Singlet Generation in Mixed State Quantum Networks
We study the generation of singlets in quantum networks with nodes initially sharing a finite number of partially entangled bipartite mixed states. We prove that singlets between arbitrary nodes in such networks can be created if and only if the initial states connecting the nodes have a particular form. We then generalize the method of entanglement percolation, previously developed for pure states, to mixed states of this form. As part of this, we find and compare different distillation protocols necessary to convert groups of mixed states shared between neighboring nodes of the network into singlets. In addition, we discuss protocols that only rely on local rules for the efficient connection of two remote nodes in the network via entanglement swapping. Further improvements of the success probability of singlet generation are developed by using particular forms of ‘quantum preprocessing’ on the network. This includes generalized forms of entanglement swapping and we show how such strategies can be embedded in regular and hierarchical quantum networks.
pacs:03.67.Bg, 03.67.-a, 64.60.ah
Quantum entanglement is one of the most notable features of quantum systems and has been accepted as a key resource for quantum information processing Nielsen and Chuang (2000). The distribution of entanglement through quantum networks is therefore essential for the future of a variety of applications, ranging from quantum cryptography to quantum teleportation and distributed quantum computing Bennett et al. (1993). However, the generation of these entangled states faces a severe obstacle. Quantum channels such as free-space transmission or optical fibers are prone to loss and decoherence. This causes the desired maximally entangled states to degrade into mixtures and limits the distance over which the quantum information can be sent directly. To overcome these problems ‘quantum repeater’ schemes have been proposed Briegel et al. (1998); Duan et al. (2001); Dür et al. (1999); Childress et al. (2005); Hartmann et al. (2007); Dorner et al. (2008) which make use of the ability to ‘purify’ Bennett et al. (1996a); Deutsch et al. (1996) and ‘swap’ Zukowski et al. (1993); Bose et al. (1999) entanglement to maintain a high fidelity throughout. Quantum repeaters are a promising tool for entanglement distribution, particularly since the amount of required physical resources increases only polynomially with the distance Dür et al. (1999), but operate in a 1D setup of network nodes. Real networks are typically two-(or higher) dimensional and it is therefore desirable to study if entanglement distribution can be made more efficient in these cases.
A scheme for entanglement distribution in higher dimensional networks was recently proposed by Acín et al. Acín et al. (2007) in which ideas from classical bond percolation have been applied to regular, i.e. lattice-shaped, quantum networks. The scheme makes use of the networks’ connectivity and allows for the generation of maximally entangled singlet states between arbitrary points of the network, with a probability that is independent of their separation. The only requirement is that the nodes are initially connected by bipartite pure states with sufficiently high entanglement. The restriction to pure states was made since a pure, partially entangled state can be converted into a singlet with finite probability via local operations and classical communication (LOCC) Vidal (1999) which is essential for the bond percolation protocol: Initially one attempts to convert all bipartite pure states into a singlet which, in each case, succeeds with a certain probability. If this singlet conversion probability (SCP) exceeds a lattice-geometry-dependent threshold, arbitrarily large clusters of singlet-connected nodes form which can successively be connected via entanglement swapping. In this way we can create a singlet between arbitrarily remote nodes in the network. However, it was pointed out in Acín et al. (2007) that this process, known as Classical Entanglement Percolation (CEP), is not optimal since certain quantum preprocessing schemes applied to the network can improve the SCP Acín et al. (2007); Perseguers et al. (2008a); Lapeyre Jr. et al. (2009); Kieling and Eisert (2009); Cuquet and Calsamiglia (2009); Perseguers et al. (2009), and thus it is possible to apply bond percolation to lattices in which this would otherwise not be possible.
Clearly, the assumption of having a pure-state network is an idealization and in any practical situation the states connecting the nodes of the network will be mixed. In Broadfoot et al. (2009) the idea of entanglement percolation was applied to mixed states for the first time. In this paper we elaborate and extend the ideas presented in Broadfoot et al. (2009). The networks we consider are composed of nodes, each of which can consist of several qubits, and may be connected by a finite number of bipartite mixed states (see Fig. 1). We aim to create a perfect singlet between two arbitrary nodes in the network using a finite amount of resources, i.e. a finite number of initial states which are converted into a singlet, which distinguishes our and other entanglement percolation schemes from, e.g., the quantum repeater protocol where one aims to generate a state with high but non-unit entanglement fidelity. Particularly we structure the paper as follows.
In Sec. II we prove a necessary and sufficient condition that a perfect singlet can be generated in a network of arbitrary geometry the nodes of which are initially connected by bipartite mixed qubit states. We show that singlet generation between two nodes is possible if and only if they are connected by at least two ‘paths’ consisting of a particular class of states. These states arise naturally in systems undergoing amplitude damping. Thus our result is not only of theoretical but also of practical relevance. Unfortunately, the proof does not deliver an efficient scheme for singlet generation. We therefore specialize in the remaining sections on networks with regular geometry, i.e. lattices in 2D and 3D and devise generalizations of entanglement percolation to the mixed states described in Sec. II.
