Simulating Open Quantum Systems with Hamiltonian Ensembles and the Nonclassicality of the Dynamics

Abstract

The incoherent dynamical properties of open quantum systems are generically attributed to an ongoing correlation between the system and its environment. Here, we propose a novel way to assess the nature of these system-environment correlations by examining the system dynamics alone. Our approach is based on the possibility or impossibility to simulate open-system dynamics with Hamiltonian ensembles. As we show, such (im)possibility to simulate is closely linked to the system-environment correlations. We thus define the nonclassicality of open-system dynamics in terms of the nonexistence of a Hamiltonian-ensemble simulation. This classifies any nonunital open-system dynamics as nonclassical. We give examples for open-system dynamics that are unital and classical, as well as unital and nonclassical.

Quantifying the nonclassicality of pure dephasing

“…Averaged dynamics of Hamiltonian ensembles. A HE {(p λ , H λ )} λ is a collection of Hermitian operators H λ acting on the same system [17,39], where each member Hamiltonian is drawn according to the probability distribution p λ ≥ 0. A system ρ 0 , isolated from any environment, is sent into a unitarily-evolving channel ρ λ (t) = U λ ρ 0 U † λ , with U λ = exp[−i H λ t/ ] for a chosen H λ according to p λ .…”

“…An autonomous system-environment arrangement is characterized by a time-independent total Hamiltonian H T and evolves unitarily with U T = exp[−i H T t]. We have shown that [17], if the total system ρ T (t) = U T ρ T (0) U † T remains at all times classically correlated between the system and its environment, displaying neither quantum discord [43,44] nor entanglement, then the reduced system dynamics ρ S (t) = E t {ρ S (0)} = Tr E [ρ T (t)] can be described by a time-independent HE equipped with a legitimate (i.e., non-negative and normalized to unity) probability distribution. Moreover, such ensemble description under classical environments in the absence of back-action has also been discussed in the literature [45][46][47].…”

“…Although being conceptually clear, as was shown in Ref. [17] for the example of an extended spin-boson model, this is technically highly nontrival in general; especially for high dimensions. For example, the closely related problem of random-unitary decomposition can in general merely be numerically implemented [18].…”

“…Alternatively, we propose to take the presence or absence of quantum correlations between system and environment as a signature for the quantum nature of the open system dynamics. As was shown recently [17], such presence or absence of nonclassical system-environment correlations is intimately linked to the (im)possibility to simulate the open system dynamics with a Hamiltonian ensemble (HE), which may thus serve as the classical strategy to witness the nonclassicality of the open system dynamics. HEs, which are also used to describe disordered quantum systems, attribute to each member of a collection of (time-independent) Hamiltonians a probability of occurrence, giving rise to an effective average dynamics.…”

Mathematical models of Markovian dephasing

“…Our formulation is closest to the one in [5]; in the case of maximal dephasing, it leads to the concept of a stable basis that recovers the "pointer basis" introduced by Zurek [8] and also embodies the simplest informationpreserving structure [9]. For more recent representative contributions on decoherence through random unitary models see for instance [10,11,12,13].…”