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We present a general formulation of Floquet states of periodically time-dependent open Markovian quasifree fermionic many-body systems in terms of a discrete Lyapunov equation. Illustrating the technique, we analyze periodically kicked XY spin-1/2 chain which is coupled to a pair of Lindblad reservoirs at its ends. A complex phase diagram is reported with reentrant phases of long range and exponentially decaying spin-spin correlations as some of the system's parameters are varied. The structure of phase diagram is reproduced in terms of counting nontrivial stationary points of Floquet quasiparticle dispersion relation.
We propose a simple complexity indicator of classical Liouvillian dynamics, namely the separability entropy, which determines the logarithm of an effective number of terms in a Schmidt decomposition of phase space density with respect to an arbitrary fixed product basis. We show that linear growth of separability entropy provides a stricter criterion of complexity than Kolmogorov-Sinai entropy, namely it requires that the dynamics be exponentially unstable, nonlinear, and non-Markovian.