Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality

Characterizing quantum correlations in terms of information-theoretic principles is a popular chapter of quantum foundations. Traditionally, the principles adopted for this scope have been expressed in terms of conditional probability distributions, specifying the probability that a black box produces a certain output upon receiving a certain input. This framework is known as "device-independent". Another major chapter of quantum foundations is the information-theoretic characterization of quantum theory, with its sets of states and measurements, and with its allowed dynamics. The different frameworks adopted for this scope are known under the umbrella term "general probabilistic theories". With only a few exceptions, the two programmes on characterizing quantum correlations and characterizing quantum theory have so far proceeded on separate tracks, each one developing its own methods and its own agenda. This paper aims at bridging the gap, by comparing the two frameworks and illustrating how the two programmes can benefit each other.Comment: 61 pages, no figures, published versio

Bell non-locality and Kochen-Specker contextuality: How are they connected?

Bell non-locality and Kochen-Specker (KS) contextuality are logically independent concepts, fuel different protocols with quantum vs classical advantage, and have distinct classical simulation costs. A natural question is what are the relations between these concepts, advantages, and costs. To address this question, it is useful to have a map that captures all the connections between Bell non-locality and KS contextuality in quantum theory. The aim of this work is to introduce such a map. After defining the theory-independent notions of Bell non-locality and KS contextuality for ideal measurements, we show that, in quantum theory, due to Neumark's dilation theorem, every matrix of quantum Bell non-local correlations can be mapped to an identical matrix of KS contextual correlations produced in a scenario with identical relations of compatibility but where measurements are ideal and no space-like separation is required. A more difficult problem is identifying connections in the opposite direction. We show that there are "one-to-one" and partial connections between KS contextual correlations and Bell non-local correlations for some KS contextuality scenarios, but not for all of them. However, there is also a method that transforms any matrix of KS contextual correlations for quantum systems of dimension $d$ into a matrix of Bell non-local correlations between two quantum subsystems each of them of dimension $d$. We collect all these connections in map and list some problems which can benefit from this map.Comment: 13 pages, 2 figure

Almost Quantum Correlations are Inconsistent with Specker's Principle

Ernst Specker considered a particular feature of quantum theory to be especially fundamental, namely that pairwise joint measurability of sharp measurements implies their global joint measurability (https://vimeo.com/52923835). To date, Specker's principle seemed incapable of singling out quantum theory from the space of all general probabilistic theories. In particular, its well-known consequence for experimental statistics, the principle of consistent exclusivity, does not rule out the set of correlations known as almost quantum, which is strictly larger than the set of quantum correlations. Here we show that, contrary to the popular belief, Specker's principle cannot be satisfied in any theory that yields almost quantum correlations.Comment: 17 pages + appendix. 5 colour figures. Comments welcom

Quantum correlations from simple assumptions

We address the problem of deriving the set of quantum correlations for every Bell and Kochen-Specker (KS) contextuality scenario from simple assumptions. We show that the correlations that are possible according to quantum theory are equal to those possible under the assumptions that there is a nonempty set of correlations for every KS scenario and a statistically independent realization of any two KS experiments. The proof uses tools of the graph-theoretic approach to correlations and deals with Bell nonlocality and KS contextuality in a unified way.Comment: 15 pages, 7 figure

Necessary and sufficient condition for contextuality from incompatibility

Measurement incompatibility is the most basic resource that distinguishes quantum from classical physics. Contextuality is the critical resource behind the power of some models of quantum computation and is also a necessary ingredient for many applications in quantum information. A fundamental problem is thus identifying when incompatibility produces contextuality. Here, we show that, given a structure of incompatibility characterized by a graph in which nonadjacent vertices represent incompatible ideal measurements, the necessary and sufficient condition for the existence of a quantum realization producing contextuality is that this graph contains induced cycles of size larger than three.Comment: 7 pages, 1 figur

Quantum Apices: Identifying Limits of Entanglement, Nonlocality, & Contextuality

This work develops analytic methods to quantitatively demarcate quantum reality from its subset of classical phenomenon, as well as from the superset of general probabilistic theories. Regarding quantum nonlocality, we discuss how to determine the quantum limit of Bell-type linear inequalities. In contrast to semidefinite programming approaches, our method allows for the consideration of inequalities with abstract weights, by means of leveraging the Hermiticity of quantum states. Recognizing that classical correlations correspond to measurements made on separable states, we also introduce a practical method for obtaining sufficient separability criteria. We specifically vet the candidacy of driven and undriven superradiance as schema for entanglement generation. We conclude by reviewing current approaches to quantum contextuality, emphasizing the operational distinction between nonlocal and contextual quantum statistics. We utilize our abstractly-weighted linear quantum bounds to explicitly demonstrate a set of conditional probability distributions which are simultaneously compatible with quantum contextuality while being incompatible with quantum nonlocality. It is noted that this novel statistical regime implies an experimentally-testable target for the Consistent Histories theory of quantum gravity.Comment: Doctoral Thesis for the University of Connecticu

General Bayesian theories and the emergence of the exclusivity principle

We construct a general Bayesian framework that can be used to organize beliefs and to update them when new information becomes available. The framework includes classical, quantum, and other alternative models of Bayesian reasoning that may arise in future physical theories. It is only based on the rule of conditional probabilities and the requirement that the agent's beliefs are consistent across time. From these requirements, we show that ideal experiments within every Bayesian theory must satisfy the exclusivity principle, which is a key to explain quantum correlations. As a consequence, the limits to the strength of correlations set by the exclusivity principle can be interpreted as the ultimate limits set by Bayesian consistency. It is an open question whether Bayesian reasoning alone is sufficient to recover all of quantum theory.Comment: 4+ 4 pages, no figure