Maps for general open quantum systems and a theory of linear quantum error correction

TL;DR

This paper develops a generalized theory of quantum error correction applicable to all linear maps describing open quantum system dynamics, including non-completely positive maps, and provides geometric tools for analyzing their positivity domains.

Contribution

It extends quantum error correction theory to non-CP maps and introduces a geometric characterization of their positivity domains.

Findings

Quantum subdynamics can be described by Hermitian maps regardless of initial states.

The generalized error correction theory applies to all linear maps, not just CP maps.

The positivity domain of general linear maps is convex, enabling boundary analysis.

Abstract

We show that quantum subdynamics of an open quantum system can always be described by a Hermitian map, irrespective of the form of the initial total system state. Since the theory of quantum error correction was developed based on the assumption of completely positive (CP) maps, we present a generalized theory of linear quantum error correction, which applies to any linear map describing the open system evolution. In the physically relevant setting of Hermitian maps, we show that the CP-map based version of quantum error correction theory applies without modifications. However, we show that a more general scenario is also possible, where the recovery map is Hermitian but not CP. Since non-CP maps have non-positive matrices in their range, we provide a geometric characterization of the positivity domain of general linear maps. In particular, we show that this domain is convex, and that this implies a simple algorithm for finding its boundary.