Singlet Generation in Mixed State Quantum Networks
aa r X i v : . [ qu a n t - ph ] D ec Singlet Generation in Mixed State Quantum Networks
S. Broadfoot, U. Dorner, and D. Jaksch
1, 2 Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom Centre for Quantum Technologies, National University of Singapore, 117543, Singapore (Dated: November 10, 2018)We study the generation of singlets in quantum networks with nodes initially sharing a finitenumber of partially entangled bipartite mixed states. We prove that singlets between arbitrarynodes in such networks can be created if and only if the initial states connecting the nodes have aparticular form. We then generalize the method of entanglement percolation, previously developedfor pure states, to mixed states of this form. As part of this, we find and compare different distillationprotocols necessary to convert groups of mixed states shared between neighboring nodes of thenetwork into singlets. In addition, we discuss protocols that only rely on local rules for the efficientconnection of two remote nodes in the network via entanglement swapping. Further improvementsof the success probability of singlet generation are developed by using particular forms of ‘quantumpreprocessing’ on the network. This includes generalized forms of entanglement swapping and weshow how such strategies can be embedded in regular and hierarchical quantum networks.
PACS numbers: 03.67.Bg, 03.67.-a, 64.60.ah
I. INTRODUCTION
Quantum entanglement is one of the most notable fea-tures of quantum systems and has been accepted as a keyresource for quantum information processing [1]. The dis-tribution of entanglement through quantum networks istherefore essential for the future of a variety of applica-tions, ranging from quantum cryptography to quantumteleportation and distributed quantum computing [2].However, the generation of these entangled states facesa severe obstacle. Quantum channels such as free-spacetransmission or optical fibers are prone to loss and de-coherence. This causes the desired maximally entangledstates to degrade into mixtures and limits the distanceover which the quantum information can be sent directly.To overcome these problems ‘quantum repeater’ schemeshave been proposed [3, 4, 5, 6, 7, 8] which make use ofthe ability to ‘purify’ [9, 10] and ‘swap’ [11, 12] entangle-ment to maintain a high fidelity throughout. Quantumrepeaters are a promising tool for entanglement distri-bution, particularly since the amount of required phys-ical resources increases only polynomially with the dis-tance [5], but operate in a 1D setup of network nodes.Real networks are typically two-(or higher) dimensionaland it is therefore desirable to study if entanglement dis-tribution can be made more efficient in these cases.A scheme for entanglement distribution in higher di-mensional networks was recently proposed by Ac´ın etal. [13] in which ideas from classical bond percolationhave been applied to regular, i.e. lattice-shaped, quan-tum networks. The scheme makes use of the networks’connectivity and allows for the generation of maximallyentangled singlet states between arbitrary points of thenetwork, with a probability that is independent of theirseparation. The only requirement is that the nodes areinitially connected by bipartite pure states with suffi-ciently high entanglement. The restriction to pure stateswas made since a pure, partially entangled state can be converted into a singlet with finite probability via lo-cal operations and classical communication (LOCC) [14]which is essential for the bond percolation protocol: Ini-tially one attempts to convert all bipartite pure statesinto a singlet which, in each case, succeeds with a cer-tain probability. If this singlet conversion probability(SCP) exceeds a lattice-geometry-dependent threshold,arbitrarily large clusters of singlet-connected nodes formwhich can successively be connected via entanglementswapping. In this way we can create a singlet betweenarbitrarily remote nodes in the network. However, it waspointed out in [13] that this process, known as ClassicalEntanglement Percolation (CEP), is not optimal sincecertain quantum preprocessing schemes applied to thenetwork can improve the SCP [13, 15, 16, 17, 18, 19],and thus it is possible to apply bond percolation to lat-tices in which this would otherwise not be possible.Clearly, the assumption of having a pure-state net-work is an idealization and in any practical situation thestates connecting the nodes of the network will be mixed.In [20] the idea of entanglement percolation was appliedto mixed states for the first time. In this paper we elab-orate and extend the ideas presented in [20]. The net-works we consider are composed of nodes, each of whichcan consist of several qubits, and may be connected bya finite number of bipartite mixed states (see Fig. 1).We aim to create a perfect singlet between two arbitrarynodes in the network using a finite amount of resources,i.e. a finite number of initial states which are convertedinto a singlet, which distinguishes our and other entan-glement percolation schemes from, e.g., the quantum re-peater protocol where one aims to generate a state withhigh but non-unit entanglement fidelity. Particularly westructure the paper as follows.In Sec. II we prove a necessary and sufficient conditionthat a perfect singlet can be generated in a network ofarbitrary geometry the nodes of which are initially con-nected by bipartite mixed qubit states. We show thatsinglet generation between two nodes is possible if andonly if they are connected by at least two ‘paths’ consist-ing of a particular class of states. These states arise nat-urally in systems undergoing amplitude damping. Thusour result is not only of theoretical but also of practicalrelevance. Unfortunately, the proof does not deliver anefficient scheme for singlet generation. We therefore spe-cialize in the remaining sections on networks with regulargeometry, i.e. lattices in 2D and 3D and devise general-izations of entanglement percolation to the mixed statesdescribed in Sec. II.In Sec. III we briefly summarize the idea of classicalentanglement percolation with pure states.In Sec. IV we extend the concept of classical entangle-ment percolation to mixed states. To this end we con-sider regular networks where each node is connected to itsneighboring nodes by a finite number of the mixed statesintroduced in Sec. II. We present two different distilla-tion protocols which are used to convert these states intoa singlet with a probability above the percolation thresh-old of a variety of lattice geometries. After the distilla-tion, clusters of singlet-connected nodes emerge and weaim to create a singlet between two nodes in such a clus-ter by successive application of entanglement swapping.By communicating classically each node can determine ifsinglets exist between it and its neighboring nodes. Thisinformation can be communicated and stored classicallyin a central data processor. Typically one would then usethis information to apply a path-finding algorithm whichlocates a suitable ‘path’ of singlets before swapping oper-ations are performed. As an alternative to this we discussa classical and a quantum protocol which merely requireclassical communication between neighboring nodes andbasic computing within each node. The quantum proto-col relies on the formation of many-qubit GHZ states vialocal operations and classical communication with neigh-boring nodes and subsequent measurements at all nodesexcept the ones to be left in the final singlet.In Sec. V we show that the idea of ‘quantum pre-processing’ as it was successfully applied in pure statenetworks can be generalized to mixed states. In par-ticular we devise a number of strategies on small net-works which improve the SCP, and we show that thesesmaller networks can be embedded into larger networksto enable CEP which would otherwise not be possible.Furthermore, we discuss ‘hierarchical schemes’, i.e. net-works which are defined iteratively and were first dis-cussed in [15, 16]. Also in these cases it turns out thatquantum methods outperform classical percolation. Fi-nally, in Sec. VI we summarize and conclude.
II. SINGLET GENERATION WITHIN ANARBITRARY MIXED STATE NETWORK
In this section we consider quantum networks of arbi-trary geometry as shown in Fig. 1 where the qubits in thenodes are ‘connected’ by bipartite mixed states to qubits
A B
FIG. 1: Mixed-state quantum network. Qubits in a node(circles) may be connected by bonds (thick lines), i.e. theyshare mixed entangled states, ‘edges’, (solid, black lines) ofqubits (black dots) with other nodes. When two ‘paths’ ofstates of the form (1) connect A and B a singlet (dashedline) can be created with finite probability. This is proven bypartitioning the nodes into two groups with one containingA (shaded region) and the other B. For it to be possible togenerate a singlet between A and B these groups must belinked by at least two states of the form (1) for all possiblepartitions. in other nodes. We will call a single bipartite mixed statean edge and the set of edges directly connecting two nodesa bond . Note that an edge connects exactly two qubitsin different nodes. In the following we will prove thatthe generation of a perfect singlet between two arbitrarynodes A and B with finite probability in such a networkis possible if and only if there are at least two paths ofstates linking A and B which have, up to local unitaries,the form ρ ( α, γ, λ ) = λ | α, γ ih α, γ | + (1 − λ ) | ih | , (1)where | α, γ i = √ α | i + √ − α − γ | i + √ γ | i and 0 ≤ λ ≤
1. We show this by separately proving a necessaryand sufficient condition which, together, prove the abovestatement.
Necessary condition.
