Information preserving structures: A general framework for quantum zero-error information

TL;DR

This paper develops a comprehensive framework to classify and analyze all types of information that can be perfectly preserved in quantum systems, providing theoretical insights and efficient algorithms for identifying such structures.

Contribution

It introduces a unified operational framework for quantum information preservation, characterizes all perfect codes as matrix algebras, and offers algorithms to find preservation structures efficiently.

Findings

All perfect quantum codes have matrix algebra structure.

Preserved information can always be corrected.

Efficient algorithms exist for identifying preservation structures.

Abstract

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., "noiseless", "unitarily correctible", etc.) and introduce two new and natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational critera, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.