bibkey: yu2010jointmeasurement authors: Sixia Yu; Nai-le Liu; Li Li; C. H. Oh year: 2010 title: “Joint measurement of two unsharp observables of a qubit” doi: 10.1103/PhysRevA.81.062116 url: https://arxiv.org/abs/0805.1538 claim: “Equation (5) defines the squared-bias square-root function used in the necessary and sufficient joint-measurability criterion.” strata_touched:

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

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DOI: https://doi.org/10.1103/PhysRevA.81.062116

Source: https://arxiv.org/abs/0805.1538

Joint measurement of two unsharp observables of a qubit

Equation (5), PDF p. 2, defines

$$F_x=\frac{\sqrt{(1+x{)}^2-m^2}+\sqrt{(1-x{)}^2-m^2}}{2}.$$

Here $m$ is the length of the Bloch vector and $x$ is the bias. The condition $|x|+m\le 1$ describes a two-outcome qubit observable. Theorem 1 gives the necessary and sufficient condition for joint measurability. Its strict violation, restricted to the parameter bodies of both observables, is the incompatible set in the probability calculation.

The Lean function hYu replaces $m$ by the norm of a vector in EuclideanSpace ℝ (Fin 3) and retains the squares on both $1+x$ and $1-x$.