In Sec. III we briefly summarize the idea of classical entanglement percolation with pure states.
In Sec. IV we extend the concept of classical entanglement percolation to mixed states. To this end we consider regular networks where each node is connected to its neighboring nodes by a finite number of the mixed states introduced in Sec. II. We present two different distillation protocols which are used to convert these states into a singlet with a probability above the percolation threshold of a variety of lattice geometries. After the distillation, clusters of singlet-connected nodes emerge and we aim to create a singlet between two nodes in such a cluster by successive application of entanglement swapping. By communicating classically each node can determine if singlets exist between it and its neighboring nodes. This information can be communicated and stored classically in a central data processor. Typically one would then use this information to apply a path-finding algorithm which locates a suitable ‘path’ of singlets before swapping operations are performed. As an alternative to this we discuss a classical and a quantum protocol which merely require classical communication between neighboring nodes and basic computing within each node. The quantum protocol relies on the formation of many-qubit GHZ states via local operations and classical communication with neighboring nodes and subsequent measurements at all nodes except the ones to be left in the final singlet.
In Sec. V we show that the idea of ‘quantum preprocessing’ as it was successfully applied in pure state networks can be generalized to mixed states. In particular we devise a number of strategies on small networks which improve the SCP, and we show that these smaller networks can be embedded into larger networks to enable CEP which would otherwise not be possible. Furthermore, we discuss ‘hierarchical schemes’, i.e. networks which are defined iteratively and were first discussed in Perseguers et al. (2008a); Lapeyre Jr. et al. (2009). Also in these cases it turns out that quantum methods outperform classical percolation. Finally, in Sec. VI we summarize and conclude.
Ii Singlet Generation within an Arbitrary Mixed State Network
In this section we consider quantum networks of arbitrary geometry as shown in Fig. 1 where the qubits in the nodes are ‘connected’ by bipartite mixed states to qubits in other nodes. We will call a single bipartite mixed state an edge and the set of edges directly connecting two nodes a bond. Note that an edge connects exactly two qubits in different nodes. In the following we will prove that the generation of a perfect singlet between two arbitrary nodes and with finite probability in such a network is possible if and only if there are at least two paths of states linking and which have, up to local unitaries, the form
where and . We show this by separately proving a necessary and sufficient condition which, together, prove the above statement.
Necessary condition. We split the network into two groups of nodes, , containing A and a finite number of other nodes, and , which consists of the rest of the network and particularly contains B. These groups are linked by a finite number of edges. A singlet can be established with finite probability, via local operations in the groups and classical communication between them, if and only if at least two of the states have the form (1). Appendix A contains a concise proof of this fact based on Jané (2002) which agrees with the result of Ref. Kent (1998), that, in general, a singlet can not be generated with a finite probability from a finite number of mixed states.
With two states of the form (1), and , we obtain a singlet with a finite probability by first performing two C-NOT gates locally, with the state’s qubits acting as the target qubits. These target qubits are then measured in the computational basis. If we find both qubits to be in the state we have generated a pure entangled state between the qubits that originally corresponded to the state. We will refer to this measurement as the pure state conversion measurement (PCM). The state formed is
i.e. is a Schmidt-coefficient that has the value
The probability that the PCM succeeds in generating this state is given by
For identical states, i.e. , the PCM already yields a singlet. Otherwise the state can be transformed into a singlet via the ‘Procrustean method’ Bennett et al. (1996b) that converts any pure 2-qubit state into a singlet with a probability . The total success probability of generating a singlet is then given by the SCP
which coincides with the optimal probability for creating a singlet from two of these states Jané (2002).
We can perform this partition of the network in an arbitrary way, as long as one group contains and the other contains . To be able to create a singlet between and via LOCC we must have at least two states of the form (1) in all possible partitions. This gives us a necessary condition that to create a singlet between two nodes with a non-zero probability there have to be at least two distinct ‘paths’ of edges of the form (1) connecting the corresponding nodes. In Fig. 1(a) this is indicated by two spatially distinct paths of bonds. The states of the qubits that are not contained in this path are irrelevant and can therefore be in arbitrary states.
