Assessing dimensions from evolution

TL;DR

This paper introduces a method using classical signal processing to determine the dimension of a quantum system and the environment's memory size from observable data, highlighting differences from classical processes.

Contribution

It provides a model-independent approach to assess quantum system dimensions and environment effects, contrasting quantum and classical process restrictions.

Findings

Quantum system dimension can be inferred from observable dynamics.

Quantum and classical processes differ significantly in their dimensionality requirements.

A Hilbert space of size D+2 suffices for any D-dimensional linear quantum evolution.

Abstract

Using tools from classical signal processing, we show how to determine the dimensionality of a quantum system as well as the effective size of the environment's memory from observable dynamics in a model-independent way. We discuss the dependence on the number of conserved quantities, the relation to ergodicity and prove a converse showing that a Hilbert space of dimension D+2 is sufficient to describe every bounded sequence of measurements originating from any D-dimensional linear equations of motion. This is in sharp contrast to classical stochastic processes which are subject to more severe restrictions: a simple spectral analysis shows that the gap between the required dimensionality of a quantum and a classical description of an observed evolution can be arbitrary large.