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Fumio Hiai

Publications and source records attributed to Fumio Hiai.

At least 55 records · Page 3Linked to original sources
arXiv2013

On matrix inequalities between the power means: counterexamples

We prove that the known sufficient conditions on the real parameters $(p,q)$ for which the matrix power mean inequality $((A^p+B^p)/2)^{1/p}\le((A^q+B^q)/2)^{1/q}$ holds for every pair of matrices $A,B>0$ are indeed best possible. The proof proceeds by constructing $2\times2$ counterexamples. The best possible conditions on $(p,q)$ for which $Φ(A^p)^{1/p}\leΦ(A^q)^{1/q}$ holds for every unital positive linear map $Φ$ and $A>0$ are also clarified.

math.FA↗
arXiv2013

Concavity of certain matrix trace and norm functions

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{Φ(A^p)^{1/2}Ψ(B^q)Φ(A^p)^{1/2}}^s$, where $Φ$ and $Ψ$ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form $||{Φ(A^p)σΨ(B^q)}^s||$ involving an operator mean $σ$. Carlen and Lieb's variational method is also used to improve the convexity property of norm functions $||Φ(A^p)^s||$.

math.FA↗
arXiv2012

Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited

Given a positive function $f$ on $(0,\infty)$ and a non-zero real parameter $θ$, we consider a function $I_f^θ(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^θ(X)$ in three matrices $A,B>0$ and $X$. In the literature $θ=\pm1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^θ$, thus unifying some known results for various quantum quantities.

math-ph↗
arXiv2011

Jensen and Minkowski inequalities for operator means and anti-norms

Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q<1 and the Minkowski functional. An interpolation theorem for the Schur multiplication is given in this setting. Two sections have been added to the previous version, devoted to means of severable variables and anti-norms.

math.FA↗
arXiv2011

Norm and anti-norm inequalities for positive semi-definite matrices

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and all symmetric norms, $\|g(A+B)\|^{1/m} \le \|g(A)\|^{1/m} + \|g(B)\|^{1/m}$. To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten $q$-norms for $q\in(0,1]$ and $q<0$. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let $f:[0,\infty) \to [0,\infty)$ be concave and $p\in(1,\infty)$. If $f^p(t)$ is superadditive, then $Tr f(A) \ge (\sum_{i=1}^m f^p(a_{ii}))^{1/p}$ for all positive $m\times m$ matrix $A=[a_{ij}]$. Furthermore, for the normalized trace $τ$, we consider functions $ϕ(t)$ and $f(t)$ for which the functional $A\mapstoϕ\circτ\circ f(A)$ is convex or concave, and obtain a simple analytic criterion.

math.FA↗
arXiv2010

Loewner matrices of matrix convex and monotone functions

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$. Similar characterizations are also obtained for matrix monotone functions on a finite interval $(a,b)$.

math.FA↗
arXiv2010

Strongly subadditive functions

Let f be a function defined on positive numbers. The subject is the trace inequality $Tr f(A) + Tr f(P_2AP_2) \le Tr f(P_{12}AP_{12}) + \Tr f(P_{23}AP_{23})$, where $A$ is a positive operator, $P_1,P_2,P_3$ are orthogonal projections such that $P_1+P_2+P_3=I$, $P_{12}=P_1+P_2$ and $P_{23}=P_2+P_3$. There are several examples of functions f satisfying the inequality (called (SSA)) and the case of equality is described.

math.FA↗
arXiv2009

Quantum hypothesis testing with group symmetry

The asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound and Stein's lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.

quant-ph↗
arXiv2009

A new approach to mutual information. II

A new concept of mutual pressure is introduced for potential functions on both continuous and discrete compound spaces via discrete micro-states of permutations, and its relations with the usual pressure and the mutual information are established. This paper is a continuation of the paper of Hiai and Petz in Banach Center Publications, Vol. 78.

math.PR↗
arXiv2008

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 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.

math-ph↗
arXiv2008

Free energy density for mean field perturbation of states of a one-dimensional spin chain

Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).

math-ph↗
arXiv2007

Orbital approach to microstate free entropy

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $χ_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $χ_orb$ with the usual free entropy $χ$ is shown. Moreover, the dimension counterpart $δ_{0,orb}$ of $χ_orb$ is discussed, and we obtain the relation of $δ_{0,orb}$ with the original free entropy dimension $δ_0$ with applications to $δ_0$ itself.

math.OA↗
arXiv2007

Quantum covariance, quantum Fisher information and the uncertainty principle

In this paper the relation between quantum covariances and quantum Fisher informations are studied. This study is applied to generalize a recently proved uncertainty relation based on quantum Fisher information. The proof given hereconsiderably simplifies the previously proposed proofs and leads to more general inequalities.

math-ph↗
arXiv2007

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 counter-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.

quant-ph↗
arXiv2007

Monotonicity for entrywise functions of matrices

We characterize real functions $f$ on an interval $(-α,α)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown.

math.FA↗
arXiv2006

A log-Sobolev type inequality for free entropy of two projections

We prove an inequality between the free entropy and the mutual free Fisher information for two projections, regarded as a free analog of the logarithmic Sobolev inequality. The proof is based on the random matrix approximation procedure via the Grassmannian random matrix model of two projections.

math.OA↗
arXiv2006

Notes on microstate free entropy of projections

We study the microstate free entropy of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of their free entropy (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to the microstate free entropy of projections.

math.OA↗