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Quantum devices are subject to natural decay. We propose to study these decayprocesses as the Markovian evolution of quantum channels, which leads us todynamical semigroups of superchannels. A superchannel is a linear map that mapsquantum channels to quantum channels, while satisfying suitable consistencyrelations. If the input and output quantum channels act on the same space, thenwe can consider dynamical semigroups of superchannels. No useful constructivecharacterization of the generators of such semigroups is known. We characterizethese generators in two ways: First, we give an efficiently checkable criterionfor whether a given map generates a dynamical semigroup of superchannels.Second, we identify a normal form for the generators of semigroups of quantumsuperchannels, analogous to the GKLS form in the case of quantum channels. Toderive the normal form, we exploit the relation between superchannels andsemicausal completely positive maps, reducing the problem to finding a normalform for the generators of semigroups of semicausal completely positive maps.We derive a normal for these generators using a novel technique, which appliesalso to infinite-dimensional systems. Our work paves the way to a thoroughinvestigation of semigroups of superchannels: Numerical studies become feasiblebecause admissible generators can now be explicitly generated and checked. Andanalytic properties of the corresponding evolution equations are now accessiblevia our normal form.

Quantum and classical dynamical semigroups of superchannels and semicausal channels