We split the network into twogroups of nodes, A , containing A and a finite numberof other nodes, and B , which consists of the rest of thenetwork and particularly contains B. These groups arelinked by a finite number of edges. A singlet can beestablished with finite probability, via local operations inthe groups and classical communication between them, ifand only if at least two of the states have the form (1).Appendix A contains a concise proof of this fact basedon [21] which agrees with the result of Ref. [22], that,in general, a singlet can not be generated with a finiteprobability from a finite number of mixed states.With two states of the form (1), ρ ( α, γ, λ ) and ρ ( β, δ, ν ), we obtain a singlet with a finite probabilityby first performing two C-NOT gates locally, with the ρ ( β, δ, ν ) state’s qubits acting as the target qubits. Thesetarget qubits are then measured in the computational ba-sis. If we find both qubits to be in the state | i we havegenerated a pure entangled state between the qubits thatoriginally corresponded to the ρ ( α, γ, λ ) state. We willrefer to this measurement as the pure state conversionmeasurement (PCM). The state formed is | α ′ i ≡ | α ′ , γ = 0 i = √ α ′ | i + √ − α ′ | i , (2)i.e. α ′ is a Schmidt-coefficient that has the value α ′ = min( α (1 − β − δ ) , β (1 − α − γ )) α (1 − β − δ ) + β (1 − α − γ ) . (3)The probability that the PCM succeeds in generating thisstate is given by p c = λν ( α (1 − β − δ ) + β (1 − α − γ )) . (4)For identical states, i.e. α = β, γ = δ , the PCM alreadyyields a singlet. Otherwise the state can be transformedinto a singlet via the ‘Procrustean method’ [23] that con-verts any pure 2-qubit state | α ′ i into a singlet | / i witha probability 2 min(1 − α ′ , α ′ ). The total success proba-bility of generating a singlet is then given by the SCP p conv = 2 λν min[ α (1 − β − δ ) , β (1 − α − γ )] (5)which coincides with the optimal probability for creatinga singlet from two of these states [21].We can perform this partition of the network in anarbitrary way, as long as one group contains A and theother contains B . To be able to create a singlet between A and B via LOCC we must have at least two states ofthe form (1) in all possible partitions. This gives us anecessary condition that to create a singlet between twonodes with a non-zero probability there have to be atleast two distinct ‘paths’ of edges of the form (1) con-necting the corresponding nodes. In Fig. 1(a) this is in-dicated by two spatially distinct paths of bonds. Thestates of the qubits that are not contained in this pathare irrelevant and can therefore be in arbitrary states. Sufficient condition.
In order to show this we makeuse of entanglement swapping. This operation can beperformed in the setup shown in Fig. 2 and consists ofperforming a measurement in the standard Bell basis onthe qubits located at C and LOCC which causes C and C to become entangled. If the edges are of theform (1), ρ ( α, γ, λ ) and ρ ( β, δ, ν ), then there are fourpossible outcomes. The probabilities to obtain measure-ment outcomes corresponding to the Bell states | Ψ ± i =( | i ± | i ) / √ | Φ ± i = ( | i ± | i ) / √ p (Ψ ± ) = 12 ( h ± λν + (1 − β − δ )(1 − λ ) ν + αλ (1 − ν )) (6) C C C
FIG. 2: Basic arrangement for entanglement swapping. En-tanglement swapping involves a measurement in the Bell basisat node C and classical communication between the nodesfollowed by local unitaries which causes C and C to becomeentangled. and p (Φ ± ) = 12 ( g ± λν +(1 − ν )(1 − αλ )+( β + δ )(1 − λ ) ν ) , (7)where h ± = αβ + (1 − α − γ )(1 − β − δ )+ ( √ αδ ± p γ (1 − β − δ )) , (8) g ± = γβ + (1 − α − γ ) δ + (1 − α − γ ) β + ( p γδ ± p α (1 − β − δ )) . (9)If we measure the qubits at B to be in the states | Ψ ± i then we actually form another state, ρ αβh ± , ( √ αδ ± p γ (1 − β − δ )) h ± , λνh ± p (Ψ ± ) ! , (10)of the form (1) between C and C . Unfortunately forthe other outcomes the states’ form is not generally main-tained. Note that if δ = γ = 0 we can discard these casesby replacing the state with | i leading to an operationthat transforms ρ ( α, , λ ) ⊗ ρ ( β, , ν ) into ρ (cid:18) αβh ± , , λνh ± (cid:19) , (11)which will be useful in Sec. V A. We can therefore createa state of the form (1) with non-zero probability betweentwo nodes of the network, e.g. A and B in Fig. 1, giventhat these nodes are connected by a path consisting ofstates of the same form. Two such states, originatingfrom two paths, can then be converted into a singlet,using a PCM and the Procrustean procedure. Unfor-tunately, this scheme leads to an exponential decreaseof entanglement fidelity [5], and thus success probabil-ity, with the number of swapping operations. Hence it isnot an effective solution to the problem of long-distanceentanglement distribution. In Sec. IV we will thereforeintroduce effective protocols which can be applied in reg-ular network geometries and succeed in creating a singletwith a probability independent of distance.Note that when entanglement swapping is done withpure states all of the outcomes can be used, and if theseoutcomes occur with probabilities p m the pure state | ˜ α i Singletsrandomly form ClusterformsEntanglementswapping
FIG. 3: Illustration of classical entanglement percolation ina square network. Pairs of qubits (black dots) in neighbor-ing nodes (circles) are in identical, pure, partially entangledstates (solid, black lines). The percolation scheme involvesthese entangled states being converted into singlets (dashedlines) with probability p . If p exceeds the percolation thresh-old these form large clusters and we can obtain a singlet be-tween any two qubits within a cluster by performing swappingoperations. with ˜ α = 12 s − αβ (1 − α )(1 − β ) p m ! (12)is recovered by using classical communication and localunitaries. III. CLASSICAL ENTANGLEMENTPERCOLATION WITH PURE STATES
In this section we will briefly review the use of perco-lation for distributing singlets in pure state networks [13,15], known as classical entanglement percolation (CEP).The procedure is based on classical bond percolation,where we consider a regular lattice of nodes connectedby identical quantum states, as shown in Fig. 3. A de-scription of classical bond percolation can be found inRef. [24]. If the nodes are connected by pure states ofthe form | α i they can be converted into singlets usingthe Procrustean method with a SCP p = 2 min( α, − α ).These singlets act as the bond in the bond percolationmodel [38] and are distributed randomly with a proba-bility p . The nodes that can be connected by a pathof singlets form a cluster. By using entanglement swap-ping (see Sec. II) we can then generate a singlet between any two nodes in the cluster. In the theoretical caseof an infinitely large lattice a cluster that is infinite inextent forms if and only if p > p c , where p c is a lattice-dependent percolation threshold. This approximates thecase for large but finite lattices where the threshold be-comes more definitive as the size of the lattice increases.Values of p c for a number of lattice geometries are givenin Table I. If each bond in a network consists of a singlepure state | α i we can calculate a threshold for α givenby 2 min( α, − α ) > p c . The probability that a node be-longs to the infinite cluster is known as the percolationprobability θ ( p ). Two randomly chosen nodes are bothpart of the infinite cluster with a probability θ ( p ) andthus can be connected over an arbitrary distance. Lattice Threshold p c
2D Square 0.52D Triangular 2 sin( π/ ≈ . − π/ ≈ . ≈ . ≈ . It has been shown that CEP using pure states is notoptimal and that by performing particular quantum pre-processing steps, particularly swapping operations onthe lattice before converting to singlets, improvementscan be achieved. These improvements include obtain-ing a geometry with a lower percolation threshold af-ter the swapping operation and splitting the lattice intotwo, so that a higher percolation probability can be ob-tained [13, 15, 16, 17, 18]. Recently, another method,that transforms the initial bipartite network into a prob-abilistic multipartite network, has also been shown toyield an improvement [19].
