"""Quantum state distinguishability
=================================

In this tutorial we are going to cover the problem of *quantum state
distinguishability* (sometimes analogously referred to as quantum state
discrimination). We are going to briefly describe the problem setting and then
describe how one may use :code:`|toqito⟩` to calculate the optimal probability
with which this problem can be solved when given access to certain
measurements.
"""
# %%
# Further information beyond the scope of this tutorial can be found in the text
# :footcite:`Watrous_2018_TQI` as well as the course :footcite:`Sikora_2019_Semidefinite`.
#
# The state distinguishability problem
# -------------------------------------
#
# The quantum state distinguishability problem is phrased as follows.
#
# 1. Alice possesses an ensemble of :math:`n` quantum states:
#
#    .. math::
#        \begin{equation}
#            \eta = \left( (p_0, \rho_0), \ldots, (p_n, \rho_n)  \right),
#        \end{equation}
#
#    where :math:`p_i` is the probability with which state :math:`\rho_i` is
#    selected from the ensemble. Alice picks :math:`\rho_i` with probability
#    :math:`p_i` from her ensemble and sends :math:`\rho_i` to Bob.
#
# 2. Bob receives :math:`\rho_i`. Both Alice and Bob are aware of how the
#    ensemble is defined but he does *not* know what index :math:`i`
#    corresponding to the state :math:`\rho_i` he receives from Alice is.
#
# 3. Bob wants to guess which of the states from the ensemble he was given. In
#    order to do so, he may measure :math:`\rho_i` to guess the index :math:`i`
#    for which the state in the ensemble corresponds.
#
#
# This setting is depicted in the following figure.
#
# .. figure:: ../../figures/quantum_state_distinguish.svg
#   :alt: quantum state distinguishability
#   :align: center
#
#   The quantum state distinguishability setting.
#
# Depending on the sets of measurements that Alice and Bob are allowed to use,
# the optimal probability of distinguishing a given set of states is characterized
# by the following image.
#
# .. figure:: ../../figures/measurement_inclusions.svg
#   :width: 200
#   :alt: measurement hierarchy
#   :align: center
#
#   Measurement hierarchy.
#
# That is,the probability that Alice and Bob are able to distinguish using PPT
# measurements is a natural upper bound on the optimal probability of
# distinguishing via separable measurements.
#
# In general:
#
# * LOCC: These are difficult objects to handle mathematically; difficult to
#   design protocols for and difficult to provide bounds on their power.
#
# * Separable: Separable measurements have a nicer structure than LOCC.
#   Unfortunately, optimizing over separable measurements in NP-hard.
#
# * PPT: PPT measurements offer a nice structure and there exists efficient
#   techniques that allow one to optimize over the set of PPT measurements via
#   semidefinite programming.
#
#
# Optimal probability of distinguishing a quantum state
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# The optimal probability with which Bob can distinguish the state he is given
# may be obtained by solving the following semidefinite program (SDP).
#
# .. math::
#    \begin{align*}
#        \text{maximize:} \quad & \sum_{i=0}^n p_i \langle M_i,
#        \rho_i \rangle \\
#        \text{subject to:} \quad & \sum_{i=0}^n M_i = \mathbb{I}_{\mathcal{X}},\\
#                                 & M_i \in \text{Pos}(\mathcal{X}).
#    \end{align*}
#
# This optimization problem is solved in :code:`|toqito⟩` to obtain the optimal
# probability with which Bob can distinguish state :math:`\rho_i`.
#
# To illustrate how we can phrase and solve this problem in :code:`|toqito⟩`,
# consider the following example. Assume Alice has an ensemble of quantum states
#
# .. math::
#    \eta = \{ (1/2, \rho_0), (1/2, \rho_1) \}
#
# such that
#
# .. math::
#    \rho_0 = | 0 \rangle \langle 0 | = \begin{pmatrix}
#                1 & 0 \\
#                0 & 0
#             \end{pmatrix} \quad \text{and} \quad
#    \rho_1 = | 1 \rangle \langle 1 | = \begin{pmatrix}
#                0 & 0 \\
#                0 & 1
#             \end{pmatrix}.
#
#
# These states are completely orthogonal to each other, and it is known that Bob
# can optimally distinguish the state he is given perfectly, i.e. with probability
# :math:`1`.
#
# Using :code:`|toqito⟩`, we can calculate this probability directly as follows:

