Measures to characterise Approximate Mutually Unbiased Bases
Ajeet Kumar, Uditanshu Sadual
TL;DR
This work develops a unified, geometry- and design-driven framework to quantify how close approximate Mutually Unbiased Bases (AMUBs) are to exact MUBs. It introduces two AMUB families—beta-AMUBs and Almost Perfect MUBs (APMUBs)—and a collection of intrinsic measures built from cross-basis overlaps, including the $t$-coherence $\Omega_t^{l,m}$, the overlap deviation $\tau^{l,m}$, the overlap spectrum $\Delta$, and the Bengtsson distance $D^2_{l,m}$, along with derived quantities like bar$D^2$. The paper derives relationships among these measures, provides asymptotic bounds for beta- and APMUBs, and gives exact expressions for APMUBs' measures in terms of $\beta$ and $d$. It further analyzes Weak MUBs and AMUBs constructed from resolvable block designs (RBDs), delivering explicit formulas for $\tau$, $\sigma$, and $\bar{D}^2$ and their asymptotics, as well as concrete examples. Collectively, the framework clarifies which MUB-optimal features survive under approximation and guides the construction of AMUBs with predictable performance in tasks like quantum state tomography, entropic uncertainty relations, and quantum key distribution.
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.