A limit of the quantum Renyi divergence

TL;DR

This paper investigates the limitations of the quantum Renyi divergence, showing it can only recover the 0-relative Renyi entropy when the operators have equal supports, highlighting fundamental constraints in quantum information measures.

Contribution

It proves that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only under support equality, clarifying the relationship between key quantum entropic quantities.

Findings

Quantum Renyi divergence cannot recover 0-relative Renyi entropy unless supports are equal.

Supports equality is necessary for deriving the 0-relative Renyi entropy from the divergence.

The results suggest focusing on two main quantum relative entropies for operational tasks.

Abstract

Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators $\rho$ and $\sigma$. It depends on a parameter $\alpha$ and acts as a parent quantity for other relative entropies which have important operational significances in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Renyi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only if $\rho$ and $\sigma$ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived - the quantum Renyi divergence with parameter $\alpha \ge 1/2$, and the $\alpha$-relative R\'enyi entropy with $\alpha\in [0,1)$.