# sphinx_gallery_start_ignore
# sphinx_gallery_thumbnail_path = 'figures/quantum_state_distinguish.svg'
# sphinx_gallery_end_ignore
import numpy as np
from toqito.states import basis
from toqito.state_opt import state_distinguishability

# Define the standard basis |0> and |1>.
e_0, e_1 = basis(2, 0), basis(2, 1)

# Define the corresponding density matrices of |0> and |1>
# given as |0><0| and |1><1|, respectively.
E_0 = e_0 @ e_0.conj().T
E_1 = e_1 @ e_1.conj().T

# Define a list of states and a corresponding list of probabilities with which those
# states are selected.
states = [E_0, E_1]
probs = [1 / 2, 1 / 2]

# Calculate the probability with which Bob can distinguish the state he is provided.
print(np.around(state_distinguishability(states, probs)[0], decimals=2))

# %%
# Specifying similar state distinguishability problems can be done so using this
# general pattern.
#
# .. _ref-label-state-dist-ppt:
#
# Optimal probability of distinguishing a state via PPT measurements
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# We may consider the quantum state distinguishability scenario under somewhat
# different and more limited set of circumstances. Specifically, we may want to
# ask the same question but restrict to enforcing that in order to determine the
# state that Bob is provided, he is limited to using a certain class of
# measurement. There are a wider class of measurements with respect to the ones
# we considered in the previous example referred to as PPT (positive partial
# transpose).
#
# The problem of state distinguishability with respect to PPT measurements can
# also be framed as an SDP and was initially presented in this manner in
# :footcite:`Cosentino_2013_PPT`
#
# .. math::
#
#    \begin{equation}
#        \begin{aligned}
#            \text{minimize:} \quad & \frac{1}{k} \text{Tr}(Y) \\
#            \text{subject to:} \quad & Y \geq \text{T}_{\mathcal{A}}
#                                      (\rho_j), \quad j = 1, \ldots, k, \\
#                                     & Y \in \text{Herm}(\mathcal{A} \otimes
#                                      \mathcal{B}).
#        \end{aligned}
#    \end{equation}
#
# Using :code:`|toqito⟩`, we can determine the optimal probability for Bob to
# distinguish a given state from an ensemble if he is only given access to PPT
# measurements.
#
# 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 shown in :footcite:`Cosentino_2013_PPT` and later extended in :footcite:`Cosentino_2014_Small` 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`.
#
# This ensemble of states and some of its properties with respect to
# distinguishability were initially considered in :footcite:`Yu_2012_Four`. In :code:`|toqito⟩`,
# we can calculate the probability with which Bob can distinguish these states
# via PPT measurements in the following manner.


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 :footcite:`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)

states = [x_1, x_2, x_3, x_4]

probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
# Calculate the PPT distinguishability value.
ppt_val, _ = ppt_distinguishability(vectors=states, probs=probs, dimensions=[2, 2, 2, 2], subsystems=[0, 2])

# Print the rounded result.
print(f"Optimal probability with PPT measurements: {np.around(ppt_val, decimals=2)}")

# %%
# Probability of distinguishing a state via separable measurements
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# As previously mentioned, optimizing over the set of separable measurements is
# NP-hard. However, there does exist a hierarchy of semidefinite programs which
# eventually does converge to the separable value. This hierarchy is based off
# the notion of symmetric extensions. More information about this hierarchy of
# SDPs can be found here :footcite:`Navascues_2008_Pure`.

# %%
#
#
# References
# ----------
#
# .. footbibliography::