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arXiv:1110.5200 · International Journal of Quantum Information

CLASSIFICATION OF ENTANGLEMENT IN SYMMETRIC STATES

Martin Aulbach

Published Oct 1, 2012• 37 citations
arXivHTMLPDF

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.

DOI
10.1142/s0219749912300045

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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.
Permutation-symmetric n qubit pure states can be represented by n points on the surface of the unit sphere by means of the Majorana representation. Here this representation is employed to characterize and compare the three entanglement classification schemes LOCC, SLOCC and the Degeneracy Configuration. Symmetric SLOCC operations are found to be described by Mobius transformations, and an intuitive visualization of their freedoms is presented. For symmetric states of up to 5 qubits explicit forms of representative states for all SLOCC classes are derived. The symmetric 4 qubit entanglement classes are compared to the entanglement families introduced in [PRA 65, 052112 (2002)], and examples are given how the SLOCC-inequivalence of symmetric states can be quickly determined from known results about Mobius transformations.
We discuss the entanglement properties of symmetric states of n qubits. The Majorana representation maps a generic such state into a system of n points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an overcomplete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well-documented three-qubits case. In the SLOCC case, a cross ratio of related Möbius transformations are shown to play a central role, exemplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.
We investigate the relationship between multipartite entanglement and symmetry, focusing on permutation symmetric states. We give a highly intuitive geometric interpretation to entanglement via the Majorana representation, where these states correspond to points on a unit sphere. We use this to show how various entanglement properties are determined by the symmetry properties of the states. The geometric measure of entanglement is thus phrased entirely as a geometric optimization and a condition for the equivalence of entanglement measures written in terms of point symmetries. Finally, we see that different symmetries of the states correspond to different types of entanglement with respect to interconvertibility under stochastic local operations and classical communication.
Quantum entanglement is a key property of quantum information theory, that is at the heart of numerous promising applications in fields such as quantum cryptography, quantum computing or quantum sensing. In the past decades, the advent of such innovative technologies has reinforced the need for a better understanding of entanglement. The aim of this thesis is to contribute to this effort through the development of new tools targeting the characterization of several features of entanglement. Concerning the issue of entanglement detection, we present an optimization of the approach that exploits the concept of generalized concurrences to solve the separability problem for pure states. We then reformulate the separability question of mixed states into a matrix analysis problem, from which we obtain general separability criteria for multipartite states of ranks two and three. We also briefly discuss some properties of separable states. In particular, we characterize optimal separable decompositions of symmetric (i.e. permutation invariant) states of two and three qubits with maximal rank properties. Regarding the quantification of entanglement, we propose a function to quantify the entanglement of symmetric multiqubit states within classes of entangled states gathering states that are stochastically equivalent through local operations assisted with classical communication. This function establishes a link between the amount of entanglement of a symmetric state and the distribution of its Majorana points on the Bloch sphere. We finally investigate the robustness of entanglement with respect to particle loss and provide a full description of all multiqubit states that are fragile for the loss of one of their qubits. For symmetric states, the fragility for the loss of one qubit is shown to be related to a particular symmetry of the Majorana points.
We analyze entanglement classes for permutation-symmetric states for nnn qudits (i.e., ddd-level systems), with respect to local unitary operations (LU equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operators is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the Greenberger-Horne-Zeilinger and WWW states for the many-particle qudit case. In the latter case, we introduce the excitation state as a natural generalization of the WWW state for d>2d>2d>2.
The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which maps an n qubit symmetric state to n points on the unit sphere. It is shown how symmetries of the point distribution can be exploited to simplify the calculation of entanglement and also help find the maximally entangled symmetric state. Using a combination of analytical and numerical results, the most entangled symmetric states for up to 12 qubits are explored and discussed. The optimization problem on the sphere presented here is then compared with two classical optimization problems on the S 2 sphere, namely Tóth's problem and Thomson's problem, and it is observed that, in general, they are different problems.