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arXiv:2402.17513 · Journal of Physics Condensed Matter

Study of quantum nonlocality by CHSH function and its extension in disordered fermions

Yoshihito Kuno

Published Sep 18, 2024• 2 citations
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Abstract

Quantum nonlocality is an important concept in quantum physics. In this work, we study the quantum nonlocality in a fermion many-body system under quasi-periodic disorders. The Clauser-Horne-Shimony-Holt (CHSH) inequality is systematically investigated, which quantifies quantum nonlocality between two sites. We find particular behaviors of the quantifiers of quantum nonlocality around the extended and critical phase transitions in the system and further that the CHSH inequality is not broken in the globally averaged picture of the maximum value of the quantum nonlocality, but the violation probability of the CHSH inequality for two site pairs in the system becomes sufficiently finite in the critical phase and on a critical phase boundary. Further, we investigate an extension of the CHSH inequality, Mermin-Klyshko-Svetlichny (MKS) polynomials, which can characterize multipartite quantum nonlocality. We also find a similar behavior to the case of CHSH inequality. In particular, in the critical regime and on a transition point, the adjacent three-qubit MKS polynomial in a portion of the system exhibits a quantum nonlocal violation regime with a finite probability in the critical phase.

DOI
10.1088/1361-648x/ad7cb4

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Quantum nonlocality is an important concept in quantum physics. In this work, we study the quantum nonlocality in a fermion many-body system under quasi-periodic disorders. The Clauser-Horne-Shimony-Holt (CHSH) inequality is systematically investigated, which quantifies quantum nonlocality between two sites. We find particular behaviors of the quantifiers of quantum nonlocality around the extended and critical phase transitions in the system and further that the CHSH inequality is not broken in the globally averaged picture of the maximum value of the quantum nonlocality, but the violation probability of the CHSH inequality for two site pairs in the system becomes sufficiently finite in the critical phase and on a critical phase boundary. Further, we investigate an extension of the CHSH inequality, Mermin-Klyshko-Svetlichny (MKS) polynomials, which can characterize multipartite quantum nonlocality. We also find a similar behavior to the case of CHSH inequality. In particular, in the critical regime and on a transition point, the adjacent three-qubit MKS polynomial in a portion of the system exhibits a quantum nonlocal violation regime with a finite probability in the critical phase.
Incompatibility and nonlocality are not only of foundational interest but also act as important resources for quantum information theory. In the Clauser-Horne-Shimony-Holt (CHSH) scenario, the incompatibility of a pair of observables is known to be equivalent to Bell nonlocality. Here, we investigate these notions in the context of qubit channels. The Bell-CHSH inequality has a greater perspective---compared to any genuine tripartite nonlocality scenario---while determining the interplay between nonlocality breaking qubit channels and incompatibility breaking qubit channels. In the Bell-CHSH scenario, we prove that if the conjugate of a channel is incompatibility breaking, then the channel is itself nonlocality breaking and vice versa. However, this equivalence is not straightforwardly generalized to multipartite systems, due to the absence of an equivalence relation between incompatibility and nonlocality in the multipartite scenario. We investigate this relation in the tripartite scenario by considering some well-known states like Greenberger-Horne-Zeilinger and WWW states and using the notion of Mermin and Svetlichny nonlocality. By subjecting the parties in question to unital qubit channels, we identify the range of state and channel parameters for which incompatibility coexists with nonlocality. Further, we identify the set of unital qubit channels that is Mermin or Svetlichny nonlocality breaking irrespective of the input state.
The characterization of the extremal points of the set of quantum correlations has attracted wide interest. In the simplest bipartite Bell scenario, a necessary and sufficient criterion for identifying extremal correlations has recently been conjectured, but extremality of tripartite correlations is not well known. In this study, we analyze tripartite extremal correlations in terms of the conjectured bipartite extremal criterion, and we demonstrate that the bipartite part of some extremal correlations satisfies the bipartite criterion, even though they violate Svetlichny’s inequality, and therefore are considered (stronger) genuine tripartite nonlocal correlations. This phenomenon arises from the fact that the conjectured extremal criterion is automatically satisfied when the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality exceeds a certain threshold, the value of which is given by the maximum CHSH violation at the edges of the probability space. This also suggests the possibility that the extremality of bipartite correlations can be certified by verifying whether the CHSH violation exceeds the threshold.
We investigate the nonlocality of Schmidt-correlated (SC) states, and present analytical expressions of the maximum violation value of Bell inequalities. It is shown that the violation of Clauser—Horne—Shimony—Holt (CHSH) inequality is necessary and sufficient for the nonlocality of two-qubit SC states, whereas the violation of the Svetlichny inequality is only a sufficient condition for the genuine nonlocality of three-qubit SC states. Furthermore, the relations among the maximum violation values, concurrence, and relative entropy entanglement are discussed.
We investigate the nonlocality of Schmidt-correlated (SC) states, and present analytical expressions of the maximum violation value of Bell inequalities. It is shown that the violation of Clauser—Horne—Shimony—Holt (CHSH) inequality is necessary and sufficient for the nonlocality of two-qubit SC states, whereas the violation of the Svetlichny inequality is only a sufficient condition for the genuine nonlocality of three-qubit SC states. Furthermore, the relations among the maximum violation values, concurrence, and relative entropy entanglement are discussed.
The Clauser-Horne-Shimony-Holt inequality was originally proposed as a Bell inequality to detect nonlocality in bipartite systems. However, it can also be used to certify the nonlocality of multipartite quantum states. We apply this to study the nonlocality of multipartite Greenberger-Horne-Zeilinger (GHZ), W, and graph states under local decoherence processes. We derive lower bounds on the critical local-noise strength tolerated by the states before becoming local. In addition, for the whole noisy dynamics, we derive lower bounds on the corresponding nonlocal content for the three classes of states. All the bounds presented can be calculated efficiently and, in some cases, provide significantly tighter estimates than with any other known method. For example, they reveal that NNN-qubit GHZ states undergoing local dephasing are, for all NNN, nonlocal throughout all the dephasing dynamics.
There exist bipartite entangled states whose violations of Clauser-Horne-Shimony-Holt (CHSH) Bell inequality can be observed by a single Alice and arbitrarily many sequential Bobs [Phys. Rev. Lett. 125, 090401 (2020)]. Here we consider its analogues for tripartite systems: a tripartite entangled state is shared among Alice, Bob and multiple Charlies. The first Charlie measures his qubit and then passes his qubit to the next Charlie who measures again with other measurements and so on. The goal is to maximize the number of Charlies that can observe some kind of nonlocality with the single Alice and Bob. It has been shown that at most two Charlies could share genuine nonlocality of the Greenberger-Horne-Zeilinger (GHZ) state via the violation of Svetlichny inequality with Alice and Bob [Quantum Inf. Process. 18, 42 (2019) and Phys. Rev. A 103, 032216 (2021)]. In this work, we show that arbitrarily many Charlies can have standard nonlocality (via violations of Mermin inequality) and some other kind of genuine nonlocality (which is known as genuinely nonsignal nonlocality) with the single Alice and single Bob.