"""The BB84 extended nonlocal game
=====================================

In our :ref:`sphx_glr_auto_examples_extended_nonlocal_games_enlg_introduction.py`
tutorial, we introduced the framework for extended nonlocal games. Now, we will
construct our first concrete example, the *BB84 extended nonlocal game*.
"""

# %%
# The *BB84 extended nonlocal game* is defined as follows. Let :math:`\Sigma_A =
# \Sigma_B = \Gamma_A = \Gamma_B = \{0,1\}`, define
#
# .. math::
#    \begin{aligned}
#        V(0,0|0,0) = \begin{pmatrix}
#                        1 & 0 \\
#                        0 & 0
#                     \end{pmatrix}, &\quad
#        V(1,1|0,0) = \begin{pmatrix}
#                        0 & 0 \\
#                        0 & 1
#                     \end{pmatrix}, \\
#        V(0,0|1,1) = \frac{1}{2}\begin{pmatrix}
#                        1 & 1 \\
#                        1 & 1
#                     \end{pmatrix}, &\quad
#        V(1,1|1,1) = \frac{1}{2}\begin{pmatrix}
#                        1 & -1 \\
#                        -1 & 1
#                     \end{pmatrix},
#    \end{aligned}
#
# define
#
# .. math::
#    V(a,b|x,y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
#
# for all :math:`a \not= b` or :math:`x \not= y`, define :math:`\pi(0,0) =
# \pi(1,1) = 1/2`, and define :math:`\pi(x,y) = 0` if :math:`x \not=y`.
#
# We can encode the BB84 game, :math:`G_{BB84} = (\pi, V)`, in :code:`numpy`
# arrays where :code:`prob_mat` corresponds to the probability distribution
# :math:`\pi` and where :code:`pred_mat` corresponds to the operator :math:`V`.

# sphinx_gallery_start_ignore
# sphinx_gallery_thumbnail_path = 'figures/extended_nonlocal_game.svg'
# sphinx_gallery_end_ignore
# Define the BB84 extended nonlocal game.
import numpy as np

from toqito.states import basis

# The basis: {|0>, |1>}:
e_0, e_1 = basis(2, 0), basis(2, 1)

# The basis: {|+>, |->}:
e_p = (e_0 + e_1) / np.sqrt(2)
e_m = (e_0 - e_1) / np.sqrt(2)

# The dimension of referee's measurement operators:
dim = 2
# The number of outputs for Alice and Bob:
a_out, b_out = 2, 2
# The number of inputs for Alice and Bob:
a_in, b_in = 2, 2

# Define the predicate matrix V(a,b|x,y) \in Pos(R)
bb84_pred_mat = np.zeros([dim, dim, a_out, b_out, a_in, b_in])

# V(0,0|0,0) = |0><0|
bb84_pred_mat[:, :, 0, 0, 0, 0] = e_0 @ e_0.conj().T
# V(1,1|0,0) = |1><1|
bb84_pred_mat[:, :, 1, 1, 0, 0] = e_1 @ e_1.conj().T
# V(0,0|1,1) = |+><+|
bb84_pred_mat[:, :, 0, 0, 1, 1] = e_p @ e_p.conj().T
# V(1,1|1,1) = |-><-|
bb84_pred_mat[:, :, 1, 1, 1, 1] = e_m @ e_m.conj().T

# The probability matrix encode \pi(0,0) = \pi(1,1) = 1/2
bb84_prob_mat = 1 / 2 * np.identity(2)

# %%
# The unentangled value of the BB84 extended nonlocal game
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# It was shown in :footcite:`Tomamichel_2013_AMonogamy` and :footcite:`Johnston_2016_Extended` that
#
# .. math::
#    \omega(G_{BB84}) = \cos^2(\pi/8).
#
# This can be verified in :code:`|toqito⟩` as follows.


