@article{12214,
  abstract     = {Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. },
  author       = {Gehér, György Pál and Titkos, Tamás and Virosztek, Daniel},
  issn         = {1469-7750},
  journal      = {Journal of the London Mathematical Society},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {3865--3894},
  publisher    = {Wiley},
  title        = {{The isometry group of Wasserstein spaces: The Hilbertian case}},
  doi          = {10.1112/jlms.12676},
  volume       = {106},
  year         = {2022},
}

@article{8373,
  abstract     = {It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.},
  author       = {Pitrik, József and Virosztek, Daniel},
  issn         = {0024-3795},
  journal      = {Linear Algebra and its Applications},
  keywords     = {Kubo-Ando mean, weighted multivariate mean, barycenter},
  pages        = {203--217},
  publisher    = {Elsevier},
  title        = {{A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means}},
  doi          = {10.1016/j.laa.2020.09.007},
  volume       = {609},
  year         = {2021},
}

@article{9036,
  abstract     = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.},
  author       = {Virosztek, Daniel},
  issn         = {0001-8708},
  journal      = {Advances in Mathematics},
  keywords     = {General Mathematics},
  number       = {3},
  publisher    = {Elsevier},
  title        = {{The metric property of the quantum Jensen-Shannon divergence}},
  doi          = {10.1016/j.aim.2021.107595},
  volume       = {380},
  year         = {2021},
}

@article{7389,
  abstract     = {Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space
W_p(R) for all p \in [1,\infty) \setminus {2}. We show that W_2(R) is also exceptional regarding the
parameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying
space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact
interval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if
p is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))
cannot be embedded into Isom(W_1(R)).},
  author       = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel},
  issn         = {10886850},
  journal      = {Transactions of the American Mathematical Society},
  keywords     = {Wasserstein space, isometric embeddings, isometric rigidity, exotic isometry flow},
  number       = {8},
  pages        = {5855--5883},
  publisher    = {American Mathematical Society},
  title        = {{Isometric study of Wasserstein spaces - the real line}},
  doi          = {10.1090/tran/8113},
  volume       = {373},
  year         = {2020},
}

@article{7618,
  abstract     = {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 ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case. },
  author       = {Pitrik, Jozsef and Virosztek, Daniel},
  issn         = {1573-0530},
  journal      = {Letters in Mathematical Physics},
  number       = {8},
  pages        = {2039--2052},
  publisher    = {Springer Nature},
  title        = {{Quantum Hellinger distances revisited}},
  doi          = {10.1007/s11005-020-01282-0},
  volume       = {110},
  year         = {2020},
}