Hidden variables simulating quantum contextuality increasingly violate the holevo bound

Abstract

In this paper we approach some questions about quantum contextuality with tools from formal logic. In particular, we consider an experiment associated with the Peres-Mermin square. The language of all possible sequences of outcomes of the experiment is classified in the Chomsky hierarchy and seen to be a regular language. We introduce a very abstract model of machine that simulates nature in a particular sense. A lower-bound on the number of memory states of such machines is proved if they were to simulate the experiment that corresponds to the Peres-Mermin square. Moreover, the proof of this lower bound is seen to scale to a certain generalization of the Peres-Mermin square. For this scaled experiment it is seen that the Holevo bound is violated and that the degree of violation increases uniformly. © 2011 Springer-Verlag Berlin Heidelberg.