ARTICLESPUBLISHED ONLINE: 16 MAY 2010 | DOI: 10.1038/NPHYS1665

Quantum random networksS. Perseguers1, M. Lewenstein2,3, A. Acín2,3 and J. I. Cirac1*Quantum mechanics offers new possibilities to process and transmit information. In recent years, algorithms and cryptographicprotocols exploiting the superposition principle and the existence of entangled states have been designed. They should allowus to realize communication and computational tasks that outperform any classical strategy. Here we show that quantummechanics also provides fresh perspectives in the field of random networks. Already the simplest model of a classical randomgraph changes markedly when extended to the quantum case, where we obtain a distinct behaviour of the critical probabilitiesat which different subgraphs appear. In particular, in a network of N nodes, any quantum subgraph can be generated by localoperations and classical communication if the entanglement between pairs of nodes scales as N−2. This result also opens upnew vistas in the domain of quantum networks and their applications.

Communication networks based on the laws of quantumphysics are expected to be developed in the near future1,2.These quantum networks offer us new opportunities and

phenomena as compared with their classical counterparts. Forinstance, two distant parties can apply a quantum-cryptographyprotocol3 and exchange information in a provable secure wayowing to quantum effects: the security is ensured because noone can eavesdrop information encoded on quantum particleswithout leaving a trace. In these networks, entanglement is apowerful but fragile resource that has to be distributed betweenthe nodes4. Entangled particles are indeed exploited in a multitudeof quantum communication protocols, such as teleportation5,entanglement-based quantum cryptography6, superdense coding7or secret sharing8, just to name a few. Lately, it has been shownthat quantum phase transitions may occur in the entanglementproperties of quantum systems defined on regular graphs andthat the use of joint operations may be beneficial, for example,for quantum teleportation between the nodes9,10. The ability tomanipulate superpositions of quantum states is indeed crucial inthat setting, and turns out to play a key role in the presentwork too.

Until now, quantum networks have been mostly modelled aslattice graphs, where neighbouring stations (vertices) are connectedby partially entangled quantum states (edges). However, realcommunication networks, such as the Internet or the publicswitched telephone network, do not exhibit any regular or periodicorganization. At first sight, their structure seems to be completelydisordered and random. The first theory tackling such systemswas developed in the 1950s and 1960s in a series of papers11–13 byErdős and Rényi. This theory of random graphs, in contrast withthat of regular graphs, explains one of the main features of realnetworks, namely the small-world property14. Various models werethen proposed to capture some other fundamental characteristics ofnetworks with a complex architecture (for example, a clustering15or a scale-free16 behaviour), but the theory of Erdős and Rényi isstill widely used as it provides a reference for empirical studies inmany fields. Here, we extend the random graphs to the quantumworld by introducing amodel of quantum random networks, a newclass of systems that exhibit some totally unexpected phenomena.We indeed obtain a completely different classification of theirproperties as comparedwith their classical counterparts.

1Max-Planck–Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany, 2ICFO—Institut de Ciències Fotòniques, MediterraneanTechnology Park, 08860 Castelldefels, Spain, 3ICREA—Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain.*e-mail: ignacio.cirac@mpq.mpg.de.

p = 0.1 p = 0.25 p = 0.5

Figure 1 | Classical random graphs. Evolution process of a random graphwith N= 10 nodes: starting from isolated nodes, we randomly add edgeswith increasing probability p, to eventually get the complete graph K10

for p= 1.

Classical and quantum random graphsA classical network is mathematically represented by a graph, whichis a pair of sets G= (V ,E), where V is a set of N nodes and E is aset of L edges connecting two nodes. The theory of random graphsconsiders graphs in which each pair of nodes i and j are joined bya link with probability pi,j . The simplest and most studied modelis the one where this probability is independent of the nodes, withpi,j = p, and the resulting graph is denoted GN ,p. The constructionof these graphs can be considered as an evolution process: startingfrom N isolated nodes, random edges are successively added andthe obtained graphs correspond to larger and larger connectionprobability, see Fig. 1. One of the main goals of random-graphtheory is then to determine at which probability p a specific propertyP of a graph GN ,p mostly arises, as N tends to infinity. Manyproperties of interest appear suddenly; that is, there exists a criticalprobability pc(N ) such that almost every graph has the property Pif p≥ pc(N ) and fails to have it otherwise; such a graph is said toby typical. For instance, the critical probability for the appearanceof a given subgraph F of n nodes and l edges in a typical randomgraph is17:

