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Four-qubit CHSH game

Joaquim Jusseau, Hamza Jaffali, Frédéric Holweck

Published Oct 14, 2025• 3 citations
Publisher

Abstract

Abstract 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 $$\vert {GHZ_4} \rangle =\dfrac{1}{\sqrt{2}}(\vert {0000} \rangle +\vert {1111} \rangle )$$ | G H Z 4 ⟩ = 1 2 ( | 0000 ⟩ + | 1111 ⟩ ) 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 $$\vert {W_4} \rangle =\dfrac{1}{2}(\vert {0001} \rangle +\vert {0010} \rangle +\vert {0100} \rangle +\vert {1000} \rangle )$$ | W 4 ⟩ = 1 2 ( | 0001 ⟩ + | 0010 ⟩ + | 0100 ⟩ + | 1000 ⟩ ) 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.

DOI
10.1007/s11128-025-04939-8

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•Oct 27, 2026•1 citation
Abstract 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 ∣GHZ4⟩=12(∣0000⟩+∣1111⟩)\vert {GHZ_4} \rangle =\dfrac{1}{\sqrt{2}}(\vert {0000} \rangle +\vert {1111} \rangle )∣GHZ4​⟩=2​1​(∣0000⟩+∣1111⟩) | G H Z 4 ⟩ = 1 2 ( | 0000 ⟩ + | 1111 ⟩ ) 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 ∣W4⟩=12(∣0001⟩+∣0010⟩+∣0100⟩+∣1000⟩)\vert {W_4} \rangle =\dfrac{1}{2}(\vert {0001} \rangle +\vert {0010} \rangle +\vert {0100} \rangle +\vert {1000} \rangle )∣W4​⟩=21​(∣0001⟩+∣0010⟩+∣0100⟩+∣1000⟩) | W 4 ⟩ = 1 2 ( | 0001 ⟩ + | 0010 ⟩ + | 0100 ⟩ + | 1000 ⟩ ) 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.
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.
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.
In the Quantum Computing field, the CHSH (Clauser-Horne-Shimony-Holt) game plays a vital role in determining whether a system is in classical or quantum domain. This quantum advantage property is applied in determining the security of the quantum systems, especially in quantum cryptography, in the form of Device-Independent (DI) secure Quantum Key Distribution (QKD). The traditional CHSH game is designed for a system with two players, where each player has access to one entangled qubit from the Bell’s state. In 2024, Hamza Jaffali and Frédéric Holweck proposed a new three-qubit CHSH game by extending the traditional CHSH game to three players using the three-qubit Greenberger–Horne–Zeilinger (GHZ) state and W state.In this paper, first, we present our findings of new CHSH quantum games with quantum advantage in the four-qubit and five-qubit systems. These CHSH quantum games, when played using their quantum strategy, are found to outperform any classical strategy. Second, we apply the quantum advantage property of the existing three-qubit game to develop the corresponding three party DIQKD protocol. Third, we then apply the quantum advantage property of the newly developed four-qubit and five-qubit games to develop their corresponding four party and five party DIQKD protocols.
Optimal, and approximately optimal, quantum strategies for XOR∗\mathrm{XOR}^{*}XOR∗ and FFL\mathrm{FFL}FFL games
We analyze optimal, and approximately optimal, quantum strategies in two non-local settings based upon previous investigations of the XOR and CHSH games. By building upon previous arguments due to Ostrev in 2016, {\color{blue}[37]}, which characterized approximately optimal, and optimal, strategies that players Alice and Bob can adopt for increasing their respective winning probabilities, we identify additional applications of the framework for analyzing prospective quantum advantage in other game-theoretic settings. In particular, while it is possible for Alice and Bob to realize quantum advantage if they adopt strategies that are dependent upon entanglement, two-dimensional resource systems, and reversible transformations, the magnitude of the duality gap between the performance of classical and quantum strategies has not yet been classified from the perspective of error bounds. Such bounds are not only dependent upon the winning probability if Alice and Bob leverage entanglement in their strategies but also upon a suitably defined representation-theoretic intertwining operation.
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.
How much entanglement does near-optimal play of a two-player nonlocal game require? We give an explicit binary linear system game, with 168716871687 equations in 199319931993 variables and at most three variables in each equation, whose quantum value is one, and in which every strategy that wins with probability 1−ε1-\varepsilon1−ε needs entanglement entropy ε−1+o(1)\varepsilon^{-1+o(1)}ε−1+o(1), hence local dimension exp⁡(ε−1+o(1))\exp(\varepsilon^{-1+o(1)})exp(ε−1+o(1)). The same bound holds for the entanglement of formation of mixed strategies, and a strategy on a maximally entangled state of local dimension exp⁡(Cε−1log⁡(1/ε))\exp(C\varepsilon^{-1}\log(1/\varepsilon))exp(Cε−1log(1/ε)) suffices, so the exponent is sharp. For explicit two-player games of quantum value one, the best previous lower bound was polynomial in 1/ε1/\varepsilon1/ε. The game comes from a finitely generated, elementary amenable group: a lamplighter whose lamps are Pauli pairs indexed by finite sets of points on the three rays on which Houghton's group acts. Exponentially many of these pairs share one central involution, yet the relation that any two of them commute can be derived from a fixed finite set of relations in a number of steps only quasi-polynomial in the number of points. Two tools may be of independent use. The first counts qubits from approximate Pauli relations measured in an arbitrary state, with the errors of different pairs adding in squares. The second identifies, in both directions and at the natural precision ε\sqrt\varepsilonε​, the near-optimal strategies of a linear system game with approximate representations of its solution group measured in the players' actual state, so that no power of ε\varepsilonε is lost by passing to a flat state. The same tools give sharp rates for two earlier examples: the CHSH(n)(n)(n) games need Θ(min⁡(n,1/ε))\Theta(\min(n,1/\varepsilon))Θ(min(n,1/ε)) ebits, and the linear system game of Slofstra and Vidick needs local dimension Θ(ε−1/2)\Theta(\varepsilon^{-1/2})Θ(ε−1/2).