Periodic and discrete Zak bases

DOI10.1088/0305-4470/39/7/011zbMATH Open1092.81021arXivquant-ph/0511234OpenAlexW2056998345WikidataQ62097214 ScholiaQ62097214MaRDI QIDQ3379338

Kean Loon Lee, Berthold-Georg Englert, M. Revzen, A. Mann

Publication date: 6 April 2006

Published in: Journal of Physics A: Mathematical and General (Search for Journal in Brave)

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/quant-ph/0511234

zbMATH Keywords

position and momentum operatorZak basis

Mathematics Subject Classification ID

Commutation relations and statistics as related to quantum mechanics (general) (81S05) Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory (81Q70)

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