Relations for certain symmetric norms and anti-norms before and after partial trace

TL;DR

This paper investigates how certain symmetric norms and anti-norms change under partial trace operations on operators in finite-dimensional quantum systems, providing bounds and inequalities relevant to quantum information theory.

Contribution

It introduces bounds on unitarily invariant norms and anti-norms under partial trace, connecting these to quantum entropy inequalities and the structure of completely positive maps.

Findings

Upper bounds on norms of partial traces in terms of original operator norms and dimensions

Inequalities for anti-norms of positive matrices under partial trace

Relations between entropies of composite systems and subsystems

Abstract

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.