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Physical Review A

Re-examining a local-realism test by modifying the classical boundary for modular phase-space measurements

Ming‐Da Huang, Ya-Fei Yu, Zhi‐Ming Zhang

Published Aug 27, 2020• 5 citations
Publisher

Abstract

In recent work, Arora and Asadian have proposed a sophisticated physical structure for testing the macroscopic Clauser-Horne-Shimony-Holt (CHSH) inequality of two entangled particles using a multislit Young experiment that employs classical-like modular variables and measurements. By imposing the algebraic CHSH constraint on the Wigner representations of these observables, they found that a strong connection exists between the non-negative Wigner function of a quantum state and having the macroscopic CHSH inequality hold. The entangled states with negative Wigner functions violate the macroscopic CHSH inequality when the number of slits is $N\ensuremath{\ge}8$ and the maximum violation, in principle, requires $N\ensuremath{\rightarrow}+\ensuremath{\infty}$. In this paper, we further study this macroscopic local-realism test by analyzing the eigenvalues of the classical-like modular variables when $N$ is finite. Through this analysis, we find that one can modify the classical boundary for the macroscopic CHSH inequality when $N$ is finite, which makes it possible to demonstrate macroscopic Bell nonlocality for entangled states with $N<8$. More interestingly, we find another form of an entangled state that violates the macroscopic CHSH inequality under the modified boundaries when the number of slits is $N=2$. But the violation of the macroscopic CHSH inequality under the modified classical boundaries with a finite $N$ does not necessarily correspond to states with a non-positive-definite Wigner function.

