Distribution of entanglement in large-scale quantum networks

While the study of quantum measurement is a broad and deep subject, here we only need to introduce a few ideas to discuss entanglement distribution. Measurements occur in the laboratory, but the computational tool to predict their result is associated with each type of measurement from a set of linear operators on the state space. The measurements described above are called projective measurements, which are defined by a collection of projectors E_j onto subspaces of the state space, each of which is associated with a possible measurement value. After measuring a value, we know that the system has collapsed into a state in the subspace corresponding to the associated projector. The projectors are orthogonal and the requirement that we must get some result implies that their sum is the identity. The maximum number of elements in a set of orthogonal projectors is equal to the dimension of the state space and corresponds to a complete measurement. On the contrary, we use an incomplete measurement, with fewer projectors, when we want to extract only partial information from a state while preserving some property common to all subspaces. Finally, we will sometimes make use of generalized measurements, in which the condition that the operators are orthogonal is relaxed. Measurements are then defined by a collection of semi-definite positive operators that sum to the identity [14].

The other component of quantum theory that transforms states is evolution. Evolution is also represented theoretically by operators, but in contrast with measurement, an initial state is transformed into a final state in a deterministic way. Operators representing evolution must have the algebraic unitarity property in order to conserve total probability. These operators are commonly referred to as unitaries. The entanglement distribution procedures in this review are built from a combination of measurements and unitaries on quantum states, together with classical communication.

To illustrate the basic idea of entanglement while making a connection with the remainder of this review, let us consider two qubits, i.e. two systems each of which consists of a two-level quantum state. Two qubits live in a four-dimensional state space whose computational basis is |00〉, |01〉, |10〉, |11〉, where the first binary digit corresponds to qubit A and the second to qubit B. Mathematically, these basis vectors are tensor products of vectors in the local spaces: |ij〉 = |i〉⊗ |j〉. If the two systems have never interacted directly or indirectly, then each one has an independent description of its state, as in (2.1). In this case, it is possible to find local bases for each qubit such that the joint state is one of the four computational basis states of the joint space. These states are called product states. Measurements and operations applied on one system have no effect on subsequent measurement outcomes on the other system. However, suppose an interaction between the systems is turned on, then off. In general, there is no local basis such that the state of the system after the interaction can be written as a product of states of the subsystems. A pure bipartite quantum state is then said to be entangled if it cannot be written as a tensor product. That is, it cannot be written as

$$\begin{equation*} {\,|{\phi_{AB}} \rangle} = {\,|{\phi_A} \rangle} \otimes {\,|{\phi_B} \rangle}. \end{equation*}$$

A famous example of entangled states are the four Bell states

$$\begin{equation*} {\,|\Phi^{\pm} \rangle} \equiv \frac{1}{\sqrt{2}}\left({\,|00 \rangle}\pm{\,|11 \rangle}\right), \end{equation*}$$

$$\begin{equation*} {\,|{\Psi^{\pm}} \rangle} \equiv \frac{1}{\sqrt{2}} \left({\,|01 \rangle}\pm{\,|10 \rangle}\right), \end{equation*}$$

which form another basis of two-qubit systems. These states are of central importance as they are maximally entangled states. The projective measurement corresponding to the four states of the Bell basis is called a Bell measurement.

Just as two two-dimensional systems form a four-dimensional system, in general, the space describing a collection of quantum systems of dimensions n₁, n₂, ..., n_N has dimension n₁ × n₂ × ... × n_N. However, generalizing the case of two qubits, we may partition any of these composite spaces into two subspaces and, viewed this way, the system is called a bipartite system. The study of bipartite entanglement, that is entanglement between the two subspaces, is far better understood than the more general case of multi-partite entanglement; we shall mostly be concerned in this review with bipartite systems.

