The Structure of State Space with Respect to Imbedding

TL;DR

This paper explores the geometric structure of quantum state space using entanglement and entropy measures, introduces a leaf structure for state classification, and discusses additivity properties with counterexamples.

Contribution

It generalizes the concept of state space leaves using entanglement and entropy, extending to infinite systems and analyzing additivity properties.

Findings

Leaves can be specified using entanglement and conditional entropy.

A generalized leaf structure applies to infinite systems.

Counterexamples challenge the additivity of conditional entropy.

Abstract

The entanglement of formation as well as the conditional entropy can be used to define leaves in the state space, given by a linear superposition of their extremal points. Examples are presented, where these leaves can be specified and can be used to calculate the entanglement respectively the conditional entropy. The definition of entanglement is generalized to infinite systems and allows again to define a leaf structure. Finally we remark on the additivity property of both expressions, offering a counter example to the additivity of the conditional entropy.