Complete positivity on the subsystems level

TL;DR

This paper proves that subsystems of a composite open quantum system can only have completely positive dynamics if the initial state is a tensor product, providing an algorithm for Kraus form and illustrating with interacting qubits.

Contribution

It establishes a necessary and sufficient condition for subsystems to exhibit completely positive dynamics in open quantum systems.

Findings

Subsystems have CP dynamics iff initial state is tensor product.

Provides an algorithm to derive Kraus form for subsystems.

Illustrates with a pair of interacting qubits.

Abstract

We consider complete positivity of dynamics regarding subsystems of an open composite quantum system, which is subject of a completely positive dynamics. By "completely positive dynamics", we assume the dynamical maps called the completely positive and trace preserving maps, with the constraint that domain of the map is the whole Banach space of the system's density matrices. We provide a technically simple and conceptually clear proof for the subsystems' completely positive dynamics. Actually, we prove that every subsystem of a composite open system can be subject of a completely positive dynamics if and only if the initial state of the composite open system is tensor-product of the initial states of the subsystems. An algorithm for obtaining the Kraus form for the subsystem's dynamical map is provided. As an illustrative example we consider a pair of mutually interacting qubits. The presentation is performed such that a student with the proper basic knowledge in quantum mechanics should be able to reproduce all the steps of the calculations.