量子信息处理中的几何代数
Geometric Algebra in Quantum Information Processing
摘要：
本文发展了一个耦合两态量子系统（量子比特）的几何模型，该模型使用几何（又名Clifford）代数来表示。它首先显示如何欧几里德旋量可以解释为实体的几何代数的欧几里德向量空间。这个代数，然后解除闵可夫斯基时空及其相关的几何代数，这提供了如何密度算子和纠缠行为下洛伦兹变换的见解进行了讨论。多个时空副本的直和在相关代数上诱导了张量积结构，其中合适的商同构于多量子比特量子力学中常规使用的矩阵代数。最后，几何代数的效用，在理解酉和非酉量子操作上的几个例子，在量子信息处理的兴趣。
英文摘要：
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.
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