Richárd Simon | Searcharxiv

arXiv2025

Isometries of the qubit state space with respect to quantum Wasserstein distances

In this paper we study isometries of quantum Wasserstein distances and divergences on the quantum bit state space. We describe isometries with respect to the symmetric quantum Wasserstein divergence $d_{sym}$, the divergence induced by all of the Pauli matrices. We also give a complete characterization of isometries with respect to $D_z$, the quantum Wasserstein distance corresponding to the single Pauli matrix $σ_z$.

math-ph↗

arXiv2023

Preservers of the $p$-power and the Wasserstein means on $2 \times 2$ matrices

In one of his recent papers \cite{ML1}, Molnár showed that if $\mathcal{A}$ is a von Neumann algebra without $I_1, I_2$-type direct summands, then any function from the positive definite cone of $\mathcal{A}$ to the positive real numbers preserving the Kubo-Ando power mean for some $0 \neq p \in (-1,1)$ is necessarily constant. It was shown in that paper, that $I_1$-type algebras admit nontrivial $p$-power mean preserving functionals, and it was conjectured, that $I_2$-type algebras admit only constant $p$-power mean preserving functionals. We confirm the latter. A similar result occurred in another recent paper of Molnár \cite{ML2} concerning the Wasserstein mean. We prove the conjecture for $I_2$-type algebras in regard of the Wasserstein mean, too. We also give two conditions that characterise centrality in $C^*$-algebras.

math.OA↗