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arXiv:1510.07431 · New Journal of Physics

An explicit classical strategy for winning a ${\mathrm{CHSH}}_{q}$ game

Matej Pivoluska, Martin Plesch

Published Feb 12, 2016• 7 citations
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Abstract

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 .

DOI
10.1088/1367-2630/18/2/025013

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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 .
The Quantum game theory is an extension of the classical game theory which describes games in which players share quantum information during the game. The use of the quantum mechanical effects enables players’ better results than they would achieve in the classical game without sharing any information at all. In this paper I would like to describe a new two player cooperative game which is based on the well known CHSH game. A quantum strategy for a new game will be introduced and it will be compared to the classical strategy and CHSH game strategy.
We show that the maximum winning probability in the repeated CHSH game for classical strategies is at most (1+54)n\left(\frac{1+\sqrt{5}}{4}\right)^n(41+5​​)n. Together with a matching lower bound by Barak et al. (FOCS'2008), this determines the asymptotic value of the repeated CHSH game exactly. We also show that, if an XOR game has a gap between classical and quantum values, there is also a gap between the asymptotic values for the repeated game.
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.
We introduce a novel generalization of the Clauser-Horne-Shimony-Holt (CHSH) game to a multiplayer setting, i.e., Hypercube game, where all mmm players are required to assign values to vertices on corresponding facets of an mmm-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 mmm increases, while the no-signalling value always remains to be one.
In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a GHZGHZGHZ-state, breaking the traditional 10\% gain observed in 2 and 3 qubit CHSH games and achieving a 22.5\% gap. In the other game, players gain a greater advantage using a WWW-state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.
Nonlocal games are an important part of quantum information processing. Recently there has been an increased interest in generalizing nonlocal games beyond the basic setup by considering games with multiple parties and/or with large alphabet inputs and outputs. We consider another interesting generalization---games with nonuniform inputs. Here we derive a tight upper bound for the classical winning probability for a family of nonlocal games with nonuniform input distribution, known as CHSHq(p){\mathrm{CHSH}}_{q}(p)CHSHq​(p), which was introduced recently in the context of relativistic bit-commitment protocols by Chakraborty et al. [Phys. Rev. Lett. 115, 250501 (2015)].