Testing microscopic discretization

TL;DR

This paper develops a mathematical framework to analyze the spectra of microscopic variables from coarse-grained measurements, revealing fundamental limits on their quantum and classical descriptions.

Contribution

It introduces algorithms to determine which Gaussian distributions can be approximated by sums of finitely-valued microscopic variables and explores unbounded and infinite scenarios.

Findings

Bipartite Gaussian states of light cannot be viewed as independent d-dimensional pairs.

Classical models of certain optical experiments require variables with infinite spectra.

Provides a new theoretical approach linking microscopic discretization to macroscopic fluctuations.

Abstract

What can we say about the spectra of a collection of microscopic variables when only their coarse-grained sums are experimentally accessible? In this paper, using the tools and methodology from the study of quantum nonlocality, we develop a mathematical theory of the macroscopic fluctuations generated by ensembles of independent microscopic discrete systems. We provide algorithms to decide which multivariate gaussian distributions can be approximated by sums of finitely-valued random vectors. We study non-trivial cases where the microscopic variables have an unbounded range, as well as asymptotic scenarios with infinitely many macroscopic variables. From a foundational point of view, our results imply that bipartite gaussian states of light cannot be understood as beams of independent d-dimensional particle pairs. It is also shown that the classical description of certain macroscopic optical experiments, as opposed to the quantum one, requires variables with infinite cardinality spectra.