pc(N )= cN−n/l (1)

with c being independent of N . It is instructive to look at theappearance of subgraphs assuming that p(N ) scales as N z , withz ∈ (−∞,0] being a tunable parameter: as z increases, more andmore complex subgraphs emerge, see Table 1.

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1665

Table 1 |Thresholds for the appearance of classical subgraphs.

z −∞ −2 −32 −

43 −1 −

23

F

Some critical probabilities, according to equation (1), at which a subgraph F appears in random graphs of N nodes connected with probability p∼Nz : cycles and trees of all orders appear at z=−1, whereascomplete subgraphs (of order four or more) appear at a higher connection probability30 .

a b

Figure 2 |Quantum random graphs. An example of a quantum randomgraph on five nodes. a, Each node is in possession of four qubits that areentangled with qubits belonging to the other nodes. All of the connectionsare identical and pure but non-maximally entangled pairs. b, Imperfectpairs can be converted into maximally entangled ones with someprobability of success p (here p=0.25); this strategy mimics the behaviourof classical random graphs.

We consider now the natural extension of the previous scenarioto a quantum context. For each pair of nodes we replace theprobability pi,j by a quantum state ρi,j of two qubits, one at eachnode. Hence every node possesses N − 1 qubits that are pairwiseentangled with the qubits of the other nodes, as depicted inFig. 2. As entangled pairs of particles are the resource for quantuminformation tasks, the goal is then to establish maximally entangledstates within certain subsets of nodes by using protocols that consistonly of local operations and classical communication18 (LOCC).Therefore, each node can apply different measurements on itsquantum system and then communicate the outcome results to therest of the network. As in the classical random graphs, we considerthat pairs of particles are identically connected, with ρi,j = ρ.Furthermore, we restrict ourselves to the simplest case of pure statesof qubits, that is, ρ = |ϕ〉〈ϕ| , because it already leads to some veryintriguing phenomena.We take these states to be:

|ϕ〉=√1−p/2 |00〉+

√p/2 |11〉 (2)

where 0 ≤ p ≤ 1 measures the degree of entanglement of thelinks. As before p scales with N and we write |GN ,p〉 forthe corresponding quantum random graph. The choice of thecoefficients in equation (2) becomes clear if one considers thefollowing simple strategy where links are treated independently:each pair of qubits tries to convert its connection by LOCC intothe maximally entangled state |8+〉∝ |00〉+ |11〉. The probabilityof a successful conversion is p, which is optimal19,20, and thereforethe task of determining the type of maximally entangled statesremaining after these conversions can be exactly mapped to theclassical problem. In that case we obtain the results of Table 1, andfor example for z ≥ −2 the probability to find a pair of nodesthat share a maximally entangled state is one, whereas that ofhaving three nodes sharing three maximally entangled states iszero unless z ≥−1.

Advantage of joint measurementsAllowing strategies that entangle the qubits within the nodes offersnew possibilities and brings powerful results. This is indeed ageneral fact in quantum information theory as, for example, in thecontext of distillation of entanglement21 or in the non-additivityof the classical and quantum capacity of quantum channels22.As a first illustration of the advantage of joint actions on qubits

Figure 3 |Graphical representation of |Kc〉. Each node possesses exactlyc− 1 orthogonal states in which c−2 qubits are |0〉 and one is |1〉; qubitsthat were originally entangled share the same state. A link denotes, here,the separable state |11〉, whereas |00〉 is not drawn, and |Kc〉 is defined asthe coherent superposition of all perfect matchings of the complete graphKc. In this example an appropriate labelling of the states leads to|K4〉∝ |1111〉+ |2222〉+ |3333〉.

