Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction

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.

Figure 1

The reversible transformations within ERL theory preserve the epistemic constraint. As a result, a transformation that reduces the uncertainty in both position and momentum of a Gaussian distribution, as illustrated, is not allowed within ERL mechanics.

Figure 2

Valid Liouville distributions corresponding to perfect knowledge of (a) position, (b) momentum, and (c) a general quadrature.

Figure 3

A transformation that shuffles uncertainty from one subsystem to another, as illustrated in (a), is allowed within ERL mechanics. However, a transformation as illustrated in (b) that concentrates uncertainty into one subsystem, such that one can have perfect knowledge of the other, is not allowed within the theory.