IV. CLASSICAL ENTANGLEMENTPERCOLATION WITH MIXED STATES
In this section we extend CEP to mixed states. Weconsider regular lattices, e.g. triangular (see Fig. 4),square, or even lattices in higher dimensions. Bondsbetween network nodes are composed of multiple edgeswhich satisfies the necessary condition proven in Sec. II.We assume that each bond is identical. When thesebonds contain at least two states of the form (1) theycan be converted into singlets by PCM followed by theProcrustean method. If the probability that a bond be-comes a singlet exceeds the percolation threshold CEP isachieved. In the remainder of the paper we will assumethat the states forming edges are of the form (1) with γ = 0. Setting γ = 0 is not a major restriction but al-lows us to keep the equations manageable. All protocolspresented in this paper can also be performed if γ = 0. FIG. 4: Triangular network. This is a simple 2D arrangementof PMSs in which CEP is possible.
A Bn ρ(α,λ)
FIG. 5: The purification setup consists of n PMSs (solid lines)shared between two nodes A and B . The aim is to distill thesestates into a singlet. We will call states of the form (1) with γ = 0, i.e. ρ ( α, λ ) ≡ ρ ( α, γ = 0 , λ ) , (13)purifiable mixed states (PMSs). Note that these statesform the states of two entangled atomic ensembles in theDLCZ quantum repeater scheme [4]. A. Distillation Procedures
1. Distillable Subspace Scheme
We assume that each pair of neighboring nodes is con-nected by n PMSs and our aim is to distill these intoa singlet. The basic setup is shown in Fig. 5. To ac-complish this we will use ideas proposed in Ref. [26].Here the concept of a distillable subspace (DSS) is in-troduced as a subspace such that the local projection ofthe system state into this space is pure and entangled.Locating the DSS involves calculating the eigenvectorsof the state with non-zero eigenvalues. To simplify no-tation we will represent the states at A and B usingthe decimal value of its binary form, i.e. for example | i A | i B = | i dA | i dB .As an example, in the case of n = 2 identical states ρ ( α, λ ) the eigenvalues and corresponding eigenvectors S C P n=2n=3n=40.20.61.0 TriangularSquareHoneycomb l FIG. 6: Singlet conversion probability for n -edged bondsusing the recycling scheme for α = 1 / n =2 , (3) , , , , , , ,
16 (bottom to top). The n = 3 line(dashed) corresponds to the DSS scheme. The percolationthresholds for triangular (T), square (S) and honeycomb (H)lattices are given by the horizontal lines. are λ : α | i dA | i dB + p α (1 − α ) | i dA | i dB + p α (1 − α ) | i dA | i dB + (1 − α ) | i dA | i dB ,λ (1 − λ ) : √ α | i dA | i dB + √ − α | i dA | i dB ,λ (1 − λ ) : √ α | i dA | i dB + √ − α | i dA | i dB , (1 − λ ) : | i dA | i dB . (14)If this is acted on by the projective measurement | i dA h | + | i dA h | at A and | i dB h | + | i dB h | at B thestate remaining is ( | i dA | i dB + | i dA | i dB ) / √
2. Both ofthese projective measurements only occur with probabil-ity p n =2 = 2 λ α (1 − α ) . (15)Note that this is the same SCP as obtained for PCM [seeEq. (3)]. For this example there is no choice betweenentangled states to project out and if the original statesare the same a maximally entangled state is automati-cally obtained. For states that are not identical this doesnot need to be the case.An extension of this scheme to n identical copies ofPMSs ρ ( α, λ ) yields the SCP p n = n X m =0 λ n − m (1 − λ ) m (cid:18) nm (cid:19) × n − m − X k =1 α n − m − k (1 − α ) k (cid:0) n − mk (cid:1) ( (cid:0) n − mk (cid:1) − (cid:0) nk (cid:1) − ! . (16)A derivation of this formula is given in Appendix B. As aparticular example it is worthwhile to discuss the case ofthree states in more detail. In this case the measurementat A is given by a Positive Operator Valued Measure ρ(α λ ), i i ρ(α λ ), i+1 i+1 ρ(½ ),1 FIG. 7: The recycling scheme consists of splitting the statesinto pairs that are then purified. If no singlets are successfullyproduced some of the states may still have been transformedinto PMSs and given enough of these the process can be re-peated. (POVM) with the elements( | i dA h | + | i dA h | ) , ( | i dA h | + | i dA h | ) / , ( | i dA h | + | i dA h | ) / , ( | i dA h | + | i dA h | ) / , ( | i dA h | + | i dA h | ) / , ( | i dA h | + | i dA h | ) / , ( | i dA h | + | i dA h | ) / . (17)The measurement at B then depends on this outcomeand creates a maximally entangled state with a certainprobability. The SCP is obtained by setting n = 3 inEq. (16) and is given by p n =3 = 3 λ α (1 − α ) . (18)Comparing this with the n = 2 case [cf. Eq. (15)] showsan increase in the success probability which can be seenin Fig. 6, where the dashed line represents the SCP forthree identical states.
2. Recycling scheme
The SCP using the DSS scheme does generally increasewith increasing n . However, the scheme does not makeuse of the available resources in the best way. Indeed,the SCP p n can be significantly improved by grouping n identical PMSs into sets of m and converting each ofthese sets into a singlet. For example for m = 2 we applythe PCM as described in Sec. II on pairs of states whichconverts them into singlets with a probability given byEq. (5). If this fails for a given pair we may still find bothmeasured qubits in the state | i and have generated an-other PMS. This PMS can then be used again in anotherpurification attempt. To be more precise, starting with n copies of a state ρ ( α, λ ) (with α ≥ /
2) we apply a 2-state purification protocol on groups of two. If no singlet S C P l FIG. 8: Success probabilities for the recycling schemes thatsplit the states into pairs (dashed lines) and sets of three(solid lines). Shown are the SCPs for 6, 9 and 12 initial states(bottom to top) for α = 1 / is obtained the procedure is repeated on the remainingPMSs as illustrated in Fig. 7. The coefficients for thePMSs after k repetitions, when no singlet is created, aregiven by α k = α k − − α k − + 2 α k − , (19) λ k = λ k − (1 − α k − + 2 α k − )1 − λ k − + 2 λ k − (1 − α k − + α k − ) , (20)where α = α and λ = λ . For states of the form ρ ( α k , λ k ) ⊗ ρ ( α k , λ k ) the probability of obtaining a PMSis c k = 1 − λ k + 2(1 − α k + α k ) λ k . If the PCM yields twoqubits that are measured in different states the purifica-tion step between the two PMSs has completely failed.The probability of this is given by f k = 2 λ k (1 − λ k ). Theprobability of not generating a singlet using this recyclingprotocol on n states of the form ρ ( α i , λ i ) is then found tobe F n ( i ) = ⌊ n ⌋ X k =0 (cid:18)(cid:18) ⌊ n ⌋ k (cid:19) f ⌊ n ⌋− ki c ki F k ( i + 1) (cid:19) , (21)where F ( i ) = 1. Consequently, the probability of suc-cessfully generating a singlet by applying the procedureto n states of the form ρ ( α, λ ) is 1 − F n (0) which is cal-culated iteratively. Examples are shown in Figs. 6 for α = 1 / B. Percolation Thresholds
Using the purification procedures described above wecan apply CEP, as described in Sec. II, for lattice net-works with multi-edged bonds. In most cases it is advan-tageous to use the two-state recycling scheme, except for n = 3 where the DSS scheme should be used. From Fig. 6it can be seen that the SCP increases with the numberof edges per bond and this allows for a larger range ofvalues for λ and α such that CEP is successful. For dou-ble edged bonds the optimal probability of generating asinglet is given by0 ≤ λ α (1 − α ) ≤ / . (22)When the bonds are composed of three edges, i.e. threePMSs between nodes, we have0 ≤ λ α (1 − α ) ≤ / . (23)By comparing these ranges to the percolation thresholdswe see that a basic successful setup is a double bondedtriangular lattice (see Fig. 4). The double bonds can beconverted to singlets and if the chance of this is largerthan the percolation threshold an infinite cluster willform. A singlet can then be created between any twonodes within the cluster. Thus percolation occurs if2 λ α (1 − α ) > π/ ≈ . . (24)However, the singlet conversion probability never exceeds1 / λ α (1 − α ) > /