we show how some relevant multipartite entangled states, suchas the three-dimensional Greenberger–Horne–Zeilinger23 (GHZ)state of four particles, can be obtained when z = −2. At eachnode we apply a generalized (or incomplete) measurement24 thatconsists of projectors Pm onto the subspaces containing exactlym qubits in |1〉:

Pm=∑πm

πm | 0···0︸︷︷︸N−m−1

1···1︸︷︷︸m

〉〈0···01···1|π †m (3)

where πm designates a permutation of the qubits. Therefore, wecount the number m of ‘excitations’, that is, links |11〉, attachedto the nodes, but without revealing their precise location. Notethat such links are separable and so do not represent the soughtconnections |8+〉. For each measurement we get a randomoutcome m with a value of either 0 or 1 because, as in the classicalcase, the probability to getm≥ 2 is zero for N tending to infinity25.In fact, setting p= 2cN−2 one finds that there are on average c nodeshaving exactly one qubit in |1〉, see the Methods section, whereasall nodes where m = 0 factor out because they are completelyuncorrelated with the rest of the system. In Fig. 3 we describe theremaining state, which we write as |Kc〉, and show that for c=4 it isa GHZ state of four qutrits. The size of |Kc〉 can then be decreasedin a deterministic fashion: one measures all qubits of a node i, andgets 0 for all outcomes except for a random one, say j. In fact,exactly one excited link is attached to i, and it equally points to allother nodes. Hence, the nodes i and j factor out and the state isprojected onto |Kc−2〉.

Collapse of the critical exponentsWe now turn to the main result of this article: for N tending toinfinity and with p∼N−2 one is able to obtain, with probabilityapproaching unity, a quantum state with the structure of anyfinite subgraph. That is, for an arbitrary subgraph F = (V ,E)composed of n vertices and l edges, one can get the state |F〉consisting of l maximally entangled pairs |8+〉 shared among nnodes, according to E :

|F〉=l⊗

i=1

|8+〉Ei (4)

This result implies that in the quantum case the structure of Table 1completely changes, as all subgraphs already appear at z = −2.

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© 2010 Macmillan Publishers Limited. All rights reserved.

NATURE PHYSICS DOI: 10.1038/NPHYS1665 ARTICLESa b

Figure 4 | Construction of quantum subgraphs. Sketch of the four steps ofthe main proof. a, In the first two steps we create the state |ϕDn 〉; here n= 2(top) and D= d2

=4 (bottom). The starting state is in a coherentsuperposition of all perfect matchings of the depicted graph, and thereforemeasuring the bottom nodes in the Fourier basis yields the state|12〉+ |13〉+···+ |34〉. b, In steps 3 and 4 the n nodes of the system havedimension D each, which corresponds to log2(D)= 2l qubits. In the currentcase n= 3 and l= 2, and appropriate Fourier measurements of half thequbits and projections of the others lead to the simplest non-trivialsubgraph |8+〉AB⊗ |8

+〉AC.

More precisely, for any subgraph F there exists a quantum randomgraph with p∼N−2 where F appears. Actually, our results are moregeneral because any state of some interest in quantum informationtheory, such as W states, Dicke states, graph states and so on, alsoarises at z =−2. Note that this exponent is optimal because theregime z < −2 is trivial. In fact, for z < −2 the overlap of thequantum random graph and the product state of all qubits in |0〉approaches unity in the limit of infinite lattices:

|〈GN ,p|0···0〉|2=(1−

p2

) N (N−1)2

' exp(−N z+2

4

)→ 1

in which case no local quantum operation is able to createentanglement between the nodes.

We give here a sketch of the proof, and refer the reader tothe Methods section for detailed calculations. The constructionof |F〉 from a typical quantum random graph is realized in foursteps, as depicted in Fig. 4. First, we create the state |Kc〉 withc = n+D, D= d2 and d = 2l : n nodes will be kept to build thefinal state and D extra nodes are needed to establish the desiredquantum correlations. Second, we measure all links correspondingto the subgraph Kn and the other D vertices are measured in theFourier basis, which leads to a highly entangled state of n nodes ofdimension D. Third, to extract the right correlations, we split eachnode into two subsystems of dimension d =

√D, measure one of

them in the Fourier basis and post-select a specific outcome pattern.Finally, we show that the resulting state, an n-partite GHZ stateof dimension d , can be converted into |F〉 by some appropriateprojections at the nodes.

