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arXiv:2208.13526

Possibilistic approach to network nonlocality

Antoine Restivo, Nicolas Brunner, Denis Rosset

Published Aug 29, 2022• 0 citations
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Abstract

The investigation of Bell nonlocality traditionally relies on joint probabilities of observing certain measurement outcomes. In this work we explore a possibilistic approach, where only patterns of possible outcomes matter, and apply it to Bell nonlocality in networks with independent sources. We present various algorithms for determining whether a given outcome pattern can be achieved via classical resources or via non-signaling resources. Next we illustrate these methods considering the triangle and square networks (with binary outputs and no inputs), identifying patterns that are incompatible with the network structure, as well as patterns that imply nonlocality. In particular, we obtain an example of quantum nonlocality in the square network with binary outcomes. Moreover, we show how to construct certificates for detecting the nonlocality of a certain pattern, in the form of nonlinear Bell-type inequalities involving joint probabilities. Finally, we show that these inequalities remain valid in the case where the sources in the network become partially correlated.

DOI
10.48550/arxiv.2208.13526

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The investigation of Bell nonlocality traditionally relies on joint probabilities of observing certain measurement outcomes. In this work we explore a possibilistic approach, where only patterns of possible outcomes matter, and apply it to Bell nonlocality in networks with independent sources. We present various algorithms for determining whether a given outcome pattern can be achieved via classical resources or via non-signaling resources. Next we illustrate these methods considering the triangle and square networks (with binary outputs and no inputs), identifying patterns that are incompatible with the network structure, as well as patterns that imply nonlocality. In particular, we obtain an example of quantum nonlocality in the square network with binary outcomes. Moreover, we show how to construct certificates for detecting the nonlocality of a certain pattern, in the form of nonlinear Bell-type inequalities involving joint probabilities. Finally, we show that these inequalities remain valid in the case where the sources in the network become partially correlated.
In recent years, the study of Bell nonlocality has been generalized to quantum networks, where multiple independent sources distribute physical systems to distant parties who perform local measurements. In this context, a central open question is to identify the minimal network configuration in which quantum resources produce Bell nonlocal correlations. Here we address this question and show that quantum nonlocality is possible in the triangle network where the parties have no input choices and produce only binary-valued outcomes. To do so, we start by identifying a family of target distributions and proving their nonlocality. Next, we construct an explicit quantum model that reproduces the target distributions to machine precision. For this, we develop an efficient method for parameterizing quantum distributions in networks, inspired by the formalism of higher-order quantum operations. When considering the number of observed variables and their cardinality, this constitutes the smallest scenario possible that supports quantum nonlocality in networks. Moreover, by analyzing the explicit quantum model, we obtain new insights into how nonlocal distributions can be generated in quantum networks.
The possibility of Bell inequality violations in quantum theory had a profound impact on our understanding of the correlations that can be shared by distant parties. Generalizing the concept of Bell nonlocality to networks leads to novel forms of correlations, the characterization of which is, however, challenging. Here, we investigate constraints on correlations in networks under the natural assumptions of no-signaling and independence of the sources. We consider the triangle network with binary outputs, and derive strong constraints on correlations even though the parties receive no input, i.e., each party performs a fixed measurement. We show that some of these constraints are tight, by constructing explicit local models (i.e. where sources distribute classical variables) that can saturate them. However, we also observe that other constraints can apparently not be saturated by local models, which opens the possibility of having nonlocal (but non-signaling) correlations in the triangle network with binary outputs.
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In the above-mentioned reference, the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios has yet to be provided. The present work characterizes correlations arising due to fixed local measurements in a triangle network under a source independence assumption. Precisely speaking, a set of criteria is framed in the form of Bell-type inequalities, each of which is necessarily satisfied by trilocal correlations. Possible quantum violation of at least one criterion from the set is analyzed, which in turn points out the utility of the set of criteria to detect nonlocality (nontrilocality) in quantum triangle networks. Interestingly, measurement on a local product state basis turns out to be sufficient to generate nontrilocal correlations in some quantum networks. Noise tolerance of the detection criteria is discussed followed by a generalization of the framework for demonstrating correlations in any nnn-sided polygon where nnn is finite.
Quantum networks allow in principle for completely novel forms of quantum correlations. In particular, quantum nonlocality can be demonstrated here without the need of having various input settings, but only by considering the joint statistics of fixed local measurement outputs. However, previous examples of this intriguing phenomenon all appear to stem directly from the usual form of quantum nonlocality, namely via the violation of a standard Bell inequality. Here we present novel examples of "quantum nonlocality without inputs," which we believe represent a new form of quantum nonlocality, genuine to networks. Our simplest examples, for the triangle network, involve both entangled states and joint entangled measurements. A generalization to any odd-cycle network is also presented.
Nonlocal correlations are usually understood through the outcomes of alternative measurements (on two or more parts of a system) that cannot altogether actually be carried out in an experiment. Indeed, a joint input-output --- e.g., measurement-setting--outcome --- behavior is nonlocal if and only if the outputs for all possible inputs cannot coexist consistently. It has been argued that this counterfactual view is how Bell's inequalities and their violations are to be seen. I propose an alternative perspective which refrains from setting into relation the results of mutually exclusive measurements, but that is based solely on data actually available. My approach uses algorithmic complexity instead of probability and randomness, and implies that nonlocality has consequences similar to those in the probabilistic view. Our view is conceptually simpler than the traditional reasoning.