Initial entanglement, entangling unitaries, and completely positive maps

TL;DR

This paper investigates how initial entanglement and specific unitaries influence whether the system's evolution remains completely positive, revealing that almost any initial entangled state can lead to CP dynamics with suitable unitaries, though such unitaries are rare.

Contribution

It analyzes the relationship between initial entangled states, applied unitaries, and resulting CP maps, providing a detailed characterization of the set of unitaries that preserve complete positivity.

Findings

Almost any initial entangled state admits infinitely many unitaries inducing CP dynamics.

The set of unitaries that generate CP maps from pure entangled states has zero measure.

The interplay between initial entanglement and unitaries determines the nature of the system's evolution.

Abstract

The problem of conditions on the initial correlations between the system and the environment that lead to completely positive (CP) or not-completely positive (NCP) maps has been studied by various authors. Two lines of study may be discerned: one concerned with families of initial correlations that induce CP dynamics under the application of an arbitrary joint unitary on the system and environment; the other concerned with specific initial states that may be highly entangled. Here we study the latter problem, and highlight the interplay between the initial correlations and the unitary applied. In particular, for almost any initial entangled state, one can furnish infinitely many joint unitaries that generate CP dynamics on the system. Restricting to the case of initial, pure entangled states, we obtain the scaling of the dimension of the set of these unitaries and show that it is of zero measure in the set of all possible interaction unitaries.