2. Analogously, CEP isalso possible in honeycomb lattices with three edges perbond.
C. Local Processing Strategies
The process of creating singlets, randomly replacingthe initial network bonds, can be run if each node canonly communicate classically with their neighbors. Eachnode then knows if a qubit that it contains is part of asinglet after this procedure has finished. This informa-tion can be stored classically within a node but after thebonds are distilled we are faced by the problem of find-ing a set of singlets that connect our requested nodes, A and B .If all of the singlet generation data is collected by a‘controller’ then an efficient path finding algorithm canbe applied to determine a suitable ‘path’ of singlets link-ing the nodes. An example of a suitable algorithm would FIG. 9: Procedure to join a node’s singlets onto a GHZ state.Here the GHZ state is represented by a dotted box. Addi-tional qubits (black dots) that are a part of the GHZ stateare linked to the dotted box by a dotted line. A CNOT gateand measurement (both are represented by a shaded oval) areperformed between the qubit already in the GHZ state andthose that are part of a singlet (dashed lines). Each mea-surement outcome needs to be sent (gray arrow) to the othersinglet qubit to perform a local unitary (shaded square). Thisextends the GHZ state to include qubits connected by edgesto the node being attached. Once this has occurred for eachqubit in a singlet the process is repeated by sending out asignal to repeat the step at each node that was linked by asinglet. be a Dijkstra scheme [27] such as the A* path findingalgorithm [28]. The path information can then be usedto instruct the correct nodes to perform swapping. Theswapping operations are performed in order from node A to B , so that the measurement outcomes only needto be communicated along the chain, between neighbor-ing nodes. However this procedure requires one classicalcomputer to have complete knowledge of the network.Instead, it is interesting to note that this does not needto be the case as there are algorithms which do not re-quire any more classical communication than this ‘con-troller’ method, indeed they do not require a central ‘con-troller’ at all. This can be done not only classically butalso via a quantum algorithm using multipartite entan-glement which we will introduce below.A classical path-finding method would use a typeof breadth-first search algorithm called a burning algo-rithm [29]. Node A sends a ‘burning’ signal to its neigh-boring nodes connected by singlets. These nodes keep arecord of where they received the signal from and sendout an identical signal to the other nodes that they areconnected to. We say that the node has ‘burned’. If ithas already received a signal from a different node thenthe additional signal is ignored. This continues outwardsfrom A , ‘burning’ the nodes. Once node B receives thesignal it replies to the node it came from with a ‘swap-ping’ message. This node can then perform a swappingoperation and send another ‘swapping’ signal, togetherwith the Bell-measurement outcome, back to the node itreceived a ‘burning’ signal from. The path can then betraced back along the nodes with swapping performed ateach step until node A is reached. Both A and B candetermine if the protocol has been successful. However, A and B may not be in the same cluster and they donot know if the protocol has failed when the network isof infinite size. This is not a problem for finite networks,containing N nodes, as A and B can time the steps takenand if these exceed 2( N −
1) they both know they are notin the same cluster.Note that no extra information actually needs to betransmitted. We can combine the burning algorithm withthe process of transmitting the distillation protocol in-formation. For example, in a double edged network ofidentical edges, A can perform her PCM and if | i is theoutcome she assumes she has a singlet and sends a burn-ing signal to the node that would contain the singlet’sother qubit. If a node receives this signal it can performits PCM and determine if there is a singlet there. Whenthere is and if it is the first instance for the node it shouldrecord that entry qubit and repeat the process, perform-ing a PCM on the remaining qubits and sending signalsto those with the | i outcome. Once B receives a signalit can check that a singlet has been created with a PCMand then send a swapping signal back as before. Duringthe swapping, a node can use the Bell-measurement infor-mation received to indicate that a swapping is requiredso no explicit ‘swapping’ signal is required either. Allof this information transfer would have been necessaryas well if a controller algorithm would have been used.Hence the generation of the singlet can be accomplishedby defining rules for each node and allowing them to runwith nearest neighbor classical communication. This isfundamentally different to the controller process and hasmade use of parallel computation to find a path that nosingle node has full knowledge of.We will now consider an alternative, quantum algo-rithm that is based on the burning algorithm and makesuse of multipartite entanglement in the network. Theprotocol starts after we attempted to convert all bondsinto singlets and every node has knowledge about its sin-glet connections to nearest neighbors. We build up aprogressively larger multi-qubit GHZ state, defined by | GHZ n i = ( | i ... | i n + | i ... | i n ) / √
2, spread betweenthe ‘burned’ nodes by adding qubits in each burning step.Building up such a state requires joining two GHZ states, | GHZ n i and | GHZ m i , to create | GHZ n + m − i (note thata singlet equals | GHZ i ). This is done by performinga CNOT gate between a qubit in | GHZ n i and a tar-get qubit in | GHZ m i , measuring the target qubit in the Z -basis, communicating the measurement result to theother qubits in | GHZ m i and performing a unitary op-eration on them depending on the outcome. Now weperform the same process as for the ‘burning algorithm’,however, as each node is ‘burned’ it is connected to theGHZ state spread over the previously burned nodes. Theprocess to do this is illustrated in Fig. 9 and consists of AB AB AB AB AB AB
FIG. 10: After the singlets are formed we can repeatedly ex-tend a GHZ state from node A . This procedure uses theoperation shown in Fig. 9 to add qubits to the GHZ state.The black squares depict qubits that are part of the GHZstate. Arrows represent a message to add nodes to the GHZstate along singlet paths. Each node keeps a record of thenode from which it received this message from, symbolizedhere by a white dot. When a node cannot extend the GHZstate any further (highlighted by a dashed outline) it mea-sures its qubits in the X basis (open squares) and sends thisinformation back towards A (thin arrows) along the routerecorded. Certain nodes are selected beforehand not to per-form the measurement (here A and B ) and these will formthe resulting GHZ state. At each node the incoming datacan be combined and sent back along one path if the routesbranch. Once this data returns to A a phase operation canbe performed on the qubit there to correct for any errors andthe final GHZ state (here a singlet between A and B ) willremain. joining the singlets partially contained in that node tothe GHZ state. Within each node one qubit is left entan-gled with the GHZ state. After this operation has beenrun for a maximum of N − A have a qubit from a single GHZstate.At each node a record is kept of the bond via whichit has been included into the GHZ state. If there is asinglet between two nodes that are being burned thenthe singlet is ignored. Furthermore we add the rule thatwhenever a node can not extend the GHZ state anymore X basis measurements are performed along the recordedpath back to A . This removes a qubit from the GHZ statebut introduces a phase error in the remaining GHZ statedepending on the outcomes of the measurement. The in-formation about these measurement outcomes has to besent back along the path to A . Whenever the route backbranches, the measurement outcome is sent in one wayand a message corresponding to ‘no phase error occur-rence’ is sent to the others. At each node the returningprocess is paused until all of the bonds it sent a burningsignal to provide it with the phase information. At nodes A and B we do not perform the X measurement. Finallyafter A receives all of the phase information a phase cor-rection can be performed and we obtain a singlet between A and B . In Fig. 10 an example is given to illustrate theprotocol. V. QUANTUM PREPROCESSING
Despite being a very effective method, it is known thatCEP in a network of pure states can be improved by cer-tain quantum ‘pre-processing’ strategies, and thereforeCEP is not optimal [13, 15, 16, 17, 18, 19]. In the fol-lowing we show that this is also the case in mixed-statenetworks.
