A Graph-Theoretic Approach to Quantum Measurement Incompatibility

TL;DR

This paper introduces a graph-theoretic framework to quantify quantum measurement incompatibility, linking it to graph invariants and applying it to large families of quantum observables.

Contribution

It develops a novel graph-based method to measure quantum incompatibility, connecting it to well-known graph invariants and analyzing specific graph families.

Findings

Incompatibility robustness is a graph invariant linked to Lovász number, clique number, and fractional chromatic number.

Spectral bounds on robustness are derived for line graphs using graph energy concepts.

Closed-form formulas are obtained for certain symmetric graph families, identifying extremal cases.

Abstract

Measurement incompatibility--the impossibility of jointly measuring certain quantum observables--is a fundamental resource for quantum information processing. We develop a graph-theoretic framework for quantifying this resource for large families of binary measurements, including Pauli observables on multi-qubit systems and $k$-body Majorana observables on $n$-mode fermionic systems. To each set of observables we associate an anti-commutativity graph, whose vertices represent observables and whose edges indicate pairs that anti-commute. In this representation, the incompatibility robustness--the minimal amount of classical noise required to render the set jointly measurable--becomes a graph invariant. We derive general bounds on this invariant in terms of the Lov\'asz number, clique number, and fractional chromatic number, and show that the Lov\'asz number yields the correct asymptotic scaling for $k$-body Majorana observables. For line graphs $L(G)$, which naturally arise in the characterisation of exactly solvable spin models, we obtain spectral bounds on the robustness expressed through the energy and skew-energy of the underlying graph $G$. These bounds become tight for highly symmetric graphs, leading to closed formulas for several graph families. Finally, we identify structural conditions under which the robustness is determined by a simple function of the graph's maximum degree and the number of vertices and edges, and show that such extremal cases occur only when combinatorial structures such as Hadamard, conference or weighing matrices exist.