Complete Positivity Violation in Higher-order Quantum Adiabatic Elimination

TL;DR

This paper investigates the limitations of adiabatic elimination in quantum systems, showing that higher-order expansions can violate complete positivity due to initial correlations, challenging previous assumptions of positivity preservation.

Contribution

It demonstrates that complete positivity can be violated at the fourth-order expansion in quantum adiabatic elimination, regardless of parametrization, due to initial correlations.

Findings

Complete positivity holds at second order but can be violated at fourth order.

Violation is caused by initial correlations in the system.

Positivity cannot be guaranteed in any parametrization of slow dynamics.

Abstract

When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complete positivity cannot be ensured in any parametrization. The violation stems from correlation in the initial state.