Witnessing irreducible dimension

TL;DR

This paper introduces the concept of irreducible dimension in quantum systems, providing a way to certify the true information processing capacity of high-dimensional quantum behaviors beyond trivial dimension bounds.

Contribution

It defines irreducible dimension and presents a witness for certifying irreducible dimension four, advancing device-independent quantum dimension certification.

Findings

Introduces the notion of irreducible dimension in quantum behaviors

Provides a witness for certifying irreducible dimension four

Highlights the difference between trivial dimension bounds and true information capacity

Abstract

The Hilbert space dimension of a quantum system is the most basic quantifier of its information content. Lower bounds on the dimension can be certified in a device-independent way, based only on observed statistics. We highlight that some such "dimension witnesses" capture only the presence of systems of some dimension, which in a sense is trivial, not the capacity of performing information processing on them, which is the point of experimental efforts to control high-dimensional systems. In order to capture this aspect, we introduce the notion of irreducible dimension of a quantum behaviour. This dimension can be certified, and we provide a witness for irreducible dimension four.