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)