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arXiv:1806.02642

Quantum Advantages in Hypercube Game

He, Xiaoyu, Kun Fang, Xiaoming Sun, Runyao Duan

Published Jun 7, 2018• 0 citations
arXivHTMLPDF

Abstract

We introduce a novel generalization of the Clauser-Horne-Shimony-Holt (CHSH) game to a multiplayer setting, i.e., Hypercube game, where all $m$ players are required to assign values to vertices on corresponding facets of an $m$-dimensional hypercube. The players win if and only if their answers satisfy both parity and consistency conditions. We completely characterize the maximum winning probabilities (game value) under classical, quantum and no-signalling strategies, respectively. In contrast to the original CHSH game designed to demonstrate the superiority of quantumness, we find that the quantum advantages in the Hypercube game significantly decrease as the number of players increase. Notably, the quantum value decays exponentially fast to the classical value as $m$ increases, while the no-signalling value always remains to be one.

DOI
10.48550/arxiv.1806.02642

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We present a generalization of the Mermin-Peres magic square game to NNN players, in which the square is replaced by an NNN-dimensional hypercube. We show that the winning probability of the optimal classical strategy remains 8/98/98/9 for all NNN, while for N≥3N \ge 3N≥3, a generalization of the quantum strategy for N=2N = 2N=2 achieves a winning probability between 1/31/31/3 and 3/43/43/4. This loss of quantum advantage for N≥3N \ge 3N≥3 is a consequence of monogamy.
Abstract The Clauser–Horne–Shimony–Holt (CHSH) game provides a captivating illustration of the advantages of quantum strategies over classical ones. In a recent study, a variant of the CHSH game leveraging a single qubit system, referred to as the CHSH* game, has been identified. We demonstrate that this mapping relationship between these two games remains effective even for a non-unitary gate. Here we delve into the breach of Tsirelson’s bound in a non-Hermitian system, predicting changes in the upper and lower bounds of the player’s winning probability when employing quantum strategies in a single dissipative qubit system. We experimentally explore the impact of the CHSH* game on the player’s winning probability in a single trapped-ion dissipative system, demonstrating a violation of Tsirelson’s bound under the influence of parity-time ( P T ) symmetry. These results contribute to a deeper understanding of the influence of non-Hermitian systems on quantum games and the behavior of quantum systems under P T symmetry, which is crucial for designing more robust and efficient quantum protocols.
An explicit classical strategy for winning a CHSHq{\mathrm{CHSH}}_{q}CHSHq​ game
A CHSH q game is a generalization of the standard two player CHSH game, with q different input and output options. In contrast to the binary game, the best classical and quantum winning strategies are not known exactly. In this paper we provide a constructive classical strategy for winning a CHSH q game, with q being a prime. Our construction achieves a winning probability better than , which is in contrast with the previously known constructive strategies achieving only the winning probability of .
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In this paper, we generalize to three players the well-known CHSH quantum game. To do so, we consider all possible 3-variables Boolean functions and search among them which ones correspond to a game scenario with a quantum advantage (for a given entangled state). In particular, we provide two new three-player quantum games where, in one case, the best quantum strategy is obtained when the players share a GHZGHZGHZ state, while in the other one, the players have a better advantage when they use a WWW state as their quantum resource. To illustrate our findings we implement our game scenarios on an online quantum computer and prove experimentally the advantage of the corresponding quantum resource for each game.