周期和离散Zak基
Periodic and discrete Zak bases
摘要：
如果基本位移的乘积等于普朗克常数，则位置和动量的外尔位移算子可交换。然后，它们的公共本征态构成Zak基，每个状态由两个相位参数指定。在对相位强制执行周期依赖性时，人们得到线上的希尔伯特空间到环面上的希尔伯特空间的一对一映射。周期Zak基的傅立叶系数构成离散Zak基。这两个基础是相互无偏的。我们详细研究了这些基础，包括简要讨论它们与Aharonov的模算子的关系，并提到它们如何与一对真正的量子比特线的单自由度相关联。
英文摘要：
Weyl's displacement operators for position and momentum commute if the product of the elementary displacements equals Planck's constant. Then, their common eigenstates constitute the Zak basis, each state specified by two phase parameters. Upon enforcing a periodic dependence on the phases, one gets a one-to-one mapping of the Hilbert space on the line onto the Hilbert space on the torus. The Fourier coefficients of the periodic Zak bases make up the discrete Zak bases. The two bases are mutually unbiased. We study these bases in detail, including a brief discussion of their relation to Aharonov's modular operators, and mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits.
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