# Quasi-Quantization: Classical Statistical Theories with an Epistemic Restriction

## Abstract

A significant part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over physical states that can be prepared. This is termed an “epistemic restriction” because it implies a fundamental limit on the amount of knowledge that any observer can have about the physical state of a classical system. This article provides an overview of *epistricted theories*, that is, theories that start from a classical statistical theory and apply an epistemic restriction. We consider both continuous and discrete degrees of freedom, and show that a particular epistemic restriction called *classical complementarity* provides the beginning of a unification of all known epistricted theories. This restriction appeals to the symplectic structure of the underlying classical theory and consequently can be applied to an arbitrary classical degree of freedom. As such, it can be considered as a kind of *quasi-quantization* scheme; “quasi” because it generally only yields a theory describing a subset of the preparations, transformations and measurements allowed in the full quantum theory for that degree of freedom, and because in some cases, such as for binary variables, it yields a theory that is a distortion of such a subset. Finally, we propose to classify quantum phenomena as weakly or strongly nonclassical by whether or not they can arise in an epistricted theory.

## Notes

### Acknowledgments

I acknowledge Stephen Bartlett and Terry Rudolph for discussions on the quadrature subtheory of quantum mechanics, Jonathan Barrett for suggesting to define the Poisson bracket in the discrete case in terms of finite differences, and Giulio Chiribella, Joel Wallman and Blake Stacey for comments on a draft of this article. Much of the work presented here summarizes unpublished results obtained in collaboration with Olaf Schreiber. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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