Revisiting the operator extension of strong subadditivity

TL;DR

This paper provides a new proof of the operator extension of strong subadditivity of von Neumann entropy, linking it to Connes' theory of spatial derivatives and generalizing it to arbitrary von Neumann algebra inclusions.

Contribution

It introduces a novel proof method based on Connes' spatial derivatives, extending the inequality to broader algebraic contexts.

Findings

Generalizes strong subadditivity to arbitrary von Neumann algebra inclusions

Connects the inequality to Connes' theory of spatial derivatives

Reduces to monotonicity of the relative modular operator in standard cases

Abstract

We give a new proof of the operator extension of the strong subadditivity of von Neumann entropy $\rho_{AB} \otimes \sigma_{C}^{-1} \leq \rho_{A} \otimes \sigma_{BC}^{-1}$ by identifying the mathematical structure behind it as Connes' theory of spatial derivatives. This immediately generalizes the inequality to arbitrary inclusions of von Neumann algebras. In the case of standard representations, it reduces to the monotonicity of the relative modular operator.