The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.

中文翻译：

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量子比特的对称性和几何结构及其用途

对称性 SU(2) 及其几何 Bloch Sphere 渲染已成功应用于单个量子位 (spin-1/2) 的研究；然而，尽管此类系统在量子信息处理中处于中心地位，但将此类对称性和几何结构扩展到多个量子位（甚至只有两个）的研究却少得多。在过去的二十年中，为此目的结合了两种具有独立起点和动机的不同方法。一种方法是开发两个或多个量子位的幺正时间演化，以研究量子相关性；通过利用相关的李代数，尤其是所涉及的哈密顿量的子代数，研究人员已经获得了与有限投影几何和组合设计的联系。独立地，几何学家，通过研究投影环线和相关的有限几何，得出了平行的结论。这篇综述汇集了量子物理学的李代数/群表示视角和几何代数视角，以及它们与复杂四元数的联系。总而言之，这可以看作是 Felix Klein 的对称性和几何学的 Erlangen 计划的进一步发展。特别是，两个量子比特的连续 SU(4) 李群的 15 个发生器可以与有限射影几何、组合 Steiner 设计和有限四元数群一一对应。我们考虑的非常不同的观点可能会提供对量子信息问题的进一步洞察。扩展被考虑用于多个量子位，以及更高自旋或更高维量子位。