Several key properties of quantum evolutions are characterized by divisibility of the corresponding dynamical maps. In particular, a Markovian evolution respects CP-divisibility, whereas breaking of P-divisibility provides a clear sign of non-Markovian effects. We analyze a class of evolutions which interpolates between CP- and P-divisible classes and is characterized by dissipativity -- a long known but so far not widely used formal concept to classify open system dynamics. By making a connection to stochastic jump unravellings of master equations, we demonstrate that there exists inherent freedom in how to divide the terms of the underlying master equation into the deterministic and jump parts for the stochastic description. This leads to a number of different unravelings, each one with a measurement scheme interpretation and highlighting different properties of the considered open system dynamics. Starting from formal mathematical concepts, our results allow us to get fundamental insights in open system dynamics and to ease their numerical simulations.
We analyze the relation between completely positive (CP) divisibility and the lack of information backflow for an arbitrary-not necessarily invertible-dynamical map. It is well known that CP divisibility always implies a lack of information backflow. Moreover, these two notions are equivalent for invertible maps. In this Letter, it is shown that for a map which is not invertible the lack of information backflow always implies the existence of a completely positive propagator which, however, needs not be trace preserving. Interestingly, for a wide class of image nonincreasing dynamical maps, this propagator becomes trace preserving as well, and hence, the lack of information backflow implies CP divisibility. This result sheds new light into the structure of the time-local generators giving rise to CP-divisible evolutions. We show that if the map is not invertible then positivity of dissipation/decoherence rates is no longer necessary for CP divisibility.
The are several nonequivalent notions of Markovian quantum evolution. In this paper we show that the one based on the so-called CP divisibility of the corresponding dynamical map enjoys the following stability property: the dynamical map ${\mathrm{\ensuremath{\Lambda}}}_{t}$ is CP divisible if and only if the second tensor power ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ is CP divisible as well. Moreover, the P divisibility of the map ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ is equivalent to the CP divisibility of the map ${\mathrm{\ensuremath{\Lambda}}}_{t}$. Interestingly, the latter property is no longer true if we replace the P divisibility of ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ by simple positivity and the CP divisibility of ${\mathrm{\ensuremath{\Lambda}}}_{t}$ by complete positivity. That is, unlike when ${\mathrm{\ensuremath{\Lambda}}}_{t}$ has a time-independent generator, positivity of ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ does not imply complete positivity of ${\mathrm{\ensuremath{\Lambda}}}_{t}$.
P-divisibility is a central concept in both classical and quantum non-Markovian processes; in particular, it is strictly related to the notion of information backflow. When restricted to a fixed commutative algebra generated by a complete set of orthogonal projections, any quantum dynamics naturally provides a classical stochastic process. It is indeed well known that a quantum generator gives rise to a
P-divisible quantum dynamics if and only if all its possible classical reductions give rise to divisible classical stochastic processes. However, this property does not hold if one operates a classical reduction of the quantum dynamical maps instead of their generators: As an example, for a unitary dynamics,
P-divisibility of its classical reduction is inevitably lost and the latter thus exhibits information backflow. Instead, for some important classes of purely dissipative qubit evolutions, quantum
P-divisibility always implies classical
P-divisibility and therefore excludes information backflow in both the quantum and classical scenarios. On the contrary, for a wide class of orthogonally covariant qubit dynamics, we show that loss of classical
P-divisibility originates from the classical reduction of a purely dissipative
P-divisible quantum dynamics as in the unitary case. Moreover, such an effect can be interpreted in terms of information backflow due to the coherences developed by the quantumly evolving classical state.
Divisibility of dynamical maps is a central notion in the study of quantum non-Markovianity, providing a natural framework to characterize memory effects via time-local master equations. In this work, we generalize the notion of divisibility of quantum dynamical maps from the Schrödinger to the Heisenberg picture. While the two pictures are equivalent at the level of physical predictions, we show that the divisibility properties of the corresponding dual maps are, in general, not equivalent. This inequivalence originates from the distinction between left and right generators of time-local master equations, which interchange roles under duality. We demonstrate that Schrödinger and Heisenberg divisibility are distinct concepts by constructing explicit dynamics divisible only in one picture. Furthermore, we introduce a quantifier for the violation of Heisenberg P-divisibility, analogous to the trace-distance-based measure of non-Markovianity, and provide it with an operational interpretation in terms of the guessing probability between effects. Our results show that Heisenberg divisibility is an independent witness of memory effects and highlight the need to consider both pictures when characterizing non-Markovian quantum dynamics.
The study of non-Markovian quantum processes has attracted significant interest in recent decades, giving rise to several competing notions of quantum non-Markovianity. These notes serve as an introduction to the topic for graduate students familiar with quantum mechanics and probability theory. Owing to the vastness of the literature, we focus on two prominent characterizations of quantum Markovianity based on the divisibility of quantum channels and monotonically decreasing state distinguishability. The correspondence between classical concepts (stochastic matrices, Chapman-Kolmogorov equation) and their quantum analogs (dynamical maps, CP-divisibility) is emphasized throughout.