量子可实现性问题的一种估计理论方法
An estimation theoretic approach to quantum realizability problems
AI总结：
本文提出一种基于估计理论的通用方法解决量子可实现性问题，通过建立量子态实现的属性值与其作为估计值之间的对应关系，利用矩映射估计推导出存在量子态共同实现任意有限矩映射集合的充要条件，并应用于量子边缘问题。
摘要：
本论文旨在开发一种解决所谓量子可实现性问题的通用方法，该问题具有以下形式：在何种条件下存在一个展现给定属性集合的量子态？本文采用的方法是利用先前为相关的属性估计问题开发的数学技术，该问题关注于学习或估计未知量子态的属性。我们的主要成果是认识到以下两者之间的对应关系：(i) 由某个量子态实现的属性值，以及 (ii) 偶尔作为一般量子态估计值而产生的属性值。第3章回顾了来自几何不变量理论和非交换优化理论的稳定性与范数最小化概念，旨在刻画量子态在约化群作用下的流。第4章演示了如何通过对大量相同副本的量子态执行协变量子测量，来估计该流的梯度（亦称矩映射）。第5章概述了矩映射值的可实现性与它作为估计值产生的渐近似然性之间的对应关系。通过适当地组合这些矩映射估计方案，我们推导出了存在一个共同实现任意有限矩映射集合的量子态的充分必要条件。第6章将这些见解应用于量子边缘问题，并且是 arXiv:2211.00685 的复本。
英文摘要：
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.
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