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International Journal of Modern Physics B

The relationship between the degree of violation of Bell-CHSH inequality and measurement uncertainty in two classes of spin squeezing mechanisms

Anas Ait Chlih, Nabil Habiballah, D. Khatib

Published Aug 17, 2023• 9 citations
Publisher

Abstract

In solid-state systems, spin squeezing states have acquired much interest in quantum information processing since they are not only essential from a basic physics standpoint but also have many applications, notably reducing noise in interferometry. This paper addresses the study of quantum nonlocality as examined by the violation of the Bell and Clauser–Horne–Shimony–Holt (CHSH) inequality in two classes of spin squeezing states and how this relates to the quantum-memory-assisted entropic uncertainty relation (QMA–EUR). The spin squeezing states in the two classes are systematically generated by two twisting nonlinear interactions. These separately describe two spin-[Formula: see text] interaction mechanisms that undergo one-axis twisting (OAT) and two-axis countertwisting (TAC). We investigate each spin squeezing mechanism in a thermal bath ([Formula: see text]) with a transverse magnetic field. Our main results indicate that the degree of violation of the Bell-CHSH inequality and the entropic uncertainty relation (EUR) is typically anticorrelated in both mechanisms. We demonstrate that the spin squeezing states in the OAT mechanism violate the Bell-CHSH inequality and thus exhibit Bell nonlocality only at low temperatures ([Formula: see text]), whereas in the TAC mechanism, increasing the strength of the spin squeezing interaction leads to a violation of the Bell-CHSH inequality even at finite temperatures. We have also shown that there is always an expansion of the EUR and a decrease in the degree of violation of the Bell-CHSH inequality in the high-temperature regime. In addition, the transverse magnetic field exhibits different behavior under the considered temperature regimes in the two classes of spin squeezing states. Other results will also be discussed.

DOI
10.1142/s0217979224503107

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In solid-state systems, spin squeezing states have acquired much interest in quantum information processing since they are not only essential from a basic physics standpoint but also have many applications, notably reducing noise in interferometry. This paper addresses the study of quantum nonlocality as examined by the violation of the Bell and Clauser–Horne–Shimony–Holt (CHSH) inequality in two classes of spin squeezing states and how this relates to the quantum-memory-assisted entropic uncertainty relation (QMA–EUR). The spin squeezing states in the two classes are systematically generated by two twisting nonlinear interactions. These separately describe two spin-[Formula: see text] interaction mechanisms that undergo one-axis twisting (OAT) and two-axis countertwisting (TAC). We investigate each spin squeezing mechanism in a thermal bath ([Formula: see text]) with a transverse magnetic field. Our main results indicate that the degree of violation of the Bell-CHSH inequality and the entropic uncertainty relation (EUR) is typically anticorrelated in both mechanisms. We demonstrate that the spin squeezing states in the OAT mechanism violate the Bell-CHSH inequality and thus exhibit Bell nonlocality only at low temperatures ([Formula: see text]), whereas in the TAC mechanism, increasing the strength of the spin squeezing interaction leads to a violation of the Bell-CHSH inequality even at finite temperatures. We have also shown that there is always an expansion of the EUR and a decrease in the degree of violation of the Bell-CHSH inequality in the high-temperature regime. In addition, the transverse magnetic field exhibits different behavior under the considered temperature regimes in the two classes of spin squeezing states. Other results will also be discussed.
Quantum systems are inherently open and fragile, leading to the loss of information and quantum properties to the environment. This study investigates the behavior of quantum entanglement (QE), quantum nonlocality (QN), and measurement uncertainty in a two-atom system interacting with a common dissipative thermal reservoir under the Markovian approximation. QE is measured using concurrence, QN through the CHSH-Bell inequality, and uncertainty through the quantum memory-assisted entropic uncertainty relation (QMA-EUR). The findings show that both QE and QN decay as the scaled time and excitation numbers increase, with QE being more resilient. In contrast, QMA-EUR shows an anticorrelation with concurrence and CHSH-Bell violation, increasing as these properties diminish. Adjusting the system initial-state parameters improves QE and CHSH degree of violation, while reducing uncertainty. Over long durations, concurrence and CHSH-Bell violations disappear, while QMA-EUR stabilizes at its maximum value. These results highlight the importance of precise parameter control to enhance quantum features and predict outcomes, especially for bipartite systems initialized in a Werner-like state.
We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.
In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nthn\mathrm{th}nth particle pair would be independent of measurement choices and outcomes for the first $(n\ensuremath{-}1)$ pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the two-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the two-sided memory loophole allows a systematic violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality when the data are analyzed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in the existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearized CHSH inequality as well as other Bell-type inequalities.
Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Although Bell nearly showed a violation using the Clauser-Horne-Shimony-Holt (CHSH) inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he does not give a quantitative estimation of the maximum achievable violation for the wave function he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of 222\sqrt{2}22​. Although Bell's wave function does not produce a large violation of the CHSH inequality, other states may yet do so.
Abstract We investigate thermal quantum correlations and the entropic uncertainty relation in the presence of quantum memory under two kinds of two-qubit spin squeezing models, namely the one-axis twisting model and two-axis countertwisting model. We provide the analytical expressions of quantum correlations and the entropic uncertainty relation in the ground state for these two models and we find a simple relationship between quantum discord and entropic uncertainty. It is found that there is a fully anti-correlated relationship between quantum correlations and entropic uncertainty. Also, the maximal quantum correlation and minimal uncertainty can be achieved when the spin squeezing parameters are large enough. Accordingly, these results may be significant in practical goals in which the maximum quantum correlation and minimum uncertainty are required such as in quantum key distribution and quantum computing.