Sufficient condition. In order to show this we make use of entanglement swapping. This operation can be performed in the setup shown in Fig. 2 and consists of performing a measurement in the standard Bell basis on the qubits located at and LOCC which causes and to become entangled. If the edges are of the form (1), and , then there are four possible outcomes. The probabilities to obtain measurement outcomes corresponding to the Bell states and are
If we measure the qubits at to be in the states then we actually form another state,
of the form (1) between and . Unfortunately for the other outcomes the states’ form is not generally maintained. Note that if we can discard these cases by replacing the state with leading to an operation that transforms into
which will be useful in Sec. V.1. We can therefore create a state of the form (1) with non-zero probability between two nodes of the network, e.g. and in Fig. 1, given that these nodes are connected by a path consisting of states of the same form. Two such states, originating from two paths, can then be converted into a singlet, using a PCM and the Procrustean procedure. Unfortunately, this scheme leads to an exponential decrease of entanglement fidelity Dür et al. (1999), and thus success probability, with the number of swapping operations. Hence it is not an effective solution to the problem of long-distance entanglement distribution. In Sec. IV we will therefore introduce effective protocols which can be applied in regular network geometries and succeed in creating a singlet with a probability independent of distance.
Note that when entanglement swapping is done with pure states all of the outcomes can be used, and if these outcomes occur with probabilities the pure state with
is recovered by using classical communication and local unitaries.
Iii Classical Entanglement Percolation with Pure States
In this section we will briefly review the use of percolation for distributing singlets in pure state networks Acín et al. (2007); Perseguers et al. (2008a), known as classical entanglement percolation (CEP). The procedure is based on classical bond percolation, where we consider a regular lattice of nodes connected by identical quantum states, as shown in Fig. 3. A description of classical bond percolation can be found in Ref. Bollobás and Riordan (2006). If the nodes are connected by pure states of the form they can be converted into singlets using the Procrustean method with a SCP . These singlets act as the bond in the bond percolation model 111Note that within the notation of this paper a ‘bond’ corresponds to a set of states connecting two nodes and not to a perfect singlet and are distributed randomly with a probability . The nodes that can be connected by a path of singlets form a cluster. By using entanglement swapping (see Sec. II) we can then generate a singlet between any two nodes in the cluster. In the theoretical case of an infinitely large lattice a cluster that is infinite in extent forms if and only if , where is a lattice-dependent percolation threshold. This approximates the case for large but finite lattices where the threshold becomes more definitive as the size of the lattice increases. Values of for a number of lattice geometries are given in Table 1. If each bond in a network consists of a single pure state we can calculate a threshold for given by . The probability that a node belongs to the infinite cluster is known as the percolation probability . Two randomly chosen nodes are both part of the infinite cluster with a probability and thus can be connected over an arbitrary distance.
|3D Simple Cubic|
|3D Face-Centered Cubic|
It has been shown that CEP using pure states is not optimal and that by performing particular quantum pre-processing steps, particularly swapping operations on the lattice before converting to singlets, improvements can be achieved. These improvements include obtaining a geometry with a lower percolation threshold after the swapping operation and splitting the lattice into two, so that a higher percolation probability can be obtained Acín et al. (2007); Perseguers et al. (2008a); Lapeyre Jr. et al. (2009); Kieling and Eisert (2009); Cuquet and Calsamiglia (2009). Recently, another method, that transforms the initial bipartite network into a probabilistic multipartite network, has also been shown to yield an improvement Perseguers et al. (2009).
Iv Classical Entanglement Percolation with mixed states
In this section we extend CEP to mixed states. We consider regular lattices, e.g. triangular (see Fig. 4), square, or even lattices in higher dimensions. Bonds between network nodes are composed of multiple edges which satisfies the necessary condition proven in Sec. II. We assume that each bond is identical. When these bonds contain at least two states of the form (1) they can be converted into singlets by PCM followed by the Procrustean method. If the probability that a bond becomes a singlet exceeds the percolation threshold CEP is achieved. In the remainder of the paper we will assume that the states forming edges are of the form (1) with . Setting is not a major restriction but allows us to keep the equations manageable. All protocols presented in this paper can also be performed if . We will call states of the form (1) with , i.e.
purifiable mixed states (PMSs). Note that these states form the states of two entangled atomic ensembles in the DLCZ quantum repeater scheme Duan et al. (2001).
iv.1 Distillation Procedures
iv.1.1 Distillable Subspace Scheme
We assume that each pair of neighboring nodes is connected by PMSs and our aim is to distill these into a singlet. The basic setup is shown in Fig. 5. To accomplish this we will use ideas proposed in Ref. Chen et al. (2002). Here the concept of a distillable subspace (DSS) is introduced as a subspace such that the local projection of the system state into this space is pure and entangled. Locating the DSS involves calculating the eigenvectors of the state with non-zero eigenvalues. To simplify notation we will represent the states at and using the decimal value of its binary form, i.e. for example .
As an example, in the case of identical states the eigenvalues and corresponding eigenvectors are
If this is acted on by the projective measurement at and at the state remaining is . Both of these projective measurements only occur with probability
Note that this is the same SCP as obtained for PCM [see Eq. (3)]. For this example there is no choice between entangled states to project out and if the original states are the same a maximally entangled state is automatically obtained. For states that are not identical this does not need to be the case.
