$n$-量子比特态空间的量子Wasserstein等距映射：一种Wigner型结果

量子态等距映射的稳定性

Stability of isometries of quantum states

本文证明正迹类锥上近似等距映射可被Wigner对称一致逼近，给出与维度无关的误差界，并揭示无穷维与有限维稳定性本质差异。

量子Wasserstein距离不同定义之间的关系

Relations between different definitions of the quantum Wasserstein distance

本文研究了量子Wasserstein距离在qubit系统中单个算符出现在成本函数时不同定义之间的关系，发现自距离等于Wigner-Yanase斜信息。

Quantum statistical learning via Quantum Wasserstein natural gradient

In this article, we introduce a new approach towards the statistical learning problem $\operatorname{argmin}_{ρ(θ) \in \mathcal P_θ} W_{Q}^2 (ρ_{\star},ρ(θ))$ to approximate a target quantum state $ρ_{\star}$ by a set of parametrized quantum states $ρ(θ)$ in a quantum $L^2$-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional $C^*$ algebras. For continuous parametric models of density operators, we pull back the quantum Wasserstein metric such that the parameter space becomes a Riemannian manifold wit…

Maps on probability measures preserving certain distances --- a survey and some new results

Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric…

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 is…

非可逆对称性的广义威金定理

Generalized Wigner theorem for non-invertible symmetries

本文探讨非可逆算符关联的守恒定律如何作为量子物理中的对称性实现的条件，提出非可逆对称性对应于项目性酉或反酉变换，扩展了量子态的传统理解。

Wasserstein空间的等距群：希尔伯特情形

The isometry group of Wasserstein spaces: the Hilbertian case

该研究刻画了可分实希尔伯特空间上所有$0<p<\infty$阶Wasserstein空间的等距群，证明了$0<p<1$时Polish空间上Wasserstein空间的等距刚性，且解决了Kloeckner的质量分裂等距存在性问题，说明刚性结果不适用于$p>1$的情形。

Wasserstein空间$\mathcal{W}_1(\mathbf{G})$在Carnot群上的等距刚性

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

本文研究了Carnot群上1-Wasserstein空间的等距刚性，证明了在水平严格凸范数下该空间的等距映射与群上的等距映射相关联。

欧几里得球面上 Wasserstein 空间的等距刚性

Isometric rigidity of Wasserstein spaces over Euclidean spheres

本文研究欧氏范数距离下球面上 Wasserstein 空间的等距群，借助均方误差函数模拟位移插值，证明二次情形及 $\mathbb{S}^1$ 上 $1\leq p<2$ 情形的等距刚性。

Quantum optimal transport with quantum channels

We propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport plans between quantum states are in natural correspondence with quantum channels, such that the transport can be interpreted as a physical operation on the system. Our main result is the proof of a modified triangle inequality for our transport distance. We also prove that the distance between a quantum state and itself is intimately connected with the Wigner-…

对基本粒子的群样描述

A groupoidal description of elementary particles

本文通过扩展时空对称性的描述框架至更灵活的运动群束，扩展了威金的程序至弯曲时空。提出基本粒子为运动群束的不可约项目表示，通过自然的威金群束定义，展示了这些表示具有与庞加莱群相似的量子数，并通过扩展麦克凯理论分类了基本粒子。