arXiv subjects

György Pál Gehér

Publications and source records attributed to György Pál Gehér.

At least 19 recordsLinked to original sources

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↗

arXiv2023

When is the variance of one observable less than or equal to that of another with respect to all quantum states?

In quantum mechanics, the well-known Loewner order expresses that one observable's expectation value is less than or equal than that of another with respect to all quantum states. In this paper we propose and study a similar order relation in terms of the variance, and we prove two theorems. Our first result states that one observable's variance is less than or equal than that of another with respect to all quantum states if and only if the former is a $1$-Lipschitz function of the latter. The other main result we prove characterises the order automorphisms with respect to this proposed order relation. It turns out that in some sense these automorphisms have a more rigid form than in the case of the Loewner order.

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↗

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↗

arXiv2021

The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner's theorem states that every bijection $ϕ\colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn's theorem generalises this result for bijective maps $ϕ$ that are only assumed to preserve the quantum angle $\fracπ{2}$ (orthogonality) in both directions. Recently, two papers, written by Li--Plevnik--Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $α$ in both directions, provided that $0 < α\leq \fracπ{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $α$ that satisfy $\fracπ{4} < α< \fracπ{2}$, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner's.

math-ph↗

arXiv2020

Maps preserving absolute continuity and singularity of positive operators

In this paper we consider the cone of all positive, bounded operators acting on an infinite dimensional, complex Hilbert space, and examine bijective maps that preserve absolute continuity in both directions. It turns out that these maps are exactly those that preserve singularity in both directions. Moreover, in some weak sense, such maps are always induced by bounded, invertible, linear- or conjugate linear operators of the underlying Hilbert space. Our result gives a possible generalization of a recent theorem of Molnar which characterizes maps on the positive cone that preserve the Lebesgue decomposition of operators.

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↗

arXiv2019

On isometric embeddings of Wasserstein spaces -- the discrete case

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of $\mathcal{X}\times(0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of $\mathcal{W}_p(\mathcal{X})$ splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that $\mathcal{W}_p(\mathcal{X})$ is isometrically rigid for all $0<p<\infty$.

math.FA↗

arXiv2017

A characterisation of isometries with respect to the Lévy-Prokhorov metric

According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L{é}vy-Prokhorov distance. Motivated by these facts, the famous Banach-Stone theorem, and some recent works related to characterisations of onto isometries of spaces of Borel probability measures, here we give a complete description of surjective isometries with respect to the L{é}vy-Prokhorov metric in case when the underlying metric space is a separable Banach space. Our result can be considered as a generalisation of L. Molnár's earlier Banach-Stone-type result which characterises onto isometries of the space of all probability distribution functions on the real line wit respect to the Lévy distance. However, the present more general setting requires the development of an essentially new technique.

math.FA↗

arXiv2017

Wigner's theorem on Grassmann spaces

Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator. This vital theorem has been generalised in various ways by several scientists. In 2001, Molnár provided a natural generalisation, namely, he provided a characterisation of (not necessarily bijective) maps which act on the Grassmann space of all rank-$n$ projections and leave the system of Jordan principal angles invariant (see [20] and [17]). In this paper we give a very natural joint generalisation of Wigner's and Molnár's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity $\mathrm{tr} PQ$ (i.e.~the sum of the squares of cosines of principal angles) for every pair of rank-$n$ projections $P$ and $Q$.

math.FA↗

arXiv2017

Surjective Kuiper isometries

Characterizations of surjective isometries with respect to the Kuiper distance on three classes of Borel probability measures of $\mathbb{R}$ (or equivalently, probability distribution functions) are presented here. These classes are the set of continuous, absolute continuous and general measures.

math.FA↗

arXiv2017

Maps on positive definite operators preserving the quantum $χ_α^2$-divergence

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $χ_α^2$-divergence for some $α\in [0,1]$. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.

math-ph↗

arXiv2016

Symmetries of projective spaces and spheres

Let $H$ be either a complex inner product space of dimension at least two, or a real inner product space of dimension at least three. Let us fix an $α\in \left(0,\tfracπ{2}\right)$. The purpose of this paper is to characterize all bijective transformations on the projective space $P(H)$ obtained from $H$ which preserves the angle $α$ between lines in both directions. (We emphasize that we do not assume anything about other angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $α$, and when $H$ is a complex inner product space of dimension at least three we describe the structure of these transformations for $α\leq\tfracπ{4}$. As an application, we give an Uhlhorn-type generalization of a famous theorem of Wigner which is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, we show that under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos^2α$ in both directions is a Wigner symmetry (i.e. it automatically preserves all transition probability), except for the case when $H=\mathbb{C}^2$ and $α= \tfracπ{4}$ where an additional possibility occurs. We note that the classical theorem of Uhlhorn is the solution for the $α= \tfracπ{2}$ case. Usually in the literature, results which are connected to Wigner's theorem are discussed under the assumption of completeness of $H$, however, here we shall remove this unnecessary hypothesis. Our main tools are a characterization of bijective maps on unit spheres of real inner product spaces which preserve an angle in both directions, and an extension of Uhlhorn's theorem for non-complete inner product spaces.

math-ph↗

arXiv2016

Isometries of Grassmann spaces

Botelho, Jamison, and Moln\' ar have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Moln\' ar, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surectivity assumption. A variety of tools is used in order to achieve our goal, such as topological, geometrical and linear algebra techniques. The famous two projections theorem for two finite rank projections will be re-proven using linear algebraic methods. A theorem of Györy and the second author on orthogonality preservers on Grassmann spaces will be strengthened as well. This latter result will be obtained by using Chow's fundamental theorem of geometry of Grassmannians.

math.FA↗

arXiv2016

Asymptotic Behaviour and Cyclic Properties of Weighted Shifts on Directed Trees

In this paper we investigate a new class of operators called weighted shifts on directed trees introduced recently in [Z. J. Jablonski, I. B. Jung and J. Stochel, A Non-hyponormal Operator Generating Stieltjes Moment Sequences, J. Funct. Anal. 262 (2012), no. 9, 3946--3980.]. This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour and similarity properties to deal with cyclicity. We also show that a weighted backward shift operator is cyclic if and only if there is at most one zero weight.

math.FA↗

arXiv2016

Is it possible to determine a point lying in a simplex if we know the distances from the vertices?

It is an elementary fact that if we fix an arbitrary set of $d+1$ affine independent points $\{p_0,\dots p_d\}$ in $\mathbb{R}^d$, then the Euclidean distances $\{|x-p_j|\}_{j=0}^d$ determine the point $x$ in $\mathbb{R}^d$ uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least $d$-dimensional, real normed spaces $(X, \|\cdot\|)$ such that for every set of $d+1$ affine independent points $\{p_0,\dots p_d\} \subset X$, the distances $\{\|x-p_j\|\}_{j=0}^d$ determines the point $x$ lying in the simplex $\mathrm{Conv}(p_0,\dots p_d)$ uniquely. Surprisingly, the characterization depends on $d$.

math.MG↗

arXiv2015

Bilateral weighted shift operators similar to normal operators

We prove that an injective, not necessarily bounded weighted bilateral shift operator on $\ell^2(\mathbb{Z})$ is similar to a normal operator if and only if it is similar to a scalar multiple of the simple (i.e. unweighted) bilateral shift operator $S$.

math.FA↗