Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

TL;DR

This paper extends the concept of quantum Markov chains and sufficiency of quantum channels to Renyi information measures, providing new characterizations and solving open questions in quantum information theory.

Contribution

It demonstrates that properties of quantum Markov chains and sufficiency extend to Renyi generalizations, offering new insights and resolving open problems in the field.

Findings

Renyi measures characterize quantum Markov chains and sufficiency.

Extended properties support Renyi measures as legitimate generalizations.

Solved open questions on matrix traces related to entropy inequalities.

Abstract

A short quantum Markov chain is a tripartite state $\rho_{ABC}$ such that system $A$ can be recovered perfectly by acting on system $C$ of the reduced state $\rho_{BC}$. Such states have conditional mutual information $I(A;B|C)$ equal to zero and are the only states with this property. A quantum channel $\mathcal{N}$ is sufficient for two states $\rho$ and $\sigma$ if there exists a recovery channel using which one can perfectly recover $\rho$ from $\mathcal{N}(\rho)$ and $\sigma$ from $\mathcal{N}(\sigma)$. The relative entropy difference $D(\rho\Vert\sigma)-D(\mathcal{N}(\rho)\Vert\mathcal{N}(\sigma))$ is equal to zero if and only if $\mathcal{N}$ is sufficient for $\rho$ and $\sigma$. In this paper, we show that these properties extend to Renyi generalizations of these information measures which were proposed in [Berta et al., J. Math. Phys. 56, 022205, (2015)] and [Seshadreesan et al., J. Phys. A 48, 395303, (2015)], thus providing an alternate characterization of short quantum Markov chains and sufficient quantum channels. These results give further support to these quantities as being legitimate Renyi generalizations of the conditional mutual information and the relative entropy difference. Along the way, we solve some open questions of Ruskai and Zhang, regarding the trace of particular matrices that arise in the study of monotonicity of relative entropy under quantum operations and strong subadditivity of the von Neumann entropy.