Quantum Communication Using Coherent Rejection Sampling

Abstract

Compression of a message up to the information it carries is key to many tasks involved in classical and quantum information theory. Schumacher [B. Schumacher, Phys. Rev. A 51, 2738 (1995)PLRAAN1050-294710.1103/PhysRevA.51.2738] provided one of the first quantum compression schemes and several more general schemes have been developed ever since [M. Horodecki, J. Oppenheim, and A. Winter, Commun. Math. Phys. 269, 107 (2007); CMPHAY0010-361610.1007/s00220-006-0118-xI. Devetak and J. Yard, Phys. Rev. Lett. 100, 230501 (2008); PRLTAO0031-900710.1103/PhysRevLett.100.230501A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, Proc. R. Soc. A 465, 2537 (2009)PRLAAZ1364-502110.1098/rspa.2009.0202]. 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, and quantum state redistribution (up to a certain optimization in the latter case). We also present a port-based teleportation scheme which uses a fewer number of ports in the presence of information about input.

Publicness, Privacy and Confidentiality in the Single-Serving Quantum Broadcast Channel

“…Then the result follows from the definition of the quantities. Proof of Lemma 14: Similar to Lemma 11 in [18], the proof follows by straightforward calculation as shown below:…”

Universal superposition codes: Capacity regions of compound quantum broadcast channel with confidential messages

“…With subsequent application of a covering principle, these codes were transformed to fulfill the security criterion in Lemma 13. As a possible alternative technique to generate such codes, we mention the rather recent "position decoding" and "convex split" techniques [5,6]. This approach proved to be powerful yet elegant and was successfully applied to determine "one-shot capacities" or "second order rates" in several scenarios.…”

“…A one-shot dynamic capacity theorem was derived for regions corresponding to tasks of common, public and private message transmission over the quantum channel in [27]. It would be interesting to see if the coding strategies used therein, derived from position based decoding (see [5,6]), can be used to design codes for the compound channel model. In the first section following this introduction, we introduce the notation used in this work.…”

One-shot coherence distillation with catalysts

“…To prove this theorem, we give two different protocols. The first one applies the convex-split lemma proposed in [52]. The second one uses an embezzling state modified from the entanglement embezzling state [51].…”

“…The convex-split lemma is first proposed in [52] as a mathematical tool inspired by classical communication theory. This lemma has also been applied to the study of catalytic decoupling [46] and quantifying resources with resource destroying maps in the assistance with catalysts [47].…”