bibkey: torii2025comparing authors: K. Torii; R. Takakura; R. Imamura year: 2025 title: “Comparing quantum incompatibility of device sets from an operational perspective” doi: 10.1103/s6mc-lnvl url: https://arxiv.org/abs/2509.08284v2 claim: “Device sets are compared by the state sets on which their incompatibility can be detected; for pairs of unbiased qubit observables with |a1|=|a2| and |b1|=|b2| in (1/√2,1], operational equivalence forces equality up to signs and exchange (Proposition 7), and whether this holds without the equal-length constraints is left open.” strata_touched:

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

Torii, Takakura and Imamura, comparing quantum incompatibility of device sets

K. Torii, R. Takakura and R. Imamura, Comparing quantum incompatibility of device sets from an operational perspective, Phys. Rev. A 112, 062228 (2025); arXiv:2509.08284v2 (v1 10 September 2025, v2 8 January 2026; quant-ph).

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DOI: 10.1103/s6mc-lnvl. Primary version read: https://arxiv.org/abs/2509.08284v2. Source: https://arxiv.org/pdf/2509.08284v2 (text read from the PDF): §III.A (Definitions 4 and 5), §IV.A (Propositions 6 and 7), §V (conclusion).

Source statements

Definition 4: “For pairs of device sets D={D_1,…,D_n} and E={E_1,…,E_m}, we write E ⪯_inc D if D is S_0-incompatible for any state set S_0 ⊂ S(H_in) such that E is S_0-incompatible, or equivalently, E is S_0-compatible for any S_0 ⊂ S(H_in) such that D is S_0-compatible.”

Definition 5: “For two device sets D={D_1,…,D_n} and E={E_1,…,E_m}, we write D ∼_inc E if both D ⪯_inc E and E ⪯_inc D are satisfied.”

Proposition 7: “Let A_ub = {A^{a_1}_ub, A^{a_2}_ub} and B_ub = {A^{b_1}_ub, A^{b_2}_ub} be pairs of unbiased qubit observables satisfying |a_1|=|a_2|=t, |b_1|=|b_2|=u for some t,u ∈ (1/√2,1]. If A_ub ∼_inc B_ub, then either (b_1,b_2) = (±a_1,±a_2) or (b_1,b_2) = (±a_2,±a_1) holds.”

§V: “Future work will also address the generalization of Proposition 7, as it remains an open problem whether the result holds without the constraints of |a_1|=|a_2| and |b_1|=|b_2|.”