An extension of this scheme to identical copies of PMSs yields the SCP
A derivation of this formula is given in Appendix B. As a particular example it is worthwhile to discuss the case of three states in more detail. In this case the measurement at is given by a Positive Operator Valued Measure (POVM) with the elements
The measurement at then depends on this outcome and creates a maximally entangled state with a certain probability. The SCP is obtained by setting in Eq. (16) and is given by
iv.1.2 Recycling scheme
The SCP using the DSS scheme does generally increase with increasing . However, the scheme does not make use of the available resources in the best way. Indeed, the SCP can be significantly improved by grouping identical PMSs into sets of and converting each of these sets into a singlet.
For example for we apply the PCM as described in Sec. II on pairs of states which converts them into singlets with a probability given by Eq. (5). If this fails for a given pair we may still find both measured qubits in the state and have generated another PMS. This PMS can then be used again in another purification attempt. To be more precise, starting with copies of a state (with ) we apply a 2-state purification protocol on groups of two. If no singlet is obtained the procedure is repeated on the remaining PMSs as illustrated in Fig. 7. The coefficients for the PMSs after repetitions, when no singlet is created, are given by
where and . For states of the form the probability of obtaining a PMS is . If the PCM yields two qubits that are measured in different states the purification step between the two PMSs has completely failed. The probability of this is given by . The probability of not generating a singlet using this recycling protocol on states of the form is then found to be
where . Consequently, the probability of successfully generating a singlet by applying the procedure to states of the form is which is calculated iteratively. Examples are shown in Figs. 6 for .
Obviously, the states do not necessarily need to be split into pairs. For example we can separate all of the states into sets of three and apply the three-state DSS distillation. In case of failure this can yield a PMS state as well, which can then be used in later distillation steps. There are a variety of ways to combine the three-state distillation with the two-state recycling scheme. Here we concentrate on the straightforward approach which only uses the three-state distillation on every level of the recycling scheme. The results are shown in Fig. 8. As can be seen in most cases the two-state recycling scheme has a higher chance of success and because of this we will focus on the pairing arrangement in this paper.
iv.2 Percolation Thresholds
Using the purification procedures described above we can apply CEP, as described in Sec. II, for lattice networks with multi-edged bonds. In most cases it is advantageous to use the two-state recycling scheme, except for where the DSS scheme should be used. From Fig. 6 it can be seen that the SCP increases with the number of edges per bond and this allows for a larger range of values for and such that CEP is successful. For double edged bonds the optimal probability of generating a singlet is given by
When the bonds are composed of three edges, i.e. three PMSs between nodes, we have
By comparing these ranges to the percolation thresholds we see that a basic successful setup is a double bonded triangular lattice (see Fig. 4). The double bonds can be converted to singlets and if the chance of this is larger than the percolation threshold an infinite cluster will form. A singlet can then be created between any two nodes within the cluster. Thus percolation occurs if
However, the singlet conversion probability never exceeds for two states. Therefore we would require more states in other geometries. For example, if we have three-edged bonds between each neighboring node we can apply CEP to a square lattice. This is because there are parameters such that . Analogously, CEP is also possible in honeycomb lattices with three edges per bond.
iv.3 Local Processing Strategies
The process of creating singlets, randomly replacing the initial network bonds, can be run if each node can only communicate classically with their neighbors. Each node then knows if a qubit that it contains is part of a singlet after this procedure has finished. This information can be stored classically within a node but after the bonds are distilled we are faced by the problem of finding a set of singlets that connect our requested nodes, and .
If all of the singlet generation data is collected by a ‘controller’ then an efficient path finding algorithm can be applied to determine a suitable ‘path’ of singlets linking the nodes. An example of a suitable algorithm would be a Dijkstra scheme Dijkstra (1959) such as the A* path finding algorithm Hart et al. (1968). The path information can then be used to instruct the correct nodes to perform swapping. The swapping operations are performed in order from node to , so that the measurement outcomes only need to be communicated along the chain, between neighboring nodes. However this procedure requires one classical computer to have complete knowledge of the network. Instead, it is interesting to note that this does not need to be the case as there are algorithms which do not require any more classical communication than this ‘controller’ method, indeed they do not require a central ‘controller’ at all. This can be done not only classically but also via a quantum algorithm using multipartite entanglement which we will introduce below.
A classical path-finding method would use a type of breadth-first search algorithm called a burning algorithm Herrmann et al. (1984). Node sends a ‘burning’ signal to its neighboring nodes connected by singlets. These nodes keep a record of where they received the signal from and send out an identical signal to the other nodes that they are connected to. We say that the node has ‘burned’. If it has already received a signal from a different node then the additional signal is ignored. This continues outwards from , ‘burning’ the nodes. Once node receives the signal it replies to the node it came from with a ‘swapping’ message. This node can then perform a swapping operation and send another ‘swapping’ signal, together with the Bell-measurement outcome, back to the node it received a ‘burning’ signal from. The path can then be traced back along the nodes with swapping performed at each step until node is reached. Both and can determine if the protocol has been successful. However, and may not be in the same cluster and they do not know if the protocol has failed when the network is of infinite size. This is not a problem for finite networks, containing nodes, as and can time the steps taken and if these exceed they both know they are not in the same cluster.
