Generalized Multiplicative Domains and Quantum Error Correction

TL;DR

This paper introduces generalized multiplicative domains for completely positive maps, extending classical concepts and providing a new algebraic framework for understanding quantum error correction.

Contribution

It defines and characterizes generalized multiplicative domains for quantum channels, extending Choi's work, and offers a novel algebraic approach to quantum error correction.

Findings

Extended Choi's characterization to generalized domains.

Provided a new algebraic description of quantum error-correcting codes.

Connected multiplicative domains with quantum error correction structures.

Abstract

Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi's characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.