Information Flow Versus Divisibility for Non-invertible Dynamical Maps
We study the equivalence between information flow and completely positive divisibility—the two main approaches to Markovianity in quantum regime. Such equivalence is well known to hold for maps which are invertible. For non-invertible maps, the problem is more subtle. We show that for a class of so-called image non-increasing dynamical maps, the equivalence still holds true. Moreover, for qubit dynamics we show that the equivalence is universal, thus providing a comprehensive theory of quantum Markovianity at least for two dimensions. In the course of our proofs, we found certain mathematical restrictions on existence and impossibility of existence of completely positive trace-preserving projectors onto subspaces of finite-dimensional operator space. We illustrate our results with appropriate examples.
KeywordsOpen quantum systems Dynamical maps Divisible maps Markovian evolution
D.C. was supported by the Polish National Science Centre project 2018/30/A/ST2/00837. A.R. acknowledges the Spanish MINECO grants FIS2015-67411 and FIS2017-91460-EXP, the CAM research consortium QUITEMAD + grant S2013/ICE-2801, and the US Army Research Office through Grant No. W911NF-14-1-0103 for partial financial support.
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