The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes

Hayashi, Masahito; Markham, Damian; Murao, Mio; Owari, Masaki; Virmani, S.

doi:10.1063/1.3271041

Cited by 44 publications

(41 citation statements)

References 28 publications

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“…Similarly, for a symmetric multipartite pure states, Hayashi et al [8] indicated that the closest product state in terms of the fidelity (which is a distance measure describing how close two given quantum states are; for details, see [49]) can be chosen as a symmetric product state for symmetric pure states whose amplitudes are all nonnegative in a computational basis. Moreover, Hübener et al [3] claimed that the closest product state to a symmetric entangled multiparticle state is also symmetric up to a phase.…”

Section: Best Complex Rank-one Approximation and Us-eigenvaluesmentioning

confidence: 99%

“…In fact, the geometric measure of entanglement problem is a multiway optimization problem, as well as a tensor decomposition problem or a rank-one approximation to high-order tensors problem [8,43,44,45]. Recently, it was shown that the geometric measure of a symmetric pure state with nonnegative amplitudes is equal to the largest Z-eigenvalue of the underlying nonnegative tensor [44].…”

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confidence: 99%

“…It has applications in various different topics, including many body physics [4], local discrimination, quantum computation, condensed matter systems, entanglement witnesses, and the study of quantum channel capacities. The geometric measure of entanglement is nothing but the injective tensor norm itself [8], which appears in the theory of operator algebra [9] and has now become increasingly important in theoretical physics-particularly in quantum channel capacities [10,11,12,13,14,15,16,17,18].…”

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confidence: 99%

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Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Bai

2014

SIAM J. Matrix Anal. & Appl.

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We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rankone approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.Key words. unitary eigenvalue (U-eigenvalue), Z-eigenvalue, symmetric real tensor, geometric measure of entanglement, the best rank-one approximation 1. Introduction. Entanglement of compound systems is a key resource in quantum information processing. In many practical applications it is of fundamental importance to know whether a state is entangled or not. However, this information is often not sufficient, and it is also required to know how much a state is entangled. A useful tool for quantifying the amount of entanglement of a state is given by the so-called entanglement measures [1, 2].A widely used entanglement measure is provided by the geometric measure of entanglement [3,4,5] that is defined for a pure state. The geometric measure of entanglement was first proposed by Shimony [6] and extended to multipartite systems by Wei and Goldbart [7]. It has applications in various different topics, including many body physics [4], local discrimination, quantum computation, condensed matter systems, entanglement witnesses, and the study of quantum channel capacities. The geometric measure of entanglement is nothing but the injective tensor norm itself [8], which appears in the theory of operator algebra [9] and has now become increasingly important in theoretical physics-particularly in quantum channel capacities [10,11,12,13,14,15,16,17,18].A tensor is a multidimensional array [19]. Tensor decompositions originated with Hitchcock in 1927 [20], and the idea of a multiway model is attributed to Cattell in 1944 [21]. Recently, interest in tensor decompositions has expanded to other fields. Examples include signal processing [22,23], numerical linear algebra [24,25], computer

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Section: Best Complex Rank-one Approximation and Us-eigenvaluesmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Bai

2014

SIAM J. Matrix Anal. & Appl.

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“…It is also related to optimal entanglement witnesses [3,6] and has nice applications in many-body physics and condensed matter systems [7][8][9][10]. Despite its usefulness, the explicit value of the geometric measure of entanglement has only been derived, so far, for a limited number of entangled states, such as N -qubit Greenberger-Horne-Zeilinger (GHZ) states [3], Dicke states [3], generalized W states [11], graph states [12], and other typical states with given symmetry properties [6,[13][14][15][16]. The geometric measure remains unknown for most of the multipartite states simply because of the definition that involves an optimization procedure over the class of separable states and this represents a formidable task in the general case even with numerical approaches.…”

Section: Introductionmentioning

confidence: 99%

Multiqubit symmetric states with high geometric entanglement

et al. 2010

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We propose a detailed study of the geometric entanglement properties of pure symmetric N -qubit states, focusing more particularly on the identification of symmetric states with a high geometric entanglement and how their entanglement behaves asymptotically for large N . We show that much higher geometric entanglement with improved asymptotical behavior can be obtained in comparison with the highly entangled balanced Dicke states studied previously. We also derive an upper bound for the geometric measure of entanglement of symmetric states. The connection with the quantumness of a state is discussed.

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“…This straightforward conjecture has been actively investigated [21,83], but a proof is far from trivial. After some special cases were proven [146,147], Hübener et al [84] were able to give a proof for the general case of pure symmetric states 5 . They showed that for n ≥ 3 qudits the CPSs of a pure symmetric state are necessarily symmetric, thus greatly reducing the complexity of finding the CPSs and the entanglement of 5 One could ask whether this result also holds for translationally invariant states (which appear in spin models), but this is not the case.…”

Section: Symmetric Statesmentioning

confidence: 99%

Classification of Entanglement in Symmetric States

Aulbach

2012

Int. J. Quantum Inform.

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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.

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The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes

Cited by 44 publications

References 28 publications

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Multiqubit symmetric states with high geometric entanglement

Classification of Entanglement in Symmetric States

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