A Mathematical Framework For Operational Fine Tunings

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Lorenzo Catani and Matthew Leifer

Institute for Quantum Studies & Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA, 92866, USA

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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.

Featured image: A schematic representation of the no fine tuning condition. The condition demands that an operational equivalence, possibly involving classical processings $f,f’$ on two experiments $E,E’$, implies a consistent ontological equivalence involving corresponding ontic classical processings $h,h’$. The classical processings are represented by the grey boxes. The above representation is meant to be read in terms of the conditional probabilities of the observed variables, i.e. $O,tilde{O},O’,tilde{O}’$ (and ontic variables, i.e. $Lambda,Lambda’,Omega,Omega’$) given the controlled variables, i.e. $C,tilde{C},C’,tilde{C}’$, associated with the experiments.

[embedded content]Superdeterminism and Retrocausality – International Centre for Philosophy, Bonn (Germany), 17-20/05/2022.

Contributed talk at Quantum physics and logic, online due to pandemic, 1-5/06/2020

Seminar at Perimeter Institute, Waterloo (Canada), 13/09/2019.

Popular summary

After about a century since the advent of quantum theory, it is still not clear what is the picture of the world that the theory entails. A promising way to answer this question is to first identify what are the features of the theory that truly resist any classical explanation. So far, the features that are universally considered as truly nonclassical are the ones coming from no-go theorems (Bell, Kochen-Specker, …).
These theorems always work as follows: one assumes a mathematical framework to model reality, termed ontological model framework, defines on this framework a precise notion of classicality, and then proves a contradiction between the statistics of this framework respecting the notion of classicality and the statistics predicted by quantum theory.

The typical lesson that has been taken from these no-go theorems is to conclude that the quantum world is described by an ontological model that violates the classical assumption in question (locality in Bell theorem and noncontextuality in Kochen-Specker theorem). However, this conclusion is problematic, because it forces one to accept that the quantum world involves fine tune properties. The latter are properties that hold at the level of the predicted statistics of quantum theory, but do not hold at the level of the model of reality of the theory (the ontological model). Their appearance at the level of the operational statistics is due to unexplained special choices of the ontological parameters, which is what is meant by a fine tuning. For instance, in the case of a violation of noncontextuality, the statistical equivalences between different procedures (e.g., different decomposition of the completely mixed quantum state of a qubit), arise as a fine tuning of distinct ontological representations. Such fine tunings seem to entail a conspiracy in nature and deny the empiricist roots of science: if two procedures are distinct, why must we experience them, in principle, as equivalent?

We argue that the presence of fine-tuned properties constitutes a serious problem for obtaining an unambiguous interpretation of the nature of the quantum reality and requires an explanation. We see two possibilities to solve the problem of fine tunings in quantum theory. The first is to explain fine tunings as emergent, i.e., provide a physical mechanism that explains their presence (for example, in the case of noncontextuality violation, a mechanism that explains why preparations that are represented as ontologically distinct are operationally equivalent). The second is to develop a new mathematical framework to model reality, different from the standard ontological model framework, that does not suffer of the no-go theorems, i.e., it is absent of fine tunings.

The research program just outlined currently lacks the main basic ingredient: a rigorous mathematical framework for defining and characterizing fine tunings. This is what we do in this work. The idea is that an ontic extension (a more general model of reality than the standard ontological model framework, in that it does not involve causal assumptions) is no fine-tuned with respect to a given property of the physical theory (defined as an operational equivalence in the theory) if such property holds in the ontic extension. Fine tunings capture the common aspect among all the features of quantum theory that are inherently nonclassical according to the no-go theorems. As such, they allow to distill the nonclassicality of quantum theory in one single notion.

Having a precise and mathematically rigorous definition of what captures the nonclassicality of quantum theory is not only crucial for the foundational reasons outlined above, but also to study what is the origin of the quantum computational speed-up. More precisely, with this framework we aim to develop a resource theory to quantify fine tunings and study their role as resources for quantum computational advantages.

► BibTeX data

@article{Catani2023mathematical, doi = {10.22331/q-2023-03-16-948}, url = {https://doi.org/10.22331/q-2023-03-16-948}, title = {A mathematical framework for operational fine tunings}, author = {Catani, Lorenzo and Leifer, Matthew}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {948}, month = mar, year = {2023}<br /> }

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Cited by

[1] Lorenzo Catani, Matthew Leifer, David Schmid, and Robert W. Spekkens, “Why interference phenomena do not capture the essence of quantum theory”, arXiv:2111.13727, (2021).

[2] Lorenzo Catani, Matthew Leifer, Giovanni Scala, David Schmid, and Robert W. Spekkens, “What aspects of the phenomenology of interference witness nonclassicality?”, arXiv:2211.09850, (2022).

[3] Lorenzo Catani, “Relationship between covariance of Wigner functions and transformation noncontextuality”, arXiv:2004.06318, (2020).

[4] Anubhav Chaturvedi and Debashis Saha, “Quantum prescriptions are more ontologically distinct than they are operationally distinguishable”, Quantum 4, 345 (2020).

[5] J. C. Pearl and E. G. Cavalcanti, “Classical causal models cannot faithfully explain Bell nonlocality or Kochen-Specker contextuality in arbitrary scenarios”, arXiv:1909.05434, (2019).

[6] Anubhav Chaturvedi, Marcin Pawłowski, and Debashis Saha, “Quantum description of reality is empirically incomplete”, arXiv:2110.13124, (2021).

[7] Lorenzo Catani, Ricardo Faleiro, Pierre-Emmanuel Emeriau, Shane Mansfield, and Anna Pappa, “Connecting XOR and XOR* games”, arXiv:2210.00397, (2022).

[8] J. C. Pearl and E. G. Cavalcanti, “Classical causal models cannot faithfully explain Bell nonlocality or Kochen-Specker contextuality in arbitrary scenarios”, Quantum 5, 518 (2021).

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