Measures to characterise Approximate Mutually Unbiased Bases

TL;DR

This paper introduces measures to quantify how close approximate mutually unbiased bases (AMUBs) are to ideal MUBs, especially in dimensions where exact constructions are unknown, aiding their characterization and application.

Contribution

It defines quantifiable measures for AMUBs based on geometric and design principles, enabling their evaluation without explicit constructions.

Findings

Measures can estimate proximity of AMUBs to true MUBs.

Measures are applicable to various AMUB classes like Weak MUBs.

Evaluation of measures is possible without exact AMUB construction details.

Abstract

Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.