A quantum prediction as a collection of epistemically restricted classical predictions

TL;DR

This paper demonstrates how quantum predictions for certain systems can be reconstructed by combining classical, epistemically restricted predictions, revealing a nuanced bridge between classical and quantum descriptions.

Contribution

It introduces a method to decompose quantum experiment descriptions into collections of epistemically restricted classical predictions with global constraints.

Findings

Quantum predictions can be reconstructed from classical predictions.

Epistemic restrictions limit classical probability distributions.

A nonclassical rule combines classical parts to recover quantum outcomes.

Abstract

Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we show how, for a certain class of quantum systems, the quantum description of an experiment can be decomposed into classical descriptions that are epistemically restricted, though in a different sense than in Spekkens' work. For each aspect of the experiment -- the preparation, the transformations, and the measurement -- the epistemic restriction limits the form of the probability distribution an imagined classical observer may use. There are also global constraints that the whole collection of classical descriptions must satisfy. Each classical description generates its own prediction regarding the outcome of the experiment. One recovers the quantum prediction via a simple but highly nonclassical rule: the "nonrandom part" of the predicted quantum probabilities is obtained by summing the nonrandom parts of the classically predicted probabilities. By "nonrandom part" we mean the deviation from complete randomness, that is, from what one would expect upon measuring the fully mixed state.