Smooth Entropy Bounds on One-Shot Quantum State Redistribution

We give upper and lower bounds on the amount of quantum communication required to perform the task of quantum state redistribution in a one-shot setting. In quantum state redistribution as considered in [Luo and Devetak (2009)] and [Devetak and Yard (2008)], there are 4 systems of interest: the A system held by Alice, the B system held by Bob, the C system that is to be transmitted from Alice to Bob, and the R system that holds a purification of the state in the ABC registers. Our bounds are in terms of smooth conditional min- and max- entropies, and the smooth max-information. The protocol for the upper bound has a clear structure, building on the work [Oppenheim (2008)]: it decomposes the quantum state redistribution task into two simpler quantum state merging tasks by introducing a coherent relay. There remains a gap between our upper and lower bounds. This gap vanishes in the independent and identical (iid) asymptotic limit and the remaining terms can then be rewritten as a quantum conditional mutual information, thus yielding an alterna-tive proof of optimality of this communication rate for iid asymptotic quantum state redistribution.