Until now, we have discussed only what are known as pure quantum states, in which the probability of measuring a value is of local quantum mechanical origin. However, the generic case is that the system is entangled with other systems which we cannot measure. It turns out that to predict local measurements in this case, we can assume that the local state is effectively a classical ensemble of pure states is known as a mixed state. These states are described not by vectors in the Hilbert space of pure states, but rather by linear operators known as density operators, which act on the same Hilbert space. For instance, the density operator corresponding to a pure state is given by the outer product of the pure state vector with itself, which is denoted by |φ〉〈φ| . In general, a density operator has the form

$$\begin{equation*} \rho = \sum_i p_i {\,|{\psi_i} \rangle}{\langle {\psi_i}|}, \end{equation*}$$

where {p_i} is a probability distribution. It is important to realize, however, that this decomposition is not unique. Rules of quantum mechanics together with classical statistics imply that density operators must be non-negative and have trace one. The set of operators on finite-dimensional Hilbert spaces may be represented by a set of matrices that depends on a chosen basis. Since all systems considered in this review are composed of locally finite-dimensional Hilbert spaces, we follow the common practice of using the term density matrix even if the choice of basis is not fixed.

The notion of entanglement can be easily generalized to mixed states [33]. In this case, a bipartite system is entangled if and only if there are no local bases in which its joint density operator ρ_AB can be written

$$\begin{equation} \rho_{AB} = \sum_i p_i\ {\,|{\psi_i^A} \rangle}{\langle {\psi_i^A}|\,} \otimes {\,|{\phi_i^B} \rangle}{\langle {\phi_i^B}|\,}, \end{equation} \tag{ 2.2 }$$

that is, it is not a mixture of product states.

Measures of entanglement. If one qubit of a Bell pair is measured in the computational Z basis, that is, if it is projected onto the eigenstates |0〉 and |1〉 of the Pauli Z matrix, then the measurement outcome is either 0 or 1 with equal probabilities. If the second qubit is then measured in the same basis, the outcome is completely determined by the first result. But, if we instead were to measure each qubit in the same arbitrary rotated basis, the same correlation between outcomes would be seen. In fact, it can be shown that the Bell states are the maximally correlated two-qubit states; they are maximally entangled. In operational terms, this means that they can be used to teleport perfectly the state of exactly one qubit. In practice, however, the entanglement between two qubits is never perfect, so that partially entangled states have to be considered. In what follows, we present some common measures of entanglement.

In the pure-state formalism, local bases may be found so that any two-qubit state is written

$$\begin{equation} {\,|{\varphi} \rangle} = \sqrt{\varphi_0}{\,|00 \rangle}+\sqrt{\varphi_1}{\,|11 \rangle}, \end{equation} \tag{ 2.3 }$$

where the two Schmidt coefficients φ₀ and φ₁ satisfy φ₀ + φ₁ = 1 and φ₀ ⩾ φ₁ (by convention). If one of the coefficients φ₀ or φ₁ vanishes, the system is separable. If φ₀ = φ₁ = 1/2, it is maximally entangled. Otherwise, the system is said to be partially or weakly entangled. An important measure of entanglement is

$$\begin{equation*} E(\varphi) = 2 \varphi_1 \in [0,1], \end{equation*}$$

which corresponds to the optimum probability of successfully converting |φ〉 into a perfect Bell pair by local operations [34]; see section 2.2.3.

Another common measure of entanglement, the concurrence [36], reads in the case of pure states:

$$\begin{equation*} C(\varphi) = 2 \sqrt{\varphi_0\varphi_1} \in [0,1]. \end{equation*}$$

