When presenting Bell’s theorem  to audiences seeing it for the first time, be they students or (surprisingly) physicists from areas with little regard to the quantum foundations, I often anticipate a reaction of shock. This is the least I would expect from those suddenly learning that seemingly metaphysical concepts like realism or even “free-will” can, to some extent, be tested in the laboratory  . Often, however, I’m the one shocked by the attitude of those listening, who misinterpret the elementary mathematics required to prove Bell’s theorem as a reflection of its unimportance. When viewed from a new perspective, however, apparent simplicity can give rise to a rich mathematical structure and a plethora of interesting novel phenomena and potential applications. That is precisely what the paper “Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes” by Wolfe et. al. does for Bell’s theorem.
Traditionally, Bell’s theorem has been understood as the need to give up at least one of our two most ingrained concepts about the world: Realism, stating that the properties of a physical system are well-defined independently of our act of measuring them. Or locality, the idea that two distant (space-like separated) events should not have any direct causal influence one over the other. Quantum correlations obtained by measurements of distant shares of an entangled state can be incompatible with the conjunction of those notions, something that can be witnessed experimentally by the violation of a Bell inequality. In fact, those violations have synonymous with Bell’s theorem, the phenomenon generally known as Bell non-locality (a nomenclature that the authors defy in favour of a more neutral choice of non-classicality). Irrespectively of personal choices of which assumption to abandon, realism or locality, the guidance provided by Bell inequalities has proven valuable over the years both in foundational and applied sides. The non-classicality exposed by Bell’s theorem has become a guiding principle for efforts to recover quantum theory from information-theoretical principles , which one might hope will provide a more palatable explanation as compared to the standard textbook postulates. In practice, applications ranging from randomness certification  to self-testing (the ability to identify quantum states and measurements from observed correlations alone)  and distributed computing  have been firmly founded on that which is now known as the device-independent framework, where tasks are accomplished without the need of a precise characterization of the physical devices.
In hindsight, it is clear that Bell’s theorem is a statement about the incompatibility of quantum theory with classical notions of causality. Ironically, the nascent mathematical theory of causality  was only first developed (mostly by the computer science and artificial intelligence community) 30 years after Bell’s seminal result. From the causal perspective, Bell’s theorem can be seen as a particular case of a causal inference problem  , a realization that has sparked a number of generalizations to causal scenarios of growing size and complexity. Conceptually, this new len’s on Bell’s theorem replaces the often hazy and much debated concepts of local causality and realism with mathematically well defined concepts given by Reichenbach’s principle (correlations have to be explained causally) and the principle of no fine-tuning (observed statistical independences are a reflection of the underlying causal structure). Critically, these causal principles can also hold quantum mechanically upon generalizing  our idea of what a common cause is.
Perspectives and debates aside, it is clear that to better understand non-locality/non-classicality we must find good ways to quantify it. To this end, a resource theory is required. In the operational approach (motivated by the device-independent paradigm) what matters is the classical data generated in a Bell experiment: the records of which measurements where chosen to perform and the corresponding outcomes of those measurements. At first, one might be tempted to consider free operations as those that do not generate Bell non-classicality   , an approach that can be problematic. For example, in entanglement theory, it is clear that separable maps cannot create entanglement. Separable operations, however, are a strict superset of a more intuitive and natural class of free operations given by local operations and classical communication (LOCCs).
In turn, the starting point of Wolfe et. al.’s paper is to derive their resource theory from first principles, based on physical/causal justifications and crucially by interpreting Bell’s theorem as a result about the nonclassicality of the common cause. The central objects in this resource theory are boxes taking classical variables as inputs and producing classical variables as outputs. In particular, Wolfe et. al. consider common cause boxes, namely those which are powered by a source that can generate correlations (classical, quantum or even post-quantum) between distant and non-communicating parties. Further, the operations allowed by the theory are what the authors call “clampings,” any process taking boxes to boxes and respecting the common-cause structure, i.e. processes which exclude communication between the distant wings. When applied to the standard Bell scenario, their framework singles out a unique set of free operations, precisely the set of local operations and shared randomness (LOSR) previously considered in the literature  . However, the authors’ framework is foundationally much more general, such that, unlike previous resource theories, it can readily be applied to analyze virtually any causal structure, including those involving communication between the parties or several independent common-cause boxes. Moreover, the authors manage to characterize the set of LOSR in terms of the convex hull of deterministic local operations. One can think of this result as a generalization of Fine’s theorem , which states that classical correlations admit description in terms of deterministic strategies. Compared with previous formulations, it offers a much more efficient and computationally automatized framework for analyzing the set of LOSR operations
In fact, one stand out application of their new framework is in regard to the interconvertibility of resources and their ordering (for which they focus on single-copy conversion under the free operations). Given a resource theory one can define quantifiers (monotones) for the non-classicality as exactly those measures that cannot increase under the actions of free-operations. As such, can we say that a given correlation is more non-classical than another? Does even make sense to try to compare them? Wolfe et. al.’s formulation casts the problem of single-copy resource convertibility as a linear program, giving necessary and sufficient conditions applicable to Bell scenarios with any number of parties, inputs and outputs.
In applications such as communication complexity  or randomness certification , typically the more a given Bell inequality is violated, the more resourceful the correlation is. In particular, in the CHSH scenario , the simplest Bell scenario with 2 distant parties measuring two possible dichotomic observables each, the famous PR-box  which maximally violates the CHSH inequality would be on top of all other correlations. However, Wolfe and co-authors show that facet-defining Bell inequalities are far from sufficient to characterize resourcefullness of correlations in a Bell scenario, even in the simple CHSH case. They identify a pair of monotones which occassionally give conflicting orderings, meaning that some Bell correlations are evidently incomparable. As a matter of fact, they show that the quantum set contains infinitely many distinct incomparable correlations. They find that at least 8 independent measures of non-classicality are needed to fully specify the ordering of correlations already for the CHSH scenario. (For comparison, the unique symmetry class of facet-defining Bell inequalities in this scenario constitutes only one such measure.)
Quoting Judea Pearl, the father of the classical theory of causality, whom in his “The Book of Why”  says “You cannot answer a question that you cannot ask, and you cannot ask a question that you have no words for.” I see the resource theory developed in this paper as precisely the new language/dictionary that was needed. It will allow us to wonder about new puzzles around Bell’s theorem and its generalizations, and hopefully lead to a deeper understanding of the role of causality in quantum theory.
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