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

Abstract

The quantum Markov property is equivalent to the strong additivity of von Neumann entropy for graded quantum systems. The additivity of von Neumann entropy for bipartite graded systems implies the statistical independence of states. However, the structure of Markov states for graded systems is different from that for tensorproduct systems which have trivial grading. For three-composed graded systems we have U͑1͒-gauge invariant Markov states whose restriction to the marginal pair of subsystems is nonseparable.

The structure of strongly additive states and Markov triplets on the CAR algebra

“…As an element in A, ρAB is the density of the state ϕ • EAB. was proved in [11]. This inequality is equivalent with S(ρ, ρBC) − S(ρAB, ρB) ≥ 0.…”

“…The Markov states for CAR algebras were studied in [4]. The strong subadditivity of entropy on CAR systems was recently shown and it was proved that strong additivity is equivalent to Markov property in the case of even states, see [11]. For noneven states, a necessary and sufficient condition for equality in (SSA) was given in [6].…”

Entropy of quantum Markov states on Cayley trees

“…According to [24] the map T Y X is the unique Umegaki conditional expectation from A Y to A X with respect to the tracial state.…”

Entropy of Quantum Markov states on Cayley trees

“…According to [26] the map T Y X is the unique Umegaki conditional expectation from A Y to A X with respect to the tracial state.…”

“…is enough (in the sense of [43,26]) for the states ϕ⌈ AΛ n+1 and ϕ⌈ AΛ [n,n+1] . Hence, by a result of [26], we obtain…”