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arXiv:1205.1288 · Physical Review A

Computability limits nonlocal correlations

Tanvirul Islam, Stephanie Wehner

Published Oct 10, 2012• 3 citations
arXivHTMLPDF

Abstract

If the no-signaling principle was the only limit to the strength of nonlocal correlations, we would expect that any form of no-signaling correlation can indeed be realized. That is, there exists a state and measurements that remote parties can implement to obtain any such correlation. Here, we show that in any theory in which some functions cannot be computed, there must be further limits to nonlocal correlations than the no-signaling principle alone. We proceed to argue that even in a theory such as quantum mechanics in which nonlocal correlations are already weaker, the question of computability imposes such limits.

DOI
10.1103/physreva.86.042109

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If the no-signaling principle was the only limit to the strength of nonlocal correlations, we would expect that any form of no-signaling correlation can indeed be realized. That is, there exists a state and measurements that remote parties can implement to obtain any such correlation. Here, we show that in any theory in which some functions cannot be computed, there must be further limits to nonlocal correlations than the no-signaling principle alone. We proceed to argue that even in a theory such as quantum mechanics in which nonlocal correlations are already weaker, the question of computability imposes such limits.
We show that, assuming that quantum mechanics holds locally, the finite speed of information is the principle that limits all possible correlations between distant parties to be quantum mechanical as well. Local quantum mechanics means that a Hilbert space is assigned to each party, and then all local positive-operator-valued measurements are (in principle) available; however, the joint system is not necessarily described by a Hilbert space. In particular, we do not assume the tensor product formalism between the joint systems. Our result shows that if any experiment would give nonlocal correlations beyond quantum mechanics, quantum theory would be invalidated even locally.
Nonlocality is arguably one of the most fundamental and counterintuitive aspects of quantum theory. Nonlocal correlations could, however, be even more nonlocal than quantum theory allows, while still complying with basic physical principles such as no-signaling. So why is quantum mechanics not as nonlocal as it could be? Are there other physical or information-theoretic principles which prohibit this? So far, the proposed answers to this question have been only partially successful, partly because they are lacking genuinely multipartite formulations. In Fritz et al. [Nat. Commun. 4, 2263 (2013)], we introduced the principle of local orthogonality (LO), an intrinsically multipartite principle which is satisfied by quantum mechanics but is violated by nonphysical correlations. Here we further explore the LO principle, presenting additional results and explaining some of its subtleties. In particular, we show that the set of no-signaling boxes satisfying LO is closed under wirings, present a classification of all LO inequalities in certain scenarios, show that all extremal tripartite boxes with two binary measurements per party violate LO, and explain the connection between LO inequalities and unextendible product bases.
It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are nonlocal, in the sense that a Bell inequality is violated. They cannot, however, be used for superluminal signaling. It is also known that it is possible to write down sets of ``superquantum'' correlations that are more nonlocal than is allowed by quantum mechanics, yet are still nonsignaling. Viewed as an information-theoretic resource, superquantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signaling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signaling polytope in the case that two observers each choose from two possible measurements with ddd outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.
The study of quantum correlations is important for fundamental reasons as well as for quantum communication and information processing tasks. On the one hand, it is of tremendous interest to derive the correlations produced by measurements on separated composite quantum systems from within the set of all correlations obeying the no-signaling principle of relativity, by means of information-theoretic principles. On the other hand, an important ongoing research program concerns the formulation of device-independent cryptographic protocols based on quantum nonlocal correlations for the generation of secure keys, and the amplification and expansion of random bits against general no-signaling adversaries. In both these research programs, a fundamental question arises: Can any measurements on quantum states realize the correlations present in pure extremal no-signaling boxes? Here, we answer this question in full generality showing that no nontrivial (not local realistic) extremal boxes of general no-signaling theories can be realized in quantum theory. We then explore some important consequences of this fact.
A central theme in quantum mechanics is nonlocality, which states that a pair of quantum systems which are shown not to be physically interacting may nevertheless be impossible to describe as independent entities. Quantum theory, however, forbids nonlocal correlations beyond a certain limit. Why is nature only so nonlocal and not more? Approaching the question from the direction of statistics and statistical inference, we identify a statistical no-signaling principle which states that no information may pass through a disconnected channel. We show this principle to be equivalent to the Tsirelson bound on nonlocality for the Bell--CHSH inequality.
The idea that nonlocal correlations stronger than quantum correlations between two no-signaling systems could “theoretically” exist is based on an incorrect statistical interpretation of the no-signaling condition. This article shows that any physically realizable no-signaling “box” involving local incompatible observables indeed requires to be described in a noncommutative, quantum-like language of operators, which leads to the derivation of the Tsirelson bound and then contradicts this idea.