bibkey: zhang2019incompatibilityprobability authors: Lin Zhang; Hua Xiang; Xiao Li-Jost; Shao-Ming Fei year: 2019 title: “Incompatibility probability of random quantum measurements” doi: 10.1103/PhysRevE.100.062139 url: https://arxiv.org/abs/1912.12321 claim: “The incompatibility probability of a pair of random qubit measurements is conjectured to equal one quarter.” strata_touched:

• D5/S3/Quantum/Measurement/OrderedRootCoordinates
• D5/S3/Quantum/Measurement/QubitIncompatibilityRadialReduction
• D5/S3/Quantum/Measurement/QubitIncompatibilityProbability license: citation-only triage: anchor

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DOI: https://doi.org/10.1103/PhysRevE.100.062139

Source: https://arxiv.org/abs/1912.12321

Incompatibility probability of random quantum measurements

Section III.C, PDF p. 5, states:

We conjecture that the incompatibility probability of a pair of random qubit measurements $\mathbf{A}$ and $\mathbf{B}$, generated by $(a_0,\mathbf{a})$ and $(b_0,\mathbf{b})$ in Eq. (4), is given by $\mathbf{Pr}[{\Theta }_{2,2}^{\mathrm{NJM}}]=\frac{1}{4}.$

The parameter body is $V=\{(x_0,\mathbf{x}):|x_0|+|\mathbf{x}|\le 1\}$. Its volume is $2\pi /3$. The incompatible set consists of pairs in $V\times V$ violating the Yu–Liu–Li–Oh inequality. The probability is the incompatible Lebesgue volume divided by $(2\pi /3{)}^2$.

The function $h$ uses $\sqrt{(1\pm x_0{)}^2-|\mathbf{x}{|}^2}$, as in Appendix C and the original Yu–Liu–Li–Oh criterion. The main-text display omits these squares. The Lean function hYu uses the squared expression.

The strict-interior coordinate maps are smooth bijections. The body boundaries, zero vectors and cosine endpoints have zero volume. These exclusions impose no extra hypothesis on the probability statement.

The formal refutation gives the exact reduction $\mathbf{Pr}=9I$ and the rational bound $\mathbf{Pr}\le 5910272/23648625<1/4$. The closed form of $I$ and higher-dimensional analogues remain open here. The source independently proves probability $3/5$ for uniform pairs of unbiased qubit measurements (Theorem 2); that result does not use the one-quarter conjecture. Other distributions or families of unbiased measurements are outside this refutation. The identity and bound do not establish publication priority; the seventh citing work in the supplied literature screen remains ASSUMED-UNVERIFIED.