A. Swapping procedure
To start with we generalize the swapping arrangementshown in Fig. 2 previously studied for pure states [12, 15].In this arrangement we have two 2-qubit states that bothhave a qubit in a common node. If the two states arepure states | α i and | β i , with α ≥ / β ≥ /
2, wecan obtain a singlet by swapping and then converting theresulting pure state into a singlet with a total probabilityof 2 min((1 − α ) , (1 − β )) which turns out to be the optimalprobability. Particularly CEP, which consists here of theProcrustean method followed by entanglement swapping,always has a smaller SCP of 4(1 − α )(1 − β ). Note thatthe optimal probability is equal to that of converting theleast entangled of the two bonds into a singlet using theprocrustean scheme [12].To generalize this to mixed states we must considerdouble-edged bonds, each consisting of two PMSs, as il-lustrated in Fig. 11, since otherwise singlet generationwould not be possible. Introducing more than one edgebetween the nodes allows us to concentrate the entangle-ment at different stages which gives rise to three differentpossibilities: I CEP - As previously described, the bonds are converted
SwappingSwappingA B C ρ(α,λ)ρ(β,ν) ρ(β,ν)ρ(α,λ)
PCM
ClassicalPercolationDirectSwappingHybridSwapping ρ( )α,λ~ ~ρ( )α,λ~ ~ρ(α, )1 ˆ ρ(α, )1~ ρ(α, )1~ ρ(½, )1 ρ(½, )1ρ(½, )1 PCM andProcrusteanPCM andProcrusteanSwapping Procrustean
FIG. 11: Three methods can be used to generate a singletbetween two nodes A and C via an intermediate node B inan arrangement with two edges per bond. The thick blacklines indicate pure but not maximally entangled states. to singlets and then swapping is performed over theresulting states. II Direct swapping - This applies entanglement swappingtwice and then the resulting states are convertedinto a singlet.
III
Hybrid swapping - Here we distill a state of higherentanglement in each bond (but not necessarilya singlet) leading, if successful, to a single (par-tially) entangled pure state in each bond. This isfollowed by entanglement swapping and the Pro-crustean scheme to create a singlet.Each of these possibilities uses the swapping operation atdifferent stages as illustrated in Fig. 11. The exact im-plementations for the procedures depend on the types ofstates used. We will first apply each of them on a networkof pure states and compare the SCPs. We then generalizeto PMSs and show that direct and hybrid swapping canoutperform CEP.
1. Pure states
If we start with bonds made of pure states | α i and | β i we must have a way to convert each bond into a singletin order to apply CEP(I). The method and highest possi-ble probability to accomplish this are given by Majoriza-tion [30] with a probability p = min(1 , − αβ )) [15, 16].0 S C P a FIG. 12: Comparison of the three methods described in thetext for creating a singlet between nodes A and C in thesetup shown in Fig. 11 for pure states. Shown are the successprobabilities if the bonds are made up initially of the states | α i and | / i for CEP (solid line), direct swapping (dottedline) and hybrid swapping (dashed line). CEP applies this operation on each bond and if bothbonds are converted into singlets swapping can be per-formed and the operation is a success. Therefore CEPsucceeds with a probability (min(1 , − αβ ))) .Our second method, direct swapping(II), is simply theapplication of the procedure for bonds containing oneedge twice. If either generates a singlet the proceduresucceeds. This gives a SCP of 1 − (1 − − α ))(1 − − β )). There are adjustments we could make, for exampleuse the results of Majorization to convert both of thestates into a singlet with the highest possible probability,however all of these have a smaller SCP than CEP for arange of parameters.Finally, the hybrid swapping(III) method concentrateseach bond to one pure state, | max(1 / , αβ ) i , with cer-tainty. This concentration procedure is also found usingresults from Majorization theory [30]. Afterwards thereis one pure state in each bond, as discussed previously,and we can then perform the strategy with optimal suc-cess probability min(1 , − αβ )), i.e. swapping over thepure states followed by the Procrustean method. We canactually consider the setup as a bipartite system between A and BC . The Majorization results then give the bestpossible probability of generating a maximally entangled2-qubit state between these systems as min(1 , − αβ ))which means that it must be the highest possible proba-bility for any method to succeed.Figure 12 shows the probabilities in all three cases andwe can see that CEP is outperformed for a vast rangeof parameters by both other strategies. In hybrid swap-ping (III), we have used multi-edged bonds to create purestates with the highest probability before applying en-tanglement swapping. We will refer to all strategies thathave this property as ‘hybrid’. This probability is unityfor initial pure states but for mixed states the initial con-version of bonds to pure states is probabilistic, so whenthe conversion fails the bond is destroyed.
2. Purifiable Mixed States
We will now investigate if similar improvements can beobtained with PMSs, i.e. if the bonds between the nodesare composed of ρ ( α, λ ) and ρ ( β, ν ). Again we will seethat hybrid swapping provides the highest SCP. I CEPThe classical percolation scheme involves perform-ing a PCM described in Sec. II followed by the pro-crustean protocol on both bonds and each succeedswith a probability given by Eq. (5) which simplifiesto p conv = 2 λν min( α (1 − β ) , β (1 − α )) . (25)To perform a swapping operation yielding a singlet,between nodes A and C we must succeed for bothbonds which gives the total chance of success p CEP = (2 λν min( α (1 − β ) , β (1 − α ))) , (26)by simply squaring Eq. (25). In this case the swap-ping operation is the final step of the protocol. II Direct swappingIn our 2-edged setup we perform the swapping op-eration introduced in Sec. II twice and there aretwo choices to do this if the states are not identical.Either we perform the swapping over the identicalstates ρ ( α, λ ) ⊗ ρ ( α, λ ) or we perform the operationon the states ρ ( α, λ ) ⊗ ρ ( β, ν ). When we swap overidentical states we obtain the state ρ (cid:18) α − α + 2 α , λ (1 − α + 2 α ) (cid:19) , (27)together with a further state where β is replacing α and ν is replacing λ . Note that Eq. (27) is obtainedby setting γ = δ = 0 and α = β, λ = ν in Eq. (11).This pair of states can then be transformed into asinglet with a probability p d ∗ = 2 λ ν min( α (1 − β ) , β (1 − α ) ) (28)which is calculated using Eq. (25). In the casewhere we swap over non-identical states ρ ( α, λ ) ⊗ ρ ( β, ν ) we obtain two states of the form (11) with γ = δ = 0. These can be converted into a singletwith probability p d = 2 λ ν αβ (1 − α )(1 − β ) . (29)This is always larger than p d ∗ and thus swappingwith non-identical states should be preferred. III
Hybrid swappingThe hybrid method requires a concentration proce-dure to be performed (yielding a single pure state1 a S C P / l FIG. 13: Success probability to generate a singlet betweenthe end nodes of the swapping setup shown in Fig. 11 for theclassical scheme (solid line), direct swapping (dotted line),and the hybrid scheme (dashed line). Each bond initiallycontains the states ρ ( α, λ ) and ρ (1 / , λ ). We have indicatedthe percolation threshold of a face-centered cubic network. in each bond) which is given here by PCM. How-ever, in contrast to the pure state case discussedabove, if α = β we obtain singlets (in which casethe method is identical to CEP) and, generally, theoperation succeeds with a finite probability givenby Eq. (4). For non-identical PMSs PCM yieldstwo non-maximally entangled pure states which arethen used for entanglement swapping followed bythe Procrustean method. The probability of suc-ceeding in converting both of the bonds to purestates is p c = λ ν ( α (1 − β ) + β (1 − α )) . (30)These pure states have largest Schmidt coefficientˆ α = max( α (1 − β ) , β (1 − α ))( α (1 − β ) + β (1 − α )) . (31)So, by using the SCP in single edged swapping withpure states we find that we can convert this pair ofstates into a singlet between the end nodes withprobability2(1 − ˆ α ) = 2 min( α (1 − β ) , β (1 − α ))( α (1 − β ) + β (1 − α )) . (32)Hence, the overall probability of succeeding withthis scheme is p h =2 λ ν [ α (1 − β ) + β (1 − α )] × min[ α (1 − β ) , β (1 − α )] . (33)If we compare the success probability of direct swap-ping, p d , to the probability of success in the classicalpercolation scheme, p CEP , it can be seen that classicalpercolation is more likely to succeed in producing a sin-glet if2 min( α (1 − β ) , β (1 − α )) > max( α (1 − β ) , β (1 − α )) . (34) FIG. 14: Illustration of entanglement percolation in a 3Dnetwork. The circles represent nodes containing qubits andthe lines represent bonds containing pairs of two-qubit entan-gled states (the edges are not shown). The 3D network canbe transformed into a Face-centered cubic network by per-forming the swapping operations (see Fig. 9) over the smallernodes. For some bond parameters the hybrid scheme allowspercolation to occur where classical percolation fails.
