On the Information of the Second Moments Between Random Variables Using Mutually Unbiased Bases

TL;DR

This paper explores how mutually unbiased bases (MUB) can be used to analyze second moments of random variables, revealing their spectral properties, and demonstrates applications in communication, signal analysis, and dimensionality reduction.

Contribution

It establishes the spectral properties of MUB for finite discrete signals and introduces novel applications in communication protocols, signal analysis, and data dimensionality reduction.

Findings

MUB spectra fully characterize correlation matrices of finite discrete signals.

MUB spectra can be computed efficiently when signal length is prime.

MUB-based analysis enhances understanding of random vectors and operators.

Abstract

The notation of mutually unbiased bases(MUB) was first introduced by Ivanovic to reconstruct density matrixes\cite{Ivanovic}. The subject about how to use MUB to analyze, process, and utilize the information of the second moments between random variables is studied in this paper. In the first part, the mathematical foundation will be built. It will be shown that the spectra of MUB have complete information for the correlation matrixes of finite discrete signals, and the nice properties of them. Roughly speaking, it will be shown that each spectrum from MUB plays an equal role for finite discrete signals, and the effect between any two spectra can be treated as a global constant shift. These properties will be used to find some important and natural characterizations of random vectors and random discrete operators/filters. For a technical reason, it will be shown that any MUB spectra can be found as fast as Fourier spectrum when the length of the signal is a prime number. In the second part, some applications will be presented. First of all, a protocol about how to increase the number of users in a basic digital communication model will be studied, which has bring some deep insights about how to encode the information into the second moments between random variables. Secondly, the application of signal analysis will be studied. It is suggested that complete "MUB" spectra analysis works well in any case, and people can just choose the spectra they are interested in to do analysis. For instance, single Fourier spectra analysis can be also applied in nonstationary case. Finally, the application of MUB in dimensionality reduction will be considered, when the prior knowledge of the data isn't reliable.