A quantum prediction as a collection of epistemically restricted classical predictions

William F. Braasch Jr.; William K. Wootters

doi:10.22331/q-2022-02-21-659

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

Published:	2022-02-21, volume 6, page 659
Eprint:	arXiv:2107.02728v5
Doi:	https://doi.org/10.22331/q-2022-02-21-659
Citation:	Quantum 6, 659 (2022).

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Abstract

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.

Featured image: A qubit undergoes a unitary transformation followed by a test for the pure state $|0\rangle$. We start at the upper left with the Wigner function of the initial state (wheat colored), which includes a negative probability. This distribution is split into three actual probability distributions (orange), one for each striation of the $2 \times 2$ phase space. The unitary transformation of each of these distributions is then split into three stochastic transformations, one for each of the three “legal” symplectic matrices. This process leads to new phase-space distributions (red). (Not shown are the uniform distributions.) Each classical probability of the state $|0\rangle$ (blue) is the sum of the left-hand column of phase space. We then sum the “nonrandom parts” of the classical probabilities to get the nonrandom part of the quantum probability ¼.

Popular summary

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.

► BibTeX data

@article{BraaschJr2022quantumpredictionas,
  doi = {10.22331/q-2022-02-21-659},
  url = {https://doi.org/10.22331/q-2022-02-21-659},
  title = {A quantum prediction as a collection of epistemically restricted classical predictions},
  author = {Braasch Jr., William F. and Wootters, William K.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {659},
  month = feb,
  year = {2022}
}

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

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