Completely positive master equation for arbitrary driving and small level spacing
Abstract
Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously noncompletelypositive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultraweak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast timedependent driving. Here we show that a recently derived Markovian coarsegrained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the timedependent CGME, is thus suitable for the analysis of fast operations in gatemodel quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarsegrained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any timedependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.01095
 Bibcode:
 2019arXiv190801095M
 Keywords:

 Quantum Physics;
 Mathematical Physics
 EPrint:
 62 pages, 16 figures