Strongly subadditive functions

TL;DR

This paper investigates a trace inequality involving positive operators and projections, characterizes functions satisfying it, and describes conditions for equality, contributing to the understanding of strongly subadditive functions.

Contribution

It introduces and analyzes the class of functions satisfying a specific trace inequality, expanding the theory of strongly subadditive functions and their equality cases.

Findings

Identifies functions satisfying the trace inequality (SSA)

Provides conditions for equality in the inequality

Enhances understanding of strongly subadditive functions

Abstract

Let f be a function defined on positive numbers. The subject is the trace inequality $Tr f(A) + Tr f(P_2AP_2) \le Tr f(P_{12}AP_{12}) + \Tr f(P_{23}AP_{23})$, where $A$ is a positive operator, $P_1,P_2,P_3$ are orthogonal projections such that $P_1+P_2+P_3=I$, $P_{12}=P_1+P_2$ and $P_{23}=P_2+P_3$. There are several examples of functions f satisfying the inequality (called (SSA)) and the case of equality is described.