Results 1  10
of
30
Local asymptotic normality in quantum statistics
, 2006
"... The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕn θ0+u / √ n consisting of joint states of n identically pr ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕn θ0+u / √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all “local parameters ” u ∈ Rm such that θ = θ0 + u / √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ Rm. In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and show that weak and strong convergence are equivalent in the case of finite number of parameters for experiments based on type I algebras with discrete center. For reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory.
monotonicity and wigneryanasedyson metrics,” Infinite Dimensional Analysis, Quantum Probability and
 Related Topics
, 2004
"... We show that, for each value of α ∈ (−1, 1), the only Riemannian metrics on the space of positive definite matrices for which the ∇(α) and ∇(−α) connections are mutually dual are matrix multiples of the WignerYanaseDyson metric. If we further impose that the metric be monotone, then this set is re ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
We show that, for each value of α ∈ (−1, 1), the only Riemannian metrics on the space of positive definite matrices for which the ∇(α) and ∇(−α) connections are mutually dual are matrix multiples of the WignerYanaseDyson metric. If we further impose that the metric be monotone, then this set is reduced to scalar multiples of the WignerYanaseDyson metric. 1
Quantum covariance, quantum Fisher information and the uncertainty principle
, 2007
"... 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 here considerably simplify the previously proposed proofs and leads to m ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
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 here considerably simplify the previously proposed proofs and leads to more general inequalities.
From quasientropy to skew information
, 2009
"... This paper gives an overview about particular quasientropies, generalized quantum covariances, quantum Fisher informations, skewinformations and their relations. The point is the dependence on operator monotone functions. It is proven that a skewinformation is the Hessian of a quasientropy. The ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
This paper gives an overview about particular quasientropies, generalized quantum covariances, quantum Fisher informations, skewinformations and their relations. The point is the dependence on operator monotone functions. It is proven that a skewinformation is the Hessian of a quasientropy. The skewinformation and some inequalities are extended to a von Neumann algebra setting.
From fdivergence to quantum quasientropies and their use
"... Csiszar's fdivergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasientropy which is related to some oth ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Csiszar's fdivergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasientropy which is related to some other important concepts as covariance, quadratic costs, Fisher information, CramerRao inequality and uncertainty relation. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting, but at the very end the von Neumann algebra approach is sketched shortly.
On the geometry of generalized Gaussian distributions
, 2007
"... In this paper we consider the space of those probability distributions which maximize the qRényi entropy. These distributions have the same parameter space for every q, and in the q = 1 case these are the normal distributions. Some methods to endow this parameter space with Riemannian metric is pre ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
In this paper we consider the space of those probability distributions which maximize the qRényi entropy. These distributions have the same parameter space for every q, and in the q = 1 case these are the normal distributions. Some methods to endow this parameter space with Riemannian metric is presented: the second derivative of the qRényi entropy, Tsallisentropy and the relative entropy give rise to a Riemannian metric, the Fisherinformation matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovrić, MinOo and Ruh. These metrics are different therefore our differential geometrical calculations based on a unified metric, which covers all the above mentioned metrics among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors and scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metric in quantum information geometry can be easily recovered from classical metrics.
From quasientropy to various quantum information quantities
 PUBL. RIMS KYOTO UNIV. 48(2012), 525–542.
, 2012
"... The subject is the applications of the use of quasientropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the fdivergence, monotone metrics, covariance an ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
The subject is the applications of the use of quasientropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the fdivergence, monotone metrics, covariance and the χ 2divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.
Riemannian geometry on positive definite matrices
, 2008
"... The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K φ ∑ D (H, K) = i,j φ(λi, λj) −1Tr PiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H, K are tangent vectors. For such kernel metrics th ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K φ ∑ D (H, K) = i,j φ(λi, λj) −1Tr PiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H, K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pullback of a kernel metric under a mapping D ↦ → G(D) is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the FisherRao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case φ(x, y) = M(x, y) θ is mostly studied when M(x, y) is a mean of the positive numbers x and y. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases.