Classification of Entanglement in Symmetric States

TL;DR

This paper classifies entanglement in symmetric quantum states, especially multi-qubit states, using the Majorana representation to visualize and analyze entanglement classes, invariants, and maximally entangled states.

Contribution

It introduces a comprehensive entanglement classification framework for symmetric states using Majorana representation and derives explicit forms for symmetric SLOCC classes up to 5 qubits.

Findings

Maximally entangled symmetric states identified up to 12 qubits.

Majorana representation facilitates visualization of SLOCC transformations.

Explicit SLOCC class representatives for states up to 5 qubits derived.

Abstract

Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis the entanglement of symmetric multipartite states is categorised, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalisation of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterisation of entanglement equivalence classes such as Stochastic Local Operations and Classical Communication (SLOCC) and the Degeneracy Configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Mobius transformations of complex analysis, which allows for a clear visualisation of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to 5 qubits are derived. Well-known entanglement classification schemes such as the 4 qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.