Markov property and Strong additivity of von Neumann entropy for graded quantum systems

Recently the equality condition of the strong subadditivity of von Neumann entropy, simply called the strong additivity of entropy, has been studied. We show that the equivalence of the quantum Markov property invented by Accardi and the strong additivity of entropy is valid for graded quantum systems as well. However, the structure of Markov states for graded systems is different from that for tensor product systems. For three-composed graded systems there are U(1)-gauge invariant Markov states whose restriction to the pair of marginal subsystems is non-separable. We also prove that if a state satisfies the additivity of entropy for a bipartite graded system, then it is a product state. (Here we do not assume the evenness of states, which is a consequence.