Small sets of complementary observables

TL;DR

This paper explores small, unextendible sets of mutually complementary quantum observables, providing explicit constructions up to dimension 16 and insights into quantum state discrimination and the structure of quantum state space.

Contribution

It constructs explicit examples of unextendible sets of complementary observables in finite dimensions and conjectures their properties in higher dimensions.

Findings

Constructed unextendible sets of three complementary observables up to dimension 16.

Most examples enable discrimination between pure and some mixed states.

Provides insights into the topology of the Bloch space in higher dimensions.

Abstract

Two observables are called complementary if preparing a physical object in an eigenstate of one of them yields a completely random result in a measurement of the other. We investigate small sets of complementary observables that cannot be extended by yet another complementary observable. We construct explicit examples of the unextendible sets up to dimension $16$ and conjecture certain small sets to be unextendible in higher dimensions. Our constructions provide three complementary measurements, only one observable away from the ultimate minimum of two observables in the set. Almost all of our examples in finite dimension allow to discriminate pure states from some mixed states, and shed light on the complex topology of the Bloch space of higher-dimensional quantum systems.