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Measurement theory based on the truth values provides the maximum violation of the Bell-Mermin inequality

Koji Nagata, Tadao Nakamura

Published Apr 1, 2016• 0 citations
OpenAlex

Abstract

We investigate the violation factor of the Bell-Mermin inequality. Until now, we use an assumption that the results of measurement are $\pm 1$. In this case, the maximum violation factor is $2^{(n-1)/2}$. The quantum predictions by $n$-partite Greenberger-Horne-Zeilinger (GHZ) state violate the Bell-Mermin inequality by an amount that grows exponentially with $n$. Recently, a new measurement theory based on the truth values is proposed. The values of measurement outcome are either $+1$ or 0. Here we use the new measurement theory. We consider multipartite GHZ state. It turns out that the Bell-Mermin inequality is violated by the amount of $2^{(n-1)/2}$. The measurement theory based on the truth values provides the maximum violation of the Bell-Mermin inequality.

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We investigate the violation factor of the Bell-Mermin inequality. Until now, we use an assumption that the results of measurement are ±1\pm 1±1. In this case, the maximum violation factor is 2(n−1)/22^{(n-1)/2}2(n−1)/2. The quantum predictions by nnn-partite Greenberger-Horne-Zeilinger (GHZ) state violate the Bell-Mermin inequality by an amount that grows exponentially with nnn. Recently, a new measurement theory based on the truth values is proposed. The values of measurement outcome are either +1+1+1 or 0. Here we use the new measurement theory. We consider multipartite GHZ state. It turns out that the Bell-Mermin inequality is violated by the amount of 2(n−1)/22^{(n-1)/2}2(n−1)/2. The measurement theory based on the truth values provides the maximum violation of the Bell-Mermin inequality.
We investigate the violation factor of the original Bell-Mermin inequality. Until now, we have used an assumption that the results of measurement are . In this case, the maximum violation factor is as follows: and . The quantum predictions by n-partite Greenberger-Horne-Zeilinger state violate the Bell-Mermin inequality by an amount that grows exponentially with n. Recently, a new measurement theory is proposed [K. Nagata and T. Nakamura, International Journal of Theoretical Physics, 49, 162 (2010)]. The values of measurement outcome are . Here we use the new measurement theory. We consider a multipartite GHZ state. We use the original Bell-Mermin inequality. It turns out that the original Bell-Mermin inequality is satisfied irrespective of the number of particles. In this case, the maximum violation factor is as follows: and . Thus the original Bell-Mermin inequality is satisfied by the new measurement theory. We propose the following conjecture: All the two-orthogonal-settings experimental correlation functions admit local realistic theories irrespective of a state if we use the new measurement theory.
Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number mmm provides an additional certification dimension complementary to the system size nnn. We show that, for the powers-of-two setting choices considered here, increasing mmm leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing mmm strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.
Introduction Since its inception, fifty years ago, Bell's theorem has had a long history not only of experimental tests but also of theoretical developments. Studying pairs of correlated quantummechanical particles separated in space, in a composite “entangled” state, Bell [1] showed that the joint ascription of hidden variables and locality to the system led to an inequality that is violated by the predictions of quantum mechanics. Fifteen years later, experiments confirmed the predictions of quantum mechanics, ruling out a large class of local realist theories. One of the most meaningful theoretical developments that followed Bell's work was the Greenberger–Horne–Zeilinger (GHZ) theorem, also known as Bell's theorem without inequalities. In 1989, the American physicists Daniel Greenberger and Michael Horne, who had been working on Bell's theorem since the late 1960s, together with the Austrian physicist Anton Zeilinger, introduced a novelty into the testing of entanglement, extending Bell's theorem in a different and interesting direction. According to Franck Laloë [2], For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism. The trio analyzed the Einstein, Podolsky, and Rosen 1935 argument once again and were able to write what are now called Greenberger–Horne–Zeilinger (GHZ) entangled states, involving three or four correlated spin-1/2 particles, leading to conflicts between local realist theories and quantum mechanics. However, unlike Bell's theorem, the conflict now was not of a statistical nature, as was the case with Bell's theorem, insofar as measurements on a single GHZ state could lead to conflicting predictions with local realistic models [3–5]. In this paper we present the history of the creation of this theorem and analyze its scope.
The existence of Greenberger-Horne-Zeilinger (GHZ) contradictions in many-qutrit systems was a long-standing theoretical question until its (affirmative) resolution in 2013. To enable experimental tests, we derive Mermin inequalities from concurrent observable sets identified in those proofs. These employ a weighted sum of observables, called M\mathcal{M}M, in which every term has the chosen GHZ state as an eigenstate with eigenvalue unity. The quantum prediction for M\mathcal{M}M is then just the number of concurrent observables, and this grows asymptotically as 2N/3{2}^{N}/32N/3 as the number of qutrits $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The maximum classical value falls short for every $N\ensuremath{\ge}3$, so that the quantum to classical ratio (starting at 1.5 when N=3N=3N=3) diverges exponentially ($\ensuremath{\sim}1.{064}^{N}$) as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, where the system is in a Schr\"odinger-cat-like superposition of three macroscopically distinct states.