arXiv:quant-ph/0511234 · Journal of Physics A Mathematical and General

Periodic and discrete Zak bases

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

Published Feb 1, 2006• 17 citations

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.

DOI: 10.1088/0305-4470/39/7/011

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arXiv:quant-ph/0511234 · Journal of Physics A Mathematical and General

Periodic and discrete Zak bases

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

Published Feb 1, 2006• 17 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/39/7/011

Related papers

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arXiv:2205.05445Journal of Physics A Mathematical and TheoreticalJun 8, 20231 citation

h

, Planck's constant.

Abstract The eigenbases of two quantum observables, <?CDATA

\{|a_i\rangle\}_{i = 1}^D

?> { | a i ⟩ } i = 1 D and <?CDATA

\{|b_j\rangle\}_{j = 1}^D

?> { | b j ⟩ } j = 1 D , form mutually unbiased bases (MUB) if <?CDATA

|\langle a_i |b_j \rangle| = 1/\sqrt{D}

?> | ⟨ a i | b j ⟩ | = 1 / D for all i and j. In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey <?CDATA

|\langle a_i |b_j \rangle| \leqslant c/\sqrt{D}

?> | ⟨ a i | b j ⟩ | ⩽ c / D , where c is some positive constant independent of D. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) on d-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter q. We solve the model analytically and observe that for prime d the eigenvectors of two QW evolution operators form AMUB. Namely, if d is prime the corresponding eigenvectors of the evolution operators, that act in the D-dimensional Hilbert space ( <?CDATA

D = 2d

?> D = 2 d ), obey <?CDATA

|\langle v_q|v^{^{\prime}}_{q^{^{\prime}}} \rangle| \leqslant \sqrt{2}/\sqrt{D}

?> | ⟨ v q | v q ′ ′ ⟩ | ⩽ 2 / D for <?CDATA $q\ne