Bounding the joint numerical range of Pauli strings by graph parameters

TL;DR

This paper links the geometric structure of the joint numerical range of Pauli strings to graph parameters, providing new bounds and counterexamples, with implications for quantum information and high-dimensional analysis.

Contribution

It introduces a novel graph parameter $eta(G)$ related to the joint numerical range of Pauli strings and resolves open questions in the field.

Findings

Counterexamples to a previous conjecture.

Introduction of the graph parameter $eta(G)$.

Connections established between graph theory and quantum measurement geometry.

Abstract

The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical range. In this work, we explore geometric properties of this construct for measurements represented by tensor products of Pauli observables, also known as Pauli strings. The structure of pairwise commutation and anticommutation relations among a set of Pauli strings determines a graph $G$, sometimes also called the frustration graph. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range. Such an outer approximation can be very practical since ellipsoids can be handled analytically even in high dimensions. We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O. G\"uhne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph parameter that we call $\beta(G)$. Besides, we develop this approach in different directions, such as comparison with graph-theoretic approaches in other fields, applications in quantum information theory, numerical methods, properties of the new graph parameter, etc. Our approach suggests many open questions that we discuss briefly at the end.