Quantum ff-divergence preserving maps on positive semidefinite operators acting on finite dimensional Hilbert spaces

We determine the structure of all bijections on the cone of positive semidefinite operators which preserve the quantum $f$-divergence for an arbitrary strictly convex function $f$ defined on the positive halfline. It turns out that any such transformation is implemented by either a unitary or an antiunitary operator.Comment: v2: some typos corrected v3: improved presentation and some new references v4: accepted manuscript versio

Some inequalities for quantum Tsallis entropy related to the strong subadditivity

In this paper we investigate the inequality $S_q(\rho_{123})+S_q(\rho_2)\leq S_q(\rho_{12})+S_q(\rho_{23}) \, (*)$ where $\rho_{123}$ is a state on a finite dimensional Hilbert space $\mathcal{H}_1\otimes \mathcal{H}_2\otimes \mathcal{H}_3,$ and $S_q$ is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of $(*)$, which is an inequality of relative quasi-entropies. We derive an inequality of the form $S_q(\rho_{123})+S_q(\rho_2)\leq S_q(\rho_{12})+S_q(\rho_{23})+f_q(\rho_{123})$, where $f_1(\rho_{123})=0$. Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that $(*)$ does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.Comment: v2: the introductory part reorganized v3: the published versio

Quantum Hellinger distances revisited

This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences, that are of the form $\phi(A,B)=\mathrm{Tr} \left((1-c)A + c B - A \sigma B \right),$ where $\sigma$ is an arbitrary Kubo-Ando mean, and $c \in (0,1)$ is the weight of $\sigma.$ 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 (DPI). 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 the work of Bhatia et al. mentioned above, is true in the case of commuting operators, but it is not correct in the general case.Comment: v2: Section 4 on the commutative case, and Subsection 5.2 on a possible measure of non-commutativity added, as well as references to the maximal quantum $f$-divergence literature; v3: Section 4 on the commutative case improved, and the proposed measure of non-commutativiy changed accordingly; v4: accepted manuscript versio

Continuous Jordan triple endomorphisms of P2\mathbb{P}_2

We describe the structure of all continuous Jordan triple endomorphisms of the set $\mathbb{P}_2$ of all positive definite $2\times 2$ matrices thus completing a recent result of ours. We also mention an application concerning sorts of surjective generalized isometries on $\mathbb{P}_2$ and, as second application, we complete another former result of ours on the structure of sequential endomorphisms of finite dimensional effect algebras

Maps on positive definite matrices preserving Bregman and Jensen divergences

In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given

Preservers of the pp-power and the Wasserstein means on 2×22 \times 2 matrices

In one of his recent papers \cite{ML1}, Moln\'ar 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 Moln\'ar \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.Comment: accepted manuscript versio

Connections between centrality and local monotonicity of certain functions on C*-algebras

We introduce a quite large class of functions (including the exponential function and the power functions with exponent greater than one), and show that for any element $f$ of this function class, a self-adjoint element $a$ of a $C^*$-algebra is central if and only if $a \leq b$ implies $f(a) \leq f(b).$ That is, we characterize centrality by local monotonicity of certain functions on $C^*$-algebras. Numerous former results (including works of Ogasawara, Pedersen, Wu, and Moln\'ar) are apparent consequences of our result.Comment: v2: major revision, stronger result, title changed. v3: minor improvements. v4: published versio

Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

Our purpose in this paper is to study isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ over the Heisenberg group $\mathbb{H}^n$ for all $p\geq1$ and for all $n\geq1$. First, we create a link between optimal transport maps in the Euclidean space $\mathbb{R}^{2n}$ and the Heisenberg group $\mathbb{H}^n$. Then we use this link to understand isometric embeddings of $\mathbb{R}$ and $\mathbb{R}_+$ into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$. That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results we determine the metric rank of $\mathcal{W}_p(\mathbb{H}^n)$. Namely, we show that $\mathbb{R}^k$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$ if and only if $k\leq n$. As a consequence, we conclude that $\mathcal{W}_p(\mathbb{R}^k)$ and $\mathcal{W}_p(\mathbb{H}^k)$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ if and only if $k\leq n$. In the second part of the paper, we study the isometry group of $\mathcal{W}_p(\mathbb{H}^n)$ for $p\geq1$. We find that these spaces are all isometrically rigid meaning that for every isometry $\Phi:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $\psi:\mathbb{H}^n\to\mathbb{H}^n$ such that $\Phi=\psi_{\#}$. Although the conclusion is the same for $p=1$ and $p>1$, the proofs are completely different, as in the $p>1$ case the proof relies on a description of complete geodesics, and such a description is not available if $p=1$.Comment: 29 page