Quantum conditional entropy from information-theoretic principles

TL;DR

This paper develops an axiomatic framework for quantum conditional entropy based on physically motivated principles, deriving key properties and operational characterizations applicable to quantum information theory.

Contribution

It introduces an axiomatic approach using monotonicity and additivity, providing new insights and a unified characterization of quantum conditional entropy.

Findings

Quantum conditional entropy can be negative on entangled states.

On maximally entangled states, it equals -log(d).

Conditional entropy is non-negative on separable states.

Abstract

We introduce an axiomatic approach for characterizing quantum conditional entropy. Our approach relies on two physically motivated axioms: monotonicity under conditional majorization and additivity. We show that these two axioms provide sufficient structure that enable us to derive several key properties applicable to all quantum conditional entropies studied in the literature. Specifically, we prove that any quantum conditional entropy must be negative on certain entangled states and must equal -log(d) on dxd maximally entangled states. We also prove the non-negativity of conditional entropy on separable states, and we provide a generic definition for the dual of a quantum conditional entropy. Finally, we develop an operational approach for characterizing quantum conditional entropy via games of chance, and we show that, for the classical case, this complementary approach yields the same ordering as the axiomatic approach.