Concurrence is also defined for mixed states. The mixed state of two qubits is represented by a four-by-four density matrix, which requires, in general, fifteen parameters. Any density matrix of two qubits σ can be transformed by local random operations to the standard form

$$\begin{equation} \rho_{{\rm W}}(x) = x {\,|{\Phi^+} \rangle\!\langle {\Phi^+}|} + \frac{1-x}{4}\ {\rm id}_4, \end{equation} \tag{ 2.4 }$$

where id₄ is the corresponding identity matrix. This process is known as depolarization; ρ_W is called a Werner state. It is important to notice that the depolarization process typically increases the entropy of the system and reduces its entanglement. One sometimes explicitly expands this expression in the Bell basis $\mathcal{B}$:

$$\begin{equation*} \rho_{{\rm W}}(F) = \left(F,\frac{1-F}{3},\frac{1-F}{3},\frac{1-F}{3}\right)_{\mathcal{B}}, \end{equation*}$$

where the components correspond to the weight of each Bell state in the mixed state decomposition of ρ_W(F). Although depolarization may decrease entanglement, it is done in a way that preserves the fidelity with respect to a preferred state [37]:

$$\begin{equation*} F(\rho_{{\rm W}}) \equiv {\langle {\Phi^+}|\,}\rho_{{\rm W}} {\,|{\Phi^+} \rangle}=(3x+1)/4. \end{equation*}$$

This fact is important because the concurrence of the Werner state ρ_W is related to the fidelity via

$$\begin{equation} C(\rho_{{\rm W}}) = \max\{0,2\,F(\rho_{{\rm W}})-1\}. \end{equation} \tag{ 2.5 }$$

The state ρ_W is entangled if and only if its concurrence is strictly positive, that is if x > 1/3; or equivalently, if

$$\begin{equation} F>F_{\min} = 1/2. \end{equation} \tag{ 2.6 }$$

Note that (2.4) emphasizes the fact that a Werner state is a mixture of a perfect quantum connection and a completely unknown state. This observation is important, for instance, when discussing the limitations of entanglement propagation in noisy networks; see section 3.4. The Werner state is an example of a full-rank state, that is, one with no vanishing eigenvalues. Full-rank states result from a general noise model and are thus important because they model any state found in a laboratory. However, the entanglement contained in these states cannot be extracted easily and special techniques have to be developed to this aim; see section 2.2.3.

The discussion to this point may leave the reader with the impression that the characterization of entanglement is a relatively uncomplicated task. In fact it is a difficult and deep question, especially for mixed and multipartite states [38]. On the one hand, a separable state can be written as a classical mixture of projectors onto product states. On the other hand, there is no general algorithm to determine whether a given density operator can be written in this form, although significant progress has been made [15, 39].

The quantum networks that we consider consist of two basic elements: nodes (or stations), each of which possesses one or more qubits and links, each of which represents entanglement between qubits on different nodes. This generalizes the one-dimensional scenario in that links may exist between all pairs of nodes, rather than only neighbouring nodes on a chain. An example of such a quantum network is shown in figure 1. It is often useful to interpret this structure in terms of graph theory, where the nodes become vertices and the links become edges. Furthermore, we sometimes use the language of statistical physics by referring to a regular graph with local connections as a lattice.

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In manipulating entanglement in a quantum system, we typically begin with a given distribution of entangled pairs of qubits and then apply a series of operations designed to distribute the entanglement in a useful way. The problems we consider naturally impose a distinction between local and distributed resources. It is important to have a clear notion of what kind of operations are allowed on these resources. In this review, we shall only consider local operations and classical communication (LOCC) on the network. It turns out that it is possible to provide an operational definition of entanglement as the quantum resource that does not increase under LOCC. It is easy to see that this definition is equivalent to the pure mathematical one given by (2.2).

It is now clear how the concept of LOCC leads to the quantum network shown in figure 1. Recall that the network is initially composed of some entangled pairs of qubits, with one party of each pair occupying a node of the network. However, several qubits from different pairs occupy a single node along with other possible resources, both quantum and classical. When we speak of local operations and resources, we mean quantum operations and resources within each node. On the other hand, we allow only classical messages to be sent between the nodes. Qubits belonging to different nodes cannot interact quantum mechanically, so that no further entanglement can be created between remote stations. However, qubits within a node may interact in any way, including with ancillary (local) resources; any measurement may be performed on the components of the node.

