Equiangular tight frames and unistochastic matrices

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.

17 pages, 3 figures. Comments are very welcome!

References in corpus (5)

• Tables of the existence of equiangular tight frames
• Quantum Conical Designs
• Maximally informative ensembles for SIC-POVMs in dimension 3
• Entanglement and Designs
• A new method to construct families of complex Hadamard matrices in even dimensions

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• Local hidden variable values without optimization procedures
• Cyclic measurements and simplified quantum state tomography
• Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces