Classifying Entanglement by Algebraic Geometry

TL;DR

This paper introduces an algebraic-geometric framework for classifying multipartite quantum entanglement, providing algorithms and tools to understand entanglement structures and transformations in complex quantum systems.

Contribution

It develops a novel classification algorithm for multipartite entanglement using algebraic geometry, including a fine-structure classification and analysis of entanglement transformations.

Findings

Algorithm for classifying entanglement via $k$-secant varieties and multilinear ranks.

Introduction of persistent tensors and bounds on their tensor rank.

Analysis of SLOCC convertibility among multipartite systems.

Abstract

Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This dissertation covers characterization of multipartite entanglement using algebraic-geometric tools. Firstly, we establish an algorithm to classify multipartite entanglement by $k$-secant varieties of the Segre variety and $\ell$-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call \emph{persistent tensors} and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.