Non-completely positive maps: properties and applications

TL;DR

This paper explores the properties and applications of non-completely positive (non-CP) maps in quantum systems, especially under initial system-environment correlations, and discusses how state preparation and noise influence their observation.

Contribution

It introduces a new classification of correlations using quantum discord, analyzes how different state preparation methods affect CP and non-CP maps, and demonstrates experimental realization of non-CP maps in quantum tomography.

Findings

Quantum discord classification shows asymmetry leading to ambiguous correlations.

Preparation procedures influence the positivity of reconstructed quantum maps.

Statistical noise can falsely indicate non-CP evolution in process tomography.

Abstract

We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.