Abstract
The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the simulability of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al. \cite{cirac}) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Ref.~\cite{Wolf2008}. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.
Highlights
The advent of quantum technologies opens questions aiming for deeper understanding of the fundamental physics beyond the idealized case of isolated quantum systems
From the slices shown above it can be noticed that every transition between the studied divisibility types is permitted. This is due to the existence of common borders between all combinations of divisibility sets; we can think of any continuous line inside the tetrahedron [6] as describing some quantum dynamical map
We studied the relations between different types of divisibility of time-discrete and time-continuous quantum processes, i.e. channels and dynamical maps, respectively
Summary
The advent of quantum technologies opens questions aiming for deeper understanding of the fundamental physics beyond the idealized case of isolated quantum systems. The well established Born-Markov approximation used to describe open quantum systems (e.g. relaxation process such as spontaneous decay) is of limited use and a more general framework of open system dynamics is demanded. Recent efforts in this area have given rise to relatively novel research subjects - non-markovianity and divisibility. We provide characterization of the subsets of qubit channels depending on their divisibility properties and implementation by means of dynamical maps.
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