The quantum subgraph |F〉 is successfully generated if we getsome specific outcomes for the various measurements describedin the proof, and therefore it appears with a non-unit probabilitypF . However, this is not a problem because pF does not dependon N but only on the subgraph F , so that the process can bearbitrarily amplified without changing z . The reason is that wecan always subdivide the N nodes into L sets of N/L nodes, withL being a constant that is much larger than 1/pF , and apply thesame construction on each set. These sets are treated as independentif we initially discard all links connecting different sets, which isdone by measuring the corresponding qubits. Consequently, |F〉can be obtained with a probability arbitrarily close to unity forN tending to infinity. We do not try to optimize the subgraphconstruction because we are mainly interested in the existence ofa finite probability pF , not in its maximal possible value. However,one notes that pF decays exponentially with the size of the subgraphF , and therefore an optimization of the procedure may be essentialfor practical purposes.

A S B

Figure 5 | Entanglement generation between two nodes. Two distantcavities are simultaneously driven by a common source S of squeezed light,and in the steady state of the system the two atoms A and Bbecome entangled.

Physical implementation andmixed-state scenarioAt this point, let us describe a possible set-up for an implementationof our ideas, where atoms store the quantum information, andthus represent qubits, whereas photons are used to create remoteentanglement. At present, this is the most promising scenario forthe realization of quantumnetworks1. In particular, we consider thecase where (continuous-variable) entanglement contained in a two-mode squeezed light is transformed into (discrete) entanglementbetween atoms trapped in distant high-quality cavities26, see Fig. 5.Assuming perfect operations, one can drive the system so thatits steady state is exactly given by the pure state described inequation (2), with p< 1 because of a finite squeezing of the light.

In realistic quantum networks there will be, of course, only afinite number of nodes, but even for very small systems one canshow that the joint measurements Pm yield better results than the‘classical’ strategy. In fact, consider a quantum random graph onthree stations, and let us construct the GHZ state |000〉+ |111〉.In the classical strategy one manipulates the atoms individually andtries to get threemaximally entangled pairs. The GHZ state betweenthe three nodes can be obtained if at least two conversions aresuccessful, which happens with probability O(p2) in the regime ofweak entanglement. Following a pure quantum strategy, however,the GHZ state can be created with probability O(p): one stationcarries out the generalized measurement Pm on its two atoms andsucceeds if the outcome ism= 1.

Let us now move towards a more realistic situation byintroducing some noise into the network. For instance, we consideran imperfect source of light that fails to emit squeezed states withsome probability ν. Equivalently, this source erroneously producesthe vacuum state |00〉with probability ν, so that the connections ofthe quantum network are in the mixture:

ρ= (1−ν) |ϕ〉〈ϕ| +ν |00〉〈00|

with |ϕ〉 defined in equation (2). For this specific error we showin the Methods section that it is still possible to construct, in theregime z =−2, mixed states that are close to arbitrary quantumsubgraphs. Therefore, the proposed construction is robust againstsome imperfections, namely a noise |00〉〈00| in the light sources.At present, it is unclear how to deal with errors involving termssuch as |01〉〈01| , |10〉〈10| or |11〉〈11| in the connections. Weare nevertheless very confident that other quantum strategies(based on purification methods for instance) will still lead tointriguing and powerful phenomena and thus stimulate furtherwork in this direction.

We have introduced a model of quantum networks with adisordered structure based on the theory of random graphs. Wehave shown that allowing joint actions on the nodes markedlychanges one of their main properties, namely the appearance ofsubgraphs according to the connection probability. In fact, allcritical exponents collapse onto the value z =−2 in the quantumcase. Three lines of research naturally follow from our findings.First, from an applied point of view, it would be interesting tosee how these effects appear in small quantum networks, openingthe way for proof-of-principle demonstrations. All of these toolswill find a natural application when practical quantum networksreach their maturity. Second, it is important to understand and