But the ratio of the success probability for the classicalscheme against the hybrid protocol, p h , is p CEP p h = 2 min( α (1 − β ) , β (1 − α ))( α (1 − β ) + β (1 − α )) . (35)Whenever α = β this is less than one and there is animprovement over the classical percolation scheme. Fur-thermore, the hybrid scheme is more likely to succeedthan direct swapping. In Fig. 13 we compare the proba-bilities of success for all schemes. As can be seen, hybridswapping leads to the highest success probability.Hybrid swapping can be used in sections of larger net-works to allow percolation to take place. A simple exam-ple is a face-centered cubic (FCC) network, where everybond is split into two 2-edged bonds (see Fig. 14). Whenthe above schemes are applied at the nodes linking two2-edged bonds the FCC network is recovered. Percola-tion is possible in these 3D networks with a thresholdof approximately ≈ .
12. Since the classical scheme al-ways gives a smaller success probability than the hybridscheme there are cases where the hybrid scheme allows2
A BC D PCM SwappingProcrustean
FIG. 15: Application of the hybrid scheme in a square net-work. This involves transforming the PMSs into pure statesprobabilistically and then applying a suitable pure state pro-cedure (see text). In the case shown all of the conversions aresuccessful. When this happens a swapping operation can beperformed and the resulting states distilled into a singlet. the percolation threshold to be exceeded but the classicalscheme does not (see Fig. 13).
B. Square Protocol
CEP can also be improved on by using the hybrid strat-egy in a 2D square network, as shown in Fig. 15. Eachbond is converted into a pure state, | ˆ α i ≡ √ ˆ α | i + √ − ˆ α | i , by using PCM which is successful with aprobability p c on each bond. If this yields only twostates | ˆ α i having a common node ( B or C ), entanglementswapping can be performed followed by the procrusteanscheme. If all four PCMs succeed the resulting states canbe connected (e.g. at nodes B and C ) via a slightly mod-ified version of entanglement swapping, the so-called XZ-swapping [15]. For this swapping operation the Bell mea-surement that usually has both qubits measured in the Z basis now measures one in the X basis. After this mea-surement unitaries are again applied to return the stateinto Schmidt form. The results of the Bell measurementhave an equal probability, p m = 1 /
4, for all outcomes m .Performing this operation twice on the square leads totwo pure states (between A and D ) of the form | ˜ α i , with˜ α = (1 + p −
16 ˆ α (1 − ˆ α ) ) /
2. These can be distilledinto a singlet with probability min[1 , − ˜ α )] by us-ing the protocol based on Majorization [30]. The overallchance of succeeding in generating a singlet is then givenby p sq = 4 p c (1 − p c )(1 − ˆ α ) + p c min(1 , − ˜ α )) . (36)When attempting to accomplish the same scheme usingCEP we succeed with a probability of ˜ p CEP = 1 − (1 − p CEP ) which can be significantly smaller than Eq. (36),as shown in Fig. 16. a S C P Triangular
FIG. 16: Comparison of singlet conversion probabilities forthe different strategies in the square configuration, i.e. ˜ p CEP (solid line) and p sq (dashed line) for λ = ν = 0 . , β = 0 . Again, this improved strategy may enable an infinitecluster to form when applied to larger networks. Anexample is shown in Fig. 17. Here the square proto-col recovers a triangular lattice. If the conversion of thesquares into singlets succeeds with a probability exceed-ing the percolation threshold an infinite cluster forms.In Fig. 16 it can be seen that the hybrid scheme exceedsthe threshold for a triangular lattice in cases where CEPdoes not.
C. Hierarchical Networks
Small networks like the square configuration discussedabove can be extended to larger networks in an iterativefashion. Networks formed in this way from pure stateswere considered in [15]. Again the probability of success-fully creating a singlet was shown to be larger when quan-tum strategies were used instead of CEP. However, thescheme with the highest probability is still unknown forthese ‘hierarchical’ networks. Here we will consider two3
AB1st
ABBA
FIG. 18: The first three iterations of a diamond lattice. Weaim to create a singlet between nodes A and B in each case. different hierarchical networks with two edges per bond.Each of these contains the square network at some iter-ation level. We determine the SCP when using CEP inboth cases which we then compare to the hybrid strategy.As it turns out, the hybrid scheme outperforms CEP.The first hierarchical network we consider is based onthe ‘Diamond’ lattice, which at each stage replaces itsbonds by the square network. The geometry for the firstthree iterations is shown in Fig. 18. The aim is to createa singlet between A and B and if we apply CEP theprobability of succeeding at each level is given by theiterative formula p Diamondi = 1 − (1 − p i − ) , (37)starting with p Diamond = p conv .The second hierarchical network we consider is the‘Tree’ network which is again built on the square con-figuration. For these networks an iteration is formed bycreating two copies of the previous iteration and linkingthe bottom-left and top-right corner of the square to sep-arate nodes A and B as shown in Fig. 19. Again, we wishto generate a singlet between the opposite corner nodes( A and B ) and CEP generates a singlet with a probability p T reei = 1 − (1 − p i − p conv ) , (38)where p = 1.Now we wish to see whether the hybrid scheme givesa larger SCP in these networks. Once again, the hy-brid scheme we consider starts by converting all of thebonds into identical non-maximally entangled pure statesprobabilistically. If the conversion fails on a bond thenthe bond is destroyed. This results in a network con-taining random pure state bonds. Each of these bonds A B
A B
A B
FIG. 19: First three iterations of the tree lattice. Each it-eration is given by repeating the previous lattice twice andlinking the pair of previous endpoints at new endpoints. Weaim to create a singlet between nodes A and B . contains one edge. Ideally we would then apply a purestate protocol yielding the highest SCP between the in-tended nodes, however, this protocol is not known in thegeneral case [15]. Instead we apply a procedure whichperforms XZ -swapping in cases when two bonds eachhave a qubit in the same node (except if these nodesare A or B ). However, we also distill pure states intostates with more entanglement whenever two edges formbetween two nodes and before performing further swap-ping. Finally, once one state is obtained between A and B , the procrustean procedure is used to create a singlet.We applied this protocol to the hierarchical diamondand tree networks. For the second and third iterationsof the diamond lattice the probabilities of creating a sin-glet are given in Fig. 20 together with the probabilitiesusing CEP. This comparison was also made for the first,second and third iteration of the tree network and the re-sults are shown in Fig. 21. These examples all illustratean improvement in the probability of forming a singletwhen using the hybrid method rather than classical per-colation. VI. CONCLUSION
We have demonstrated that within lattice networks,where the nodes are connected by multiple bipartitemixed states, percolation strategies can be applied for4 S C P a FIG. 20: Probability of succeeding in generating a singletbetween the endpoints of a diamond lattice for the 2nd and3rd iterations (dashed lines). These give higher probabilitiesthan the classical protocol (solid lines). The bonds containtwo edges with parameters λ = ν = 0 . , β = 0 . S C P a FIG. 21: Probability of succeeding in generating a singletbetween the endpoints of the 1st, 2nd and 3rd iterations ofthe tree lattice (dashed lines). These also outperform theclassical protocol (solid lines). The bonds contain two edgeswith parameters λ = ν = 0 . , β = 0 . distributing entanglement. This is reliant on the statesbeing PMSs, which are known from the DLCZ repeaterscheme and arise as a result of amplitude damping. Toshow this we have introduced some new purification pro-tocols designed to maintain the form of these states orgenerate singlets. Like in the pure state case, a higherprobability of distributing a singlet can be obtained,when the states in a bond are not identical. The ques-tion of whether quantum strategies can outperform CEPwhen each edge in a bond is identical is still open. Sincewe have shown that classical entanglement percolation isonly possible for a specific class of bipartite states, en-tanglement distribution in a network which is subject tomore general forms of noise needs to make use of othermethods. The development of these methods is one ofthe most important goals for future work. These willnot produce perfectly entangled states, however, the re-sulting state fidelity may be independent of distance andsufficient for purification. An example of such a strategy is given in Ref. [31] for a bit-flip noise model. Progress inthis direction has also been accomplished by generating3D thermal cluster states using Werner states [32, 33]. Acknowledgments
This research was supported by the EPSRC (UK)through the QIP IRC (GR/S82176/01) and the ESFproject EuroQUAM (EPSRC grant EP/E041612/1).
