Private algebras in quantum information and infinite-dimensional complementarity

TL;DR

This paper develops a generalized framework for private quantum codes using von Neumann algebras, establishing a broad complementarity theorem applicable to finite and infinite-dimensional quantum systems, with applications to bosonic and Gaussian channels.

Contribution

It introduces a new algebraic approach to private quantum codes and complementarity, extending the theory to infinite-dimensional settings and providing concrete examples with Gaussian channels.

Findings

Established a generalized complementarity theorem for private and correctable subalgebras.

Applied the framework to linear bosonic channels and Gaussian quantum channels.

Demonstrated the utility of von Neumann algebras in quantum information theory.

Abstract

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.