Structure of the sets of mutually unbiased bases forNqubits

For a system of $N$ qubits, living in a Hilbert space of dimension $d=2^N$, it is known that there exists $d+1$ mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of $N$ qubits is outlined.