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 $\lceil \frac{2^n}{n+1}\rceil$ entanglement families. By using another invariant, $\ell$-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.

• Received 23 October 2019
• Accepted 31 August 2020
• Corrected 17 November 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043003

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Published by the American Physical Society