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Monotonicity of a relative Rényi entropy
Rupert L. Frank, Elliott H. Lieb
TL;DR
The paper addresses whether the modified relative Rényi entropy is monotone under completely positive, trace-preserving maps across the conjectured parameter range. It proves this monotonicity and establishes joint convexity using a central concavity/convexity proposition.
Problem
A modified relative Rényi entropy had monotonicity under completely positive, trace-preserving maps only in a limited parameter range, leaving a broader conjecture unresolved.
Method
The proofs reduce both theorems to a proposition establishing joint concavity or convexity, using representation formulas and convexity arguments.
Results
The paper proves monotonicity for 1/2 ≤ α ≤ ∞ and joint convexity for 1/2 ≤ α ≤ 1.
Takeaways & Limitations
The monotonicity theorem proves the conjecture concerning the modified relative Rényi entropy, while the same framework also yields joint convexity results.
Abstract
from arXiv · showhide
We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al.