Note that no extra information actually needs to be transmitted. We can combine the burning algorithm with the process of transmitting the distillation protocol information. For example, in a double edged network of identical edges, can perform her PCM and if is the outcome she assumes she has a singlet and sends a burning signal to the node that would contain the singlet’s other qubit. If a node receives this signal it can perform its PCM and determine if there is a singlet there. When there is and if it is the first instance for the node it should record that entry qubit and repeat the process, performing a PCM on the remaining qubits and sending signals to those with the outcome. Once receives a signal it can check that a singlet has been created with a PCM and then send a swapping signal back as before. During the swapping, a node can use the Bell-measurement information received to indicate that a swapping is required so no explicit ‘swapping’ signal is required either. All of this information transfer would have been necessary as well if a controller algorithm would have been used. Hence the generation of the singlet can be accomplished by defining rules for each node and allowing them to run with nearest neighbor classical communication. This is fundamentally different to the controller process and has made use of parallel computation to find a path that no single node has full knowledge of.
We will now consider an alternative, quantum algorithm that is based on the burning algorithm and makes use of multipartite entanglement in the network. The protocol starts after we attempted to convert all bonds into singlets and every node has knowledge about its singlet connections to nearest neighbors. We build up a progressively larger multi-qubit GHZ state, defined by , spread between the ‘burned’ nodes by adding qubits in each burning step. Building up such a state requires joining two GHZ states, and , to create (note that a singlet equals ). This is done by performing a CNOT gate between a qubit in and a target qubit in , measuring the target qubit in the -basis, communicating the measurement result to the other qubits in and performing a unitary operation on them depending on the outcome.
Now we perform the same process as for the ‘burning algorithm’, however, as each node is ‘burned’ it is connected to the GHZ state spread over the previously burned nodes. The process to do this is illustrated in Fig. 9 and consists of joining the singlets partially contained in that node to the GHZ state. Within each node one qubit is left entangled with the GHZ state. After this operation has been run for a maximum of times all of the nodes in the cluster containing have a qubit from a single GHZ state.
At each node a record is kept of the bond via which it has been included into the GHZ state. If there is a singlet between two nodes that are being burned then the singlet is ignored. Furthermore we add the rule that whenever a node can not extend the GHZ state anymore basis measurements are performed along the recorded path back to . This removes a qubit from the GHZ state but introduces a phase error in the remaining GHZ state depending on the outcomes of the measurement. The information about these measurement outcomes has to be sent back along the path to . Whenever the route back branches, the measurement outcome is sent in one way and a message corresponding to ‘no phase error occurrence’ is sent to the others. At each node the returning process is paused until all of the bonds it sent a burning signal to provide it with the phase information. At nodes and we do not perform the measurement. Finally after receives all of the phase information a phase correction can be performed and we obtain a singlet between and . In Fig. 10 an example is given to illustrate the protocol.
V Quantum Preprocessing
Despite being a very effective method, it is known that CEP in a network of pure states can be improved by certain quantum ‘pre-processing’ strategies, and therefore CEP is not optimal Acín et al. (2007); Perseguers et al. (2008a); Lapeyre Jr. et al. (2009); Kieling and Eisert (2009); Cuquet and Calsamiglia (2009); Perseguers et al. (2009). In the following we show that this is also the case in mixed-state networks.
v.1 Swapping procedure
To start with we generalize the swapping arrangement shown in Fig. 2 previously studied for pure states Perseguers et al. (2008a); Bose et al. (1999). In this arrangement we have two 2-qubit states that both have a qubit in a common node. If the two states are pure states and , with and , we can obtain a singlet by swapping and then converting the resulting pure state into a singlet with a total probability of which turns out to be the optimal probability. Particularly CEP, which consists here of the Procrustean method followed by entanglement swapping, always has a smaller SCP of . Note that the optimal probability is equal to that of converting the least entangled of the two bonds into a singlet using the procrustean scheme Bose et al. (1999).
To generalize this to mixed states we must consider double-edged bonds, each consisting of two PMSs, as illustrated in Fig. 11, since otherwise singlet generation would not be possible. Introducing more than one edge between the nodes allows us to concentrate the entanglement at different stages which gives rise to three different possibilities:
CEP - As previously described, the bonds are converted to singlets and then swapping is performed over the resulting states.
Direct swapping - This applies entanglement swapping twice and then the resulting states are converted into a singlet.
