Sufficient Conditions and Constraints for Reversing General Quantum Errors

TL;DR

This paper investigates conditions for reversing quantum errors beyond completely positive maps, establishing when quantum error correction conditions can be applied while maintaining positivity of the quantum state.

Contribution

It extends quantum error correction theory to non-completely positive evolutions, providing necessary and sufficient conditions for reversibility and positivity.

Findings

Quantum error correction conditions may not apply if the map's output is not positive.

The paper proves that standard conditions can lead to non-positive outputs in non-CP maps.

Provides sufficient conditions ensuring the applicability of error correction conditions with positivity.

Abstract

Reversing the effects of a quantum evolution, for example as is done in error correction, is an important task for controlling quantum systems in order to produce reliable quantum devices. When the evolution is governed by a completely positive map, there exist reversibility conditions, known as the quantum error correcting code conditions, which are necessary and sufficient conditions for the reversibility of a quantum operation on a subspace, the code space. However, if we suppose that the evolution is not described by a completely positive map, necessary and sufficient conditions are not known. Here we consider evolutions that do not necessarily correspond to a completely positive map. We prove that the completely positive map error correcting code conditions can lead to a code space that is not in the domain of the map, meaning that the output of the map is not positive. A corollary to our theorem provides a class of relevant examples. Finally, we provide a set of sufficient conditions that will enable the use of quantum error correcting code conditions while ensuring positivity.