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).