Special entangled quantum systems and the Freudenthal construction

Abstract

We consider special quantum systems containing both distinguishable and identical constituents. It is shown that for these systems the Freudenthal construction based on cubic Jordan algebras naturally defines entanglement measures invariant under the group of stochastic local operations and classical communication (SLOCC). For this type of multipartite entanglement the SLOCC classes can be explicitly given. These results enable further explicit constructions of multiqubit entanglement measures for distinguishable constituents by embedding them into identical fermionic ones. We also prove that the Plücker relations for the embedding system provide a sufficient and necessary condition for the separability of the embedded one. We argue that this embedding procedure can be regarded as a convenient representation for quantum systems of particles which are really indistinguishable but for some reason they are not in the same state of some inner degree of freedom.

The black-hole/qubit correspondence: an up-to-date review

“…The antisymmetric bilinear form, quartic norm and triple product are then simply given by symmetrizing those of F(J 3C ). See [139,182] for details.…”

“…Recall, Str(J ABC ) and Str 0 (J ABC ) generate [SO(2, C)] 3 and [SO(2, C)] 2 , respectively the structure and reduced structure groups of J C . [182].…”

“…Another useful aspect of this correspondence is that the classification problem of certain types of black hole can be mapped to the classification problem of entanglement types of qubit systems [2,3,[21][22][23] .…”

“…The techniques employed in such cases have originated from the string theoretical side. For example, although Freudenthal triple systems (FTS) [34][35][36] have been well-known to the supergravity community [37][38][39], their relevance to special entangled systems [20,21,40] has only recently been realised within the framework of the BHQC. This, in turn, inspired new applications of the Freudenthal triple system to black holes in the form of Freudenthal and Jordan dualities [41,42].…”

Classification of multipartite systems featuring only $| W\rangle $ and $| {GHZ}\rangle $ genuine entangled states

“…The classification of entanglement classes of multipartite systems under SLOCC has been investigated in the last 10 years by many authors 2,3,6,[11][12][13]18,19,[30][31][32][33]36,39,40 . Interestingly under SLOCC some 3,6,11,13,30,36,39 of these entangled systems is featuring two genuine types of entanglement.…”