A quantum prediction as a collection of epistemically restricted classical predictions

William F. Braasch Jr.1 and William K. Wootters2

1Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
2Department of Physics, Williams College, Williamstown, Massachusetts 01267, USA

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Spekkens has introduced an $\textit{epistemically restricted classical theory}$ 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.

Probabilities in quantum theory are normally computed in a way that no nineteenth-century physicist would ever have dreamed of, with complex numbers playing a central role. In this paper, we show how quantum probabilities can alternatively be computed by first imagining many classical versions of the experiment under consideration and, in each case, computing the probabilities in a very ordinary way. Typically, none of the classical calculations correctly predicts the probabilities we would observe, but the classical predictions can be combined to produce the correct quantum prediction.

There is, however, no avoiding the weirdness of quantum theory. In our formulation, this weirdness manifests itself in two ways: (i) each of our imagined classical observers is subject to a limitation on their knowledge, and (ii) the rule by which the classical predictions are to be combined, though mathematically simple, is different from any way in which we would normally combine probabilities. Specifically, we obtain a quantum probability from the corresponding classical probabilities by summing their deviations from total randomness. This strange rule makes it difficult to tell a single coherent story about what is going on, even though such a story could easily be told within each of our imagined classical worlds.

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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.

[2] William F. Braasch and William K. Wootters, "A Classical Formulation of Quantum Theory?", Entropy 24 1, 137 (2022).

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