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1665

quantify the effect of noise in these systems. The model that wehave introduced is simple in the sense that the connections arerepresented by pure states, whereas more realistic set-ups involvemixed states. We have discussed this problem for a specific typeof imperfections and proved the robustness of our construction.Thus, we are confident of the persistence of unexpected andintriguing phenomena in real quantum networks. Finally, from amore abstract perspective, it is desirable to extend our model tonetworks of richer topology, that is, to complex networks. Thetheory of complex networks, which describes a wide variety ofsystems in nature and society27–29, has attracted a lot of attentionin the scientific community in the past few years, and we expect alarge variety of new phenomena in, for instance, quantum modelsof small-world or scale-free networks. We hope that our results willshed some light on the very active domain of complex networks andinspire work in all of these directions.

MethodsIn this section we make the mathematical statements of the main text more precise,starting with the generalized measurements Pm defined in equation (3). Clearly,they satisfy the completeness relation

∑mPm =1 at each node. Furthermore, the

outcome distribution of these measurements in the network follows the degreedistribution of a classical random graph GN ,p′ , with p′ = p/2. In fact, the onlyinformation we get from |GN ,p〉 by applying Pm is the number of excitations |11〉attached to a given node, and each such link gets a weight p/2 in the outcomeprobabilities. This distribution can be well approximated by the one that we get ifthe nodes are considered to be independent, so that the expected number xm ofoutcomesm reads, setting p=2cN z and for z<−1:

E(xm) =N !

m! (N −1−m)!

( p2

)m(1−

p2

)N−1−m'

cm

m!Nm(z+1)+1

In the text we often mention the act of measuring a node, or more generally asystem of dimension d : if no basis is specified we mean a measurement in thecomputational basis; that is, we project the system onto the states |1〉,..., |d〉.We also use the expression ‘to measure a link’ to indicate that one of its qubits ismeasured in the computational basis. Finally, we introduce the notation

|8k,dn 〉=

1√d

d∑j=1

e2π id jk|jj ···j︸︷︷︸

n

〉, k ∈ {1,...,d}

for the Fourier transform of |8dn〉≡ |8

d,dn 〉. The latter state is referred to in the text

as a GHZ state of dimension d (on n nodes), and ameasurement in the Fourier basisof a system of dimension d is its projection onto the states |8k,d

1 〉.

Proof of themain result. We detail here the four steps of the construction of a state|F〉, starting from |GN ,p〉, using only LOCC and for p=2cN−2; see also Fig. 4.

In the first step, we create the state |Kc 〉, with c = n+D, D= d2 and d = 2l ,which is achieved with a probability approaching unity in the limit of infiniteN . Indeed, in this limit and for sufficiently large c , the number of nodes withoutcome m= 1 can be approximated by a Gaussian distribution of mean c andstandard deviation

√c . Thus, we can always choose another constant value

c ′ & c+√c and apply the projections Pm in such a way that the probability

of getting a state |Kc̃ 〉 with c̃ ≥ c is arbitrarily close to one. Then, we use theprocedure described in the text to deterministically decrease the size of theremaining system from c̃ to c .

In the second step, we remove all connections shared between n nodes of |Kc 〉,that is, we measure the n(n−1)/2 concerned links, and the operation is successful ifall outcomes are 0. In that way we build a state that is the coherent superposition, asdescribed in Fig. 3, of all perfect matchings of the join graph of KD and of the emptygraph on n nodes. We further measure these D nodes, but this time in the Fourierbasis { |8k,c−1

1 〉,1≤ k ≤ c−1} in order not to reveal where the links |11〉 lie. As thenodes can communicate their measurement outcomes, we can correct the possiblyintroduced phases and the resulting (unnormalized) state reads:

|ϕDn 〉=

D∑i1 6=i2 ···6=in=1

|i1i2 ···in〉 (5)

In the third step, it is not convenient to deal with sums with indices that aresubject to constraints, so we develop equation (5) to let the sums freely run from 1toD. For example, the state on three nodes is expressed as:

|ϕD3 〉=

D∑i,j,k=1

|ijk〉−D∑

i,j=1

( |iij〉+ |iji〉+ |jii〉)+2D∑i=1

|iii〉

More generally, this leads to a weighted and symmetric superposition of states ofthe form

