Remarks on Quantum Markov States

The definition of a quantum Markov state was given by Accardi in 1975. For the classical case, this definition gives hidden Markov measures, which, generally speaking, are not Markov measures. We can use a nonnegative transfer matrix to define a Markov measure. We use a positive semidefinite transfer matrix and select a class of quantum Markov states (in the sense of Accardi) on the inductive limit of the $C^*$-algebras $M_{d^n}$. An entangled quantum Markov state in the sense of Accardi and Fidaleo is a quantum Markov state in our sense. For the case where the transfer matrix has rank $1$, we calculate the eigenvalues and the eigenvectors of the density matrices determining the quantum Markov state. The sequence of von Neumann entropies of the density matrices of this state is bounded.