arXiv subjects

Tamás Titkos

Publications and source records attributed to Tamás Titkos.

At least 19 recordsLinked to original sources

arXiv2026

Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits

We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.

math-ph↗

arXiv2026

Wasserstein distances and divergences of order $p$ by quantum channels

We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.

math-ph↗

arXiv2025

The swap transpose on couplings translates to Petz' recovery map on quantum channels

In [Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234], De Palma and Trevisan described a one-to-one correspondence between quantum couplings and quantum channels realizing transport between states. The aim of this short note is to demonstrate that taking the Petz recovery map for a given channel and initial state is precisely the counterpart of the swap transpose operation on couplings. That is, the swap transpose of the coupling $Π_Φ$ corresponding to the channel $Φ$ and initial state $ρ$ is the coupling $Π_{rec}$ corresponding to the Petz recovery map $Φ_{rec}.$

math-ph↗

arXiv2025

Wasserstein Rigidity over $\mathbb{R}^n$ with smooth norms

We study $p-$Wasserstein spaces $ \mathcal{W}_p(\mathbb{R}^n, d_N)$ over $\mathbb{R}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p \neq 2$. We also show that, even when $p=2$, we can recover the isometric rigidity of the Wasserstein space $\mathcal{W}_2(\mathbb{R}^n, d_N)$ when $N$ is an $l_q-$norm and $q>2$.

math.MG↗

arXiv2025

Rigid and flexible Wasserstein spaces

In this paper, we study isometries of $p$-Wasserstein spaces. In our first result, for every complete and separable metric space $X$ and for every $p\geq1$, we construct a metric space $Y$ such that $X$ embeds isometrically into $Y$, and the $p$-Wasserstein space over $Y$ admits mass-splitting isometries. Our second result is about embeddings into rigid constructions. We show that any complete and separable metric space $X$ can be embedded isometrically into a metric space $Y$ such that the $1$-Wasserstein space is isometrically rigid.

math.MG↗

arXiv2025

Isometric rigidity of the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups

This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are the Heisenberg group $\mathbb{H}^n$ endowed with the Heisenberg-Korányi norm, or with the Naor-Lee norm; and $H$-type Iwasawa groups endowed with a Korányi-type norm. We prove that on a general Carnot group there always exists a horizontally strictly convex norm. The main result of the paper says that if $(\mathbf{G},N_{\mathbf{G}})$ is a Carnot group where $N_{\mathbf{G}}$ is a horizontally strictly convex norm on $\mathbf{G}$, then the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ is isometrically rigid. That is, for every isometry $Φ:\mathcal{W}_1(\mathbf{G})\to\mathcal{W}_1(\mathbf{G})$ there exists an isometry $ψ:\mathbf{G}\to \mathbf{G}$ such that $Φ=ψ_{\#}$.

math.MG↗

arXiv2025

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 $p>1$ and for all $n\geq 1$. 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 $p>1$. We find that these spaces are all isometrically rigid meaning that for every isometry $Φ:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $ψ:\mathbb{H}^n\to\mathbb{H}^n$ such that $Φ=ψ_{\#}$.

math.MG↗

arXiv2025

Isometric rigidity of the Wasserstein space over the plane with the maximum metric

We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb{R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid.

math.MG↗

arXiv2025

On the metric property of quantum Wasserstein divergences

Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels, and they are conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure, and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general, for an arbitrary choice of states.

