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

arXiv comment11 pages, 2 figures, Minor changes, Published version

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• Classification of four qubit states and their stabilisers under SLOCC operations
• Non-gaussian Entanglement Swapping between Three-Mode Spontaneous Parametric Down Conversion and Three Qubits
• Geometric genuine multipartite entanglement for four-qubit systems
• Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities
• Classifying Entanglement by Algebraic Geometry
• Persistent Tensors and Multiqudit Entanglement Transformation
• Entangled Subspaces through Algebraic Geometry
• Special core tensors of multi-qubit states and the concurrency of three lines
• Brief Theory of Multiqubit Measurement
• Absolutely maximally entangled pure states of multipartite quantum systems