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arXiv:1901.05746 · Reviews in Mathematical Physics

On Howland time-independent formulation of CP-divisible quantum evolutions

Krzysztof Szczygielski, Robert Alicki

Published Dec 26, 2019• 4 citations
arXivHTMLPDF

Abstract

We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.

DOI
10.1142/s0129055x2050021x

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A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.
The description of the dynamics of an open quantum system in the presence of initial correlations with the environment needs different mathematical tools than the standard approach to reduced dynamics, which is based on the use of a time-dependent completely positive trace preserving (CPTP) map. Here, we take into account an approach that is based on a decomposition of any possibly correlated bipartite state as a conical combination involving statistical operators on the environment and general linear operators on the system, which allows one to fix the reduced-system evolution via a finite set of time-dependent CPTP maps. In particular, we show that such a decomposition always exists, also for infinite dimensional Hilbert spaces, and that the number of resulting CPTP maps is bounded by the Schmidt rank of the initial global state. We further investigate the case where the CPTP maps are semigroups with generators in the Gorini-Kossakowski-Lindblad-Sudarshan form; for two simple qubit models, we identify the positivity domain defined by the initial states that are mapped into proper states at any time of the evolution fixed by the CPTP semigroups.
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We study a system qubit repeatedly interacting with the same environmental qubit, with a reservoir acting on the environment between collisions via a completely positive, trace-preserving map. We show that complete suppression of system--environment correlations uniquely requires a full environmental reset, recovering a semi group dynamics with a time-independent Gorini--Kossakowski--Sudarshan--Lindblad generator, whereas a partial reset yields a continuous transition between Markovian and non-Markovian regimes governed by a single dimensionless relaxation parameter. For a resonant excitation-exchange interaction, we obtain exact closed-form expressions for the Bloch-vector dynamics for both a generalized depolarizing channel and a generalized amplitude-damping channel acting as the reservoir-induced map. Using the Breuer--Laine--Piilo measure and a Choi-matrix CP-divisibility witness, we identify three distinct dynamical regimes across the parameter space: CP-divisible Markovian dynamics, CP-indivisible but P-divisible dynamics, and non-P-divisible non-Markovian dynamics. The boundaries between these regimes, and the structural differences between uniform and anisotropic environmental relaxation, are characterized numerically.