Almost any state of any amplitude Markov chain is recurrent

We introduce ''amplitude Markov chains" associated to the matrix elements, in a fixed basis, of a unitary operator (discrete quantum dynamics). We prove the amplitude analogue of the relation between the recurrence probability of a state of a classical Markov chain and its first return probability. This formula is then used to prove the universal property, mentioned in the title, which emphasizes a striking difference between the amplitude Markov chains and their classical analogues. This property is probably the statistical reflex of the reversibility of the quantum evolution. Finally note that, in the finite dimensional case, which is the most important one for the applications to quantum information, the word ''almost" in the title can be omitted.