Minimal sufficient positive-operator valued measure on a separable Hilbert space

Abstract

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM a sufficient statistic called a Lehmann-Scheffé-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.

25 pages. The main result is improved, and a new appendix is added

References in corpus (3)

• Sharp and fuzzy observables on effect algebras
• Classicality condition on a system's observable in a quantum measurement and relative-entropy conservation law
• Construction of the least informative observable conserved by a given quantum instrument

Cited by in corpus (12)

• Simulability of observables in general probabilistic theories
• A Resource Theory of Quantum Measurements
• Optimal quantum observables
• Post-processing of quantum instruments
• Quantum Measurements in the Light of Quantum State Estimation
• Order preserving maps on quantum measurements
• Minimal sufficient statistical experiments on von Neumann algebras
• The compatibility dimension of quantum measurements
• The unavoidable information flow to environment in quantum measurements
• Post-processing minimal joint observables
• Entropic partial orderings of quantum measurements
• Infinite dimensionality of the post-processing order of measurements on a general state space