标题： 关于多量子比特、赛格嵌入和科克斯特室 显示英文标题

标题： On Multiquantum Bits, Segre Embeddings and Coxeter Chambers

摘要： 本工作探讨了量子信息论、代数几何和数论之间的相互作用，重点关注多量子比特系统、它们的纠缠结构以及通过几何嵌入进行的分类。 代数几何中的基本构造——Segre嵌入，提供了一个代数框架以区分可分态和纠缠态，在射影几何中编码量子关联。 我们系统地研究了量子比特模空间，通过超立方体构造和Coxeter室分解展示了纠缠的几何结构。 我们建立了张量积射影空间的Segre嵌入与长度为$n-1$的二进制词之间的一一对应关系，这些二进制词被构造成一个$(n-1)$维的超立方体，其中相邻性对应于单个Segre操作。 这揭示了嵌入层次结构下的组合结构，直接关系到量子纠错方案。 Segre簇在类型为$A$的Coxeter群下的对称性使我们可以通过反射群来分析量子态和错误，视可分态为Segre簇上不同Coxeter室中的点。 置换群在这类室上的传递作用提供了一种自然的方法来追踪量子态中的错误并可能逆转它们。 除了基础方面，我们还强调了Segre簇与Dixon椭圆曲线之间的联系，揭示了纠缠与数论之间的关系。 显示英文摘要

摘要： This work explores the interplay between quantum information theory, algebraic geometry, and number theory, with a particular focus on multiqubit systems, their entanglement structure, and their classification via geometric embeddings. The Segre embedding, a fundamental construction in algebraic geometry, provides an algebraic framework to distinguish separable and entangled states, encoding quantum correlations in projective geometry. We develop a systematic study of qubit moduli spaces, illustrating the geometric structure of entanglement through hypercube constructions and Coxeter chamber decompositions. We establish a bijection between the Segre embeddings of tensor products of projective spaces and binary words of length $n-1$, structured as an $(n-1)$-dimensional hypercube, where adjacency corresponds to a single Segre operation. This reveals a combinatorial structure underlying the hierarchy of embeddings, with direct implications for quantum error correction schemes. The symmetry of the Segre variety under the Coxeter group of type $A$ allows us to analyze quantum states and errors through the lens of reflection groups, viewing separable states as lying in distinct Coxeter chambers on a Segre variety. The transitive action of the permutation group on these chambers provides a natural method for tracking errors in quantum states and potentially reversing them. Beyond foundational aspects, we highlight relations between Segre varieties and Dixon elliptic curves, drawing connections between entanglement and number theory.