A geometric criterion for optimal measurements in multiparameter quantum metrology

• URL: http://arxiv.org/abs/2601.21801v1
• Date: Thu, 29 Jan 2026 14:44:07 GMT
• Title: A geometric criterion for optimal measurements in multiparameter quantum metrology
• Authors: Jing Yang, Satoya Imai, Luca Pezzè,
• Abstract summary: We establish an equivalence between QCRB saturation and the simultaneous hollowization of a set of traceless operators associated with the estimation model.<n>We then identify conditions under which the partial commutativity condition proposed in [Phys. Rev. A 100, 032104( 2019)] becomes necessary and sufficient for the saturation of the QCRB.
• Score: 3.2237900102017503
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Determining when the multiparameter quantum Cramér--Rao bound (QCRB) is saturable with experimentally relevant single-copy measurements is a central open problem in quantum metrology. Here we establish an equivalence between QCRB saturation and the simultaneous hollowization of a set of traceless operators associated with the estimation model, i.e., the existence of complete (generally nonorthogonal) bases in which all corresponding diagonal matrix elements vanish. This formulation yields a geometric characterization: optimal rank-one measurement vectors are confined to a subspace orthogonal to a state-determined Hermitian span. This provides a direct criterion to construct optimal Positive Operator-Valued Measures(POVMs). We then identify conditions under which the partial commutativity condition proposed in [Phys. Rev. A 100, 032104(2019)] becomes necessary and sufficient for the saturation of the QCRB, demonstrate that this condition is not always sufficient, and prove the counter-intuitive uselessness of informationally-complete POVMs.

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