Uncertainty principle, Shannon-Nyquist sampling and beyond

TL;DR

This paper explores the connection between the uncertainty principle, Shannon-Nyquist sampling, and information recovery, revealing that both classical and quantum systems can recover information below the uncertainty limit using a unified sampling criterion.

Contribution

It introduces a new signal recovery formula based on Shannon-Nyquist sampling that parallels Donoho-Stark's method and extends the concept to quantum state recovery from below the uncertainty limit.

Findings

Shannon-Nyquist sampling uses the same mechanism as Donoho-Stark recovery.

A new formula for signal recovery analogous to Donoho-Stark is proposed.

Quantum state recovery below the uncertainty limit is theoretically possible.

Abstract

Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of view and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal recovery formula, which is analogous to Donoho-Stark formula, is given using the idea of Shannon-Nyquist sampling; in this formulation, the smearing of information below the uncertainty limit as well as the recovery of information with specified bandwidth take place. We also discuss the recovery of states from the domain below the uncertainty limit of coordinate and momentum in quantum mechanics and show that in principle the state recovery works by assuming ideal measurement procedures. The recovery of the lost information in the sub-uncertainty domain means that the loss of information in such a small domain is not fatal, which is in accord with our common understanding of the uncertainty principle, although its precise recovery is something we are not used to in quantum mechanics. The uncertainty principle provides a universal sampling criterion covering both the classical Shannon-Nyquist sampling theorem and the quantum mechanical measurement.