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arXiv:2003.10050 · Quantum

A mathematical framework for operational fine tunings

Lorenzo Catani, Matthew Leifer

Published Mar 16, 2023• 15 citations
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Abstract

In the framework of ontological models, the inherently nonclassical features of quantum theory always seem to involve properties that are fine tuned, i.e. properties that hold at the operational level but break at the ontological level. Their appearance at the operational level is due to unexplained special choices of the ontological parameters, which is what we mean by a fine tuning. Famous examples of such features are contextuality and nonlocality. In this article, we develop a theory-independent mathematical framework for characterizing operational fine tunings. These are distinct from causal fine tunings – already introduced by Wood and Spekkens in [NJP,17 033002(2015)] – as the definition of an operational fine tuning does not involve any assumptions about the underlying causal structure. We show how known examples of operational fine tunings, such as Spekkens' generalized contextuality, violation of parameter independence in Bell experiment, and ontological time asymmetry, fit into our framework. We discuss the possibility of finding new fine tunings and we use the framework to shed new light on the relation between nonlocality and generalized contextuality. Although nonlocality has often been argued to be a form of contextuality, this is only true when nonlocality consists of a violation of parameter independence. We formulate our framework also in the language of category theory using the concept of functors.

DOI
10.22331/q-2023-03-16-948

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In the framework of ontological models, the inherently nonclassical features of quantum theory always seem to involve properties that are fine tuned, i.e. properties that hold at the operational level but break at the ontological level. Their appearance at the operational level is due to unexplained special choices of the ontological parameters, which is what we mean by a fine tuning. Famous examples of such features are contextuality and nonlocality. In this article, we develop a theory-independent mathematical framework for characterizing operational fine tunings. These are distinct from causal fine tunings – already introduced by Wood and Spekkens in [NJP,17 033002(2015)] – as the definition of an operational fine tuning does not involve any assumptions about the underlying causal structure. We show how known examples of operational fine tunings, such as Spekkens' generalized contextuality, violation of parameter independence in Bell experiment, and ontological time asymmetry, fit into our framework. We discuss the possibility of finding new fine tunings and we use the framework to shed new light on the relation between nonlocality and generalized contextuality. Although nonlocality has often been argued to be a form of contextuality, this is only true when nonlocality consists of a violation of parameter independence. We formulate our framework also in the language of category theory using the concept of functors.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextuality, on the other hand, is on a more controversial foundation. In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality. First, I show that Bell inequalities can be derived solely from the assumptions of no-signalling and no-fine-tuning of the causal model. This removes two extra assumptions from a recent result from Wood and Spekkens, and remarkably, does not require any assumption related to independence of measurement settings -- unlike all other derivations of Bell inequalities. I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning. Thus the quantum violation of classical causality goes beyond the case of space-like separated systems, and manifests already in scenarios involving single systems.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and they are key resources for quantum advantage in information-processing tasks.Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure.Contextuality, on the other hand, is on a more controversial foundation.In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality.First, I show that Bell inequalities can be derived solely from the assumptions of no signaling and no fine-tuning of the causal model.This removes two extra assumptions from a recent result from Wood and Spekkens and, remarkably, does not require any assumption related to independence of measurement settings-unlike all other derivations of Bell inequalities.I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning.Thus, the quantum violation of classical causality goes beyond the case of spacelike-separated systems and already manifests in scenarios involving single systems.
A key aim of quantum foundations is to characterize the sense in which nature goes beyond classical physics. Understanding nonclassicality is one of our best avenues towards finding a satisfactory interpretation of quantum theory. By determining which classical principles cannot be satisfied in any empirically adequate physical theory, we begin to see which principles can be preserved, which in turn gives us insight into the ontology of the world. These insights then guide us in determining which questions to ask and which experiments to perform next. Furthermore, it is these nonclassical aspects of nature that give rise to new technologies such as the speed-ups of quantum computation or the security of quantum key distribution. \n \nThe gold standard for establishing that a phenomenon is truly nonclassical is to prove that it violates the principle of local causality or the principle of noncontextuality. Much of this thesis reports on my research relating to these two principles. This research primarily involves (i) finding new justifications for our notions of nonclassicality; (ii) refining their fundamental definitions; (iii) quantifying and characterizing their various manifestations; and (iv) finding applications where nonclassical phenomena act as resources for information processing. \n \nUltimately, all of this work on nonclassicality is woven together into a novel framework for physical theories introduced by myself, John Selby, and Rob Spekkens. Its main advantage over preexisting frameworks is that it maintains a clear distinction between which elements of a given physical theory directly describe causal processes, and which refer only to one’s inferences about causal processes. This clarifies a number of confusions in the literature which arose precisely because previous frameworks scrambled causal and inferential concepts. Furthermore, local causality and noncontextuality emerge in this framework as the assumptions that the causal and inferential structures (respectively) that are operationally observed must be respected in the underlying ontology. This work constitutes a first step in developing a new interpretation of quantum theory—the first interpretation designed to satisfy the spirit of both local causality and noncontextuality.
Quantum theory departs from classical physics in its treatment of correlations, most prominently through the phenomena of contextuality and nonlocality. Once regarded primarily as foundational curiosities, these effects are now understood as key operational resources for quantum computation, communication, and simulation. Although traditionally investigated in distinct settings, recent theoretical and experimental advances have revealed deep conceptual, mathematical, and operational connections between them. This review presents a unified perspective on these developments based on sheaf-theoretic and graph-theoretic frameworks, which provide theory-independent characterizations of statistical correlations. These approaches clarify the structural relationship between contextuality and nonlocality, facilitate the formulation of experimentally testable inequalities, and guide implementations in realistic physical platforms, with particular emphasis on photonic systems. By bridging abstract theoretical structures and concrete experimental realizations, this review sheds light on the nonclassical foundations of quantum correlations and their emerging role in quantum technologies.
An operational concept of locality whose quantum violation is indicated independently of any other assumption(s) seems to be lacking in the quantum foundations literature so far. Bell's theorem only shows that quantum correlations violate the conjunction of the ontological assumptions of localism and determinism. Taking a cue from computational complexity theory, here we define such a concept of locality in terms of a class of operational decision problems, and propose that Einstein-Podolsky-Rosen (EPR) steering in its basic asymmetric formulation gives a specific realization of this class. Various nonclassical (convex) operational theories, including quantum mechanics, are shown to be nonlocal according to this operational criterion. We discuss several ramifications arising from this result. It indicates a basic difference between quantum no-signaling and relativistic no-signaling, and suggests an information theoretic derivation of the former from other basic principles in the convex framework. It elucidates the connection between two different bounds on nonlocality, due to Oppenheim and Wehner (2010) and Banik et al. (2015), thereby highlighting the interplay and contrast between uncertainty and measurement incompatibility. Our result provides an argument supporting the realist interpretation of the quantum state, and helps clarify why Spekkens' toy model (2007) features steering but not Bell nonlocality.