The maximally entangled set of 4-qubit states (Q2814212)

From the standpoint of \textit{quantum information} entanglemente seems to be well understood in the case of \(2\)-qubits systems (the direct offsprings of the Bell original model), but as soon as we move on to the \(n\geq3\) qubits instances difficulties grow more than exponentially and -- as the authors remark at the very beginning -- ``we are far from a full-fledged theory of multipartite entanglement''. Witness the length of this weighty, but admittedly ``very tedious'' (p.\ \(12\)) paper fulfilling the vow of a previous, much slender, article [\textit{J. I. de Vicente} et al., ``Maximally entangled set of multipartite quantum states'', Phys. Rev. Lett. 111, No. 11, Article ID 110502, 5 p. (2013; \url{doi:10.1103/PhysRevLett.111.110502})], namely to ``investigate all possible state transformations within the 4-qubit case.'' This is important, despite its enormous complexity, because ``multipartite states occur in many applications of quantum information, such as one-way quantum computing, quantum error correction, and quantum secret sharing.''NEWLINENEWLINEIn order to find ``classes of states which are indicated to be particularly relevant in an operationally meaningful sense'' the authors, within the well-known paradigm of Local Operations and Classical Communication (LOCC), ``introduced recently the concept of the maximally entangled set (MES) of multipartite states ... Considering entanglement as a resource, a maximally entangled state ought to be a state which can be transformed into any other state deterministically via LOCC.'' For \(n\geq3\) however the MES does not reduce to a single maximally entangled state. We rather have sets MES\(_n\) such that ``(i) no state in MES\(_n\) can be obtained from any other \(n\)-partite state via LOCC ... and (ii) for any truly \(n\)-partite entangled pure state, \(|\Phi\rangle\notin\mathrm{MES}_n\), there exists a state in MES\(_n\) from which \(|\Phi\rangle\) can be obtained via LOCC.''NEWLINENEWLINEIn the quoted [loc. cit.] the authors already gave a complete characterization of MES\(_3\) for the \(3\)-qubit case, but considered solely the generic \(4\)-qubit states of MES\(_4\). In the present paper then they complete their task by considering all the remaining non-generic SLOCC (stochastic LOCC) classes which can be grouped into nine families which show a distinct behavior w.r.t.\ the generic SLOCC classes. They add that ``the existence of these classes is precisely the reason for investigating in detail the very involved case of four qubits. It is our intention to investigate in the future whether this different behavior might be useful for certain applications. Moreover, their existence also shows once again the notoriously difficult structure of multipartite LOCC transformations.'' It is also remarked that the authors ``present here all the details of the derivations as they can be used directly to study arbitrary LOCC transformations among four qubits.''NEWLINENEWLINEIt would be very difficult to further summarize the remarkable results presented here without inappropriately going at length into cumbersome details and elaborate notations, and hence we prefer to leave this task to the informed and interested reader.