Entropic uncertainty relations for extremal unravelings of super-operators

TL;DR

This paper develops entropic uncertainty relations for super-operators using extremal unravelings, showing that certain entropies are minimized by specific unravelings and deriving new inequalities applicable to quantum measurements.

Contribution

It introduces a novel approach to entropic uncertainty relations based on extremal unravelings of super-operators, extending the theoretical framework of quantum uncertainty principles.

Findings

Tsallis and Renyi entropies are minimized by the same unraveling.

New uncertainty inequalities for pairs of generalized resolutions of identity.

Application of relations to complementary observables and angular momentum in quantum systems.

Abstract

A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For given input state, different effects of each unraveling result in some probability distribution at the output. As it is shown, all Tsallis' entropies of positive order as well as some of Renyi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Renyi and Tsallis entropies. The inequality with Renyi's entropies is an improvement of the previous one, whereas the inequality with Tsallis' entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a $d$-level system and for the angle and the angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.