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

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.

v4: 26 pages, 1 figure; reorganized and one open question solved with Choi's inequality (at the suggestion of an anonymous referee)

References in corpus (6)

• The Quantum Chernoff Bound
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Recoverability in quantum information theory
• Renyi generalizations of the conditional quantum mutual information
• A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
• Remainder Terms for Some Quantum Entropy Inequalities

Cited by in corpus (10)

• Recoverability in quantum information theory
• Universal recovery map for approximate Markov chains
• Fundamental work cost of quantum processes
• Preservation of a quantum Renyi relative entropy implies existence of a recovery map
• Renyi relative entropies of quantum Gaussian states
• Operational meaning of quantum measures of recovery
• Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree
• Markovian and non-Markovian quantum measurements
• Multipartite quantum states over time from two fundamental assumptions
• On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches