"""# The CHSH extended nonlocal game

Constructs the CHSH extended nonlocal game and calculates its unentangled,
standard quantum, and non-signaling values using toqito. Compares the
properties of this game to the BB84 extended nonlocal game.

"""

# %%
# Let $\Sigma_A = \Sigma_B = \Gamma_A = \Gamma_B = \{0,1\}$, define a
# collection of measurements $\{V(a,b|x,y) : a \in \Gamma_A, b \in
# \Gamma_B, x \in \Sigma_A, y \in \Sigma_B\} \subset \text{Pos}(\mathcal{R})$
# such that
#
# $$
# \begin{equation}
# \begin{aligned}
# V(0,0|0,0) = V(0,0|0,1) = V(0,0|1,0) = \begin{pmatrix}
# 1 & 0 \\
# 0 & 0
# \end{pmatrix}, \\
# V(1,1|0,0) = V(1,1|0,1) = V(1,1|1,0) = \begin{pmatrix}
# 0 & 0 \\
# 0 & 1
# \end{pmatrix}, \\
# V(0,1|1,1) = \frac{1}{2}\begin{pmatrix}
# 1 & 1 \\
# 1 & 1
# \end{pmatrix}, \\
# V(1,0|1,1) = \frac{1}{2} \begin{pmatrix}
# 1 & -1 \\
# -1 & 1
# \end{pmatrix},
# \end{aligned}
# \end{equation}
# $$
#
# define
#
# $$
# V(a,b|x,y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
# $$
#
# for all $a \oplus b \not= x \land y$, and define $\pi(0,0) =
# \pi(0,1) = \pi(1,0) = \pi(1,1) = 1/4$.
#
# In the event that $a \oplus b \not= x \land y$, the referee's measurement
# corresponds to the zero matrix. If instead it happens that $a \oplus b =
# x \land y$, the referee then proceeds to measure with respect to one of the
# measurement operators. This winning condition is reminiscent of the standard
# CHSH nonlocal game.
#
# We can encode $G_{CHSH}$ in a similar way using `numpy` arrays as
# we did for $G_{BB84}$.
#

# Define the CHSH extended nonlocal game.
import numpy as np

# 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)
chsh_pred_mat = np.zeros([dim, dim, a_out, b_out, a_in, b_in])

# V(0,0|0,0) = V(0,0|0,1) = V(0,0|1,0).
chsh_pred_mat[:, :, 0, 0, 0, 0] = np.array([[1, 0], [0, 0]])
chsh_pred_mat[:, :, 0, 0, 0, 1] = np.array([[1, 0], [0, 0]])
chsh_pred_mat[:, :, 0, 0, 1, 0] = np.array([[1, 0], [0, 0]])

# V(1,1|0,0) = V(1,1|0,1) = V(1,1|1,0).
chsh_pred_mat[:, :, 1, 1, 0, 0] = np.array([[0, 0], [0, 1]])
chsh_pred_mat[:, :, 1, 1, 0, 1] = np.array([[0, 0], [0, 1]])
chsh_pred_mat[:, :, 1, 1, 1, 0] = np.array([[0, 0], [0, 1]])

# V(0,1|1,1)
chsh_pred_mat[:, :, 0, 1, 1, 1] = 1 / 2 * np.array([[1, 1], [1, 1]])

# V(1,0|1,1)
chsh_pred_mat[:, :, 1, 0, 1, 1] = 1 / 2 * np.array([[1, -1], [-1, 1]])

# The probability matrix encode \pi(0,0) = \pi(0,1) = \pi(1,0) = \pi(1,1) = 1/4.
chsh_prob_mat = np.array([[1 / 4, 1 / 4], [1 / 4, 1 / 4]])

# %%
# ## Example: The unentangled value of the CHSH extended nonlocal game
#
# Similar to what we did for the BB84 extended nonlocal game, we can also compute
# the unentangled value of $G_{CHSH}$.
#

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

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define an ExtendedNonlocalGame object based on the CHSH game.
chsh = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat)

# The unentangled value is 3/4 = 0.75
print("The unentangled value is ", np.around(chsh.unentangled_value(), decimals=2))

# %%
# We can also run multiple repetitions of $G_{CHSH}$.

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

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define the CHSH game for two parallel repetitions.
chsh_2_reps = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat, 2)

# The unentangled value for two parallel repetitions is (3/4)**2 \approx 0.5625
print("The unentangled value for two parallel repetitions is ", np.around(chsh_2_reps.unentangled_value(), decimals=2))

# %%
# Note that strong parallel repetition holds as
#
# $$
# \omega(G_{CHSH})^2 = \omega(G_{CHSH}^2) = \left(\frac{3}{4}\right)^2.
# $$
#
# ## Example: The non-signaling value of the CHSH extended nonlocal game
#
# To obtain an upper bound for $G_{CHSH}$, we can calculate the
# non-signaling value.
#

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

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

# Define an ExtendedNonlocalGame object based on the CHSH game.
chsh = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat)

# The non-signaling value is 3/4 = 0.75
print("The non-signaling value is ", np.around(chsh.nonsignaling_value(), decimals=2))
# mkdocs_gallery_thumbnail_path = 'figures/chsh_game.png'

# %%
# As we know that $\omega(G_{CHSH}) = \omega_{ns}(G_{CHSH}) = 3/4$ and that
#
# $$
# \omega(G) \leq \omega^*(G) \leq \omega_{ns}(G)
# $$
#
# for any extended nonlocal game, $G$, we may also conclude that
# $\omega^*(G_{CHSH}) = 3/4$.
#
# So far, both the BB84 and CHSH examples have demonstrated cases where the
# unentangled and standard quantum values are equal. In the next tutorial, [An extended nonlocal game with quantum advantage](enlg_mub.md) we
# will explore a game based on mutually unbiased bases that exhibits a strict
# quantum advantage, where $\omega(G) < \omega^*(G)$.
#