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arXiv:2202.05980 · Modern Physics Letters A

Robust violation of a multipartite Bell inequality from the perspective of a single-system game

Gang-Gang He, Xing-Yan Fan, Fulin Zhang

Published Apr 20, 2022• 2 citations
arXivHTMLPDF

Abstract

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].

DOI
10.1142/s0217732322500821

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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].
The [Formula: see text]-qubit Greenberger–Horne–Zeilinger (GHZ) states are the maximally entangled states of [Formula: see text] qubits, which have had many important applications in quantum information processing, such as quantum key distribution and quantum secret sharing. Thus how to distinguish the GHZ states from other quantum states becomes a significant problem. In this work, by presenting a family of the generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, we show that the [Formula: see text]-qubit GHZ states can be indeed identified by the maximal violations of the generalized CHSH inequality under some specific measurement settings. The generalized CHSH inequality is simple and contains only four correlation functions for any [Formula: see text]-qubit system, thus has the merit of facilitating experimental verification. Furthermore, we present a quantum phenomenon of robust violations of the generalized CHSH inequality in which the maximal violation of Bell’s inequality can be robust under some specific noises adding to the [Formula: see text]-qubit GHZ states.
Alice and Bob each have half of a pair of entangled qubits. Bob measures his half and then passes his qubit to a second Bob who measures again and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. This scenario was introduced in [R. Silva et al., Phys. Rev. Lett. 114, 250401 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.250401] where the authors mentioned evidence that when the Bobs act independently and with unbiased inputs then at most two of them can expect to violate the CHSH inequality with Alice. Here we show that, contrary to this evidence, arbitrarily many independent Bobs can have an expected CHSH violation with the single Alice. Our proof is constructive and our measurement strategies can be generalized to work with a larger class of two-qubit states that includes all pure entangled two-qubit states. Since violation of a Bell inequality is necessary for device-independent tasks, our work represents a step towards an eventual understanding of the limitations on how much device-independent randomness can be robustly generated from a single pair of qubits.
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.
Abstract Bell non-locality is closely related with device independent quantum randomness. In this paper, we present a kind of sum-of-squares (SOS) decomposition for general Bell inequalities in two qubits systems. By using the obtained SOS decomposition, we can then find the measurement operators associated with the maximal violation of considered Bell inequality. We also practice the SOS decomposition method by considering the (generalized) Clauser-Horne-Shimony-Holt (CHSH) Bell inequality, the Elegant Bell inequality, the Gisin inequality and the Chained Bell inequality as examples. The corresponding SOS decompositions and the measurement operators that cause the maximum violation values of these Bell inequalities are derived, which are consistent with previous results. We further discuss the device independent quantum randomness by using the SOS decompositions of Bell inequalities. We take the generalized CHSH inequality with the maximally entangled state and the Werner state that attaining the maximal violations as examples. Exact value or lower bound on the maximal guessing probability using the SOS decomposition are obtained. For Werner state, the lower bound can supply a much precise estimation of quantum randomness when p tends to 1.
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.
Recently the authors of Phys. Rev. Lett. 125, 090401 (2020) considered the following scenario: Alice and Bob each have half of a pair of entangled qubit states. Bob measures his half and then passes his part to a second Bob who measures again, and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality with the single Alice. By taking the maximally entangled pure two-qubit state $|\ensuremath{\phi}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})$ as an example, it has been constructively proved that arbitrarily many independent Bobs can share the nonlocality with the single Alice. Here we demonstrate that arbitrarily many independent observers can share the nonlocality of a single arbitrary dimensional bipartite entangled but not necessary two-qubit entangled state. Furthermore, taking the generalized Greenberger-Horne-Zeilinger (GHZ) states as an example, we show that at most two Charlies can share the genuine nonlocality of a single generalized GHZ state with an Alice and a Bob.