Validity and failure of some entropy inequalities for CAR systems

TL;DR

This paper investigates the validity of entropy inequalities in CAR systems, showing they hold for even states but can fail for noneven states, and explores the implications for state extension and correlations.

Contribution

It demonstrates that certain entropy inequalities hold for even states in CAR systems and constructs counterexamples for noneven states, revealing limitations in state extension.

Findings

Inequalities hold for even states in CAR systems.

Counterexamples show inequalities can fail for noneven states.

Symmetric purification ensures inequality validity for even states.

Abstract

Basic properties of von Neumann entropy such as the triangle inequality and what we call MONO-SSA are studied for CAR systems. We show that both inequalities hold for any even state. We construct a certain class of noneven states giving counter examples of those inequalities. It is not always possible to extend a set of prepared states on disjoint regions to some joint state on the whole region for CAR systems. However, for every even state, we have its `symmetric purification' by which the validity of those inequalities is shown. Some (realized) noneven states have peculiar state correlations among subsystems and induce the failure of those inequalities.