Superadditivity of Quantum Relative Entropy for General States

The property of superadditivity of the quantum relative entropy states that, in a bipartite system H-AB = H-A circle times H-B, for every density operator rho AB, one has D(rho AB parallel to sigma(A) circle times sigma B) >= D(rho(A)parallel to sigma(A)) + D(rho B parallel to sigma B). In this paper, we provide an extension of this inequality for arbitrary density operators sigma(AB). More specifically, we prove that alpha(sigma(AB)) center dot D(rho(AB)parallel to sigma(AB)) >= D(rho(A)parallel to sigma(A))+ D(rho B parallel to sigma(B)) holds for all bipartite states rho(AB) and sigma(AB), where alpha(sigma(AB)) = 1 + 2 parallel to sigma(-1/2)(A) circle times sigma(-1/2)(B) sigma(AB) sigma(-1/2)(A) circle times sigma(-1/2)(B) -1(AB)parallel to infinity.