Uncertainty Relation for the Discrete Fourier Transform

Details Uncertainty Relation for the Discrete Fourier Transform Uncertainty Relation for the Discrete Fourier Transform

2007-10-03

arxiv.org/abs/0710.0723v2

Phys. Rev. Lett. 100, 190401 (2008)

Physics

Quantum Physics

Extended Version; 13 pages; In press in Phys. Rev. Lett

Scientific paper

10.1103/PhysRevLett.100.190401

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=exp[i phi] VU. Its most important application is to constrain how much a quantum state can be localised simultaneously in two mutually unbiased bases related by a Discrete Fourier Transform. It provides an uncertainty relation which smoothly interpolates between the well known cases of the Pauli operators in 2 dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the Harmonic oscillator equation.

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