Completely positive, trace-preserving, non-signalling, non-Markovian dynamics

Kraus' theorem, a cornerstone of quantum theory, characterizes all linear, completely positive maps that are allowed on single quantum states. In the continuous-time limit, such transformations may be viewed as Markovian dynamics, which is characterized by the Lindblad equation. In this work, we develop a general framework for completely positive, trace-preserving non-Markovian dynamics that also satisfy the non-signaling condition. We generalize the Kraus-style formulation to maps that act on multiple quantum states, whose time-continuous limit can model non-Markovian dynamics. The non-signaling constraint restricts these maps to be linear in their arguments. Within this framework, we a unique non-Markovian Lindblad-type equation that ensures completely positive evolution and express it in a physically transparent way, directly analogous to the familiar Markovian case.

We illustrate the formalism with explicit examples, including the dynamics of a qubit and harmonic oscillators coupled to a non-Markovian environment. For these systems, we show how both states and observables can be consistently constructed, providing a fully rigorous and physically meaningful description of non-Markovian quantum dynamics.