math-ph↗

arXiv2024

Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ over the sphere endowed with the distance inherited from the norm of $\mathbb{R}^{n+1}$. We prove that $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ is isometrically rigid, meaning that its isometry group is isomorphic to that of $\left(\mathbb{S}^n,\|\cdot\|\right)$. This is in striking contrast to the non-rigidity of its ambient space $\mathcal{W}_2\left(\mathbb{R}^n,\|\cdot\|\right)$ but in line with the rigidity of the geodesic space $\mathcal{W}_2\left(\mathbb{S}^n,\sphericalangle\right)$. One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$. A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of $\mathcal{W}_p\left(\mathbb{S}^1, \|\cdot\|\right)$ for $1 \leq p<2.$

math.MG↗

arXiv2024

Isometries of Lipschitz-free Banach spaces

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes e.g. Lipschitz-free spaces over any graph. We define the notion of a Lipschitz-free rigid metric space whose Lipschitz-free space only admits surjective linear isometries coming from surjective dilations (i.e. rescaled isometries) of the metric space itself. We show this class of metric spaces is surprisingly rich and contains all $3$-connected graphs as well as geometric examples such as non-abelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz-free rigid space that has only three more points.

math.FA↗

arXiv2022

Quantum Wasserstein isometries on the qubit state space

We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. We derive a Wigner-type result for the cost operator involving all the Pauli matrices: in this case, the isometry group consists of unitary or anti-unitary conjugations. In the Bloch sphere model, this means that the isometry group coincides with the classical symmetry group $\mathbf{O}(3).$ On the other hand, for the cost generated by the qubit "clock" and "shift" operators, we discovered non-surjective and non-injective isometries as well, beyond the regular ones. This phenomenon mirrors certain surprising properties of the quantum Wasserstein distance.

math-ph↗

arXiv2022

Isometric rigidity of Wasserstein tori and spheres

We prove isometric rigidity for $p$-Wasserstein spaces over finite-dimensional tori and spheres for all $p$. We present a unified approach to proving rigidity that relies on the robust method of recovering measures from their Wasserstein potentials.

math.MG↗

arXiv2022

Isometric rigidity of Wasserstein spaces: the graph metric case

The aim of this paper is to prove that the $p$-Wasserstein space $\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a countable graph metric space. As a consequence, we obtain that for every countable group $H$ and any $p\geq 1$ there exists a $p$-Wasserstein space whose isometry group is isomorphic to $H$.

math.CA↗

arXiv2021

The isometry group of Wasserstein spaces: the Hilbertian case

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$, we describe the isometry group $\mathrm{Isom}\left(\mathcal{W}_p (E)\right)$ for all parameters $0 < p < \infty$ and for all separable real Hilbert spaces $E.$ In particular, we show that $\mathcal{W}_p(X)$ is isometrically rigid for all Polish space $X$ whenever $0 1$, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries.

math.MG↗

arXiv2020

Operators on anti-dual pairs: self-adjoint extensions and the Strong Parrott Theorem

The aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott's theorem on contractive completion of $2$ by $2$ block operator-valued matrices. To exhibit the applicability in noncommutative integration, we characterize hermitian extendibility of symmetric functionals defined on a left ideal of a $C^{*}$-algebra.

math.FA↗

arXiv2020

Operators on anti-dual pairs: Generalized Schur complement

The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs. As a byproduct, we obtain a natural generalization of the parallel sum and parallel difference, as well as the Lebesgue-type decomposition. To demonstrate how this operator approach works in application, we derive the corresponding results for operators acting on rigged Hilbert spaces, and for representable functionals of ${}^{*}$-algebras.

math.FA↗

arXiv2020

Isometric study of Wasserstein spaces --- the real line

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$. 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 $\mathrm{Isom}\left(\mathcal{W}_p(\mathbb{R})\right)$, the isometry group of the Wasserstein space $\mathcal{W}_p(\mathbb{R})$ for all $p \in [1, \infty)\setminus\{2\}$. We show that $\mathcal{W}_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: $\mathcal{W}_p(\mathbb{R})$ is isometrically rigid if and only if $p\neq 2$. Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, $\mathcal{W}_p\left([0,1]\right)$ is isometrically rigid if and only if $p\neq1$. Moreover, $\mathcal{W}_1\left([0,1]\right)$ admits isometries that split mass, and $\mathrm{Isom}\left(\mathcal{W}_1\left([0,1]\right)\right)$ cannot be embedded into $\mathrm{Isom}\left(\mathcal{W}_1(\mathbb{R})\right).$

math.MG↗