The CHSH extended nonlocal game

Following our analysis of the BB84 game, let us now define another important extended nonlocal game, \(G_{CHSH}\). This game is defined by a winning condition reminiscent of the standard CHSH 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

\[\begin{split}\begin{aligned} V(0,0|0,0) = V(0,0|0,1) = V(0,0|1,0) = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \\ V(1,1|0,0) = V(1,1|0,1) = V(1,1|1,0) = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \\ V(0,1|1,1) = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \\ V(1,0|1,1) = \frac{1}{2} \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}, \end{aligned}\end{split}\]

define

\[\begin{split}V(a,b|x,y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\end{split}\]

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))

The unentangled value is  0.75

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))

The unentangled value for two parallel repetitions is  0.56

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))

/home/docs/checkouts/readthedocs.org/user_builds/toqito/envs/latest/lib/python3.11/site-packages/scs/__init__.py:83: UserWarning: Converting A to a CSC (compressed sparse column) matrix; may take a while.
  warn(
The non-signaling value is  0.75

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 \(\omega^*(G) = 3/4\).

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 we will explore a game based on mutually unbiased bases that exhibits a strict quantum advantage, where \(\omega(G) < \omega^*(G)\).

References