Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry

TL;DR

This paper introduces a detailed classification method for multiqubit entanglement using algebraic geometry, revealing numerous entanglement families and subfamilies that are meaningful for quantifying entanglement as a resource.

Contribution

It develops a novel algebraic-geometry based framework for fine-structure classification of multiqubit entanglement under SLOCC, identifying multiple entanglement families and subfamilies.

Findings

Number of entanglement families for n-qubits is approximately 2^n/(n+1).

The classification method is operationally meaningful for quantifying entanglement.

The approach employs algebraic invariants like secant varieties and multilinear ranks.

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 $\lceil\frac{2^{n}}{n+1}\rceil$ 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.