Uniqueness of Quantum States Compatible with Given Measurement Results

TL;DR

This paper explores the conditions under which quantum states are uniquely determined by measurement results, providing bounds on the number of observables needed and analyzing implications for multipartite systems and symmetry considerations.

Contribution

It establishes new bounds on the number of observables required for unique state determination and links different notions of uniqueness, extending to low rank states and symmetry scenarios.

Findings

4d-5 observables determine any pure state uniquely (case 1)

5d-7 observables suffice for unique determination among all states (case 2)

Almost all pure states are determined by RDMs with improved bounds

Abstract

We discuss the uniqueness of quantum states compatible with given results for measuring a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it is known that for a d-dimensional Hilbert space, there exists a set of 4d-5 observables that uniquely determines any pure state. We show that for case (2), 5d-7 observables suffice to uniquely determine any pure state. Thus there is a gap between the results for (1) and (2), and we give some examples to illustrate this. The case of observables corresponding to reduced density matrices (RDMs) of a multipartite system is also discussed, where we improve known bounds on local dimensions for case (2) in which almost all pure states are uniquely determined by their RDMs. We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low rank quantum states.