Hybrid swapping - Here we distill a state of higher entanglement in each bond (but not necessarily a singlet) leading, if successful, to a single (partially) entangled pure state in each bond. This is followed by entanglement swapping and the Procrustean scheme to create a singlet.
Each of these possibilities uses the swapping operation at different stages as illustrated in Fig. 11. The exact implementations for the procedures depend on the types of states used. We will first apply each of them on a network of pure states and compare the SCPs. We then generalize to PMSs and show that direct and hybrid swapping can outperform CEP.
v.1.1 Pure states
If we start with bonds made of pure states and we must have a way to convert each bond into a singlet in order to apply CEP(I). The method and highest possible probability to accomplish this are given by Majorization Nielsen and Vidal (2001) with a probability Perseguers et al. (2008a); Lapeyre Jr. et al. (2009). CEP applies this operation on each bond and if both bonds are converted into singlets swapping can be performed and the operation is a success. Therefore CEP succeeds with a probability .
Our second method, direct swapping(II), is simply the application of the procedure for bonds containing one edge twice. If either generates a singlet the procedure succeeds. This gives a SCP of . There are adjustments we could make, for example use the results of Majorization to convert both of the states into a singlet with the highest possible probability, however all of these have a smaller SCP than CEP for a range of parameters.
Finally, the hybrid swapping(III) method concentrates each bond to one pure state, , with certainty. This concentration procedure is also found using results from Majorization theory Nielsen and Vidal (2001). Afterwards there is one pure state in each bond, as discussed previously, and we can then perform the strategy with optimal success probability , i.e. swapping over the pure states followed by the Procrustean method. We can actually consider the setup as a bipartite system between and . The Majorization results then give the best possible probability of generating a maximally entangled 2-qubit state between these systems as which means that it must be the highest possible probability for any method to succeed.
Figure 12 shows the probabilities in all three cases and we can see that CEP is outperformed for a vast range of parameters by both other strategies. In hybrid swapping (III), we have used multi-edged bonds to create pure states with the highest probability before applying entanglement swapping. We will refer to all strategies that have this property as ‘hybrid’. This probability is unity for initial pure states but for mixed states the initial conversion of bonds to pure states is probabilistic, so when the conversion fails the bond is destroyed.
v.1.2 Purifiable Mixed States
We will now investigate if similar improvements can be obtained with PMSs, i.e. if the bonds between the nodes are composed of and . Again we will see that hybrid swapping provides the highest SCP.
The classical percolation scheme involves performing a PCM described in Sec. II followed by the procrustean protocol on both bonds and each succeeds with a probability given by Eq. (5) which simplifies to
To perform a swapping operation yielding a singlet, between nodes and we must succeed for both bonds which gives the total chance of success
by simply squaring Eq. (25). In this case the swapping operation is the final step of the protocol.
In our 2-edged setup we perform the swapping operation introduced in Sec. II twice and there are two choices to do this if the states are not identical. Either we perform the swapping over the identical states or we perform the operation on the states . When we swap over identical states we obtain the state
together with a further state where is replacing and is replacing . Note that Eq. (27) is obtained by setting and in Eq. (11). This pair of states can then be transformed into a singlet with a probability
This is always larger than and thus swapping with non-identical states should be preferred.
The hybrid method requires a concentration procedure to be performed (yielding a single pure state in each bond) which is given here by PCM. However, in contrast to the pure state case discussed above, if we obtain singlets (in which case the method is identical to CEP) and, generally, the operation succeeds with a finite probability given by Eq. (4). For non-identical PMSs PCM yields two non-maximally entangled pure states which are then used for entanglement swapping followed by the Procrustean method. The probability of succeeding in converting both of the bonds to pure states is
These pure states have largest Schmidt coefficient
So, by using the SCP in single edged swapping with pure states we find that we can convert this pair of states into a singlet between the end nodes with probability
Hence, the overall probability of succeeding with this scheme is
If we compare the success probability of direct swapping, , to the probability of success in the classical percolation scheme, , it can be seen that classical percolation is more likely to succeed in producing a singlet if
But the ratio of the success probability for the classical scheme against the hybrid protocol, , is
Whenever this is less than one and there is an improvement over the classical percolation scheme. Furthermore, the hybrid scheme is more likely to succeed than direct swapping. In Fig. 13 we compare the probabilities of success for all schemes. As can be seen, hybrid swapping leads to the highest success probability.
