# Quantum Hellinger distances revisited

@article{Pitrik2019QuantumHD,
title={Quantum Hellinger distances revisited},
author={J. Pitrik and D'aniel Virosztek},
journal={Letters in Mathematical Physics},
year={2019},
pages={1-14}
}
• Published 2019
• Mathematics, Physics
• Letters in Mathematical Physics
This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form $$\phi (A,B)=\mathrm {Tr} \left( (1-c)A + c B - A \sigma B \right) ,$$ ϕ ( A , B ) = Tr ( 1 - c ) A + c B - A σ B , where $$\sigma$$ σ is an arbitrary Kubo–Ando mean, and $$c \in (0,1)$$ c ∈ ( 0 , 1 ) is the weight of $$\sigma… Expand 5 Citations Generalized Hellinger metric and Audenaert's in-betweenness • Mathematics • 2020 Abstract Let σ and τ be Kubo-Ando means [1] . In this article we consider the in-betweenness property [2] for τ with respect to the generalized Hellinger metric induced by σ. That is, we show thatExpand Divergence Radii and the Strong Converse Exponent of Classical-Quantum Channel Coding With Constant Compositions • Computer Science, Physics • IEEE Transactions on Information Theory • 2021 It is shown that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényu divergences, has the same operational interpretation in the strong converse problem of constant composition classical-quantum channel coding. Expand Quantum divergences with p-power means • Mathematics • 2021 Abstract We study several properties of a family of quantum divergences with p-power means. We also show that the Hellinger distance with log Euclidean mean and the quantum Jensen-Shannon divergenceExpand Some notes on quantum Hellinger divergences with Heinz means • Mathematics • 2020 The information geometry, convexity, in-betweenness property and the barycenter problem of quantum Hellinger divergences with Heinz means is studied. The limiting cases are also considered. Correction to: Matrix versions of the Hellinger distance • Mathematics • Letters in Mathematical Physics • 2019 Theorem 9 in our paper [1] is wrong. The statement should be replaced by the following. #### References SHOWING 1-10 OF 33 REFERENCES A new quantum version of f-divergence This paper proposes and studies new quantum version of f-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relativeExpand Matrix versions of the Hellinger distance • Physics, Mathematics • 2019 On the space of positive definite matrices, we consider distance functions of the form$$d(A,B)=\left[ \mathrm{tr}\mathcal {A}(A,B)-\mathrm{tr}\mathcal {G}(A,B)\right]Expand
Different quantum f-divergences and the reversibility of quantum operations
• Mathematics, Physics
• 2016
This paper compares the standard and the maximal $f-divergences regarding their ability to detect the reversibility of quantum operations, and studies the monotonicity of the Renyi divergences under the special class of bistochastic maps that leave one of the arguments of theRenyi divergence invariant. Expand On the Bures-Wasserstein distance between positive definite matrices • Mathematics • 2017 The metric$d(A,B)=\left[ \tr\, A+\tr\, B-2\tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$on the manifold of$n\times n$positive definite matrices arises in various optimisation problems, in quantumExpand Quantum$f$-divergences in von Neumann algebras II. Maximal$f$-divergences As a continuation of the paper [20] on standard$f$-divergences, we make a systematic study of maximal$f$-divergences in general von Neumann algebras. For maximal$f\$-divergences, apart from theirExpand
Divergence Radii and the Strong Converse Exponent of Classical-Quantum Channel Coding With Constant Compositions
• Computer Science, Physics
• IEEE Transactions on Information Theory
• 2021
It is shown that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényu divergences, has the same operational interpretation in the strong converse problem of constant composition classical-quantum channel coding. Expand
Concavity of certain maps on positive definite matrices and applications to Hadamard products
Abstract If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Lowner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the followingExpand
On the Joint Convexity of the Bregman Divergence of Matrices
• Mathematics, Physics
• 2015
We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantumExpand
Matrix power means and the Karcher mean
• Mathematics
• 2012
We define a new family of matrix means {Pt(ω;A)}t∈[−1,1], where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these means exceptExpand
Monotone Riemannian metrics and relative entropy on noncommutative probability spaces
• Mathematics, Physics
• 1999
We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0,∞) satisfying g(1)=0. We show that these convex operator functions can beExpand