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A subalgebra Mo 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 Mo. The main result says that Mo is weakly sufficient for (φ, ω) if and only if Mo contains the RadonNikodym cocycle [Dφ,Dω]t. Other conditions are(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)
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)
The quantum analogue of the Fisher information metric of a probability simplex is searched and several Riemannian 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)
Entries of a random matrix are random variables but a random matrix is equivalently considered as a probability measure on the set of matrices. A simple example of random matrix has independent identically distributed entries. In this paper random unitary matrices are studied whose entries must be correlated. A unitary matrix U = (Uij) is a matrix with(More)
This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several(More)
Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert-Schmidt inner product). When both subalgebras are maximal Abelian, then the concept(More)
Quantum f -divergences are a quantum generalization of the classical notion of f divergences, and are a special case of Petz’ quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f -divergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and(More)
We prove that for the relative entropy of faithful normal states φ and ω on the von Neumann algebra M the formula S(φ,ω) = sup{ω(Λ)-logφ(/): h = h*eM] holds. In general von Neumann algebras the relative entropy was defined and investigated by Araki [1, 3]. After Lieb had proved the joint convexity of the relative entropy in the type / case [10] several(More)