APPENDIX A: PROOF OF SINGLETDISTILLATION REQUIREMENT
In this appendix we give a concise proof for a neces-sary and sufficient condition to be able to create a singletout of entangled mixed states using LOCC. We allow thestates to be arbitrary bipartite states which are sharedbetween two nodes and all operations are LOCC. A sim-ilar proof, but partly restricted to identical states, wasgiven in [21].
Lemma A.1.
If a quantum state ρ ab can be distilled toa pure states, | Ψ i , then any state with the same range R ( ρ ab ) is also distillable to this state with non-zero prob-ability.Proof. The general form of the state is ρ ab = N X i =1 p i | ψ i ih ψ i | , (A1)with p i > P i p i = 1 and | ψ i i ∈ H A ⊗ H B . If thestate is distillable to a pure state, | Ψ i , there exist linearoperators M A and N B , with M A M † A ≤ I, N B N † B ≤ I ,such that M A ⊗ N B ρ ab M † A ⊗ N † B = p | Ψ ih Ψ | (A2) ⇒ M A ⊗ N B | ψ i ih ψ i | M † A ⊗ N † B ∝ | Ψ ih Ψ | (A3)or M A ⊗ N B | ψ i ih ψ i | M † A ⊗ N † B = 0 . (A4)This can be summarized as ⇒ M A ⊗ N B | ψ i i = q i | Ψ i , (A5)where at least one q i is non-zero as otherwise the oper-ator fails to distill ρ ab . If this condition is satisfied theoperation distills the mixed state into | Ψ i . Now givenanother state ˜ ρ ab with the same range as ρ ab . We havethat ˜ ρ ab = M X i =1 ˜ p i | ˜ ψ i ih ˜ ψ i | (A6)5with | ˜ ψ i i = P Nj =0 a i,j | ψ j i and | ψ i i = P Mj =0 b i,j | ˜ ψ i i . Thisthen gives M A ⊗ N B | ˜ ψ i i = M A ⊗ N B N X j =0 a i,j | ψ j i (A7)= N X j =0 a i,j q j | Ψ i = ˜ q i | Ψ i . (A8)The value of one ˜ q i must be non-zero as otherwise all q i are zero and this contradicts the fact that the operatordistills ρ ab . Hence the protocol also distills ˜ ρ ab . Lemma A.2.
For n If a 2-qubit state has a range that can be spannedby product states then a separable state with this rangeexists. If there are n states each with a range spannedby product states the system state would have a rangeequivalent to a separable state formed by all of these 2-qubit separable states. Since it is impossible to distill apure entangled state from any separable state it is impos-sible to distill a pure entangled state from n two qubitstates each with a range spanned by product states. Sim-ilarly, if one of the 2-qubit states does not have a rangespanned by product states, but all of the other states do,the range is equivalent to the range formed from a sep-arable state and one mixed 2-qubit state. This can notbe distilled into a pure singlet as it would contradict theresult in [22]. Hence at least two states can not have arange spanned by product states to be able to distill apure entangled state.We now need to look at the two qubit states that sat-isfy this property. The states with rank one are alreadypure and if they have rank four the range can be spannedby product states. Similarly, if the state has rank threeit can also be spanned by product states. This can beseen by considering the subspace orthogonal to a gen-eral state √ α | i + √ − α | i . This space is spannedby {| i , | i , ( √ − α | i − √ α | i )( | i + | i ) / √ } andthese are all product states. The last states to con-sider are those of rank 2, which fall into two cate-gories [34]. The range is either spanned by productstates {| i , ( √ λ | i − √ − λ | i )( √ µ | i + √ − µ | i ) } or {| i , ( √ α | i + √ β | i )+ √ − β − α | i ) } . Hence onlystates that have a range containing one product stateare the mixed states satisfying the condition. All mixedrank two states of two qubits can be considered to be themixed state formed by tracing out a third qubit from apure three qubit system. The classifications of these 3qubit systems is given in [35, 36, 37] and for the range ofthe mixed system to contain one product state the threequbit state belongs to the W class. This class can alwaysbe written as √ λ | Φ i| i + √ − λ | i| i with | Φ i = √ α | i + p β | i + √ γ | i , α + β + γ = 1 . (A9) By tracing out one qubit and using local operations the2-qubit state that can not be spanned by product stateshas the form ρ = λ | ψ ih ψ | + (1 − λ ) | ih | ) , (A10)where | ψ i = √ α | i + p β | i + √ γ | i , α + β + γ = 1 . (A11)So the only states that can be purified into a perfectsinglet, given finite copies, are of this form.If there are two states of this form we know that thesystem is distillable since the procedure given in Sec. IIsucceeds in the distillation. APPENDIX B: THE DISTILLABLE SUBSPACESCHEME
To extend the DSS scheme to n PMSs, ρ ( α, λ ), wefirst need to describe the 2 n non-zero eigenvalues andtheir eigenvectors. These correspond to different combi-nations of n − l | α i terms and l | i terms. Then takingthe decimal representation of the local states we can la-bel each of these eigenvectors by the decimal differencebetween the values at each location. This difference y inbinary gives the location of the | i terms. For example,in the case of two identical PMSs these are y = 0 = 00 : | α i| α i ,y = 2 = 10 : | i| α i ,y = 1 = 01 : | α i| i ,y = 3 = 11 : | i| i (B1)and y takes all of the values from 0 to 2 n −
1. Nowwe define m ( x ) to be the number of 1s in the binaryrepresentation of x and T y = { x : x ∧ y = 0 , ≤ x < n , x ∈ N } (‘ ∧ ’ is the bitwise AND operation).Then l = m ( y ) and all of the terms in a non-zero eigen-vector are of the form X x ∈ T y q α n − m ( x ) − l (1 − α ) m ( x ) | x i dA | x + y i dB , (B2)with eigenvalue λ n − l (1 − λ ) l .From this structure we can project out an entangledstate if we measure the operator ( | c i dA h c | dA + | d i dA h d | dA ) at A and then ( | c + y i dB h c + y | dB + | d + y i dB h d + y | dB ) at B ,when c ∈ T y , d ∈ T y , d > c and as long as there areno other terms of the form | c i dA | d + y i dB + | d i dA | c + y i dB , | c i dA | d + y i dB or | d i dA | c + y i dB in any non-zero eigenstate.The term | c i dA | d + y i dB + | d i dA | c + y i dB can not appearin one eigenstate since all of the terms must have thesame y value and this would require c to be equal to d .The state | c i dA | d + y i dB lies in one if and only if ∃ w ∈ N ,0 ≤ w ≤ n − c + w = d + y and c ∈ T w .Similarly for | d i dA | c + y i dB but this case can not occur6since c ∈ T y and d ∈ T y means that w ≥ y and d + w >c + y . If we assume that ∃ w ∈ N , 0 ≤ w ≤ n − c + w = d + y and c ∈ T w this would mean that d = c + k and that c ∧ k = 0 for some k >
