Convertibility of Observables

TL;DR

This paper explores the conditions under which one set of quantum observables can be transformed into another via completely positive maps, addressing a dual problem to quantum state convertibility.

Contribution

It introduces the observable convertibility problem and provides necessary and sufficient conditions for specific cases of this problem.

Findings

Derived conditions for observable convertibility in special cases.

Established a dual framework to quantum state convertibility.

Contributed to understanding quantum measurement transformations.

Abstract

Some problems of quantum information, cloning, estimation and testing of states, universal coding etc., are special example of the following `state convertibility' problem. In this paper, we consider the dual of this problem, 'observable conversion problem'. Given families of operators $\{L_\theta}\}_{\theta\in\Theta}$ and $\{M_\theta}\}_{\theta\in\Theta}$ , we ask whether there is a completely positive (sub) unital map which sends $\{L_\theta}\}$ to $\{M_\theta}\}$ for each {\theta}. We give necessary and sufficient conditions for the convertibility in some special cases.