Complete Positivity on the Subsystems Level

Abstract

We provide a conceptually clear and technically simple presentation of certain subtleties of the concept of complete positivity of the quantum dynamical maps. The presentation is performed by addressing complete positivity of dynamics of certain subsystems of an open composite system, which is subject of a completely positive map. 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. A general algorithm for obtaining the Kraus form for a subsystem's dynamical map is designed for the finite-dimensional systems. As an illustrative example we consider a pair of mutually interacting qubits.