^{k}}(\sigma_{AB^{k}}),\\
               T_{B_{1\cdots j}}(\sigma_{AB_{1\cdots j}})\geq 0 \quad \forall j\leq k
           \end{array}\right\}
       \end{multline}

   :math:`\sqrt{\widetilde{F}_s^1}(\rho_{AB})` is the quantity to be
   approximated but this function returns
   :math:`\widetilde{F}_s^1(\rho_{AB})`.

   :math:`\operatorname{Re}[\operatorname{Tr}[X_{AB}]]` is the maximization problem subject to PPT & k-extendibile
   state constraints.

   Here, :math:`\mathcal{L}(\mathcal{H}_{AB})` is the space of linear operators over space :math:`\mathcal{H}_{AB}`.

   :math:`\sigma_{AB^{k}}` is a k-extension of :math:`\rho_{AB}`.

   :math:`\mathcal{P}_{B^{k}}` is the permutation operator among systems
   :math:`B_1, B_2,  \ldots , B_{k}` which has no effect on the k-extended
   state :math:`\sigma_{AB^{k}}`.

   The other constraints are due to the PPT condition :cite:`Peres_1996_Separability`.

   .. rubric:: Examples

   Let's consider a density matrix of a state that we know is pure and separable; :math:`|00 \rangle = |0 \rangle
   \otimes |0 \rangle`.

   The expected approximation of fidelity of separability is the maximum value possible i.e. very close to 1.

   .. math::
       \rho_{AB} = |00 \rangle \langle 00|

   >>> from toqito.state_metrics import fidelity_of_separability
   >>> from toqito.matrix_ops import tensor
   >>> from toqito.states import basis
   >>> state = tensor(basis(2, 0), basis(2, 0))
   >>> rho = state @ state.conj().T
   >>> expected_value = fidelity_of_separability(rho, [2, 2])
   >>> '%.2f' % expected_value
   '1.00'

   .. note::
       You do not need to use `'%.2f' %` when you use this function.

       We use this to format our output such that `doctest` compares the calculated output to the
       expected output upto two decimal points only. The accuracy of the solvers can calculate the
       `float` output to a certain amount of precision such that the value deviates after a few digits
       of accuracy.

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param input_state_rho: the density matrix for the bipartite state of interest.
   :param input_state_rho_dims: the dimensions of System A & B respectively in
       the input state density matrix. It is assumed that the first
       quantity in this list is the dimension of System A.
   :param k: value for k-extendibility.
   :param verbosity_option: Parameter option for `picos`. Default value is
       `verbosity = 0`. For more info, visit
       https://picos-api.gitlab.io/picos/api/picos.modeling.options.html#option-verbosity.
   :param solver_option: Optimization option for `picos` solver. Default option is
       `solver_option="cvxopt"`. For more info, visit
       https://picos-api.gitlab.io/picos/api/picos.modeling.options.html#option-solver.
   :raises AssertionError: If the provided dimensions are not for a bipartite density matrix.
   :raises ValueError: If the matrix is not a density matrix (square matrix that
       is PSD with trace 1).
   :raises ValueError: the input state is entangled.
   :raises ValueError: the input state is a mixed state.
   :return: Optimized value of the SDP when maximized over a set of linear operators subject
       to some constraints.