bibkey: debievre2023incompatibility authors: Stephan De Bièvre year: 2023 title: “Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality” doi: 10.1063/5.0110267 url: https://arxiv.org/abs/2207.07451v1 claim: “For every dimension d ≥ 4, noncommutativity of every pair of nontrivial coordinate projectors does not imply complete incompatibility of two orthonormal bases (§5, conjecture following Proposition 7).” strata_touched:

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

Noncommuting coordinate projectors and complete incompatibility

Stephan De Bièvre, J. Math. Phys. 64, 022202 (2023), arXiv:2207.07451v1. The page numbers below refer to the arXiv version.

Section 3, p. 5 defines the coordinate orthogonal projectors by

$${\Pi }_{\mathcal{A}}(S)=\sum _{i\in S}|a_i⟩⟨a_i|,{\Pi }_{\mathcal{B}}(T)=\sum _{j\in T}|b_j⟩⟨b_j|.$$

Definition 4, p. 14:

We say that two bases $\mathcal{A}$ and $\mathcal{B}$ are completely incompatible (COINC) if and only if all index sets $S,T$ in $[[1,d]]$ for which $|S|+|T|\le d$ have the property that ${\Pi }_{\mathcal{A}}(S)\mathcal{H}\cap {\Pi }_{\mathcal{B}}(T)\mathcal{H}=\{0\}$.

Proposition 7(ii), p. 16:

For all $S,T\subset [[1,d]]$, with $1\le |S|,|T|<d$, $[{\Pi }_{\mathcal{A}}(S),{\Pi }_{\mathcal{B}}(T)]\ne 0$.

Section 5, p. 16:

We conjecture it is true that (ii) does not imply (i) in all dimensions $d\ge 4$, but we have not produced such examples in other dimensions than $4$ and $6$.

The Hilbert space is complex and has dimension $d$. The encoding indexes its orthonormal bases by Fin d, relabelling the source’s indices $1,\dots ,d$ as $0,\dots ,d-1$. The projector images are the complex spans of the selected basis vectors. Complete incompatibility includes empty index sets; condition (ii) quantifies only over nonempty proper index sets.

Verified locator

• DOI: https://doi.org/10.1063/5.0110267
• URL: https://arxiv.org/abs/2207.07451v1
• The arXiv source is COINC_preprint_20220708.tex; the projector definitions are in §3, Definition 4 is labelled def:COINC, and Proposition 7 is labelled prop:incompatibilities. The conjecture immediately follows that proposition in §5. The journal text has not been checked for preservation of this sentence.