A Resource Framework for Quantum Shannon Theory
I. Devetak, A. W. Harrow, A. Winter
TL;DR
This paper introduces a unified resource-based framework for quantum Shannon theory, focusing on bipartite, unidirectional, memoryless settings. It formalizes information-processing resources and resource inequalities, constructing a general resource calculus that enables composition, reduction, and coherification to derive and relate coding theorems. By recasting major results (Schumacher compression, HSW, NSD coding, state merging, teleportation, entanglement distillation, and more) as resource inequalities, the authors build a scalable family of protocols and two-dimensional trade-off curves that capture optimal rates for multiple resources simultaneously. The approach clarifies structural dependencies among coding theorems, provides systematic proofs of new trade-offs, and suggests paths to single-letter characterizations, with potential extensions to more complex, multi-user, or bidirectional scenarios. The framework thus offers both a conceptual unification and practical tools for deriving and understanding quantum Shannon limits in a broad range of settings.
Abstract
Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional and memoryless. We formalize two principles that have long been tacitly understood. First, we describe how the Church of the larger Hilbert space allows us to move flexibly between states, channels, ensembles and their purifications. Second, we introduce finite and asymptotic (quantum) information processing resources as the basic objects of quantum Shannon theory and recast the protocols used in direct coding theorems as inequalities between resources. We develop the rules of a resource calculus which allows us to manipulate and combine resource inequalities. This framework simplifies many coding theorem proofs and provides structural insights into the logical dependencies among coding theorems. We review the above-mentioned basic coding results and show how a subset of them can be unified into a family of related resource inequalities. Finally, we use this family to find optimal trade-off curves for all protocols involving one noisy quantum resource and two noiseless ones.