[Paper Review] Coding Theorems of Quantum Information Theory

Andreas Winter|ArXiv.org|Jul 24, 1999

Quantum Computing Algorithms and Architecture52 references100 citations

TL;DR

This paper establishes fundamental coding theorems and strong converses for quantum information theory, proving the quantum source coding theorem with a strong converse and providing a new proof of the Holevo bound. It derives the capacity region for the quantum multiple access channel and proposes a program to assign operational meaning to quantum conditional entropy, advancing the mathematical foundations of quantum information processing and communication.

ABSTRACT

Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the extended model of nonstationary channels. As a by-product we obtain a new proof of the famous Holevo bound. Then multi-user systems are investigated, and the capacity region for the quantum multiple access channel is determined. The last chapter contains a preliminary discussion of some models of compression of correlated quantum sources, and a proposal for a program to obtain operational meaning for quantum conditional entropy. An appendix features the introduction of a notation and calculus of entropy in quantum systems.

Motivation & Objective

• To establish the strong converse for quantum source coding, extending Schumacher's quantum coding theorem.
• To provide a new proof of the Holevo bound for classical information transmission over quantum channels.
• To determine the capacity region of the quantum multiple access channel under general nonstationary and memoryless models.
• To initiate a program for assigning operational meaning to quantum conditional entropy through correlated source compression models.
• To develop a unified mathematical framework for quantum entropy and information using $*$-algebras and quantum operations.

Proposed method

• Uses typical subspaces and quantum typicality to analyze asymptotic quantum source coding, extending Schumacher's approach.
• Applies the method of types and large deviation bounds to prove the strong converse for memoryless quantum channels.
• Employs quantum operations and compatible $*$-subalgebras to define conditional entropy and mutual information in non-commutative settings.
• Derives the capacity region of the quantum multiple access channel using outer bounds and constructive coding schemes.
• Introduces a novel formalism for quantum entropy and divergence using operator algebras and subalgebra compatibility.
• Leverages Fano-type inequalities and information-theoretic duality to bound error probabilities and uncertainty in quantum measurements.

Experimental results

Research questions

• RQ1 What is the operational meaning of quantum conditional entropy, and can it be given a physical interpretation through source coding?
• RQ2 What is the exact capacity region for the quantum multiple access channel, and how does it depend on entanglement and input constraints?
• RQ3 Can the Holevo bound be derived independently from the quantum channel coding theorem, and what are the implications for classical information transmission?
• RQ4 Under what conditions does the strong converse hold for nonstationary quantum channels, and how does it relate to error exponents?
• RQ5 How can quantum sources with correlated or side-informed structures be optimally compressed, and what role does separability play in this?

Key findings

• The strong converse for quantum source coding is proven, showing that any rate exceeding the von Neumann entropy leads to exponentially vanishing fidelity in the asymptotic limit.
• A new proof of the Holevo bound is derived as a by-product of the channel coding analysis, establishing a fundamental limit on classical information transmission over quantum channels.
• The capacity region of the quantum multiple access channel is fully characterized, showing that the sum rate cannot exceed the sum of individual capacities under certain constraints.
• The conditional entropy $ H({rak{X}}|{rak{Y}}) $ is shown to be non-negative when the joint state is separable with respect to compatible subalgebras, supporting a conjecture on the operational meaning of quantum conditional entropy.
• The Fano inequality is generalized to the quantum setting, showing that uncertainty in a quantum observable is bounded by the error probability and the logarithm of the number of outcomes.
• The paper establishes that knowledge about a system reduces uncertainty, formalized as $ H( ho| heta) o H( ho| heta imes ext{additional knowledge}) $, which generalizes the classical data processing inequality to quantum operations.