bibkey: mcnulty2025pathrobustness authors: Daniel McNulty year: 2025 title: “A Graph-Theoretic Approach to Quantum Measurement Incompatibility” doi: null url: https://arxiv.org/abs/2511.15954 claim: “Conjecture 1 asserts that every path P_{2n}, path P_{2n+1} and even cycle C_{2n+2} has incompatibility robustness (2/(2n+2)) csc(pi/(2n+2)), for n >= 1.” strata_touched:

• D5/S3/Quantum/Measurements/CliffordJointMeasurability/McNultyPathRobustness license: citation-only triage: anchor

Path robustness of binary quantum observables

Verified locator

DOI: null

Source: https://arxiv.org/abs/2511.15954

The source is arXiv:2511.15954v1. Definition 1 and the robustness definition, Eqs. (8)–(10), are on p. 3. Conjecture 1 is on p. 9, Section V.B.5, “Paths”. Crossref’s title-and-author search has no matching record for this preprint.

Definition 1, p. 3:

The anti-commutativity graph $G=(V,E)$ of a set of observables $\mathcal{A}$ is defined by $\{v,v^{\prime }\}\in E⟺A_v{A}_{v^{\prime }}=-A_{v^{\prime }}{A}_v,$ so that adjacent vertices correspond to anti-commuting observables, and non-adjacent vertices correspond to commuting ones.

Binary effects, p. 2, Eq. (3):

$M_v(\pm )=\frac{1}{2}(1\pm A_v).$

Joint measurability, p. 3, Section II.C:

A collection of measurements are jointly measurable if there exists a parent POVM such that each measurement can be recovered as one of its marginals, or equivalently, if their statistics can be obtained from the parent after classical post-processing [47, 48].

Robustness, p. 3, Eqs. (8)–(9):

$\eta (G)\equiv \max \{\eta \in [0,1]|{\mathcal{A}}^{\eta }\text{ is jointly measurable}\},$ where $\mathcal{A}^{\eta}={A_v^{\eta},|,v\in V},\quad A_v^{\eta}=\eta A_v+\frac{\operatorname{tr}[A_v]}{d}(1-\eta)\mathbb{1},$ are noisy (unsharp) versions of the original observables with anti-commutativity graph $G$.

Conjecture 1, p. 9:

For all $n\ge 1$, paths satisfy $\eta(P_{2n})=\eta(P_{2n+1})=\eta(C_{2n+2})= \frac{2}{2n+2}\csc\Big(\frac{\pi}{2n+2}\Big),.$

The observables are Hermitian involutions on a finite-dimensional complex Hilbert space. The matrix encoding uses Fin d coordinates with 1 ≤ d, Mathlib’s pathGraph and cycleGraph, and the literal trace term in the noisy observable. A joint parent is indexed by Boolean assignments, with true denoting the positive sign; both signed marginals are exact. Greatest feasible visibility is expressed by IsGreatest on the feasible subset of [0,1].