Entanglement and nonclassical properties of hypergraph states

Gühne, Otfried; Cuquet, Martí; Steinhoff, F. E. S.; Moroder, Tobias; Rossi, Matteo A. C.; Bruß, Dagmar; Kraus, Barbara; Macchiavello, Chiara

doi:10.1088/1751-8113/47/33/335303

Cited by 63 publications

(116 citation statements)

References 38 publications

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“…The numerical result suggests that this is a general feature of four-qubit systems. We also tested special examples of highly entangled four-qubit states, such as the cluster state, classes of hypergraph states [34], the Higuchi-Sudbery |M 4 state [35] or the |χ -state [13,36]. While many of theses states can be shown to be outside of Q 2 , we were not able to prove analytically or with the help of the semidefinite program that they have a finite distance to Q 2 .…”

Section: Numerical Resultsmentioning

confidence: 99%

Characterizing Ground and Thermal States of Few-Body Hamiltonians

Huber

Gühne

2016

Phys. Rev. Lett.

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The question whether a given quantum state is a ground or thermal state of a few-body Hamiltonian can be used to characterize the complexity of the state and is important for possible experimental implementations. We provide methods to characterize the states generated by two-and, more generally, k-body Hamiltonians as well as the convex hull of these sets. This leads to new insights into the question which states are uniquely determined by their marginals and to a generalization of the concept of entanglement. Finally, certification methods for quantum simulation can be derived.PACS numbers: 03.65. Ud, 03.67.Mn Introduction.-Interactions in quantum mechanics are described by Hamilton operators. The study of their properties, such as their symmetries, eigenvalues, and ground states, is central for several fields of physics. Physically relevant Hamiltonians, however, are often restricted to few-body interactions, as the relevant interaction mechanisms are local. But the characterization of generic few-body Hamiltonians is not well explored, since in most cases one starts with a given Hamiltonian and tries to find out its properties.In quantum information processing, ground and thermal states of local Hamiltonians are of interest for several reasons: First, if a desired state is the ground or thermal state of a sufficiently local Hamiltonian, it might be experimentally prepared by engineering the required interactions and cooling down or letting thermalise the physical system [1]. For example, one may try to prepare a cluster state, the resource for measurement-based quantum computation, as a ground state of a local Hamiltonian [2]. Second, on a more theoretical side, ground states of k-body Hamiltonians are completely characterized by their reduced k-body density matrices. The question which states are uniquely determined by their marginals has been repeatedly studied and is a variation of the representability problem, which asks whether given marginals can be represented by a global state [3]. It has turned out that many pure states have the property to be uniquely determined by a small set of their marginals [4,5], and for practical purposes it is relevant that often entanglement or non-locality can be inferred by considering the marginals only [6].In this paper we present a general approach to characterize ground and thermal states of few-body Hamiltonians. We use the formalism of exponential families, a concept first introduced for classical probability distributions by Amari [7] and extended to the quantum setting in Refs. [8][9][10][11]. This offers a systematic characterization of the complexity of quantum states in a conceptionally pleasing way. We derive two methods that can be used to compute various distances to thermal states of kbody Hamiltonians: The first method is general and uses semidefinite programming, while the second method isSchematic view of the state space, the exponential families Q1 and Q2, and their convex hulls. While the whole space of mixed states is convex, the exponential famil...

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Section: Numerical Resultsmentioning

confidence: 99%

Characterizing Ground and Thermal States of Few-Body Hamiltonians

Huber

Gühne

2016

Phys. Rev. Lett.

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“…weighted graph states [26,27] and hypergraph states [28][29][30]. Another possible extension of these results is to consider the representation of graph states in a hybrid basis, i.e.…”

Section: Discussionmentioning

confidence: 98%

Group structures and representations of graph states

Kampermann

Bruß

2015

Phys. Rev. A

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A special configuration of graph state stabilizers, which contains only Pauli σX operators, is studied. The vertex sets ξ associated with such configurations are defined as what we call X-chains of graph states. The X-chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the X-basis via the so-called X-chain factorization diagram. We show that graph states with different X-chain groups can have different probability distributions of X-measurement outcomes, which allows one to distinguish certain graph states with X-measurements. We provide an approach to find the Schmidt decomposition of graph states in the X-basis. The existence of X-chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of X-chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X-chains, while its computational complexity with other known methods increases exponentially.

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“…This is in contrast to previous approaches that were either too general, e.g. Bell inequalities for general multiparticle states [19,20], or too restricted, considering only few specific examples of hypergraph states and leading to non robust criteria [9]. The violation of local realism is the key to further applications in information processing: Indeed, it is well known that violation of a Bell inequality leads to advantages, in distributed computation scenarios [21,22].…”

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confidence: 91%

“…Recently, this family of states has been generalized to hypergraph states [6][7][8][9][10][11]. These states have been recognized as special cases of the so-called locally maximally entangleable (LME) states [6].…”

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Extreme Violation of Local Realism in Quantum Hypergraph States

2016

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Hypergraph states form a family of multiparticle quantum states that generalizes the well-known concept of Greenberger-Horne-Zeilinger states, cluster states, and more broadly graph states. We study the nonlocal properties of quantum hypergraph states. We demonstrate that the correlations in hypergraph states can be used to derive various types of nonlocality proofs, including Hardytype arguments and Bell inequalities for genuine multiparticle nonlocality. Moreover, we show that hypergraph states allow for an exponentially increasing violation of local realism which is robust against loss of particles. Our results suggest that certain classes of hypergraph states are novel resources for quantum metrology and measurement-based quantum computation.PACS numbers: 03.65.Ta, 03.65.UdIntroduction.-Multiparticle entanglement is central for discussions about the foundations of quantum mechanics, protocols in quantum information processing, and experiments in quantum optics. Its characterization has, however, turned out to be difficult. One problem hindering the exploration of multiparticle entanglement is the exponentially increasing dimension of the Hilbert space. This implies that making statements about general quantum states is difficult. So, one has to concentrate on families of multiparticle states with an easier-tohandle description. In fact, symmetries and other kinds of simplifications seem to be essential for a state to be a useful resource. Random states can often be shown to be highly entangled, but useless for quantum information processing [1].

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Entanglement and nonclassical properties of hypergraph states

Cited by 63 publications

References 38 publications

Characterizing Ground and Thermal States of Few-Body Hamiltonians

Characterizing Ground and Thermal States of Few-Body Hamiltonians

Group structures and representations of graph states

Extreme Violation of Local Realism in Quantum Hypergraph States

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