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 non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained 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 time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model 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 coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence 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.

Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation

“…The Born-Markov approximation is justified if τ r τ c . For any given initial state of the system, the error bound of the Redfield equation can be expressed in terms of the properties of the heat bath only, [47]…”

“…Error bound exists for the state error added by coarse-graining, and increases linearly with the coarsegraining time. [47] 3.2 Rotating wave and CGSE approximations The RWA follows from Eqs. 23, 25 and 26 in the limit T 0 1/|ω nm |, ∀ω nm = 0.…”

“…18, because it is a higher level of approximation relative to the Redfield equation, in terms of accuracy. [22,47] So we can directly compare GAME and TDC-Redfield equation, along the diagonal arrow in the diagram in Fig. 2.…”

“…At g tot = 0.92 and 0.0092, τ opt = 1.3T f m and 16T f m , respectively, in good agreement with square root dependence on g, which makes it consistent with the coarse-graining approximation. [45,47] The corresponding values of √ τ c τ r (the optimum coarse-graining time in the CGSE, see Sec. 7.4) are approximately factor of 10 smaller than the values we find.…”

“…Since this time scales exponentially with system size, the RWA does not work well for large many-body quantum systems that generally have exponentially suppressed level spacings with increasing size of the system. More recently, coarse-grained GKSL master equations have been derived from first principles [45][46][47] capable of capturing correlation effects that the RWA cannot. Another notable completely positive approximation is the partial-RWA (PRWA).…”