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Superquantum Probabilities: Why No PR-Box?

W. M. Stuckey, Michael Silberstein, Timothy J. McDevitt

Published Jul 26, 2024• 0 citations
Publisher

Abstract

Abstract Chapter 8 introduces Popescu and Rohrlich’s no-signalling PR-box of superquantum joint probabilities violating the CHSH Bell inequality more strongly than QM. We present Mermin’s derivation of the CHSH Bell inequality. Using Tanya and Jeff Bub’s quantum guessing game, we show how the PR-box violates conservation per NPRF whence Bell state entanglement. This is shown to be the reason the PR-box violates the CHSH-Bell inequality more strongly than QM. Therefore, we conclude that the PR-box is akin to a perpetual motion machine.

DOI
10.1093/9780198919698.003.0009

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Abstract Chapter 8 introduces Popescu and Rohrlich’s no-signalling PR-box of superquantum joint probabilities violating the CHSH Bell inequality more strongly than QM. We present Mermin’s derivation of the CHSH Bell inequality. Using Tanya and Jeff Bub’s quantum guessing game, we show how the PR-box violates conservation per NPRF whence Bell state entanglement. This is shown to be the reason the PR-box violates the CHSH-Bell inequality more strongly than QM. Therefore, we conclude that the PR-box is akin to a perpetual motion machine.
Abstract Objectives: To derive assumptions of Bell Inequalities from one single postulate – Absolute Postulate of Universe – as mentioned by the author S.Bagaria. Method: It focuses on the assumptions – locality, reality, measurement independence and counter-factual definiteness - made in proving Bell’s Inequality and derive them from Absolute Postulate of the Universe. Then, its consequences are studied like PR Boxes being real, DI-QKD, quantum information theory, randomness and foundations of quantum mechanics. Findings: From Absolute Postulate of the Universe, it has been showed that there is absolute determinism in world, resulting in zero theoretical uncertainty. Thus, the algebraically maximal correlations (S = 4) are possible. Thus, the PR-boxes can be real. But, Tsirelson Bound can’t be violated in quantum computing as HUP can’t be violated in Quantum Physics. Novelty: The four assumptions of Bell Inequality are derived from Absolute Postulate of the Universe. The correlations in the universe can be maximal yet unverifiable experimentally. EPR paradox is resolved without assuming faster than light transfer of information as the world is shown to be “non-local with no-signalling”. Keywords: Bell inequality, Popescu–Rohrlich box, Tsirelson bound, Quantum nonlocality, No-signalling, CHSH inequality, Quantum violation, absolute determinism
We study the consequences of superquantum nonlocal correlations as represented by the PR-box model of Popescu and Rohrlich, and show that PR boxes can enhance the capacity of noisy interference channels between two senders and two receivers. PR-box correlations violate Bell and CHSH inequalities and are thus stronger---more nonlocal---than quantum mechanics, yet weak enough to respect special relativity in prohibiting faster-than-light communication. Understanding their power will yield insight into the nonlocality of quantum mechanics. We exhibit two proof-of-concept channels: First, we show a channel between two sender-receiver pairs where the senders are not allowed to communicate, for which a shared superquantum bit (a PR box) allows perfect communication. This feat is not achievable with the best classical (senders share no resources) or quantum-entanglement-assisted (senders share entanglement) strategies. Second, we demonstrate a class of channels for which a tunable parameter $\ensuremath{\epsilon}$ achieves a double separation of capacities; for some range of $\ensuremath{\epsilon}$, the superquantum-assisted strategy does better than the entanglement-assisted strategy, which in turn does better than the classical one.
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We quantify the amount of non-locality contained in n noisy versions of the so-called Popescu–Rohrlich boxes ( PRBs ), i.e. bipartite systems violating the CHSH Bell inequality maximally. Following the approach by Elitzur, Popescu and Rohrlich, we measure the amount of non-locality of a system by representing it as a convex combination of a local behaviour, with maximal possible weight, and a non-signalling system. We show that the local part of n systems, each of which approximates a PRB with probability 1 − ε, is of order Θ(ε ⌈ n /2⌉ ) in the isotropic, and equal to (3ε) n in the maximally biased case.