Entanglement Distribution

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Transcript of Entanglement Distribution

Entanglement Distribution
February 6, 2014

qubits (twolevel quantum systems: spin 1/2, photonpolarization)
multiple qubits and classical resources at each node (vertex) links (edges): bipartite entangled (pure/mixed) twobit 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 longdistance 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 networkbasederrorcorrection

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 nonzero 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 errorcorrection 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 maximallyentangled states afterLOCC is equivalent to the classical random graph problem

purestate entanglement percolation on a general quantumcomplex network: purification works as on regular lattices,swapping is not obvious because swapping protocols dependon the lattice structure
qswap protocol has been applied with success to complexnetworks with doublebond, pure, and partiallyentangledstates
at a qstar, instead of using purification on each doublebond,we perform swapping at the central node, which is thenreplaced by a qcylce
in certain complex networks (the Bethe lattice, ER, andscalefree), the qswap lowered the critical threshold comparedwith CEP
in some cases, however, the classical purification schemeyields a larger giant component that a qswap

we use purestate 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 shortestpath problems
differences between quantum and classical networks: nocloning theorem, teleportation
so far, there are models exploring some shortestpathalgorithm applications to quantum networks, but with limitedsuccess, in particular, mixedstate, 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 largescale quantum networksReports on Progress in Physics 76 (2013).