Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry

Abstract

We present a fine-structure entanglement classification under stochastic local operation and classical communication (SLOCC) for multiqubit pure states. To this end, we employ specific algebraic-geometry tools that are SLOCC invariants, secant varieties, to show that for n-qubit systems there are ⌈2n/(n+1)⌉ entanglement families. By using another invariant, ℓ-multilinear ranks, each family can be further split into a finite number of subfamilies. Not only does this method facilitate the classification of multipartite entanglement, but it also turns out to be operationally meaningful as it quantifies entanglement as a resource.