Symmetry and Classification of Multipartite Entangled States

TL;DR

This paper explores the classification and properties of multipartite entangled states, emphasizing symmetry, knot theory connections, and new constructions of highly entangled states, advancing understanding of complex quantum entanglement structures.

Contribution

It introduces novel classifications, constructions, and verification methods for multipartite entangled states, linking entanglement with symmetry and knot theory.

Findings

Established a connection between entanglement classification and knot theory

Constructed new Absolutely Maximally Entangled (AME) states

Developed a polynomial-based method for state equivalence verification

Abstract

One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.