Tasks and premises in quantum state determination

TL;DR

This paper explores how prior information and specific task constraints, especially regarding the rank of quantum states, influence the design of measurement schemes in quantum tomography, with a focus on covariant phase space observables.

Contribution

It characterizes the structure of quantum observables suitable for rank-constrained state determination and analyzes their effectiveness, including effects of noise, in a comprehensive framework.

Findings

Characterization of observables for rank-specific state determination

Identification of covariant phase space observables capable of these tasks

Discussion of noise impact on measurement effectiveness

Abstract

The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (POVMs) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.