⊗ri=1 |8

Dλi〉 for all partitions (λ1,λ2,...,λr ) of n, where

∑ri=1λi = n (a

partition of a positive integer is a way of writing it as a sum of positive integers).We want to remove all terms of this sum except the last one, which will allow usto obtain |F〉. To that purpose we use the fact that |8D

n 〉= |8d2

n 〉= |8dn〉⊗2 for all

n and d , split each node into two subsystems of dimension d and measure one ofthem in the Fourier basis. This operation is successful if all outcomes are k = 1,except the last one, which should be k = d−n+1. In fact, to see what happensto a state |8d

m〉 we note that 〈8k,d1 |8

j,dm 〉 ∝ |8

j−k,dm−1 〉 if m> 1, and δj,k otherwise.

Therefore, by sequentially measuring 〈81,d1 | , a state |8

dm〉 shared among any

m< n nodes transforms as:

|8dm〉≡ |8

d,dm 〉 7→ |8

d−1,dm−1 〉 7→ ··· 7→ |8

d−m+1,d1 〉 7→ 0

as long as d >m. However, this is always the case because we consider connectedsubgraphs, so that l ≥n−1 and thus d ≥ 2l ≥ 2n−1≥n>m. Hence, all terms vanishexcept the last one, the GHZ state |8d

n〉.In the fourth step, we want to transform |8d

n〉 into |F〉. To that end, we firstexpand equation (4), that is, write explicitly all its terms in the computationalbasis, and group the qubits according to the connections E . This leads to a sumof product states of the form

⊗nj=1 |ϕi,j 〉 with i= 1,...,2l and j = 1,...,n. As

we have chosen d = 2l , we can apply the measurement element∑d

i=1 |ϕi,j 〉〈i| oneach node j of |8d

n〉, which achieves the desired transformation and thereforeconcludes the proof.

The mixed-state scenario. Let us show that the construction described in theprevious section is robust against a noise |00〉〈00| in the connections of thenetwork. In the first step of the proof, the measurements Pm are not affected bythe imperfect source because no extra excitation |1〉 is added: only the numberof outcomes m= 1 slightly decreases from c to (1−ν)c . However, as before,one can choose a larger constant c ′ and get with certainty c outcomes m= 1.For c = 4 for instance, the remaining nodes are in a mixture of |K4〉 and somecompletely separable states:

ρK4= x |K4〉〈K4| +1−x3

3∑i=1

|iiii〉〈iiii|

with x = (1−ν)2 for infinite N . Despite the presence of separable states, ρK4 isuseful for quantum information tasks because it is distillable for all ν < 1; thatis, the coefficient x can be brought arbitrarily close to unity if one possesses alarge number of copies of it. However, in the regime z =−2 it is impossible to getseveral copies of ρK4 on the same four nodes, but this is not a problem because,alternatively, we can repeat the construction 1/x times so that any use of |K4〉 isstill achieved with high probability. More generally, the state ρKc is a mixture of|Kc 〉 and some partially separable states, with x equal to (1−ν)c/2. This structureis maintained throughout the construction of the quantum subgraphs (steps twoto four of the proof), so that with a strictly positive probability x we create thequantum subgraph |F〉.

Received 22 July 2009; accepted 12 April 2010; published online16 May 2010

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AcknowledgementsWe acknowledge support from the EU projects ‘AQUTE’ and ‘QAP’, the ERCgrant ‘PERCENT’ and ‘QUAGATUA’, the DPG excellence cluster ‘Munich Centerfor Advanced Photonics’ and FOR 635, the Spanish MEC Consolider QOIT andFIS2007-60182 and FIS2008-00784 projects, la Generalitat de Catalunya and CaixaManresa, the Alexander von Humboldt Foundation and the QCCC programme of theElite Network of Bavaria.

Author contributionsAll authors have equally contributed to this work.

Additional informationThe authors declare no competing financial interests. Reprints and permissionsinformation is available online at http://npg.nature.com/reprintsandpermissions.Correspondence and requests formaterials should be addressed to J.I.C.

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