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- Dénes Petz, Csaba Sudár
- 1996

The quantum analogue of the Fisher information metric of a probability simplex is searched and several Rie-mannian metrics on the set of positive deenite density matrices are studied. Some of them appeared in the literature in connection with Cram er-Rao type inequalities or the generalization of the Berry phase to mixed states. They are shown to be… (More)

- Denes Petz, D. Petz
- 1986

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)

- Dénes Petz
- 2002

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)

- Dénes Petz, Catalin Ghinea
- 2010

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)

- F. Hiai, M. Mosonyi, D. Petz, C. Beny
- 2010

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 distin-guishability 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)

- Mark Fannes, Dénes Petz
- 2001

Let H 0 be an arbitrary self-adjoint n × n matrix and H(n) be an n × n (random) Wigner matrix. We show that t → Tr exp(H(n)−itH 0) is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that t → τ (exp(a − itb)) is positive definite whenever the noncommutative random… (More)

- Dénes Petz, Sándor Szabó
- 2008

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)

- Dénes Petz
- 2002

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)

- Anna Jenčová, Dénes Petz
- 2008

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)