DOI
10.1103/physreva.102.022229

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In recent work, Arora and Asadian have proposed a sophisticated physical structure for testing the macroscopic Clauser-Horne-Shimony-Holt (CHSH) inequality of two entangled particles using a multislit Young experiment that employs classical-like modular variables and measurements. By imposing the algebraic CHSH constraint on the Wigner representations of these observables, they found that a strong connection exists between the non-negative Wigner function of a quantum state and having the macroscopic CHSH inequality hold. The entangled states with negative Wigner functions violate the macroscopic CHSH inequality when the number of slits is N\ensuremath≥8N\ensuremath{\ge}8N\ensuremath≥8 and the maximum violation, in principle, requires N\ensuremath→+\ensuremath∞N\ensuremath{\rightarrow}+\ensuremath{\infty}N\ensuremath→+\ensuremath∞. In this paper, we further study this macroscopic local-realism test by analyzing the eigenvalues of the classical-like modular variables when NNN is finite. Through this analysis, we find that one can modify the classical boundary for the macroscopic CHSH inequality when NNN is finite, which makes it possible to demonstrate macroscopic Bell nonlocality for entangled states with N<8N<8N<8. More interestingly, we find another form of an entangled state that violates the macroscopic CHSH inequality under the modified boundaries when the number of slits is N=2N=2N=2. But the violation of the macroscopic CHSH inequality under the modified classical boundaries with a finite NNN does not necessarily correspond to states with a non-positive-definite Wigner function.
The assumption of local realism imposes constraints, such as Bell inequalities, on quantities obtained from measurements. In recent years, various tests of local realism have gained popularity in undergraduate laboratories, giving students the exciting opportunity to experimentally contradict this philosophical assumption. The standard test of the CHSH (Clauser–Horne–Shimony–Holt) Bell inequality requires 16 measurements, whereas a test of Freedman's inequality requires only three measurements. The calculations required to test Freedman's inequality are correspondingly simpler and the theory is less abstract. We suggest that students may benefit from testing Freedman's inequality before proceeding to the CHSH inequality and other more complicated experiments. Our measured data violated Freedman's inequality by more than six standard deviations.
Abstract The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.
Clauser-Horne (CH) inequality, Eberhard inequality, and Clauser-Horne-Shimony-Holt (CHSH) inequality are used to determine whether quantum entanglement can contradict local realism. However, the loophole is known to allow local realism to violate these inequalities if a sample size is small and not enough [1]. Remarkably though, this paper shows that this loophole can still cause a violation in these inequalities even with a very large sample size, e.g. a 2.4 sigma violation of CH inequality and Eberhard inequality was achieved despite 12,000,000 total trials in a Monte Carlo simulation of a local realist photonic experiment based on Malus' law. In addition, this paper shows how Eberhard inequality is especially vulnerable to this loophole when combined with an improper statistical analysis and incorrect singles counts, e.g. a 13.0 sigma violation was achieved with the same large sample size, and furthermore, a 26.6 sigma violation was produced when a small, acceptable 0.2% production rate loophole was applied. Supplementally, this paper demonstrates how the finite statistics loophole allows a bigger violation in a smaller sample size despite the sample size being enough, e.g. a CHSH violation of 4.4 sigma (2.43 +/- 0.10) was achieved with 280 total trials, and 4.0 sigma (2.16 +/- 0.04) with 3,000 total trials. This paper introduces the aforementioned loopholes as plausible local realist explanations to two observed violations reported by Giustina, et al. [2], and Hensen, et al. [3].
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.
A finite non-classical framework for qubit physics is described that challenges the conclusion that the Bell Inequality has been shown to have been violated experimentally, even approximately. This framework postulates the primacy of a fractal-like ‘invariant set’ geometry I U in cosmological state space, on which the universe evolves deterministically and causally, and from which space-time and the laws of physics in space-time are emergent. Consistent with the assumed primacy of I U , a non-Euclidean (and hence non-classical) metric g p is defined in cosmological state space. Here, p is a large but finite integer (whose inverse may reflect the weakness of gravity). Points that do not lie on I U are necessarily g p -distant from points that do. g p is related to the p-adic metric of number theory. Using number-theoretic properties of spherical triangles, the Clauser-Horne-Shimony-Holt (CHSH) inequality, whose violation would rule out local realism, is shown to be undefined in this framework. Moreover, the CHSH-like inequalities violated experimentally are shown to be g p -distant from the CHSH inequality. This result fails in the singular limit p = ∞ , at which g p is Euclidean and the corresponding model classical. Although Invariant Set Theory is deterministic and locally causal, it is not conspiratorial and does not compromise experimenter free will. The relationship between Invariant Set Theory, Bohmian Theory, The Cellular Automaton Interpretation of Quantum Theory and p-adic Quantum Theory is discussed.
Macroscopic realism (MR) per se specifies that a system which has two macroscopically distinct states available to it (such as a cat being dead or alive) is at all times predetermined to be in one or other of those two states. A minimal assumption of a macroscopic realistic theory therefore is the validity of a hidden variable ${\ensuremath{\lambda}}_{M}$ that predetermines the outcome (whether dead or alive) of a measurement $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ distinguishing the two states. Proposals to test MR generally introduce a second premise to further qualify the meaning of MR. Thus, we consider a model, macroscopic local realism (MLR), where the second premise is that measurements at one location cannot cause an instantaneous macroscopic change $\ensuremath{\delta}$ to the results of measurements made on a second system at another location. To provide a practical test, we define the intermediate concept of $\ensuremath{\delta}$-scopic local realism ($\ensuremath{\delta}$-LR), where $\ensuremath{\delta}\ensuremath{\ne}0$ can be quantified, but need not be macroscopic. By considering the amplification of quantum fluctuations, we show how negation of $\ensuremath{\delta}$-LR is possible using fields violating a continuous variable Bell inequality. A modified Bell-Clauser-Horne-Shimony-Holt inequality is derived that tests $\ensuremath{\delta}$-LR, and a quantitative proposal given for experiments based on polarization entanglement. In the proposal, $\ensuremath{\delta}$ is the magnitude of the quantum noise scaled by an adjustable coherent amplitude $\ensuremath{\alpha}$ that can also be considered part of the measurement apparatus. Thus, $\ensuremath{\delta}$ is large in an absolute sense, but scales inversely with the square root of the system size, which is proportional to ${|\ensuremath{\alpha}|}^{2}$. We discuss how the proposed experiment gives a realization of a type of Schr\"odinger-cat experiment without problems of decoherence.