A general bound for the dimension of quantum behaviours in the prepare-and-measure scenario

TL;DR

This paper establishes a universal lower bound on the quantum system dimension needed for arbitrary behaviors in prepare-and-measure scenarios, applicable even with shared randomness, and uses it to analyze random access codes.

Contribution

It introduces a general matrix-based lower bound on quantum dimension in prepare-and-measure setups, applicable with shared randomness, and constructs dimension witnesses.

Findings

Derived a universal lower bound on quantum dimension

Bound the success probability of random access codes based on dimension

Constructed new dimension witnesses for quantum systems

Abstract

The prepare-and-measure scenario offers the possibility to infer the dimension of an unknown physical system in a device-independent way, i.e. using only raw measurement data with apparatuses regarded as black boxes. We provide here a general lower bound on the dimension necessary to observe arbitrary quantum behaviours in this scenario based on simple matrix analysis. This bound holds even if the preparer and measurer share randomness. This is relevant in scenarios were the parties are free to access this resource or it is not safe to assume that the devices are not correlated. We further use this result to bound the success probability of random access codes in general as a function of the dimension of the quantum systems sent from one party to another and we provide constructions of dimension witnesses.