Entanglement Distribution

of 16 /16
Entanglement Distribution February 6, 2014

Embed Size (px)

Transcript of Entanglement Distribution

  • Entanglement Distribution

    February 6, 2014

  • qubits (two-level quantum systems: spin 1/2, photonpolarization)

    multiple qubits and classical resources at each node (vertex) links (edges): bi-partite entangled (pure/mixed) two-bit states goal: entangle pairs of qubits between distant nodes quantum operations: local, within nodes Local Operations and Classical Communications (LOCC)

  • local operations: entanglement swapping and entanglementpurification

    used by quantum repeaters for 1D long-distance entanglement

    limitation: entanglement decays with the number ofoperations

    limitation: noise due to imperfect gate (imperfect operation)and/or imperfect measurements

    use quantum networks: for specific graphs only, purificationcan compensate for the loss of entanglement due to swapping

    better approach: entanglement percolation and network-basederror-correction

  • entanglement in 2D graphs with predetermined sequence oflocal operations

    an extension to quantum repeaters: effective with mixedstates in certain graphs or pure states in lattices of highconnectivity

  • more techniques in the same direction: multipartiteentanglement

    better graph or lattice transformation, same limitations

    (a) Some nodes (filled circles) transform two links into amultipartite entangled state on three nodes. No operation isperformed at the other nodes (empty circles).

    (b) Nodes represent the multipartite entangled states thathave been created; links show two adjacent multipartite states.

  • in a multipartite entanglement graph, two or moremultipartite states can be merged into a larger structure

    this process is a transition from bond percolation to sitepercolation, yielding a giant multipartite entangled statespanning the lattice

  • so far: pure states only, in reality: states experience noisewhich causes them to become mixed

    unlike pure states, we need at least two copies of a certainmixed state to get a state of higher fidelity (by LOCC)

    so in quantum networks, we need at least two disjoint pathsof purifiable mixed states between two stations in order tohave a non-zero probability of a Bell pair between them

    different sequences of local operations need to end withpurification

    these sequences depend on the structure of the network(usually a lattice or a specific class of graphs) and the initialstate

    there has been limited success, in certain restricted cases suchas the square lattice

  • entanglement percolation protocols depend on the initialentanglement of the network and the network itself, but anoptimal method is not yet known

    in the case of pure states, it is not known if there is aminimum entanglement value below which it is impossible toentangle two arbitrary qubits using LOCC

    for mixed states, the classical threshold for bond percolation isa lower bound since below that threshold, the whole systemwould consist of a classical mixture of lattices whose links areeither perfect Bell pairs or completely separable states

    since none of these lattices possess an infinite cluster or Bellpairs in the limit, we are only left with separable states, andno local operation can create entanglement from separablestates, so the threshold above is a lower bound

  • noisy network error-correction consists of a bond percolationthat induces a multipartite entangled state shared by all nodesin the network

    local measurements on all but two distant qubits partiallyreveal at which places the noise altered the creation of themultipartite state

    a global analysis of the measurement outcomes determines theoperations that have to be applied to the remaining twoqubits in order to get useful entanglement

    current protocols are based on theoretical results for latticesof infinite size, their efficiency for realistic networks of finitesize is not known

  • so far: quantum networks with a regular structure, but withthe advances of quantum information technology, quantumnetworks will evolve in complexity, like the internet

    simplest example of quantum complex networks are quantumrandom graphs, in which probability of an edge connectingany two nodes is replaced by a quantum state of two qubits

    working with pure states only, the state of the quantumrandom graph is the coherent superposition of all possiblesimple graphs, weighted by the number of entangled pairsthat they have

    determining the type of maximally-entangled states afterLOCC is equivalent to the classical random graph problem

  • pure-state entanglement percolation on a general quantumcomplex network: purification works as on regular lattices,swapping is not obvious because swapping protocols dependon the lattice structure

    q-swap protocol has been applied with success to complexnetworks with double-bond, pure, and partially-entangledstates

    at a q-star, instead of using purification on each double-bond,we perform swapping at the central node, which is thenreplaced by a q-cylce

    in certain complex networks (the Bethe lattice, ER, andscale-free), the q-swap lowered the critical threshold comparedwith CEP

    in some cases, however, the classical purification schemeyields a larger giant component that a q-swap

  • we use pure-state entanglement to gain perspective and ideas,but we are really interested in the mixed state entanglementpercolation in a quantum complex network

    main limitation: exponential decay of entanglement due toswapping

    one way to tackle this: concentrate entanglement throughever larger number of paths between two nodes

    most detailed results in this direction: single purificationprotocol (SPP)

    the shortest path between A and B is identified and then ashortest path between two nodes on that path is found. Bothpaths can be used to achieve a final entanglement between Aand B that is larger than that achieved using the first pathonly

  • in the SPP setup, we start with a complex network whoselinks are mixed states

    it is impossible to extract a pure state from a finite number ofmixed states, so we try to find the most highly entangledmixed states possible

    in the SPP setup, we swap along alternate paths and thenperform a purification along the resulting parallel states, thusconcentrating entanglement between A and B

    note: although treating mixed states, we have assumedperfect operations

    if we further assume noisy operations, the SPP is found to besensitive to noise, the advantage of SPP is destroyed for noiselevels larger than a few per cent

  • many questions in quantum complex networks can betranslated in terms of classical shortest-path problems

    differences between quantum and classical networks: nocloning theorem, teleportation

    so far, there are models exploring some shortest-pathalgorithm applications to quantum networks, but with limitedsuccess, in particular, mixed-state, noisy networks have notbeen explored from that perspective, and the efficiency of anyknown protocol is limited both theoretically and practically

  • Conclusions:

    entanglement distribution in quantum networks is related tothe network structure, as well as to the physical nature ofentanglement

    the basic concepts (direct transmission, swapping,purification, error correction) have been known, theirapplication is still in an exploratory phase

    two most important applications have been demonstrated:quantum key distribution and teleportation

    exponential decay of fidelity has not been solved methods using the geometry of the network to effectively

    concentrate entanglement show promise

    they show strong connections with the theory of classicalnetworks and graphs, in particular with percolation theory

    as the number of nodes increases to the point that statisticalmethods can be applied, we expect to see vigorous activity inthe theory of complex quantum networks

  • S. Persequers, G. Lapeyre, D. Calvalcanti, M Lewenstein.

    Distribution of entanglement in large-scale quantum networksReports on Progress in Physics 76 (2013).