From Markovian semigroup to non-Markovian quantum evolution

Abstract

We provided a class of legitimate memory kernels leading to completely positive trace preserving dynamical maps. Our construction is based on a simple normalization procedure. Interestingly, when applied to the celebrated Wigner-Weisskopf theory it gives the standard Markovian evolution governed by the local master equation.

Admissible memory kernels for random unitary qubit evolution

“…This approach resembles very much the semi-Markov construction [23,28]: for any f (t) ≥ 0 satisfying ∞ 0 f (t)dt ≤ 1 the memory kernel (40) with…”

“…The structure and the properties of (4) were carefully analyzed in [20][21][22][23][24][25][26][27][28][29]. In particular the generalization of Markovian evolution to the so-called semi-Markov was investigated within the memory kernel approach by Budini [21] and Breuer and Vacchini [23] (see also discussion in [28]). …”

Interaction free and decoherence free states

“…In this section we generalize the IFE states defined for unitary evolution to the evolution corresponding to quantum Markovian semigroup [19,20,21,22,23] (see also recent review [24]). A general quantum evolution of a system living in the Hilbert space H is described by a dynamical map, that is, a family of completely positive trace-preserving maps (CPTP) Λ t : B(H) → B(H) such that Λ 0 = 1l (an identity map).…”

Exploring complete positivity in hierarchy equations of motion

“…However, the addition of memory to the system dynamics gives rise to substantial complications [17] in particular with respect to complete positivity. Most approaches to this problem focus on integral equations over a memory kernel and for certain classes like semi-Markov processes [18,19], collision-model-based approaches [20] or continuous time quantum random walks [21] conditions for complete positivity have been found. Recently, a more general set of conditions for CP dynamics has been obtained [22] and eventually the post-Markovian master equation [23] is capable to describe CP non-Markovian dynamics by interpolating between the generalised measurement interpretation of the Kraus operator sum and the notion of a continuous measurement for Markovian processes but ultimately also imposes conditions on the memory kernel.…”