Quantum Implementation of Non-Positive-Operator-Valued Measurements in General Probabilistic Theories by Post-Selected POVMs

TL;DR

This paper demonstrates how non-positive measurements in general probabilistic theories can be implemented in quantum mechanics using post-selected POVMs, revealing new links between GPTs and quantum measurement techniques.

Contribution

It introduces a constructive method to realize N-POVM measurements via post-selected POVMs in quantum theory within restricted state domains.

Findings

N-POVM measurements can be implemented using post-selected POVMs.

Post-selected POVMs are equivalent to N-POVMs in certain domains.

Provides a new connection between GPTs and quantum measurement implementations.

Abstract

It is important problem to clarify the class of implementable quantum measurements from both fundamental and applicable viewpoints. Positive-Operator-Valued Measure (POVM) measurements are implementable by the indirect measurement methods, and the class is the largest class determined by the mathematical structure of Hilbert space. However, if we assume probabilistic consistency in our operations instead of the structure of Hilbert space, we can deal with Non-Positive-Operator-Valued Measure (N-POVM) measurements in the framework of General Probabilistic Theories (GPTs). N-POVM measurements are not considered as implementable, but this paper gives a constructive way to implement N-POVM measurements by POVM measurements and post-selection in quantum theory when we restrict the domain of target states. Besides, we show that a post-selected POVM measurement is regarded as an N-POVM measurement in a restricted domain. These results provide a new relationship between N-POVM measurements in GPTs and post-selection.