发表机构
Instituto de Matemática Pura e Aplicada; Department of Mathematics and Statistics, University of Ottawa（巴西纯粹与应用数学研究所; 渥太华大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立量子最优传输重心框架，证明存在性与对偶性，并揭示高斯刚性：若至少一个高斯输入忠实，则2-量子Wasserstein重心在所有量子态中唯一且为高斯。
AI 中文摘要
我们为量子态发展了一个量子最优传输（QOT）重心框架，作为Wasserstein重心~\\(\cite{AguCar}\\)的类比，并在可分Hilbert空间上为一大类可能无界的传输成本建立了存在性和对偶性结果。我们的框架特别地，通过特化到与Caglioti--Golse--Mouhot--Paul~\\(\cite{caglioti2021towards,Golse-Mouhot-Paul:2016}\\)和De Palma--Trevisan~\\(\cite{DPaTre19}\\)的2-量子Wasserstein距离相关联的典型二次成本算子，在量子态和量子信道两种表述中统一处理了2-量子Wasserstein（QW）重心。核心结果涉及高斯输入态，特别是高斯刚性：当最小化在所有量子态上进行时，高斯输入态是否强制2-QW重心本身为高斯且唯一确定。我们首先证明重心问题存在高斯极小元，并归结为协方差矩阵上的有限维凸优化问题。主要困难在于，最优协方差的唯一性一般不蕴含底层量子态的唯一性。我们通过协方差互补松弛条件下的态重构原理弥合了这一差距，将最优协方差的唯一性提升为完整量子态的唯一性，从而证明了一个全局刚性定理：若至少有一个高斯输入是忠实的，则重心在所有量子态中是唯一的且必然为高斯。忠实性充分但不必要：纯输入族仍确定唯一重心，而部分纯的非忠实高斯输入可能允许多个重心。
英文摘要
We develop a Quantum Optimal Transport (QOT) barycenter framework for quantum states, as an analog of Wasserstein barycenters~\cite{AguCar}, and establish existence and duality results for a broad class of possibly unbounded transport costs on separable Hilbert spaces. Our framework provides, in particular, a unified treatment of $2$-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the $2$-quantum Wasserstein distances of Caglioti--Golse--Mouhot--Paul~\cite{caglioti2021towards,Golse-Mouhot-Paul:2016} and De Palma--Trevisan~\cite{DPaTre19}. The central results concern Gaussian input states and, in particular, Gaussian rigidity: whether Gaussian input states force the 2-QW barycenter itself to be Gaussian and uniquely determined when the minimization is taken over all quantum states. We first show that the barycenter problem admits a Gaussian minimizer and reduces to a finite-dimensional convex optimization problem over covariance matrices. The main difficulty is that uniqueness of the optimal covariance does not, in general, imply uniqueness of the underlying quantum state. We bridge this gap through a state-reconstruction principle under covariance complementary slackness that upgrades uniqueness of the optimal covariance to uniqueness of the full quantum state, and thereby prove a global rigidity theorem: if at least one Gaussian input is faithful, then the barycenter is unique among all quantum states and is necessarily Gaussian. Faithfulness is sufficient but not necessary: families of pure inputs still determine a unique barycenter, whereas partially pure nonfaithful Gaussian inputs may admit multiple barycenters.
