Entanglement is one of the most intriguing features of quantum mechanics. It is widely used in quantum communication and information processing and plays a key role in quantum computation. At the same time, entanglement is not fully understood. It is deeply rooted into the linearity of quantum theory and in the superposition principle and (for pure states) is essentially and intuitively related to the impossibility of factorizing the state of the total system in terms of states of its constituents.
The characterization and quantification of entanglement is an open and challenging problem. It is possible to give a good definition of bipartite entanglement [1] in terms of the von Neumann entropy and the entanglement of formation. The problem of defining multipartite entanglement is more difficult and no unique definition exists [2].
I introduce the notion of maximally multipartite entangled states (MMES) [3] of $n$ qubits as a generalization of the bipartite case. Their bipartite entanglement does not depend on the bipartition and is maximal for all possible bipartitions. Some examples of MMES for small $n$ are investigated, both analytically and numerically. These states are the solutions of an optimization problem, that can be recast in terms of statistical mechanics [4].

\
[1] W. K. Wootters, Quantum Inf. Comput., \\textbf{1}, 27 (2001); L.\
Amico, R. Fazio, A. Osterloh and V. Vedral ``Entanglement in\
Many-Body Systems," arXiv\:quant-ph/0703044 (Rev. Mod. Phys., in\
print).\
\
[2] V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A\
\\textbf{61}, 052306 (2000); A. Wong and N. Christensen, Phys. Rev. A\
\\textbf{63}, 044301 (2001); D. Bruss, J. Math. Phys. \\textbf{43},\
4237 (2002); D.A. Meyer and N.R. Wallach, J. Math. Phys.\
\\textbf{43}, 4273 (2002);\
V. I. Man'ko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria, J. Phys.\
A\: Math. Gen. \\textbf{35}, 7137 (2002)\
\
[3] P. Facchi, G. Florio, G. Parisi and S. Pascazio, ``Maximally\
multipartite entangled states" arXiv\:0710.2868 [quant-ph].\
\
[4] P. Facchi, G. Florio, U. Marzolino, G. Parisi and S. Pascazio,\
``Statistical mechanics of multipartite entanglement"\
arXiv\:0803.4498 [quant-ph].\
\