  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.
