A lower bound on the dimension of a quantum system given measured data

TL;DR

This paper establishes a fundamental lower bound on the dimension of a quantum system's Hilbert space based on measurement data, linking it to quantum random access codes and exploring implications for hidden variable models and Bell inequalities.

Contribution

It introduces a simple lower bound on quantum system dimension from measurement data, connecting it to quantum random access codes and applications in quantum information theory.

Findings

Derived a lower bound on Hilbert space dimension from measurement probabilities

Linked the bound to quantum random access codes and their known limits

Discussed implications for hidden variable models and Bell inequalities

Abstract

We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation rho the experimenter can determine, given enough time, the probability of a given outcome a: p(a|M,rho). How large does the Hilbert space of the quantum system have to be in order to allow us to find density matrices and measurement operators that will reproduce the given probability distribution? In this note, we prove a simple lower bound for the dimension of the Hilbert space. The main insight is to relate this problem to the construction of quantum random access codes, for which interesting bounds on Hilbert space dimension already exist. We discuss several applications of our result to hidden variable, or ontological models, to Bell inequalities and to properties of the smooth min-entropy.