The aim of this paper is to show some properties of translation-invariant quantum Markov states introduced by Accardi and Frigerio. The KMS condition and the mean entropies of translation-invariant quantum Markov states are also considered.
Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of (1∣ , were inti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems. In (3), using our entanglement criterion for pure states (16) which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI† are entangled.