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Chinese Physics Letters

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

Xiao 潇 Song 宋, Teng Liu, Ji Bian, Pengfei Lu, Yang 泱 Liu 刘, Feng 峰 Zhu 朱, Le Luo

Published May 8, 2024• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/41/6/060301

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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.
Recently, Fan et al. [Mod. Phys. Lett. A 36, 2150223 (2021)] presented a generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, to identify [Formula: see text]-qubit Greenberger–Horne–Zeilinger states. They showed an interesting phenomenon that the maximal violation of the generalized CHSH inequality is robust under some specific noises. In this work, we map the inequality to the CHSH game, and consequently to the CHSH* game in a single-qubit system. This mapping provides an explanation for the robust violations in [Formula: see text]-qubit systems. Namely, the robust violations, resulting from the degeneracy of the generalized CHSH operators correspond to the symmetry of the maximally entangled two-qubit states and the identity transformation in the single-qubit game. This explanation enables us to exactly demonstrate that the degeneracy is [Formula: see text].
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.
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.
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.
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.