Please use this identifier to cite or link to this item: https://doi.org/10.1142/S0219749912300045
Title: Classification of entanglement in symmetric states
Authors: Aulbach, M. 
Keywords: degeneracy conguration
geometric measure
global entanglement
Majorana representation
maximally entangled
Mobius transformation
Platonic solids; LMG model
SLOCC
Symmetric states
Issue Date: Oct-2012
Citation: Aulbach, M. (2012-10). Classification of entanglement in symmetric states. International Journal of Quantum Information 10 (7) : -. ScholarBank@NUS Repository. https://doi.org/10.1142/S0219749912300045
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 categorized, 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 generalization 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 characterization 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 Möbius transformations of complex analysis, which allows for a clear visualization 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 five qubits are derived. Well-known entanglement classification schemes such as the four 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. © 2012 World Scientific Publishing Company.
Source Title: International Journal of Quantum Information
URI: http://scholarbank.nus.edu.sg/handle/10635/115027
ISSN: 02197499
DOI: 10.1142/S0219749912300045
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