Fumio Hiai

The proper formula for relative entropy and its asymptotics in quantum probability

Umegakis relative entropyS(ω,ϕ)=TrDω(logDω−logDϕ) (of states ω and ϕ with density operatorsDω andDϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegakis one. As a byproduct, the inequality TrA logAB ≧TrA(logA+logB) is obtained for positive definite matricesA andB.

Log majorization and complementary Golden-Thompson type inequalities

Abstract We obtain a log majorization result for power means of positive semidefinite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to the Golden-Thompson one. Other log majorization results are also obtained. We give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations.

The Golden-Thompson trace inequality is complemented

We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB)2/p⩽ Tr eA+B holds for all p > 0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p−1Tr X log Yp/2XpYp/2⩽ Tr X(log X+log Y) ⩽ p−1Tr X log Xp/2YpXp/2 for all p > 0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.

On the Quantum Rényi Relative Entropies and Related Capacity Formulas

Following Csiszárs approach in classical information theory, it is shown that the quantum α-relative entropies with parameter α ∈ (0,1) can be represented as generalized cutoff rates, and hence a direct operational interpretation of the quantum α-relative entropies are provided. It is also shown that various generalizations of the Holevo capacity, defined in terms of the α-relative entropies, coincide for the parameter range α ∈ (0,2], and an upper bound on the one-shot ε-capacity of a classical-quantum channel in terms of these capacities is given.

Introduction to matrix analysis and applications

Fundamentals of operators and matrices.- Mappings and algebras.- Functional calculus and derivation.- Matrix monotone functions and convexity.- Matrix means and inequalities.- Majorization and singular values.- Some applications.

AMENABILITY AND STRONG AMENABILITY FOR FUSION ALGEBRAS WITH APPLICATIONS TO SUBFACTOR THEORY

The relationship between index theory and random walks on fusion algebras is discussed. Popas notion of amenability is reformulated as a property of fusion algebras, and several equivalent conditions of amenability are obtained. A ratio limit theorem is proved as a characterization of amenability. A number of conditions, all equivalent to Popas notion of strong amenability in the case of subfactors, of a pair of a fusion algebra and a probability measure are proposed, and their relationship is studied from the viewpoint of random walks and entropic densities. Fusion algebra homomorphisms and free products of fusion algebras are also discussed.

Log-majorizations and norm inequalities for exponential operators

Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii–Wielandt and the Gelfand– Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm inequalities of Golden–Thompson type and its complementary type for exponential operators on a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula as well as for that involving operator means.

Operator log-convex functions and operator means

We study operator log-convex functions on (0, ∞), and prove that a continuous nonnegative function on (0, ∞) is operator log-convex if and only if it is operator monotone decreasing. Several equivalent conditions related to operator means are given for such functions. Operator log-concave functions are also discussed.

Error exponents in hypothesis testing for correlated states on a spin chain

We study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum Chernoff and Hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain. Our results apply to states satisfying a certain factorization property; typical examples are the global Gibbs states of translation-invariant finite-range interactions as well as certain finitely correlated states.

Large deviations and Chernoff bound for certain correlated states on a spin chain

In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.