Exponential complexity and ontological theories of quantum mechanics

TL;DR

This paper proves that any Markovian ontological theory of quantum mechanics must use an exponentially large number of variables, highlighting fundamental limitations in classical descriptions of quantum systems.

Contribution

It provides the first proof that the number of variables in Markovian ontological theories must grow exponentially with the system size.

Findings

Number of variables ≥ 2N-2 for a system with Hilbert space dimension N

Exponential growth of variables required in ontological Markovian theories

Implications for quantum Monte Carlo methods

Abstract

Ontological theories of quantum mechanics describe a single system by means of well-defined classical variables and attribute the quantum uncertainties to our ignorance about the underlying reality represented by these variables. We consider the general class of ontological theories describing a quantum system by a set of variables with Markovian (either deterministic or stochastic) evolution. We provide the first proof that the number of continuous variables can not be smaller than 2N-2, N being the Hilbert space dimension. Thus, any ontological Markovian theory of quantum mechanics requires a number of variables which grows exponentially with the physical size. This result is relevant also in the framework of quantum Monte Carlo methods.