Equiangular tight frames and unistochastic matrices. (English) Zbl 1369.81020
Summary: We demonstrate that a complex equiangular tight frame composed of \(N\) vectors in dimension \(d\), denoted ETF \((d, N)\), 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 ETFs in infinitely many dimensions and to derive a method to introduce non-trivial free parameters in ETFs. We present an explicit six-parametric family of complex ETF(6,16), which defines a family of symmetric POVMs. Minimal and maximal possible average entanglement of the vectors within this qubit-qutrit family are described. Furthermore, we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix, which we apply to find all existing classes of complex ETFs containing up to 20 vectors.
MSC:
| 81P40 | Quantum coherence, entanglement, quantum correlations |
| 15B51 | Stochastic matrices |
| 46G10 | Vector-valued measures and integration |
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