Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1601.08095 · Physical Review A

Tight bound on the classical value of generalized Clauser-Horne-Shimony-Holt games

Matej Pivoluska, Marcin Pawłowski, Martin Plesch

Published Aug 29, 2016• 3 citations
arXivHTMLPDF

Abstract

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 ${\mathrm{CHSH}}_{q}(p)$, which was introduced recently in the context of relativistic bit-commitment protocols by Chakraborty et al. [Phys. Rev. Lett. 115, 250501 (2015)].

DOI
10.1103/physreva.94.022338

Related papers

Generalized xor games with d outcomes and the task of nonlocal computation

Ravishankar Ramanathan, Remigiusz Augusiak, Gláucia Murta

Two-party xor games (correlation Bell inequalities with two outcomes per party) are the most studied Bell inequalities, and one of the few classes for which the optimal quantum value is known to be exactly calculable. We study a natural generalization of the binary xor games to the class of linear games with $d>2$ outcomes, and propose an easily computable bound on the quantum value of these games. Many interesting properties such as the impossibility of a quantum strategy to win these games, and the quantum bound on the CHSH game generalized to $d$ outcomes are derived. We also use the proposed bound to prove a large-alphabet generalization of the principle of no quantum advantage in nonlocal computation, showing that quantum theory provides no advantage in the task of nonlocal distributed computation of a class of functions with $d$ outcomes for prime $d$, while general no-signaling boxes do. This task is one of the information-theoretic principles attempting to characterize the set of quantum correlations from amongst general no-signaling ones.

arXiv:1502.02974•Physical Review A•Feb 29, 2016•16 citations

Biased nonlocal quantum games

Thomas Lawson, Noah Linden, Sandu Popescu

We address the question of when quantum entanglement is a useful resource for information processing tasks by presenting a new class of nonlocal games that are simple, direct, generalizations of the Clauser Horne Shimony Holt game. For some ranges of the parameters that specify the games, quantum mechanics offers an advantage, while, surprisingly, for others quantum mechanics is no more powerful than classical mechanics in performing the nonlocal task. This sheds new light on the difference between classical, quantum and super-quantum correlations.

arXiv:1011.6245•Nov 29, 2010•12 citations

Maximal Clauser-Horne-Shimony-Holt violation for qubit-qudit states

Alexander Bernal, J.A. Casas, Jesús M. Moreno

We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.

arXiv:2404.02092•Physical Review A•Jul 22, 2025•4 citations

Quantum Advantages in Hypercube Game

He, Xiaoyu, Kun Fang, Xiaoming Sun +1 more

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.

arXiv:1806.02642•Jun 7, 2018•0 citations

Reexamination of a multisetting Bell inequality for qudits

Yeong-Cherng Liang, Chu-Wee Lim, Dong‐Ling Deng

The class of $d$-setting, $d$-outcome Bell inequalities proposed by Ji and co-workers [Phys. Rev. A 78, 052103 (2008)] are reexamined. For every positive integer $d>2$, we show that the corresponding nontrivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with $d$ inputs and $d$ outputs. We also demonstrate that the general classical upper bounds given by Ji et al. are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the actual classical upper bound for $d\ensuremath{\le}13$ (including nonprime values of $d$). We further determine that for prime value $d$ in this range, most of these probability and correlation inequalities are tight, i.e., facet-inducing for the respective classical correlation polytope. Stronger lower and upper bounds on the quantum violation of these inequalities are obtained. In particular, we prove that once the probability inequalities are given, their correlation counterparts given by Ji and co-workers are no longer relevant in terms of detecting the entanglement of a quantum state.

arXiv:0903.4964•Physical Review A•Nov 25, 2009•62 citations

Non-Hermitian CHSH* Game with a Single Trapped-Ion Qubit

Xiao 潇 Song 宋, Teng Liu, Ji Bian +4 more

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.

Chinese Physics Letters•May 8, 2024•2 citations
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)].
Two-party xor games (correlation Bell inequalities with two outcomes per party) are the most studied Bell inequalities, and one of the few classes for which the optimal quantum value is known to be exactly calculable. We study a natural generalization of the binary xor games to the class of linear games with d>2d>2d>2 outcomes, and propose an easily computable bound on the quantum value of these games. Many interesting properties such as the impossibility of a quantum strategy to win these games, and the quantum bound on the CHSH game generalized to ddd outcomes are derived. We also use the proposed bound to prove a large-alphabet generalization of the principle of no quantum advantage in nonlocal computation, showing that quantum theory provides no advantage in the task of nonlocal distributed computation of a class of functions with ddd outcomes for prime ddd, while general no-signaling boxes do. This task is one of the information-theoretic principles attempting to characterize the set of quantum correlations from amongst general no-signaling ones.
We address the question of when quantum entanglement is a useful resource for information processing tasks by presenting a new class of nonlocal games that are simple, direct, generalizations of the Clauser Horne Shimony Holt game. For some ranges of the parameters that specify the games, quantum mechanics offers an advantage, while, surprisingly, for others quantum mechanics is no more powerful than classical mechanics in performing the nonlocal task. This sheds new light on the difference between classical, quantum and super-quantum correlations.
We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.
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.
The class of ddd-setting, ddd-outcome Bell inequalities proposed by Ji and co-workers [Phys. Rev. A 78, 052103 (2008)] are reexamined. For every positive integer d>2d>2d>2, we show that the corresponding nontrivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with ddd inputs and ddd outputs. We also demonstrate that the general classical upper bounds given by Ji et al. are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the actual classical upper bound for $d\ensuremath{\le}13$ (including nonprime values of ddd). We further determine that for prime value ddd in this range, most of these probability and correlation inequalities are tight, i.e., facet-inducing for the respective classical correlation polytope. Stronger lower and upper bounds on the quantum violation of these inequalities are obtained. In particular, we prove that once the probability inequalities are given, their correlation counterparts given by Ji and co-workers are no longer relevant in terms of detecting the entanglement of a quantum state.
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.