[Paper Review] Coding Theorems for Quantum Channels

A. S. Holevo|arXiv (Cornell University)|Sep 10, 1998

Quantum Computing Algorithms and ArchitectureComputer Science24 references158 citations

TL;DR

This paper provides a comprehensive, self-contained treatment of classical information transmission over quantum channels, establishing quantum analogues of classical coding theorems. It proves the achievability of the quantum entropy bound for general channels, including infinite and continuous alphabets, and derives the capacity of quantum Gaussian channels, highlighting fundamental differences from classical information theory while emphasizing recent advances and open problems in quantum information capacity.

ABSTRACT

The more than thirty years old issue of the (classical) information capacity of quantum communication channels was dramatically clarified during the last years, when a number of direct quantum coding theorems was discovered. The present paper gives a self contained treatment of the subject, following as much in parallel as possible with classical information theory and, on the other side, stressing profound differences of the quantum case. An emphasis is made on recent results, such as general quantum coding theorems including cases of infinite (possibly continuous) alphabets and constrained inputs, reliability function for pure state channels and quantum Gaussian channel. Several still unsolved problems are briefly outlined.

Motivation & Objective

• To establish a rigorous, self-contained framework for classical information transmission over quantum channels, parallel to classical information theory.
• To resolve long-standing questions about the information capacity of quantum channels, particularly the achievability of the entropy bound.
• To analyze quantum Gaussian channels and channels with infinite or continuous alphabets, extending classical coding theorems to the quantum domain.
• To highlight profound differences between classical and quantum information theory, especially in the context of superadditivity and reliability.
• To identify and outline key open problems in quantum information capacity, including the direct coding theorem for quantum state transmission.

Proposed method

• Uses the formalism of density operators and quantum states in Hilbert space to model quantum channels as affine mappings from input to output states.
• Applies the Born rule and quantum decision rules (positive operator-valued measures) to model quantum measurements and detection.
• Derives the quantum mutual information and uses it as the key quantity to determine channel capacity, analogous to classical mutual information.
• Applies asymptotic analysis and large-deviation techniques to prove coding theorems, particularly for memoryless and Gaussian channels.
• Considers the limit of continuous-time quantum Gaussian processes, modeling signals and noise via quantum stochastic processes with canonical commutation relations.
• Relies on the spectral theorem and decomposition of density operators into pure states to analyze superadditivity and capacity bounds.

Experimental results

Research questions

• RQ1 What is the maximum rate at which classical information can be reliably transmitted over a quantum channel?
• RQ2 Can the quantum entropy bound be achieved as a capacity limit, even for channels with infinite or continuous alphabets?
• RQ3 How does the capacity of a quantum Gaussian channel compare to its classical counterpart, and what is its exact expression?
• RQ4 What is the role of superadditivity in quantum channel capacity, and can it be exploited in practical coding schemes?
• RQ5 What are the fundamental limits on error probability in quantum channels, and how do they relate to the reliability function?

Key findings

• The quantum entropy bound is achievable as the channel capacity for general quantum channels, including those with infinite or continuous alphabets.
• The capacity of the quantum Gaussian channel is derived in the limit of large time, showing asymptotic equivalence to a parallel channel decomposition.
• The strict superadditivity of classical information in quantum channels is confirmed, implying that entangled inputs can increase capacity beyond the sum of individual channel capacities.
• For pure state channels, the reliability function is bounded below, and a lower bound on error probability is conjectured to exist, analogous to the classical sphere-packing bound.
• The direct coding theorem for reliable transmission of entire quantum states remains unproven, though a tentative converse has been proposed.
• The quantum Gaussian waveform channel lacks a rigorous proof of capacity derivation via classical reduction methods due to the non-diagonalizability of the quantum commutator form.