Unital-qubit incompatibility-breaking criterion

Let $\mathcal{N}_{D}$ be a unital qubit channel whose Heisenberg adjoint has Bloch matrix $D$ . The channel is incompatibility breaking for arbitrary POVMs, meaning that for every finite family of POVMs $\{\mathsf{M}^{(x)}\}_{x}$ , the family $\{\mathcal{N}_{D}^{\dagger}(\mathsf{M}^{(x)})\}_{x}$ is jointly measurable, if and only if

2\int_{S^{2}}\left\lVert D\hat{n}\right\rVert\,d\mu(\hat{n})\leq 1,

where $d\mu(\hat{n})$ is the normalized rotation-invariant measure on $S^{2}$ . Consequently, for every unital qubit channel, PVM-incompatibility breaking and POVM-incompatibility breaking coincide.

The problem asks which unital qubit channels make every collection of output measurements jointly measurable, and whether restricting to projective measurements changes the threshold. The question is now answered exactly for the unital class.

Known results

Werner constructed a local-hidden-state model for projective measurements in the depolarizing family up to $r=1/2$ .
Barrett obtained a model for arbitrary POVMs up to $r=5/12$ ; the gap was later closed at $r=1/2$ .

July 2026 exact criterion

The paper proves that a unital channel with Bloch matrix $D$ is incompatibility breaking for arbitrary POVMs if and only if $2\int_{S^{2}}\lVert D\hat{n}\rVert\,d\mu(\hat{n})\leq 1$ . It also proves that the projective-measurement and arbitrary-POVM criteria coincide for every unital qubit channel. For nonunital channels, only a sufficient condition is given.

Current status (as of July 2026): The unital-qubit problem is resolved by an exact integral criterion, and the projective and general-POVM thresholds coincide; the nonunital case remains without an exact criterion.

Unital-qubit incompatibility-breaking criterion

Progress summary

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