Compression of a message up to the information it carries is key to many tasks involved in classical and quantum information theory. Schumacher provided one of the first quantum compression schemes and several more general schemes have been developed ever since. However, the one-shot characterization of these quantum tasks is still under development, and often lacks a direct connection with analogous classical tasks. Here we show a new technique for the compression of quantum messages with the aid of entanglement. We devise a new tool that we call the convex split lemma, which is a coherent quantum analogue of the widely used rejection sampling procedure in classical communication protocols. As a consequence, we exhibit new explicit protocols with tight communication cost for quantum state merging, quantum state splitting. We also present a port-based teleportation scheme which uses less number of ports in presence of information about input.
Very recently our framework has found applications in several important settings in quantum network theory, such as a quantum version of the Gel’fand-Pinsker channel; the quantum broadcast channel and to obtain a new achievability bound on quantum state redistribution, in terms of smooth-max information and hypothesis testing relative entropy. Convex-split lemma has also found applications in the context of catalytic decoupling; privacy in quantum communication (the wiretap channel); a generalized quantum Slepian-Wolf result ; a bound for the important and consequential task of measurement compression using classical shared randomness and to obtain optimal bounds on the classical capacity of entanglement-assisted compound channels.
Given the broad applicability of the convex-split technique as exhibited in these recent works, we expect more applications in quantum network theory in the future.
Paper at : http://www.comp.nus.edu.sg/~rahul/allfiles/Coherent_rejection_sampling.pdf