标题： 用于六维欧几里得向量空间几何代数中的两个量子比特的布洛赫球型模型 显示英文标题

标题： A Bloch-Sphere-Type Model for Two Qubits in the Geometric Algebra of a 6-D Euclidean Vector Space

摘要： 几何代数是一种在任何度量向量空间中固有的数学结构，其定义要求度量张量由向量乘积的标量部分给出。 它提供了一个自然的框架，用于将经典群表示为旋转群的子群，以及它们的李代数。 在本文中，我们展示了六维实欧几里得向量空间的几何代数如何自然地允许构造一个两量子比特（量子位）希尔伯特空间上的特殊酉群，这种方式类似于在单个量子比特的已建立的布洛赫球模型中所使用的方法。 然后，这被用来说明四维特殊酉群的卡丹分解和子代数，这些最近被J. Zhang、J. Vala、S. Sastry和K. B. Whaley [Phys. Rev. A 67, 042313, 2003]用于研究两量子比特酉算子的纠缠能力。 显示英文摘要

摘要： Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional special unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.