Unified-entropy trade-off relations for a single quantum channel

TL;DR

This paper extends the concept of receiver entropy to unified entropies and derives new lower bounds on the sum of map and receiver entropies for quantum channels, enhancing understanding of their informational properties.

Contribution

It introduces unified entropies into the analysis of quantum channels and establishes new bounds on their combined entropic measures, advancing theoretical understanding.

Findings

Derived lower bounds on the sum of map and receiver $(q,s)$-entropies.

Extended receiver entropy to the family of unified entropies.

Utilized inequalities with Schatten norms and anti-norms for derivations.

Abstract

Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamio{\l}kowski-Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix obtained as reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver $(q,s)$-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.