Special functions in quantum phase estimation

TL;DR

This paper explores the use of special functions, specifically prolate spheroidal wave and Mathieu functions, to improve quantum phase estimation by optimizing probability bounds and characterizing uncertainty relations.

Contribution

It introduces the application of prolate spheroidal wave and Mathieu functions to quantum phase estimation, providing analytical tools for optimal estimation under constraints.

Findings

Prolate spheroidal wave functions approximate maximum probability for small estimation errors.

Mathieu functions provide exact optimal estimation under energy constraints.

Characterization of uncertainty relations for periodic quantum functions.

Abstract

This paper explains existing results for the application of special functions to phase estimation, which is a fundamental topic in quantum information. We focus on two special functions. One is prolate spheroidal wave function, which approximately gives the maximum probability that the difference between the true parameter and the estimate is smaller than a certain threshold. The other is Mathieu function, which exactly gives the optimum estimation under the energy constraint. It also characterizes the uncertainty relation for the position and the momentum for periodic functions.