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

摘要

How would the world appear to us if its ontology was that of classical mechanics but every agent faced anrestriction on how much they could come to know about the classical state? We show that in most respects itnwould appear to us as quantum. The statistical theory of classical mechanics, which specifies how probabilityndistributions over phase space evolve under Hamiltonian evolution and under measurements, is typically callednLiouville mechanics, so the theory we explore here is Liouville mechanics with an epistemic restriction. Thenparticular epistemic restrictionwe posit as our foundational postulate specifies two constraints. The first constraintnis a classical analog of Heisenberg’s uncertainty principle; the second-order moments of position and momentumndefined by the phase-space distribution that characterizes an agent’s knowledge are required to satisfy the samenconstraints as are satisfied by the moments of position and momentum observables for a quantum state. Thensecond constraint is that the distribution should have maximal entropy for the given moments. Starting fromnthis postulate, we derive the allowed preparations, measurements, and transformations and demonstrate thatnthey are isomorphic to those allowed in Gaussian quantum mechanics and generate the same experimentalnstatistics. We argue that this reconstruction of Gaussian quantum mechanics constitutes additional evidence innfavor of a research program wherein quantum states are interpreted as states of incomplete knowledge and thatnthe phenomena that do not arise in Gaussian quantum mechanics provide the best clues for how one mightnreconstruct the full quantum theory.