Abstract. Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well-established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five-and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states.
Quantum networks are essential to quantum information distributed applications, and communicating over them is a key challenge. Complex networks have rich and intriguing properties, which are as yet unexplored in the quantum setting. Here, we study the effect of entanglement percolation as a means to establish long-distance entanglement between arbitrary nodes of quantum complex networks. We develop a theory to analytically study random graphs with arbitrary degree distribution and give exact results for some models. Our findings are in good agreement with numerical simulations and show that the proposed quantum strategies enhance the percolation threshold substantially. Simulations also show a clear enhancement in small-world and other real-world networks. DOI: 10.1103/PhysRevLett.103.240503 PACS numbers: 03.67.Bg, 64.60.ah, 89.75.Hc A broad variety of natural and socioeconomic phenomena such as protein-protein interactions, the brain, the power-grid, friendship networks, the Internet, and foodweb, can all be modeled by graphs, i.e., by sets of nodes and edges representing the relation between them. Complex networks (CN) cover the wide range between regular lattices and completely random graphs. Their ubiquity has triggered an intense research activity on CN involving applied works, but also fundamental studies in theoretical physics and mathematics that aim at unveiling their underlying principles. Understanding structural properties of CN is very important as they crucially affect their functionality. For instance, the topology of a social network affects the spread of information or diseases, and the architecture of a computer network determines its robustness under router failures. The appearance of a giant connected component is a critical phenomenon related to percolation theory [1,2]. If, say, the infection probability of a disease exceeds a critical value, an outburst of the disease occurs infecting most of the population, whereas below the critical value only a vanishing amount of nodes is affected. This threshold and the size of the outburst are key quantities that strongly depend on the CN structure.This Letter merges the field of CN with the also new and rapidly developing field of quantum information (QI). Networks already have a prominent role in QI, e.g., in the context of measurement-based quantum computation [3], in the characterization of graph states [4], and, more naturally, as the physical substrate of nodes and channels for multipartite quantum communication protocols. Such a ''quantum internet '' [5] supports QI applications that fill the technological gap between the already available bipartite applications like quantum key distribution, and the appealing but still remote quantum computer. The reason for studying quantum CN, as opposed to regular lattices considered so far, is threefold. First, it is very plausible that future quantum communication networks acquire a complex topology resembling that of existing networks. This can certainly be the case if methods are developed to use...
Entanglement is the resource to overcome the natural limitations of spatially separated parties restricted to Local Operations assisted by Classical Communications (LOCC). Recently two new classes of operational entanglement measures, the source and the accessible entanglement, for arbitrary multipartite states have been introduced. Whereas the source entanglement measures from how many states the state of interest can be obtained via LOCC, the accessible entanglement measures how many states can be reached via LOCC from the state at hand. We consider here pure bipartite as well as multipartite states and derive explicit formulae for the source entanglement. Moreover, we obtain explicit formulae for a whole class of source entanglement measures that characterize the simplicity of generating a given bipartite pure state via LOCC. Furthermore, we show how the accessible entanglement can be computed numerically. For generic four-qubit states we first derive the necessary and sufficient conditions for the existence of LOCC transformations among these states and then derive explicit formulae for their accessible and source entanglement.
We introduce two operational entanglement measures which are applicable for arbitrary multipartite (pure or mixed) states. One of them characterizes the potentiality of a state to generate other states via local operations assisted by classical communication (LOCC) and the other the simplicity of generating the state at hand. We show how these measures can be generalized to two classes of entanglement measures. Moreover, we compute the new measures for pure few-partite systems and use them to characterize the entanglement contained in a three-qubit state. We identify the GHZand the W-state as the most powerful pure three-qubit states regarding state manipulation.PACS numbers: 03.67. Mn, 03.67.Bg Entanglement is of paramount importance in many fields of science. Due to its existence, applications such as teleportation, quantum computation, quantum simulation, and quantum error correction, to name a few, are feasible [1]. Moreover, the application of entanglement theory in other fields of science, most prominently condensed matter physics, has opened new routes towards the understanding of quantum many-body systems [2]. Due to its importance, an enormous effort has been made to qualify and quantify multipartite entanglement. Different entanglement classes have been identified and several entanglement measures have been introduced [3]. Some of them originated from analyzing the potentiality of a state for a particular realization of an application, such as the localizable entanglement [4], some others arose from the generalization of classical correlation measures, such as the generalization of the squashed entanglement [3,5].Despite these results, we are still far from completely understanding multipartite entanglement. The lack of knowledge stems on the one hand from the fact that the number of non-local parameters scales exponentially with the number of subsystems and, on the other hand, from the fact that the operations which are central in the investigation of entanglement, the local operations assisted by classical communication (LOCC), are notoriously difficult to be analyzed in general [6]. The importance of LOCC in this context is due to the fact that LOCC corresponds to those operations which can be implemented without consuming entanglement. This implies that entanglement is the resource to overcome the restriction to LOCC and that the sole condition a function has to fulfill to be a valid entanglement measure is that it is nonincreasing under LOCC [3], [20]. For the bipartite case a simple criterion for pure state transformations via LOCC has been presented [7]. These results do not only allow to identify the state |Φ + ∝ i |ii as the maximally entangled state, which can be transformed into any other bipartite state deterministically via LOCC, but also allowed to introduce new entanglement measures. Due to the existence of different SLOCC classes in the multipartite setting [8,9], i.e. the existence of pairs of states which cannot even probabilistically be transformed locally into each other, there d...
We propose a scheme to distribute graph states over quantum networks in the presence of noise in the channels and in the operations. The protocol can be implemented efficiently for large graph sates of arbitrary (complex) topology. We benchmark our scheme with two protocols where each connected component is prepared in a node belonging to the component and subsequently distributed via quantum repeaters to the remaining connected nodes. We show that the fidelity of the generated graphs can be written as the partition function of a classical Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear cluster and results for its decay rate in random graphs with arbitrary (uncorrelated) degree distributions.Comment: 16 pages, 7 figure
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