Unsharp measurements and joint measurability

TL;DR

This paper reviews joint unsharp measurements of non-commuting observables via POVMs, highlighting their role in entropic uncertainty relations, and shows that incompatible measurements are necessary at Bob's end to improve outcome predictability.

Contribution

It demonstrates the necessity of incompatible POVMs in quantum memory to surpass entropic uncertainty bounds and explores the link between joint measurability and classical moment matrices.

Findings

Incompatible POVMs are essential to beat the entropic uncertainty bound.

Compatible POVMs are required for a valid classical moment matrix.

Joint measurability impacts the existence of joint probability distributions.

Abstract

We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence, predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to {\em validate} a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix -- which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables -- could be realized if and only if compatible POVMs are employed.