Requiring linearity leads to complete positivity

TL;DR

This paper shows that requiring linearity in the reduced dynamics of an open quantum system ensures complete positivity, clarifying conditions for physical consistency in quantum dynamics.

Contribution

It establishes that linearity of reduced dynamics under arbitrary unitary evolutions and initial states guarantees complete positivity, a key property for quantum physical validity.

Findings

Linearity of reduced dynamics implies complete positivity.

Complete positivity depends on linearity conditions.

Conditions for physical consistency in quantum systems.

Abstract

The reduced dynamics of an open quantum system $S$, interacting with its environment $E$, is not completely positive, in general. In this paper, we demonstrate that if the two following conditions are satisfied, simultaneously, then the reduced dynamics is completely positive: (1) the reduced dynamics of the system is linear, for arbitrary system-environment unitary evolution $U$; and (2) the reduced dynamics of the system is linear, for arbitrary initial state of the system $\rho_S$.