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A subalgebra M o of a von Neumann algebra M is called weakly sufficient with respect to a pair (φ, ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions to M o. The main result says that M o is weakly sufficient for (φ, ω) if and only if M o contains the Radon-Nikodym cocycle [Dφ,Dω] t. Other conditions(More)
The quantum analogue of the Fisher information metric of a probability simplex is searched and several Rie-mannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér-Rao type inequalities or the generalization of the Berry phase to mixed states. They are shown to be(More)
Variance and Fisher information are ingredients of the Cramér-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence(More)
The subject of this paper is a mathematical transition from the Fisher information of classical statistics to the matrix formalism of quantum theory. If the monotonicity is the main requirement, then there are several quantum versions parametrized by a function. In physical applications the minimal is the most popular. There is a one-to-one correspondence(More)
This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew-informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew-information is the Hessian of a quasi-entropy. The skew-information and some inequalities are extended to a(More)
Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden-Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation(More)