Necessary and sufficient condition for a realistic theory of quantum systems

TL;DR

This paper proves that any classical-like realistic model of quantum systems must involve negative probabilities or a large number of variables, showing limitations of classical descriptions for quantum states.

Contribution

It establishes a necessary and sufficient condition for realistic quantum theories, demonstrating the impossibility of nonnegative classical models with small phase-space dimensions.

Findings

Classical probability distributions for quantum states must be negative or complex.

Realistic models with nonnegative probabilities require exponentially many variables.

A simple model with a large enough phase space can replicate quantum states.

Abstract

We study the possibility to describe pure quantum states and evens with classical probability distributions and conditional probabilities and show that the distributions and/or conditional probabilities have to assume negative values, except for a simple model whose realistic space dimension is not smaller than the Hilbert space dimension of the quantum system. This gives a negative answer to a question proposed by Montina [Phys.Rev.Lett.{\bf 97}, 180401 (2006)] whether or not does there exist a classical theory whose phase-space dimension is much smaller than the Hilbert space dimension for any quantum system. Thus, any realistic theory of quantum mechanics with nonnegative probability distributions and conditional probabilities requires a number of variables grows exponentially with the physical size.