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arXiv:2209.09590

On The Complete Description Of Entangled Systems Part I: Hidden Variables And The Context Communication Cost of Simulating Quantum Correlations

Karl Svozil

Published Sep 20, 2022• 0 citations
arXivHTMLPDF

Abstract

Some forms of classical simulations of quantum type probabilities and correlations are capable of violating Boole's conditions of possible experience, such as the Clauser-Horne-Shimony-Holt inequality, even beyond the Tsirelson bound. This can be achieved by communicating a single bit encoding the measurement context.
DOI
10.48550/arxiv.2209.09590

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Abstract In this article, we investigate classical simulations of quantum-type probabilities and correlations that defy Boole’s conditions of possible experience, specifically the Clauser-Horne-Shimony-Holt inequality, and even surpass the Tsirelson bound. We demonstrate that such violations can be achieved by encoding a single bit to communicate the measurement context. Our findings highlight the crucial role of context communication in reproducing quantum correlations that are stronger than classical limits, providing insights into the fundamental principles underlying entangled systems.
We present a simple hidden-variable model for the singlet state of a pair of qubits, characterized by two kinds of hierarchically ordered hidden variables. The model is such that, averaging over both types of variables, one reproduces all the quantum mechanical correlations of the singlet state. In contrast, averaging only over the hidden variables of the lower level, one obtains a general formal theoretical scheme exhibiting correlations stronger than the quantum ones, but with faster-than-light communication forbidden. By varying the parameters of the model, it is possible to obtain an arbitrary violation of the Clauser-Horne-Shimony-Holt and Tsirelson bounds. This result is interesting by itself since it shows that a violation of the quantum bound for nonlocal correlations can be implemented in a precise physical manner and not only mathematically and it suggests that resorting to two levels of nonlocal hidden variables might lead to a deeper understanding of the physical principles at the basis of quantum nonlocality. It is useful to mention that a model having features similar to ours has already been considered in the literature. However, from the point of view of its nonlocal aspects, it is physically much less appealing since averaging over the lower-level hidden variables yields correlations that can take only a discrete set of values for any choice of the upper-level variables and not the continuous set of values characterizing our approach. We discuss this problem in the concluding section of our work.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
The validity of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for hidden-variable models of the Bell type and the quantum prediction that said inequality can be violated are two indisputable assertions. However, one can prove wrong the assertion that no hidden-variable model is able to reproduce quantum predictions about the Bell-CHSH test. Here, we reproduce all quantum predictions of Bell-CHSH tests by means of the Bell-Kochen-Specker hidden-variable model for a qubit plus Einstein-Podolsky-Rosen “criterion of physical reality”.
The growing recognition that entanglement is not exclusively a quantum property, and does not even originate with Schrödinger's famous remark about it [Proc. Camb. Phil. Soc. 31, 555 (1935)], prompts examination of its role in marking the quantum-classical boundary. We have done this by subjecting correlations of classical optical fields to new Bell-analysis experiments, and report here values of the Bell parameter greater than B=2.54{\cal B} = 2.54B=2.54. This is many standard deviations outside the limit B=2{\cal B} = 2B=2 established by the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [Phys. Rev. Lett. 23, 880 (1969)], in agreement with our theoretical classical prediction, and not far from the Tsirelson limit B=2.828...{\cal B} = 2.828...B=2.828.... These results cast a new light on the standard quantum-classical boundary description, and suggest a reinterpretation of it.
In this work we aim to analyze the Clauser-Horne-Shimony-Holt CHSH inequality strictly in the context of probability theory. In the course of assembling inequality we have to take care not to produce assumptions a priori, that is, physically or intuitively accepted suppositions. Of course, this does not mean that after these considerations, we put the adequate physical conditions suitable and generally accepted in these contexts. This allows us to clearly visualize the possibility of finding a greater inequality than that of CHSH in which it is included. This inequality does not contradict the CHSH inequality. This result is suported by using a robust computational simulation, showing the possibility for obtaining an inequality for quantum mechanics that is not violated and allows random hidden variables without any conflict with Bell's inequality.