Quantum Complementarity and Operator Structures

TL;DR

This paper explores the operator algebra structures underlying quantum channels, error correction, and privacy, revealing their complementarity and deriving new dimension inequalities related to quantum state privatization.

Contribution

It introduces new operator algebra identities and dimension inequalities that deepen understanding of quantum channel complementarity and error correction.

Findings

Established operator structure identities for quantum channels and codes

Derived dimension inequalities as operator algebra uncertainty relations

Clarified the relationship between private and correctable operator algebras

Abstract

We establish operator structure identities for quantum channels and their error-correcting and private codes, emphasizing the complementarity relationship between the two perspectives. Relevant structures include correctable and private operator algebras, and operator spaces such as multiplicative domains and nullspaces of quantum channels and their complementary maps. For the case of privatizing to quantum states, we also derive related dimension inequalities that may be viewed as operator algebra uncertainty relations.