Geometric measure of entanglement for symmetric states

Hübener, Robert; Kleinmann, Matthias; Wei, Tzu-Chieh; González-Guillén, Carlos E.; Gühne, Otfried

doi:10.1103/physreva.80.032324

Cited by 129 publications

(173 citation statements)

References 31 publications

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Section: Numerical Resultsmentioning

confidence: 99%

Characterizing Entanglement and Quantum Correlations Constrained by Symmetry

Brugués¹

2017

Springer Theses

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EnglishEntanglement and nonlocal correlations constitute two fundamental resources for quantum information processing, as they allow for novel tasks that are otherwise impossible in a classical scenario. However, their elusive characterization is still a central problem in Quantum Information Theory. The main reason why such a fundamental issue remains a formidable challenge lies in the exponential growth in complexity of the Hilbert space, as well as the space of nonlocal correlations. Physical systems of interest, on the other hand, display symmetries that can be exploited to reduce this complexity, opening the possibility that, for such systems, some of these questions become tractable.This PhD Thesis is dedicated to the study and characterization of entanglement and nonlocal correlations constrained under symmetries. It contains original results in these four threads of research: PPT entanglement in the symmetric states, nonlocality detection in many-body systems, the non-equivalence between entanglement and nonlocality and elemental monogamies of correlations.First, we study PPT entanglement in fully symmetric n-qubit states. We solve the open question on the existence of four-qubit PPT entangled states of these kind, providing constructive examples and methods. Furthermore, we develop criteria for separability, edgeness and the Schmidt number of PPT entangled symmetric states. Geometrically, we focus on the characterization of extremal states of this family and we provide an algorithm to find states with such properties.Second, we study nonlocality in many-body systems. We consider permutationally and translationally invariant Bell inequalities consisting of two-body correlators. These constitute the first tools to detect nonlocality in many-body systems in an experimentally-friendly way with our current technology. Furthermore, we show how these Bell inequalities detect nonloi cality in physically relevant systems such as ground states of Hamiltonians that naturally arise e.g., in nuclear physics. We provide analytical classes of Bell inequalities and we analytically characterize which states and measurements are best suited for them. We show that the method we introduce can be fully generalized to correlators of any order in any Bell scenario. Finally, we provide some feedback from a more experimental point of view.Third, we demonstrate that entanglement and nonlocality are inequivalent concepts in general; a question that remained open in the multipartite case. We show that the strongest form of entanglement, genuinely multipartite entanglement, does not imply the strongest form of nonlocality, genuinely multipartite nonlocality, in any case. We give a constructive method that, starting from a multipartite genuinely multipartite state admitting a K-local model, extends it to a genuinely multipartite entangled state of any number of parties while preserving the degree of locality.Finally, we show that nonlocal correlations are monogamous in a much stronger sense than the typical one, in which th...

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Section: Numerical Resultsmentioning

confidence: 99%

Characterizing Entanglement and Quantum Correlations Constrained by Symmetry

Brugués¹

2017

Springer Theses

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“…It is known that the separable state closest to a symmetric state in terms of the Bures distance is also symmetric [28]. However for other distances this may not be the case.…”

Section: Definitions a Entanglement And Quantumnessmentioning

confidence: 99%

Tensor eigenvalues and entanglement of symmetric states

2016

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Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric state or a spin state, and we investigate to what extent the corresponding tensor eigenvalues contain information about the multipartite entanglement (or, equivalently, the classicality) of the state. This extends previous results connecting entanglement to spectral properties related to the state. While for spin-1 states the positivity of the smallest tensor eigenvalue is equivalent to separability, we show that for higher values of the angular momentum there is a correlation between entanglement and the value of the smallest tensor eigenvalue.

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“…Various results on entanglement for symmetric states have been obtained in the literature [5][6][7][8][9]. For instance, criteria for certifying separability in symmetric mixed states of N qubits were found in [10].…”

Section: Introductionmentioning

confidence: 99%

“…Then "sol" contains a list of Bloch vectors of the pure states that give a decomposition into separable states as in Eq. (6). The vector of weights w i can then be easily calculated.…”

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confidence: 99%

Entanglement and the truncated moment problem

2017

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We map the quantum entanglement problem onto the mathematically well-studied truncated moment problem. This yields a necessary and sufficient condition for separability that can be checked by a hierarchy of semi-definite programs. The algorithm always gives a certificate of entanglement if the state is entangled. If the state is separable, typically a certificate of separability is obtained in a finite number of steps and an explicit decomposition into separable pure states can be extracted.

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Geometric measure of entanglement for symmetric states

Cited by 129 publications

References 31 publications

Characterizing Entanglement and Quantum Correlations Constrained by Symmetry

Characterizing Entanglement and Quantum Correlations Constrained by Symmetry

Tensor eigenvalues and entanglement of symmetric states

Entanglement and the truncated moment problem

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