标题： 一种估计理论方法用于量子可实现性问题 显示英文标题

标题： An estimation theoretic approach to quantum realizability problems

摘要： 本论文旨在开发一种通用方法，用于解决所谓的量子可实现性问题，这些问题的形式如下：在什么条件下存在一个表现出给定性质集合的量子态？ 本论文采用的方法是利用之前为相关的问题——属性估计而开发的数学技术，该问题涉及学习或估计未知量子态的属性。 我们的主要结果是认识到（i）由某些量子态实现的属性值，与（ii）作为通用量子态估计偶尔产生的属性值之间的对应关系。 第3章回顾了几何不变量理论和非交换优化理论中的稳定性与范数最小化概念，目的是表征量子态在约化群作用下的流动。 第4章展示了如何通过对大量相同量子态副本进行协变量子测量来估计该流动的梯度，也称为矩图。 第5章概述了矩图值的可实现性与作为估计产生的渐近可能性之间的对应关系。 通过适当组合这些矩图估计方案，我们推导出联合实现任何有限个矩图的量子态存在的必要且充分条件。 第6章将这些见解应用于量子边缘问题，并重复了arXiv:2211.00685的内容。 显示英文摘要

摘要： This thesis seeks to develop a general method for solving so-called quantum realizability problems, which are questions of the following form: under which conditions does there exist a quantum state exhibiting a given collection of properties? The approach adopted by this thesis is to utilize mathematical techniques previously developed for the related problem of property estimation which is concerned with learning or estimating the properties of an unknown quantum state. Our primary result is to recognize a correspondence between (i) property values which are realized by some quantum state, and (ii) property values which are occasionally produced as estimates of a generic quantum state. Chapter 3 reviews concepts of stability and norm minimization from geometric invariant theory and non-commutative optimization theory for the purposes of characterizing the flow of a quantum state under the action of a reductive group. Chapter 4 demonstrates how the gradient of this flow, also called the moment map, can be estimated by performing a covariant quantum measurement on a large number of identical copies of the quantum state. Chapter 5 outlines the correspondence between between the realizability of a moment map value on one hand and the asymptotic likelihood it is produced as an estimate on the other hand. By appropriately composing these moment map estimation schemes, we derive necessary and sufficient conditions for the existence of a quantum state jointly realizing any finite collection of moment maps. Chapter 6 applies these insights to the quantum marginal problem and is a duplication of arXiv:2211.00685.