bibkey: debrota2020varieties authors: John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey year: 2020 title: “The Varieties of Minimal Tomographically Complete Measurements” doi: 10.1142/S0219749920400055 url: https://arxiv.org/abs/1812.08762v5 claim: “For an orthonormal basis of C^d, the orthocross MIC has elements E_alpha = Omega^(-1/2) Pi_alpha Omega^(-1/2), where Pi_alpha are the d^2 projectors onto |j>, (|j> + |k>)/sqrt(2) and (|j> + i|k>)/sqrt(2) (j < k) and Omega is their sum; its Gram matrix is G_ij = tr(E_i E_j); the authors conjecture that for any orthocross MIC the entries of G^(-1) are integers or half-integers; Conjecture 1: A rank-1 MIC in dimension 3 can have no more than 7 pairs of orthogonal elements; Conjecture 2: the entries in G for orthocross MICs can become arbitrarily small with increasing d, but no two elements of an orthocross MIC can be exactly orthogonal.” strata_touched:

• D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger
• D5/S3/Quantum/Measurement/QutritRankOneMicOrthogonalPairs
• D5/S3/Quantum/Measurement/OrthocrossNonorthogonality license: citation-only triage: anchor

The Varieties of Minimal Tomographically Complete Measurements

J. B. DeBrota, C. A. Fuchs and B. C. Stacey, arXiv:1812.08762 (v1 2018-12-20, v5 2020-09-21); Int. J. Quantum Inf. 19 (2021) 2040005, published online 2020-10-27. Subject: quant-ph.

The paper studies minimal informationally complete measurements (MICs) and their Gram matrices [G]_{ij} := tr E_i E_j. The orthocross MICs, introduced by Caves, Fuchs and Schack for the quantum de Finetti theorem, are built from an orthonormal basis {|j⟩}: the projectors Γ_jj, ½(|j⟩ + |k⟩)(⟨j| + ⟨k|) and ½(|j⟩ + i|k⟩)(⟨j| − i⟨k|) for j < k, their sum Ω, and E_α := Ω^{−1/2} Π_α Ω^{−1/2}. The paper computes Ω (diagonal d, ½(1 − i) above the diagonal) and its eigenvalues, and states:

The following conjectures about orthocross MICs have been motivated by numerical investigations. We suspect that their proofs will be relatively straightforward, but so far they have eluded us. […] Conjecture. For any orthocross MIC, the entries in G^{−1} are integers or half-integers.

Verified locator

• DOI: https://doi.org/10.1142/S0219749920400055 (metadata from Crossref; the journal text was not read).
• URL: https://arxiv.org/abs/1812.08762v5 (source retrieved 2026-09-30): mic_facts.tex, the definition of the Gram matrix in the introduction and the subsection on orthocross MICs (construction, Ω, and the two conjectures that close the subsection).

Rank-one qutrit MICs and Conjecture 1

On printed pages 1-2 the paper defines the objects:

Let ℋ_d be a d-dimensional complex Hilbert space, and let {E_i} be a set of positive semidefinite operators on that space which sum to the identity: ∑_(i=1)^N E_i = I.

The set {E_i} is a positive-operator-valued measure (POVM), which is the mathematical representation of a measurement process in quantum theory.

A POVM is said to be informationally complete (IC) if the operators {E_i} span ℒ(ℋ_d), the space of Hermitian operators on ℋ_d, and an IC POVM is said to be minimal if it contains exactly d² elements.

The Gram matrix is defined on printed page 2 by “[G]_{ij} := tr E_i E_j”. On printed page 6, immediately after the seven-pair example, Conjecture 1 says:

A rank-1 MIC in dimension 3 can have no more than 7 pairs of orthogonal elements.

The preceding example says:

When multiplied by 1/3, the following is a rank-1 unbiased MIC in dimension 3 with 7 orthogonal pairs.

The conjecture itself imposes no unbiasedness restriction. Increasing pairs a < b count distinct unordered pairs, excluding diagonal entries. In dimension three minimality gives nine effects; informational completeness means their real span is the Hermitian matrices. The new qutrit construction has unequal traces and refutes this literal statement. It does not certify a refutation restricted to unbiased MICs. The source locator is https://arxiv.org/abs/1812.08762v5, mic_facts.tex, Conjecture 1 and the preceding example (7orthopairs).

Orthocross Conjecture 2

The same subsection of mic_facts.tex (v5), between the sentence on numerical motivation quoted above and the half-integrality conjecture, states:

\begin{conjecture} The entries in $G$ for orthocross MICs can become arbitrarily small with increasing $d$, but no two elements of an orthocross MIC can be exactly orthogonal. \end{conjecture}

The paper also computes the frame operator: “The elements along the diagonal are all equal to $d$, the elements above the diagonal are all equal to $\frac{1}{2}(1-i)$, and the rest are $\frac{1}{2}(1+i)$”, with eigenvalues ${\lambda }_m=d+\frac{1}{2}\left(\cot \frac{\pi (4m+1)}{4d}-1\right)$, $m=0,\dots ,d-1$.