Group structures and representations of graph states

Wu, Jun-Yi; Kampermann, Hermann; Bruß, Dagmar

doi:10.1103/physreva.92.012322

Cited by 6 publications

(11 citation statements)

References 29 publications

(66 reference statements)

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“…Consider the vertex S (South Africa) in the quantum network shown in figure 1 and assume that the number of repeater stations is even on each edge (odd numbers can be treated in an analogous way). Take the product of the stabiliser generators starting from S and for every second repeater station, until reaching the neighbours T, M, P, I, and R. Due to the definition of g i in equation (2) and the fact that =  Z 2 , this product of stabilisers contains only X-operators at S and the mentioned repeater stations, and Z-operators on the neighbouring network nodes of S in the network (see also [40]). We call this stabiliser operator the main stabiliser centred on S. Measuring all repeater stations in the X-basis projects the state onto one stabilised by X Z Z Z Z Z S T M P I R in the Hilbert space of the network nodes only.…”

Section: From Graphs To Quantum Repeater Networkmentioning

confidence: 99%

Large-scale quantum networks based on graphs

2016

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Society relies and depends increasingly on information exchange and communication. In the quantum world, security and privacy is a built-in feature for information processing. The essential ingredient for exploiting these quantum advantages is the resource of entanglement, which can be shared between two or more parties. The distribution of entanglement over large distances constitutes a key challenge for current research and development. Due to losses of the transmitted quantum particles, which typically scale exponentially with the distance, intermediate quantum repeater stations are needed. Here we show how to generalise the quantum repeater concept to the multipartite case, by describing large-scale quantum networks, i.e. network nodes and their long-distance links, consistently in the language of graphs and graph states. This unifying approach comprises both the distribution of multipartite entanglement across the network, and the protection against errors via encoding. The correspondence to graph states also provides a tool for optimising the architecture of quantum networks. IntroductionQuantum entanglement is one of the pillars of quantum information processing. Distribution of entanglement among two or more spatially separated parties is a necessary ingredient for many tasks in quantum information theory, including distributed quantum computing [1], blind quantum computing [2], teleportation [3], telecloning [4], secret sharing [5] and quantum cryptography schemes [6][7][8]. Multipartite entanglement enables a violation of Bell inequalities that grows exponentially with the number of parties [9]. However, the controlled distribution of entanglement, in particular of multipartite entanglement, over long distances is a major challenge, due to unavoidable imperfections such as particle losses and decoherence.The seminal idea of quantum repeaters [10, 11] is based on the distribution of short-range entanglement between intermediate repeater stations (thus avoiding losses that grow typically exponentially with the distance) and subsequent entanglement swapping, which connects the short links along a line to long-range bipartite entanglement. Several theoretical variations have been proposed: some of them are based on entanglement distillation [12][13][14] and others are based on forward error correction [15][16][17][18]. Much experimental progress towards the realisation of a quantum repeater has been made [19][20][21][22][23][24][25].'Partially quantum' networks are considered in the so-called trusted node scenario [26], while fully quantum networks have been investigated in the context of network routing [27][28][29][30] and coding [31-33] strategies and heterogeneous network technologies [34].Here we propose a general multipartite quantum network architecture, where the long-distance links are bridged by quantum repeater stations. This idea is illustrated in figure 1 for the long-term vision of a 'world-wide quantum web'. This network contains nodes (labelled by letters), which receive, measure and send pa...

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Section: From Graphs To Quantum Repeater Networkmentioning

confidence: 99%

Large-scale quantum networks based on graphs

2016

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“…All repeaters and routers measure their qudits in the X-basis. The post-measurement state can be found by looking at the main stabilizer operators, which are constructed from chains of X-operators obtained by multiplying powers of the generators of equation (2), where the exponents are chosen such that the Zoperators cancel out each other [45]. Note that the weight of the edge enters as a multiplicative factor in the exponent of the Z-operators, see equation (2).…”

Section: Graph States and Quantum Networkmentioning

confidence: 99%

“…Consider a stabilizer operator of a graph state which does not contain Z-operators. These operators are called X-chains [45], which form a group with the usual multiplication. Any error that does not commute with an X-chain is detected [45], because the product of the measurement outcomes (to the appropriate power) on an X-chain is 1 in case of no errors.…”

Section: Robustness Of Quantum Network Codingmentioning

confidence: 99%

Robust entanglement distribution via quantum network coding

2016

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Many protocols of quantum information processing, like quantum key distribution or measurementbased quantum computation, 'consume' entangled quantum states during their execution. When participants are located at distant sites, these resource states need to be distributed. Due to transmission losses quantum repeater become necessary for large distances (e.g. 300 km). Here we generalize the concept of the graph state repeater to D-dimensional graph states and to repeaters that can perform basic measurement-based quantum computations, which we call quantum routers. This processing of data at intermediate network nodes is called quantum network coding. We describe how a scheme to distribute general two-colourable graph states via quantum routers with network coding can be constructed from classical linear network codes. The robustness of the distribution of graph states against outages of network nodes is analysed by establishing a link to stabilizer error correction codes. Furthermore we show, that for any stabilizer error correction code there exists a corresponding quantum network code with similar error correcting capabilities.

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“…From these stabilizers, the representation of graph states in the X-basis [14] and overlaps of two graph states can be determined. The vertex subsets, which induce such special graph state stabilizers, are called X-chains.…”

Section: Review: Graph States and X-chainsmentioning

confidence: 99%

“…The set of X-chains is shown to be a group with the group operation ∆ (symmetric difference of sets) [14]. An X-chain group is denoted by Γ G with its generating set being written as Γ G .…”

Section: Review: Graph States and X-chainsmentioning

confidence: 99%

DeterminingX-chains in graph states

Wu¹,

Kampermann²,

Bruß³

2015

J. Phys. A: Math. Theor.

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Abstract. The representation of graph states in the X-basis as well as the calculation of graph state overlaps can efficiently be performed by using the concept of X-Chains [Phys. Rev. A 92 (1) 012322]. We present a necessary and sufficient criterion for X-chains and show that they can efficiently be determined by Bareiss algorithm. An analytical approach for searching X-chain groups of a graph state is proposed. Furthermore we generalize the concept of X-chains to so-called Euler chains, whose induced subgraphs are Eulerian. This approach helps to determine if a given vertex set is an X-chain and we show how Euler chains can be used in the construction of multipartite Bell inequalities for graph states.

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Group structures and representations of graph states

Cited by 6 publications

References 29 publications

Large-scale quantum networks based on graphs

Large-scale quantum networks based on graphs

Robust entanglement distribution via quantum network coding

DeterminingX-chains in graph states

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