Fine-structure classification of multiqubit entanglement by algebraic geometry

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 ⌈2/ⁿn +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.