# Calculate the unentangled value of the BB84 extended nonlocal game.
import numpy as np

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define an ExtendedNonlocalGame object based on the BB84 game.
bb84 = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat)

# The unentangled value is cos(pi/8)**2 \approx 0.85356
print("The unentangled value is ", np.around(bb84.unentangled_value(), decimals=2))

# %%
# The BB84 game also exhibits strong parallel repetition. We can specify how many
# parallel repetitions for :code:`|toqito⟩` to run. The example below provides an
# example of two parallel repetitions for the BB84 game.

# The unentangled value of BB84 under parallel repetition.
import numpy as np

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define the bb84 game for two parallel repetitions.
bb84_2_reps = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat, 2)

# The unentangled value for two parallel repetitions is cos(pi/8)**4 \approx 0.72855
print("The unentangled value for two parallel repetitions is ", np.around(bb84_2_reps.unentangled_value(), decimals=2))

# %%
# It was shown in :footcite:`Johnston_2016_Extended` that the BB84 game possesses the property of strong
# parallel repetition. That is,
#
# .. math::
#    \omega(G_{BB84}^r) = \omega(G_{BB84})^r
#
# for any integer :math:`r`.
#
# The standard quantum value of the BB84 extended nonlocal game
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# We can calculate lower bounds on the standard quantum value of the BB84 game
# using :code:`|toqito⟩` as well.

# Calculate lower bounds on the standard quantum value of the BB84 extended nonlocal game.
import numpy as np

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define an ExtendedNonlocalGame object based on the BB84 game.
bb84_lb = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat)

# The standard quantum value is cos(pi/8)**2 \approx 0.85356
print("The standard quantum value is ", np.around(bb84_lb.quantum_value_lower_bound(), decimals=2))

# %%
# From :footcite:`Johnston_2016_Extended`, it is known that :math:`\omega(G_{BB84}) =
# \omega^*(G_{BB84})`, however, if we did not know this beforehand, we could
# attempt to calculate upper bounds on the standard quantum value.
#
# There are a few methods to do this, but one easy way is to simply calculate the
# non-signaling value of the game as this provides a natural upper bound on the
# standard quantum value. Typically, this bound is not tight and usually not all
# that useful in providing tight upper bounds on the standard quantum value,
# however, in this case, it will prove to be useful.
#
# The non-signaling value of the BB84 extended nonlocal game
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# Using :code:`|toqito⟩`, we can see that :math:`\omega_{ns}(G) = \cos^2(\pi/8)`.

# Calculate the non-signaling value of the BB84 extended nonlocal game.
import numpy as np

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define an ExtendedNonlocalGame object based on the BB84 game.
bb84 = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat)

# The non-signaling value is cos(pi/8)**2 \approx 0.85356
print("The non-signaling value is ", np.around(bb84.nonsignaling_value(), decimals=2))

# %%
# So we have the relationship that
#
# .. math::
#    \omega(G_{BB84}) = \omega^*(G_{BB84}) = \omega_{ns}(G_{BB84}) = \cos^2(\pi/8).
#
# It turns out that strong parallel repetition does *not* hold in the
# non-signaling scenario for the BB84 game. This was shown in :footcite:`Russo_2017_Extended`, and we
# can observe this by the following snippet.

# The non-signaling value of BB84 under parallel repetition.
import numpy as np

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define the bb84 game for two parallel repetitions.
bb84_2_reps = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat, 2)

# The non-signaling value for two parallel repetitions is cos(pi/8)**4 \approx 0.73825
print(
    "The non-signaling value for two parallel repetitions is ", np.around(bb84_2_reps.nonsignaling_value(), decimals=2)
)

# %%
# Note that :math:`0.73825 \geq \cos(\pi/8)^4 \approx 0.72855` and therefore we
# have that
#
# .. math::
#    \omega_{ns}(G^r_{BB84}) \not= \omega_{ns}(G_{BB84})^r
#
# for :math:`r = 2`.
#
# Next, we will explore another well-known example, :ref:`sphx_glr_auto_examples_extended_nonlocal_games_enlg_chsh.py`, and see how its properties compare.

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