Classification of Entanglement in Symmetric States

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.

PhD thesis, University of Leeds, July 2011. 170 pages

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Cited by in corpus (24)

• Nonlocality of Symmetric States
• Multiqubit symmetric states with maximally mixed one-qubit reductions
• A comparison of old and new definitions of the geometric measure of entanglement
• Decoherence Effects on the Non-locality of Symmetric States
• Entanglement classes of permutation-symmetric qudit states: symmetric operations suffice
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• Nonlocality and Entanglement for Symmetric States
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• Separable Schmidt modes of a non-separable state
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• Symmetric Multiqudit States: Stars, Entanglement, Rotosensors
• An Algebraic-Geometric Characterization of Tripartite Entanglement
• Symmetric 3 Qubit State Invariants
• State transformations within entanglement classes containing permutation-symmetric states
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• Generalized Weyl-Heisenberg algebra, qudit systems and entanglement measure of symmetric states via spin coherent states
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• Orthonormal bases of extreme quantumness
• Experimental construction of a symmetric three-qubit entangled state and its utility in testing the violation of a Bell inequality on an NMR quantum simulator
• Entanglement Classification via Single Entanglement Measure
• Classifying Entanglement by Algebraic Geometry
• Local Unitary Equivalent Classes of Symmetric N-Qubit Mixed States
• Entanglement in the symmetric subspace: mapping multipartite to bipartite states
• Non-Gaussianity from superselection rules