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Transactions on Machine Learning and Artificial Intelligence

Super-Quantum Correlations: How to Interpret the No-Signaling Condition?

Pierre Uzan

Published Jul 7, 2022• 0 citations

Abstract

This article deals with the question of the maximal correlation degree of two intelligent machines that cannot exchange any signals. After reminding the reader of the incorrectness of the mainstream statistical interpretation of the “no-signaling” condition, its informational meaning is explored. It is emphasized that if Pawlowski et al.’s Information Causality Principle correctly expresses (and generalizes) the no-signaling condition, its application is, for now, based on a specific scenario (suggested by van Dam) and a no less specific (and simplified) relationship between mutual information and correlators. A more general informational interpretation of the no-signaling condition from which the Tsirelson bound can be derived is then formulated in terms of correlational independence.

DOI: 10.14738/tmlai.103.12580

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This article deals with the question of the maximal correlation degree of two intelligent machines that cannot exchange any signals. After reminding the reader of the incorrectness of the mainstream statistical interpretation of the “no-signaling” condition, its informational meaning is explored. It is emphasized that if Pawlowski et al.’s Information Causality Principle correctly expresses (and generalizes) the no-signaling condition, its application is, for now, based on a specific scenario (suggested by van Dam) and a no less specific (and simplified) relationship between mutual information and correlators. A more general informational interpretation of the no-signaling condition from which the Tsirelson bound can be derived is then formulated in terms of correlational independence.

This article shows that the mainstream discussion about super-quantum correlations is skewed by an incorrect interpretation of the “no-signaling” condition (NS). Referring to counter-examples, it shows that the usual probabilistic interpretation of (NS), whose link with relativistic causality is doubtful, is too weak to assert the absence of any exchange of information between the parties. A relevant informal interpretation of (NS) is nothing but a particular specification of Pawlowski’s Information Causality principle, which rules out the possibility of correlations stronger than the strongest quantum correlations.

We introduce a generalization of the quantum mutual information between a classical system and a quantum system into the mutual information between a classical system and a system described by general probabilistic theories. We apply this generalized mutual information (GMI) to a derivation of Tsirelson's bound from information causality, and prove that Tsirelson's bound can be derived from the chain rule of the GMI. By using the GMI, we formulate the "no-supersignalling condition" (NSS), that the assistance of correlations does not enhance the capability of classical communication. We prove that NSS is never violated in any no-signalling theory.

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.

In order to analyze an information theoretical derivation of Tsirelson's bound based on information causality, we introduce a generalized mutual information (GMI), defined as the optimal coding rate of a channel with classical inputs and general probabilistic outputs. In the case where the outputs are quantum, the GMI coincides with the quantum mutual information. In general, the GMI does not necessarily satisfy the chain rule. We prove that Tsirelson's bound can be derived by imposing the chain rule on the GMI. We formulate a principle, which we call the no-supersignaling condition , which states that the assistance of nonlocal correlations does not increase the capability of classical communication. We prove that this condition is equivalent to the no-signaling condition. As a result, we show that Tsirelson's bound is implied by the nonpositivity of the quantitative difference between information causality and no-supersignaling.