Hybrid swapping can be used in sections of larger networks to allow percolation to take place. A simple example is a face-centered cubic (FCC) network, where every bond is split into two 2-edged bonds (see Fig. 14). When the above schemes are applied at the nodes linking two 2-edged bonds the FCC network is recovered. Percolation is possible in these 3D networks with a threshold of approximately . Since the classical scheme always gives a smaller success probability than the hybrid scheme there are cases where the hybrid scheme allows the percolation threshold to be exceeded but the classical scheme does not (see Fig. 13).
v.2 Square Protocol
CEP can also be improved on by using the hybrid strategy in a 2D square network, as shown in Fig. 15. Each bond is converted into a pure state, , by using PCM which is successful with a probability on each bond. If this yields only two states having a common node ( or ), entanglement swapping can be performed followed by the procrustean scheme. If all four PCMs succeed the resulting states can be connected (e.g. at nodes and ) via a slightly modified version of entanglement swapping, the so-called XZ-swapping Perseguers et al. (2008a). For this swapping operation the Bell measurement that usually has both qubits measured in the basis now measures one in the basis. After this measurement unitaries are again applied to return the state into Schmidt form. The results of the Bell measurement have an equal probability, , for all outcomes . Performing this operation twice on the square leads to two pure states (between and ) of the form , with . These can be distilled into a singlet with probability by using the protocol based on Majorization Nielsen and Vidal (2001). The overall chance of succeeding in generating a singlet is then given by
Again, this improved strategy may enable an infinite cluster to form when applied to larger networks. An example is shown in Fig. 17. Here the square protocol recovers a triangular lattice. If the conversion of the squares into singlets succeeds with a probability exceeding the percolation threshold an infinite cluster forms. In Fig. 16 it can be seen that the hybrid scheme exceeds the threshold for a triangular lattice in cases where CEP does not.
v.3 Hierarchical Networks
Small networks like the square configuration discussed above can be extended to larger networks in an iterative fashion. Networks formed in this way from pure states were considered in Perseguers et al. (2008a). Again the probability of successfully creating a singlet was shown to be larger when quantum strategies were used instead of CEP. However, the scheme with the highest probability is still unknown for these ‘hierarchical’ networks. Here we will consider two different hierarchical networks with two edges per bond. Each of these contains the square network at some iteration level. We determine the SCP when using CEP in both 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 on the ‘Diamond’ lattice, which at each stage replaces its bonds by the square network. The geometry for the first three iterations is shown in Fig. 18.
The aim is to create a singlet between and and if we apply CEP the probability of succeeding at each level is given by the iterative formula
starting with .
The second hierarchical network we consider is the ‘Tree’ network which is again built on the square configuration. For these networks an iteration is formed by creating two copies of the previous iteration and linking the bottom-left and top-right corner of the square to separate nodes and as shown in Fig. 19. Again, we wish to generate a singlet between the opposite corner nodes ( and ) and CEP generates a singlet with a probability
Now we wish to see whether the hybrid scheme gives a larger SCP in these networks. Once again, the hybrid scheme we consider starts by converting all of the bonds into identical non-maximally entangled pure states probabilistically. If the conversion fails on a bond then the bond is destroyed. This results in a network containing random pure state bonds. Each of these bonds contains one edge. Ideally we would then apply a pure state protocol yielding the highest SCP between the intended nodes, however, this protocol is not known in the general case Perseguers et al. (2008a). Instead we apply a procedure which performs -swapping in cases when two bonds each have a qubit in the same node (except if these nodes are or ). However, we also distill pure states into states with more entanglement whenever two edges form between two nodes and before performing further swapping. Finally, once one state is obtained between and , the procrustean procedure is used to create a singlet.
We applied this protocol to the hierarchical diamond and tree networks. For the second and third iterations of the diamond lattice the probabilities of creating a singlet are given in Fig. 20 together with the probabilities using CEP. This comparison was also made for the first, second and third iteration of the tree network and the results are shown in Fig. 21. These examples all illustrate an improvement in the probability of forming a singlet when using the hybrid method rather than classical percolation.
We have demonstrated that within lattice networks, where the nodes are connected by multiple bipartite mixed states, percolation strategies can be applied for distributing entanglement. This is reliant on the states being PMSs, which are known from the DLCZ repeater scheme and arise as a result of amplitude damping. To show this we have introduced some new purification protocols designed to maintain the form of these states or generate singlets. Like in the pure state case, a higher probability of distributing a singlet can be obtained, when the states in a bond are not identical. The question of whether quantum strategies can outperform CEP when each edge in a bond is identical is still open. Since we have shown that classical entanglement percolation is only possible for a specific class of bipartite states, entanglement distribution in a network which is subject to more general forms of noise needs to make use of other methods. The development of these methods is one of the most important goals for future work. These will not produce perfectly entangled states, however, the resulting state fidelity may be independent of distance and sufficient for purification. An example of such a strategy is given in Ref. Perseguers et al. (2008b) for a bit-flip noise model. Progress in this direction has also been accomplished by generating 3D thermal cluster states using Werner states Raussendorf et al. (2005); Perseguers (2009).
Acknowledgements.This research was supported by the EPSRC (UK) through the QIP IRC (GR/S82176/01) and the ESF project EuroQUAM (EPSRC grant EP/E041612/1).
