An extended nonlocal game with quantum advantage¶

In the previous tutorials on The BB84 extended nonlocal game and The CHSH extended nonlocal game, we saw examples where the standard quantum and unentangled values were equal (\(\omega(G) = \omega^*(G)\)). Here, we will construct an extended nonlocal game where the standard quantum value is strictly higher than the unentangled value, demonstrating a true quantum advantage.

A monogamy-of-entanglement game with mutually unbiased bases¶

Let \(\zeta = \exp(\frac{2 \pi i}{3})\) and consider the following four mutually unbiased bases:

\[ \begin{equation}\label{eq:MUB43} \begin{aligned} \mathcal{B}_0 &= \left\{ e_0,\: e_1,\: e_2 \right\}, \\ \mathcal{B}_1 &= \left\{ \frac{e_0 + e_1 + e_2}{\sqrt{3}},\: \frac{e_0 + \zeta^2 e_1 + \zeta e_2}{\sqrt{3}},\: \frac{e_0 + \zeta e_1 + \zeta^2 e_2}{\sqrt{3}} \right\}, \\ \mathcal{B}_2 &= \left\{ \frac{e_0 + e_1 + \zeta e_2}{\sqrt{3}},\: \frac{e_0 + \zeta^2 e_1 + \zeta^2 e_2}{\sqrt{3}},\: \frac{e_0 + \zeta e_1 + e_2}{\sqrt{3}} \right\}, \\ \mathcal{B}_3 &= \left\{ \frac{e_0 + e_1 + \zeta^2 e_2}{\sqrt{3}},\: \frac{e_0 + \zeta^2 e_1 + e_2}{\sqrt{3}},\: \frac{e_0 + \zeta e_1 + \zeta e_2}{\sqrt{3}} \right\}. \end{aligned} \end{equation} \]

Define an extended nonlocal game \(G_{MUB} = (\pi,R)\) so that

\[ \pi(0) = \pi(1) = \pi(2) = \pi(3) = \frac{1}{4} \]

and \(R\) is such that

\[ \{ R(0|x), R(1|x), R(2|x) \} \]

represents a measurement with respect to the basis \(\mathcal{B}_x\), for each \(x \in \{0,1,2,3\}\).

Taking the description of \(G_{MUB}\), we can encode this as follows.

import numpy as np

from toqito.states import basis

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

# Define the monogamy-of-entanglement game defined by MUBs.
prob_mat = 1 / 4 * np.identity(4)

dim = 3
e_0, e_1, e_2 = basis(dim, 0), basis(dim, 1), basis(dim, 2)

eta = np.exp((2 * np.pi * 1j) / dim)
mub_0 = [e_0, e_1, e_2]
mub_1 = [
    (e_0 + e_1 + e_2) / np.sqrt(3),
    (e_0 + eta**2 * e_1 + eta * e_2) / np.sqrt(3),
    (e_0 + eta * e_1 + eta**2 * e_2) / np.sqrt(3),
]
mub_2 = [
    (e_0 + e_1 + eta * e_2) / np.sqrt(3),
    (e_0 + eta**2 * e_1 + eta**2 * e_2) / np.sqrt(3),
    (e_0 + eta * e_1 + e_2) / np.sqrt(3),
]
mub_3 = [
    (e_0 + e_1 + eta**2 * e_2) / np.sqrt(3),
    (e_0 + eta**2 * e_1 + e_2) / np.sqrt(3),
    (e_0 + eta * e_1 + eta * e_2) / np.sqrt(3),
]

# List of measurements defined from mutually unbiased basis.
mubs = [mub_0, mub_1, mub_2, mub_3]

num_in = 4
num_out = 3
pred_mat = np.zeros([dim, dim, num_out, num_out, num_in, num_in], dtype=complex)

pred_mat[:, :, 0, 0, 0, 0] = mubs[0][0] @ mubs[0][0].conj().T
pred_mat[:, :, 1, 1, 0, 0] = mubs[0][1] @ mubs[0][1].conj().T
pred_mat[:, :, 2, 2, 0, 0] = mubs[0][2] @ mubs[0][2].conj().T

pred_mat[:, :, 0, 0, 1, 1] = mubs[1][0] @ mubs[1][0].conj().T
pred_mat[:, :, 1, 1, 1, 1] = mubs[1][1] @ mubs[1][1].conj().T
pred_mat[:, :, 2, 2, 1, 1] = mubs[1][2] @ mubs[1][2].conj().T

pred_mat[:, :, 0, 0, 2, 2] = mubs[2][0] @ mubs[2][0].conj().T
pred_mat[:, :, 1, 1, 2, 2] = mubs[2][1] @ mubs[2][1].conj().T
pred_mat[:, :, 2, 2, 2, 2] = mubs[2][2] @ mubs[2][2].conj().T

pred_mat[:, :, 0, 0, 3, 3] = mubs[3][0] @ mubs[3][0].conj().T
pred_mat[:, :, 1, 1, 3, 3] = mubs[3][1] @ mubs[3][1].conj().T
pred_mat[:, :, 2, 2, 3, 3] = mubs[3][2] @ mubs[3][2].conj().T

Now that we have encoded \(G_{MUB}\), we can calculate the unentangled value.

import numpy as np

from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame

g_mub = ExtendedNonlocalGame(prob_mat, pred_mat)
unent_val = g_mub.unentangled_value()
print("The unentangled value is ", np.around(unent_val, decimals=2))

Out:

The unentangled value is  0.65

That is, we have that

\[ \omega(G_{MUB}) = \frac{3 + \sqrt{5}}{8} \approx 0.65409. \]

However, if we attempt to run a lower bound on the standard quantum value, we obtain.

g_mub = ExtendedNonlocalGame(prob_mat, pred_mat)
q_val = g_mub.quantum_value_lower_bound()
print("The standard quantum value lower bound is ", np.around(q_val, decimals=2))
# mkdocs_gallery_thumbnail_path = 'figures/logo.png'

Out:

/home/runner/work/toqito/toqito/.venv/lib/python3.12/site-packages/scs/__init__.py:83: UserWarning: Converting A to a CSC (compressed sparse column) matrix; may take a while.
  warn(
The standard quantum value lower bound is  0.65

Note that as we are calculating a lower bound, it is possible that a value this high will not be obtained, or in other words, the algorithm can get stuck in a local maximum that prevents it from finding the global maximum.

It is uncertain what the optimal standard quantum strategy is for this game, but the value of such a strategy is bounded as follows

\[ 2/3 \geq \omega^*(G) \geq 0.6609. \]

For further information on the \(G_{MUB}\) game, consult ¹.

Total running time of the script: ( 0 minutes 1.210 seconds)

Download Python source code: enlg_mub.py

Download Jupyter notebook: enlg_mub.ipynb

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Vincent Russo. Extended nonlocal games. 2017. arXiv:1704.07375. ↩