0. Both ofthese results then give that k ∧ y = 0 and w = k + y . So,if w = y + k = d + y − c such that c ∧ k = 0 can not besatisfied we create a maximally entangled state.Now we have a choice of ways of creating these mea-surements. One particular way involves the definition ofsets S k = { x : m ( x ) = k , 0 ≤ x < n , x ∈ N } and J a,b = { x : x ∧ ( a OR b ) = 0, 0 ≤ x < n , x ∈ N } . Thenthe protocol consists of performing a POVM P k,a,b = C k ( | a i dA h a | dA + | b i dA h b | dA ) at location A with a, b ∈ S k , a = b and 0 < k < n . For k = 0 and n we define P k,a,b = | k − i dA h k − | dA and when these outcomesoccur the procedure has failed. Here C k is a factor toensure that k = n X a,b ∈ S k ,k =0 P k,a,b = I. (B3)With this outcome at location A another POVM is doneat location B given by the operators Q d = C k ( | a + d i dB h a + d | dB + | b + d i dB h b + d | dB ) ( d ∈ J a,b ) and F = I − P d ∈ J a,b Q d . If the outcome here is F the protocolhas failed, otherwise we have obtained a maximally en-tangled state. This protocol works since a, b ∈ T y for all y ∈ J a,b but there is no w = k + y such that k ∧ a = 0and b + y = a + y + k , since if there were we would have b = a + k but m ( a + k ) = m ( b ).The probability of succeeding is given by Eq. (16)which comes from considering a particular eigenstatewith parameter y . In this eigenstate there are N = (cid:18) n − m ( y ) k (cid:19) (cid:18)(cid:18) n − m ( y ) k (cid:19) − (cid:19) / | a i dA | a + y i dB + | b i dA | b + y i dB with a, b ∈ S k . The number of possible measured operatorsfrom this eigenstate is given by N = (cid:18) n − m ( y ) k (cid:19) (cid:18)(cid:18) nk (cid:19) − (cid:19) . (B5)Note that the pairings in the eigenstate are twice as likelyto occur than the ones with just an overlap and these havebeen counted twice in this sum. The probability thatstarting with an eigenstate (parameter y ) we succeed isthen 2 N N = (cid:0) n − m ( y ) k (cid:1) − (cid:0) nk (cid:1) − , (B6)given that we have measured the operator S k and theprobability of this was P k = (cid:18) n − m ( y ) k (cid:19) α n − m ( y ) − k (1 − α ) k . (B7)By summing over these we have, given we start with aneigenstate with m ( y ) = m , the probability of succeedingto be n − m − X k =1 α n − m − k (1 − α ) k (cid:0) n − mk (cid:1) (cid:0)(cid:0) n − mk (cid:1) − (cid:1)(cid:0) nk (cid:1) − λ n − m (1 − λ ) m (cid:18) nm (cid:19) . (B9)We have not counted k = 0 , n − m since they never con-tribute to the success probability. Then by summing overall of these we get the result in Eq. (16). [1] M. Nielsen and I. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[2] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[3] H.-J. Briegel, W. D¨ur, J. I. Cirac, and P. Zoller, Phys.Rev. Lett. , 5932 (1998).[4] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Na-ture , 413 (2001).[5] W. D¨ur, H.-J. Briegel, J. I. Cirac, and P. Zoller, Phys.Rev. A , 169 (1999).[6] L. I. Childress, J. M. Taylor, A. Sorensen, and M. D.Lukin, Phys. Rev. A , 052330 (2005).[7] L. Hartmann, B. Kraus, H.-J. Briegel, and W. D¨ur, Phys.Rev. A , 032310 (2007).[8] U. Dorner, A. Klein, and D. Jaksch, Quantum. Inf. Com-put. , 0468 (2008).[9] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. , 722 (1996).[10] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello,S. Popescu, and A. Sanpera, Phys. Rev. Lett. , 2818(1996).[11] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert,Phys. Rev. Lett. , 4287 (1993).[12] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A ,194 (1999).[13] A. Ac´ın, J. I. Cirac, and M. Lewenstein, Nat. Phys. ,256 (2007).[14] G. Vidal, Phys. Rev. Lett. , 1046 (1999).[15] S. Perseguers, J. I. Cirac, A. Ac´ın, M. Lewenstein, andJ. Wehr, Phys. Rev. A , 022308 (2008).[16] G. J. Lapeyre Jr., J. Wehr, and M. Lewenstein, Phys.Rev. A , 042324 (2009).[17] K. Kieling and J. Eisert, Quantum and Semi-classical Percolation and Breakdown in Disordered Solids (Springer, Berlin, 2009), vol. 762 of
Lecture Notesin Physics , pp. 287–319, also available at arXiv:0712.1836v1 [quant-ph]. [18] M. Cuquet and J. Calsamiglia, Phys. Rev. Lett. ,240503 (2009).[19] S. Perseguers, D. Cavalcanti, G. J. Lapeyre Jr, M. Lewen-stein, and A. Acin (2009), arXiv:0910.2438v1 [quant-ph].[20] S. Broadfoot, U. Dorner, and D. Jaksch, EuroPhys. Lett. , 50002 (2009).[21] E. Jan´e, Quantum. Inf. Comput. , 348 (2002).[22] A. Kent, Phys. Rev. Lett. , 2839 (1998).[23] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu-macher, Phys. Rev. A , 2046 (1996).[24] B. Bollob´as and O. Riordan, Percolation (CambridgeUniversity Press, Cambridge, 2006).[25] C. D. Lorenz and R. M. Ziff, Phys. Rev. E , 230 (1998).[26] P. X. Chen, L. M. Liang, C. Z. Li, and M. Q. Huang,Phys. Rev. A , 022309 (2002).[27] E. W. Dijkstra, Numer. Math. , 269 (1959).[28] P. E. Hart, N. J. Nilsson, and B. Raphael, IEEE Trans.Syst. Sci. Cybern. , 100 (1968).[29] H. J. Herrmann, D. C. Hong, and H. E. Stanley, J. Phys.A , L261 (1984). [30] M. Nielsen and G. Vidal, Quantum. Inf. Comput. , 76(2001).[31] S. Perseguers, L. Jiang, N. Schuch, F. Verstraete, M. D.Lukin, J. I. Cirac, and K. G. H. Vollbrecht, Phys. Rev.A , 062324 (2008).[32] R. Raussendorf, S. Bravyi, and J. Harrington, Phys. Rev.A , 062313 (2005).[33] S. Perseguers (2009), arXiv:0910.1459v1 [quant-ph].[34] A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A ,826 (1998).[35] A. Ac´ın, A. Andrianov, L. Costa, E. Jan´e, J. I. Latorre,and R. Tarrach, Phys. Rev. Lett. , 1560 (2000).[36] W. Dur, G. Vidal, and J. I. Cirac, Phys. Rev. A ,062314 (2000).[37] A. Ac´ın, A. Andrianov, E. Jan´e, and R. Tarrach, J. Phys.A34