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arXiv:1203.3093 · Physical Review A

Arbitrary violation of the Tsirelson bound

G. C. Ghirardi, Raffaele Romano

Published Aug 23, 2012• 2 citations

Abstract

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.

DOI: 10.1103/physreva.86.022116

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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.

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.

Nonlocality is one of the main characteristic features of quantum systems involving more than one spatially separated subsystem. It is manifested theoretically as well as experimentally through violation of some local realistic inequality. On the other hand, classical behavior of all physical phenomena in the macroscopic limit gives a general intuition that any physical theory for describing microscopic phenomena should resemble classical physics in the macroscopic regime, the so-called macrorealism. In the 2-2-2 scenario (two parties, with each performing two measurements and each measurement having two outcomes), contemplating all the no-signaling correlations, we characterize which of them would exhibit classical (local realistic) behavior in the macroscopic limit. Interestingly, we find correlations which at the single-copy level violate the Bell-Clauser-Horne-Shimony-Holt inequality by an amount less than the optimal quantum violation (i.e., Cirel'son bound

2\sqrt{2}

), but in the macroscopic limit gives rise to a value which is higher than

2\sqrt{2}

. Such correlations are therefore not considered physical. Our study thus provides a sufficient criterion to identify some of unphysical correlations.

Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value

2\sqrt{2}

, known as Tsirelson's bound. In this paper, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables

{X}_{1},\dots{},{X}_{n}

and

{Y}_{1},\dots{},{Y}_{n}

with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$ we have $\ensuremath{\mid}⟨{X}_{1}{Y}_{1}⟩+⟨{X}_{2}{Y}_{1}⟩+⟨{X}_{2}{Y}_{2}⟩+⟨{X}_{3}{Y}_{2}⟩+\ensuremath{\cdots}+⟨{X}_{n}{Y}_{n}⟩\ensuremath{-}⟨{X}_{1}{Y}_{n}⟩\ensuremath{\mid}\ensuremath{\leqslant}2n\phantom{\rule{0.2em}{0ex}}\mathrm{cos}[\ensuremath{\pi}∕(2n)]$. It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.

Local hidden-variable model of singlet-state correlations discussed in M. Czachor, Arithmetic loophole in Bell's Theorem: Overlooked threat to entangled-state quantum cryptography, Acta Phys. Polon. A 139, 70-83 (2021), is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities is shown to be a `confusion of languages' problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.