Uncertainty Relation for the Discrete Fourier Transform

TL;DR

This paper establishes an uncertainty relation for unitary operators related by a phase, constraining quantum state localization in mutually unbiased bases connected via the Discrete Fourier Transform, with implications for quantum information and signal processing.

Contribution

It introduces a new uncertainty relation for unitary operators with a phase-dependent commutation relation, bridging finite and continuous variable cases and exploring minimum uncertainty states.

Findings

Derives an uncertainty relation for unitary operators with phase-dependent commutation.

Provides bounds on quantum state localization in mutually unbiased bases.

Identifies minimum uncertainty states as solutions to a discrete harmonic oscillator.

Abstract

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.