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arXiv:2107.02728 · Quantum

A quantum prediction as a collection of epistemically restricted classical predictions

William F. Braasch Jr., William K. Wootters

Published Feb 21, 2022• 3 citations
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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.

DOI
10.22331/q-2022-02-21-659

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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.
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 start with a theory resembling Spekkens's model: we imagine a classical observer describing an experiment in terms of discrete phase space, and for each aspect of the experiment -- the state preparation, the transformations, and the measurement -- there is an epistemic restriction limiting the form of the probability distribution the classical observer may use. We show that for a certain class of systems, one can obtain the correct quantum mechanical prediction for the experiment by combining the (typically weaker) predictions of all the conceivable classical observers. The rule for effecting this combination is highly nonclassical: the ``nonrandom part of the quantum prediction is obtained by summing the ``nonrandom parts of the classical predictions.
We explore a particular way of reformulating quantum theory in classical terms, starting with phase space rather than Hilbert space, and with actual probability distributions rather than quasiprobabilities. The classical picture we start with is epistemically restricted, in the spirit of a model introduced by Spekkens. We obtain quantum theory only by combining a collection of restricted classical pictures. Our main challenge in this paper is to find a simple way of characterizing the allowed sets of classical pictures. We present one promising approach to this problem and show how it works out for the case of a single qubit.
How would the world appear to us if its ontology was that of classical mechanics but every agent faced a restriction on how much they could come to know about the classical state? We show that in most respects it would appear to us as quantum. The statistical theory of classical mechanics, which specifies how probability distributions over phase space evolve under Hamiltonian evolution and under measurements, is typically called Liouville mechanics, so the theory we explore here is Liouville mechanics with an epistemic restriction. The particular epistemic restriction we posit as our foundational postulate specifies two constraints. The first constraint is a classical analog of Heisenberg's uncertainty principle; the second-order moments of position and momentum defined by the phase-space distribution that characterizes an agent's knowledge are required to satisfy the same constraints as are satisfied by the moments of position and momentum observables for a quantum state. The second constraint is that the distribution should have maximal entropy for the given moments. Starting from this postulate, we derive the allowed preparations, measurements, and transformations and demonstrate that they are isomorphic to those allowed in Gaussian quantum mechanics and generate the same experimental statistics. We argue that this reconstruction of Gaussian quantum mechanics constitutes additional evidence in favor of a research program wherein quantum states are interpreted as states of incomplete knowledge and that the phenomena that do not arise in Gaussian quantum mechanics provide the best clues for how one might reconstruct the full quantum theory.
Abstract Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an “ontic extension” defines a nonseparable (global) random variable that generates physical correlations, and an “epistemic restriction” constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck’s constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.
When does a decohering quantum system become classical? Generalised noncontextuality gives one criterion, for scenarios rich enough in preparations: a system is classical once its statistics admit an underlying model in which operationally indistinguishable procedures are physically identical. Where it crosses depends not on the state and channel alone but on which preparations, transformations and measurements the experiment offers, and if every measurement is allowed there is no crossing. For any pairwise unbiased bases in any dimension, given preparations holding their eigenstates and surrounding the maximally mixed state, the question reduces to a finite linear program; for a qudit of odd prime dimension ddd restricted to the stabiliser subtheory, with one depolarised magic state, we solve it in closed form. With preparations and measurements alone the qutrit strange state is classical at zero noise; with the Clifford gates added, each channel represented once, it crosses at 3/53/53/5; and with a random gate represented, as Spekkens' treatment asks, by the matching mixture of stochastic maps, at 3/43/43/4. A Clifford orbit of assignments supports such a representation only if it has d2d^2d2 members, leaving, in every odd prime dimension, only Gross's discrete Wigner function: the crossing sits where it stops being negative.
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an "ontic extension" defines a nonseparable (global) random variable that generates physical correlations, and an "epistemic restriction" constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck's constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.