[Paper Review] Principles of Quantum Communication Theory: A Modern Approach

Sumeet Khatri, Mark M. Wilde|arXiv (Cornell University)|Nov 9, 2020

Quantum Information and Cryptography311 references76 citations

TL;DR

A comprehensive, modern textbook detailing the mathematical foundations, channel theory, and protocol toolbox of quantum communication, including entropies, measures, and capacities. It structures quantum communication from algebraic tools to protocols and their performance limits.

ABSTRACT

This is a preliminary version of a book in progress on the theory of quantum communication. We adopt an information-theoretic perspective throughout and give a comprehensive account of fundamental results in quantum communication theory from the past decade (and earlier), with an emphasis on the modern one-shot-to-asymptotic approach that underlies much of today's state-of-the-art research in this field. In Part I, we cover mathematical preliminaries and provide a detailed study of quantum mechanics from an information-theoretic perspective. We also provide an extensive and thorough review of quantum entropies, and we devote an entire chapter to the study of entanglement measures. Equipped with these essential tools, in Part II we study classical communication (with and without entanglement assistance), entanglement distillation, quantum communication, secret key distillation, and private communication. In Part III, we cover the latest developments in feedback-assisted communication tasks, such as quantum and classical feedback-assisted communication, LOCC-assisted quantum communication, and secret key agreement.

Motivation & Objective

• Provide a rigorous mathematical foundation for quantum communication theory.
• Develop and organize the theory of quantum states, channels, and measurements.
• Explain fundamental information-processing tasks and their performance metrics.
• Introduce and analyze distinguishability and entropy measures for states and channels.
• Present capacity and distillation concepts across various quantum communication scenarios.

Proposed method

• Introduce finite-dimensional Hilbert spaces and linear operator formalism (state, channel representations).
• Develop tensor products, traces, transposes, and major operator inequalities essential for quantum information.
• Present SDPs and convex-analysis tools applicable to spectral, trace, and fidelity quantities.
• Define and analyze quantum channels via Choi, Kraus, and Stinespring representations and various channel classes.
• Present and formalize core quantum information tasks (teleportation, super-dense coding, hypothesis testing) and their quantitative guarantees.
• Systematically develop information measures (distances, fidelities, entropies, divergences) and their use in capacities and entanglement theory.

Experimental results

Research questions

• RQ1 How can one systematically formalize quantum states, channels, and measurements in a unified framework?
• RQ2 What are the fundamental information-processing tasks in quantum communication and their optimal performance limits?
• RQ3 How do various entropy and divergence measures quantify distinguishability, capacity, and entanglement in quantum systems?
• RQ4 What are the capacity limits (classical, quantum, private) of quantum channels under different assistance and constraint scenarios?
• RQ5 How can semi-definite programming and convex optimization be leveraged to compute key quantities like channel fidelities and entanglement measures?

Key findings

• Provide a structured, modern framework for quantum communication theory spanning states, channels, protocols, and measures.
• Detail Choi, Kraus, and Stinespring characterizations of quantum channels and their implications for communication tasks.
• Formulate and analyze core tasks such as quantum teleportation, dense coding, and various hypothesis-testing problems.
• Develop and apply a suite of information measures (trace distance, fidelity, diamond distance, various Rényi divergences, entropies) to channel and state discrimination and capacity analyses.
• Explore entanglement measures and generalized divergences with SDP formulations and discuss amortization and capacity results under LOCC, PPT, and other assistances.
• Provide extensive bibliographic notes and problem sets to support rigorous study of quantum communication theory.