[Paper Review] Quantum Shannon Theory

John Preskill|arXiv (Cornell University)|Apr 25, 2016

Quantum Information and Cryptography40 references22 citations

TL;DR

This paper presents a comprehensive overview of Quantum Shannon Theory, unifying quantum information theory with classical information theory to establish fundamental limits on quantum communication and data processing. It introduces key protocols like quantum teleportation, superdense coding, and quantum error correction, demonstrating that quantum channels can transmit quantum information reliably up to their capacity, defined by the quantum capacity formula C = max_{ρ} I(A;B) where I is the coherent information.

ABSTRACT

This is the 10th and final chapter of my book on Quantum Information, based on the course I have been teaching at Caltech since 1997. An earlier version of this chapter (originally Chapter 5) has been available on the course website since 1998, but this version is substantially revised and expanded. Topics covered include classical Shannon theory, quantum compression, quantifying entanglement, accessible information, and using the decoupling principle to derive achievable rates for quantum protocols. This is a draft, pre-publication copy of Chapter 10, which I will continue to update. See the URL on the title page for further updates and drafts of other chapters, and please send me an email if you notice errors.

Motivation & Objective

• To unify classical and quantum information theory by establishing a theoretical framework for quantum communication and data processing.
• To identify fundamental limits on the transmission of quantum information over noisy quantum channels.
• To develop protocols such as quantum teleportation and superdense coding that exploit quantum entanglement for enhanced communication.
• To formalize the concept of quantum capacity and derive bounds on reliable quantum communication rates.
• To provide a pedagogical foundation for researchers in quantum information science through a structured, accessible treatment of core concepts.

Proposed method

• Formalizing quantum communication using the quantum channel model, where input and output systems are described by density operators.
• Applying the coherent information I(A;B) = S(B) - S(AB) as a key measure to quantify the quantum capacity of a channel.
• Introducing the quantum reverse Shannon theorem to relate classical and quantum communication resources under shared entanglement.
• Using quantum error-correcting codes to protect quantum states against decoherence, enabling reliable transmission over noisy channels.
• Demonstrating that quantum teleportation enables perfect state transfer using one ebit of entanglement and two classical bits.
• Applying the concept of entanglement distillation to purify noisy entangled states for use in quantum communication protocols.

Experimental results

Research questions

• RQ1 What is the maximum rate at which quantum information can be reliably transmitted over a noisy quantum channel?
• RQ2 How can quantum entanglement be used to enhance classical and quantum communication beyond classical limits?
• RQ3 What is the relationship between classical communication, quantum communication, and shared entanglement in quantum networks?
• RQ4 How do quantum error-correcting codes enable reliable quantum communication in the presence of noise?
• RQ5 What are the fundamental limits of quantum communication when shared entanglement is available?

Key findings

• The quantum capacity of a channel is given by the regularization of the coherent information, providing a fundamental upper bound on reliable quantum communication rates.
• Quantum teleportation enables the transfer of an unknown quantum state using only two classical bits and one ebit of entanglement, achieving perfect state transfer.
• Superdense coding allows the transmission of two classical bits using only one qubit and one ebit of entanglement, demonstrating the power of quantum resources.
• The quantum reverse Shannon theorem establishes that entanglement and classical communication can simulate any quantum channel, with rates determined by the entanglement cost and classical capacity.
• Quantum error-correcting codes can protect quantum information against arbitrary local noise, enabling fault-tolerant quantum computation and communication.
• Entanglement distillation protocols can purify mixed entangled states into maximally entangled states, which are essential for long-distance quantum communication.