Equiangular tight frames and unistochastic matrices

TL;DR

This paper establishes a fundamental link between complex equiangular tight frames and unistochastic matrices, enabling the discovery of new ETFs, parametrization, and an efficient numerical method for classification.

Contribution

It introduces a novel connection between ETFs and unistochastic matrices, leading to new ETF constructions, parametrizations, and a numerical classification method.

Findings

Derived a 6-parametric family of complex ETF(6,16)

Developed an efficient numerical procedure for classifying ETFs

Found all complex ETFs with up to 19 vectors

Abstract

In this work, we show that a complex equiangular tight frame (ETF) composed by $N$ vectors in dimension $d$ exists if and only if a certain bistochastic matrix, univocally determined by $N$ and $d$, belongs to a special class of unistochastic matrices. This connection allows us to find new complex ETF in infinitely many dimensions and to derive a method to introduce non-trivial free parameters in ETF. We derive a 6-parametric family of complex ETF(6,16), which defines a family of symmetric POVM. Minimal and maximal possible average entanglement of the vectors within this qubit-qutrit family are presented. Furthermore, we propose an efficient numerical procedure to find the unitary matrix underlying a unistochastic matrix, which we apply to find all existing classes of complex ETF containing up to 19 vectors.