Representation Theory of the Unitary and Symmetric Group with a view towards the Quantum Marginal Problem

量子边际问题要求量子系统的一组部分描述具有兼容性。这个问题是量子物理许多领域的基础，例如凝聚态物理和量子信息论。它的费米子版本被称为N表示问题，这与量子化学有关，在量子化学中，有效的解决方案将允许计算分子的性质，如它们的结合能。最近已经证明，量子边际问题的某些实例与李群和有限群的表示理论中出现的问题等价。这个项目的目的有两个。首先，我们希望对文献中讨论的实例提供统一的处理，并处理在此背景下出现的表示论猜想。其次，在将我们的发现应用于量子信息论和更广泛的量子物理之前，我们希望将量子边际问题和表象理论之间的联系从特定实例扩展到任意实例。

The quantum marginal problem asks for the compatibility of a set of partial descriptions of a quantum system. This problem is fundamental to many areas of quantum physics such as condensed matter physics and quantum information theory. Its fermionic version known as the N-representability problem is relevant to quantum chemistry where an efficient solution would allow for the calculation of properties of molecules such as their binding energies. Recently it has been shown that certain instances of the quantum marginal problem are equivalent to questions arising in the representation-theory of Lie groups and finite groups. The aim of this project is two-fold. Firstly, we wish to provide a unified treatment of the instances that have been discussed in the literature and tackle the representation-theoretic conjectures that have arisen in this context. Secondly, we wish to extend the connection between the quantum marginal problem and representation theory from specific instances to arbitrary instances, before applying our findings in quantum information theory and quantum physics more generally.