Testing incompatibility of quantum devices with few states

TL;DR

This paper introduces two measures of quantum device incompatibility, the incompatibility and compatibility dimensions, which quantify the minimal states needed to detect or observe incompatibility, revealing unexpected behaviors under noise.

Contribution

It proposes novel quantifications of quantum device incompatibility, providing new insights into their behavior with respect to noise and state selection.

Findings

Incompatibility and compatibility dimensions are defined and analyzed.

These dimensions exhibit unexpected behavior when noise is introduced.

Concrete examples illustrate the properties of these incompatibility measures.

Abstract

When observations must come from incompatible devices and cannot be produced by compatible devices? This question motivates two integer valued quantifications of incompatibility, called incompatibility dimension and compatibility dimension. The first one quantifies how many states are minimally needed to detect incompatibility if the test states are chosen carefully, whereas the second one quantifies how many states one may have to use if they are randomly chosen. With concrete examples we show that these quantities have unexpected behaviour with respect to noise.