Symmetry operations and critical behavior in classical to quantum stochastic processes

Recently it has been shown how to construct quantum analogs of classical stochastic processes by replacing random "which path" decisions with appropriately chosen superpositions. The resulting quantum processes are typically generating and destroying coherence at similar rates. Here we use this scheme to generate a large class of self-contained quantum extensions of a classical Markov chain process using symmetry operations. We show that the relaxation processes unfold very differently for the different quantum extensions. This is supported by monitoring the coherence, the probability of reaching the equilibrium, the decay of the number of domain walls, and the purity. We study the relation between the coherence measure based on the L1 norm and the speed of the relaxation process. For the coherence measure, we find that the finite-size scaling exists, where the corresponding critical exponents are different for short and long times.