Minimal covariant observables identifying all pure states

TL;DR

This paper investigates the existence of minimal covariant observables capable of uniquely identifying all pure states in a quantum system, revealing that their existence depends on the system's dimension and highlighting asymmetries in quantum state and observable dual pairs.

Contribution

It provides a detailed analysis of covariant observables with minimal outcomes for pure state identification and shows their existence is dimension-dependent.

Findings

Existence of such covariant observables depends on the Hilbert space dimension.

In some dimensions, these minimal covariant observables do not exist.

The asymmetry between pure states and states in the dual pair affects observable construction.

Abstract

It has been recently shown that an observable that identifies all pure states of a d-dimensional quantum system has minimally 4d-4 outcomes or slightly less (the exact number depending on the dimension d). However, no simple construction of this type of observable with minimal number of outcomes is known. In this work we investigate the possibility to have a covariant observable that identifies all pure states and has minimal number of outcomes for this purpose. It is shown that the existence of these kind of observables depends on the dimension of the Hilbert space. The fact that these kind of observables fail to exist in some dimensions indicates that the dual pair of observables -- pure states lacks the symmetry that the dual pair of observables -- states has.