Appendix A Proof of singlet distillation requirement
In this appendix we give a concise proof for a necessary and sufficient condition to be able to create a singlet out of entangled mixed states using LOCC. We allow the states to be arbitrary bipartite states which are shared between two nodes and all operations are LOCC. A similar proof, but partly restricted to identical states, was given in Jané (2002).
If a quantum state can be distilled to a pure states, , then any state with the same range is also distillable to this state with non-zero probability.
The general form of the state is
with , and . If the state is distillable to a pure state, , there exist linear operators and , with , such that
This can be summarized as
where at least one is non-zero as otherwise the operator fails to distill . If this condition is satisfied the operation distills the mixed state into . Now given another state with the same range as . We have that
with and . This then gives
The value of one must be non-zero as otherwise all are zero and this contradicts the fact that the operator distills . Hence the protocol also distills . ∎
For 2-qubit states to be distillable into a pure singlet at least two 2-qubit states cannot have a range spanned by product states.
If a 2-qubit state has a range that can be spanned by product states then a separable state with this range exists. If there are states each with a range spanned by product states the system state would have a range equivalent to a separable state formed by all of these 2-qubit separable states. Since it is impossible to distill a pure entangled state from any separable state it is impossible to distill a pure entangled state from two qubit states each with a range spanned by product states. Similarly, if one of the 2-qubit states does not have a range spanned by product states, but all of the other states do, the range is equivalent to the range formed from a separable state and one mixed 2-qubit state. This can not be distilled into a pure singlet as it would contradict the result in Kent (1998). Hence at least two states can not have a range spanned by product states to be able to distill a pure entangled state. ∎
We now need to look at the two qubit states that satisfy this property. The states with rank one are already pure and if they have rank four the range can be spanned by product states. Similarly, if the state has rank three it can also be spanned by product states. This can be seen by considering the subspace orthogonal to a general state . This space is spanned by and these are all product states. The last states to consider are those of rank 2, which fall into two categories Sanpera et al. (1998). The range is either spanned by product states or . Hence only states that have a range containing one product state are the mixed states satisfying the condition. All mixed rank two states of two qubits can be considered to be the mixed state formed by tracing out a third qubit from a pure three qubit system. The classifications of these 3 qubit systems is given in Acín et al. (2000); Dur et al. (2000); Acín et al. (2001) and for the range of the mixed system to contain one product state the three qubit state belongs to the W class. This class can always be written as with
By tracing out one qubit and using local operations the 2-qubit state that can not be spanned by product states has the form
So the only states that can be purified into a perfect singlet, given finite copies, are of this form.
If there are two states of this form we know that the system is distillable since the procedure given in Sec. II succeeds in the distillation.
Appendix B The Distillable Subspace Scheme
To extend the DSS scheme to PMSs, , we first need to describe the non-zero eigenvalues and their eigenvectors. These correspond to different combinations of terms and terms. Then taking the decimal representation of the local states we can label each of these eigenvectors by the decimal difference between the values at each location. This difference in binary gives the location of the terms. For example, in the case of two identical PMSs these are
and takes all of the values from 0 to . Now we define to be the number of 1s in the binary representation of and : (‘’ is the bitwise AND operation).
Then and all of the terms in a non-zero eigenvector are of the form
with eigenvalue .
From this structure we can project out an entangled state if we measure the operator at and then at , when , , and as long as there are no other terms of the form , or in any non-zero eigenstate.
The term can not appear in one eigenstate since all of the terms must have the same value and this would require to be equal to . The state lies in one if and only if , such that and . Similarly for but this case can not occur since and means that and . If we assume that , such that and this would mean that and that for some . Both of these results then give that and . So, if such that can not be satisfied we create a maximally entangled state.
Now we have a choice of ways of creating these measurements. One particular way involves the definition of sets : , and : ( OR ) = 0, . Then the protocol consists of performing a POVM at location with , and . For and we define and when these outcomes occur the procedure has failed. Here is a factor to ensure that
With this outcome at location another POVM is done at location given by the operators () and . If the outcome here is the protocol has failed, otherwise we have obtained a maximally entangled state. This protocol works since for all but there is no such that and , since if there were we would have but .
The probability of succeeding is given by Eq. (16) which comes from considering a particular eigenstate with parameter . In this eigenstate there are
different pairing terms, with . The number of possible measured operators from this eigenstate is given by
Note that the pairings in the eigenstate are twice as likely to occur than the ones with just an overlap and these have been counted twice in this sum. The probability that starting with an eigenstate (parameter ) we succeed is then
given that we have measured the operator and the probability of this was
By summing over these we have, given we start with an eigenstate with , the probability of succeeding to be
and these eigenstates occur with probability
We have not counted since they never contribute to the success probability. Then by summing over all of these we get the result in Eq. (16).
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