Local Additivity Revisited

TL;DR

This paper simplifies and extends the proof of local additivity of minimum output entropy in quantum channels, using advanced mathematical tools to streamline the argument and explore implications for quantum channel capacity.

Contribution

It introduces a simplified proof framework for local additivity and extends results to maximum relative entropy, impacting quantum channel capacity analysis.

Findings

Simplified proof of local additivity using integral and functional calculus.

Extended additivity results to maximum relative entropy.

Implications for superadditivity of quantum channel capacity.

Abstract

We make a number of simplifications in Gour and Friedland's proof of local additivity of minimum output entropy of a quantum channel. We follow them in reframing the question as one about entanglement entropy of bipartite states associated with a $d_B \times d_E $ matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their underlying strategy. We also extend this result to the maximum relative entropy with respect to a fixed reference state which has important implications for studying the superadditivity of the capacity of a quantum channel to transmit classical information.