Optimal covariant quantum measurements

• URL: http://arxiv.org/abs/2009.14080v1
• Date: Tue, 29 Sep 2020 15:08:07 GMT
• Title: Optimal covariant quantum measurements
• Authors: Erkka Haapasalo, Juha-Pekka Pellonp\"a\"a
• Abstract summary: We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.

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