Entanglement classification schemes : comparison between Majorana representation and algebraic geometry approaches

Quantum entanglement can be of different kinds [1] and classifying the
quantum states in this respect may represent a difficult challenge in
general multipartite systems. In particular, entanglement classes that
are inequivalent under stochastic local operations and classical communication (SLOCC) are of fundamental importance. For 𝑁-qubit
systems with 𝑁 > 3, there is an infinity of such SLOCC entanglement
classes [1] and it makes sense to gather them into a finite number
of families, as was done for symmetric states in Refs. [2,3] using two
distinct approaches (Majorana representation and algebraic geometry
tools, respectively). Here, we compare these two structures and identify whether they can be embedded into one another or not. To do
so, we formulate the structure of Ref. [2] in terms of 𝑘-secants and 𝑘-
tangents (𝑘 a positive integer) of the Veronese variety [3] and we prove
that only the 𝑘-tangent structuration provides a coherent structure
compatible with that of Ref. [3].
[1] W. Dür et al., Phys. Rev. A 62, 062314 (2000). [2] T. Bastin et
al., Phys. Rev. Lett. 103, 070503 (2009). [3] M. Sanz et al., J. Phys.
A: Math. Theor. 50, 195303 (2017)