Necessary criterion for approximate recoverability

Abstract

A tripartite state ${\rho }_{ABC}$ forms a Markov chain if there exists a recovery map ${\mathcal{R}}_{B\to BC}$ acting only on the $B$-part that perfectly reconstructs ${\rho }_{ABC}$ from ${\rho }_{AB}$. To achieve an approximate reconstruction, it suffices that the conditional mutual information $I(A:C|B{)}_{\rho }$ is small, as shown recently. Here we ask what conditions are necessary for approximate state reconstruction. This is answered by a lower bound on the relative entropy between ${\rho }_{ABC}$ and the recovered state ${\mathcal{R}}_{B\to BC}({\rho }_{AB})$. The bound consists of the conditional mutual information and an entropic correction term that quantifies the disturbance of the $B$-part by the recovery map.

v2: 18 pages, final version published in Annales Henri Poincaré

References in corpus (4)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Renyi generalizations of the conditional quantum mutual information
• Multivariate Trace Inequalities
• Relative Entropy Bounds on Quantum, Private and Repeater Capacities

Cited by in corpus (9)

• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• A chain rule for the quantum relative entropy
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Continuity of quantum entropic quantities via almost convexity
• Approximate quantum Markov chains
• Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Classical and quantum parts of conditional mutual information for open quantum systems
• Virtual Quantum Markov Chains