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

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 $N$ is sufficient for two states ρ and σ if there exists a recovery channel using which one can perfectly recover ρ from $N\left(\rho \right)$ and σ from $N\left(\sigma \right).$ The relative entropy difference $D(\rho \parallel \sigma )-D\left(N\right(\rho )\parallel N(\sigma \left)\right)$ is equal to zero if and only if $N$ is sufficient for ρ and σ. In this paper, we show that these properties extend to Rényi generalizations of these information measures which were proposed in (Berta et al 2015 J. Math. Phys. 56 022205; Seshadreesan et al 2015 J. Phys. A: Math. Theor. 48 395303), 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 Rényi 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. (paper)