Uncertainty Relation for the Discrete Fourier Transform

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form $UV=e^{i\varphi }VU$. 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 well-known 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.

Figure 1

Minimum uncertainty $\Delta U^2$ as a function of dimension $d$ when one imposes that $\Delta U^2=\Delta V^2$. The upper (red and continuous) curve is the exact bound on $\Delta U^2$. It is obtained from the smallest eigenvalue of the operator Eq. (21) when $\theta =\pi /4$. Note that when $d=2$ and $d=4$ the exact bound is $\Delta U^2=1/2$ and that when $d=3$ the exact bound is larger than $1/2$, as noted in 13. The lower (blue and dashed) curve is the bound obtained from the bound Eq. (19) upon imposing that $\Delta U^2=\Delta V^2$. The two curves coincide when $d=2$ and have the same asymptotic behavior $\Delta U^2\ge \pi /d$ for large $d$.