Uncertainty Relation for the Discrete Fourier Transform
Abstract
We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e^{iϕ}VU. Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform. It provides an uncertainty relation which smoothly interpolates between the wellknown cases of the Pauli operators in two 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.
 Publication:

Physical Review Letters
 Pub Date:
 May 2008
 DOI:
 10.1103/PhysRevLett.100.190401
 arXiv:
 arXiv:0710.0723
 Bibcode:
 2008PhRvL.100s0401M
 Keywords:

 03.65.Ta;
 03.67.Lx;
 Foundations of quantum mechanics;
 measurement theory;
 Quantum computation;
 Quantum Physics
 EPrint:
 Extended Version