We now address the task of concentrating or purifying the entanglement of two (or more) weakly entangled states into a pair with higher entanglement, using LOCC only. Suppose that these states are arranged in a parallel fashion so that local operations may be performed jointly on all left-hand members and on all right-hand members as shown in figure 2. An important task is to find criteria determining when and how a given set of states can be transformed into a more highly entangled target state.

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Pure states. Majorization theory was developed to answer the question: what does it mean to say one probability distribution is more disordered than another? One application to quantum mechanics is via the connection between disorder (of the Schmidt coefficients) and entanglement. Consider an initial pure state |α〉 and a target state |β〉 in a bipartite system. Here, we do not require that the states have dimension 2 as is the case for qubits, but rather allow them to have an dimension d ⩾ 2. Denoting by α the unit vector of the d Schmidt coefficients of |α〉 sorted in decreasing order (and similarly for |β〉), Nielsen showed in [42] that a deterministic LOCC transformation from |α〉 to |β〉 is possible if and only if the inequalities

$$\begin{equation*} \sum_{i=0}^n \alpha_i \leq \sum_{i=0}^n \beta_i \end{equation*}$$

hold for all n ∈ {0, ..., d − 1}. In this case, α is said to be majorized by β, which is denoted by α ≺ β; see [43] for more details on this topic. Note, for instance, that the maximally entangled state can be deterministically transformed into any other pure state. As another example, consider setting ρ₀ ⩾ ρ'₀ for the two pure states |ρ〉 and |ρ'〉 depicted in figure 2, so that one finds α = (ρ₀ρ'₀, ρ₀ρ'₁, ρ₁ρ'₀, ρ₁ρ'₁). In this case, the two pairs of qubits can be transformed into a single connection |σ〉 if and only if ρ₀ρ'₀ ⩽ σ₀.

Moving now to non-deterministic transformations, the optimal probability for a successful LOCC conversion is [34]:

$$\begin{equation*} {\rm prob}({\,|{\alpha} \rangle}\mapsto{\,|{\beta} \rangle}) = \min_{n} \left\{\frac{\sum_{i=n}^{d-1} \alpha_i}{\sum_{i=n}^{d-1} \beta_i}\right\}. \end{equation*}$$

Using this formula, it is trivial to check that a two-qubit pure state |φ〉 can be transformed into a Bell pair with optimal probability 2φ₁, as stated in section 2.1.2. Explicitly, this result is obtained by performing on one of the qubits a generalized measurement defined by the operators

$$\begin{eqnarray} &&M_1 = \sqrt{\frac{\alpha_1}{\alpha_0}} {\,|{0} \rangle\!\langle {0}|\,} + {\,|{1} \rangle\!\langle {1}|},\\ &&M_2 = \sqrt{1-\frac{\alpha_1}{\alpha_0}} {\,|{0} \rangle\!\langle {0}|}, \end{eqnarray} \tag{ 2.7 }$$

which is known as the 'Procrustean method' of entanglement concentration [44].

Mixed states. The purification of mixed states is a somewhat more difficult problem than that of pure states. Many techniques have been developed for doing mixed-state entanglement purification in connection to quantum error-correction [35]. For our purpose, it is sufficient to notice that:

• (i)  
  in contrast with the case of pure states, at least two copies of a Werner state are needed to get, by LOCC and with finite probability, a state of higher fidelity [46].
• (ii)  
  Perfect Bell pairs can be obtained from N Werner states only in the limit N → ∞ [47].

The first purification scheme, which was proposed by Bennett et al in [37], is depicted in figure 2: the two states ρ = ρ_W(x) and ρ' = ρ_W(x') are purified into the state σ = ρ_W(x''), with

$$\begin{equation} x'' = \frac{x+x'+4 x x'}{3 + 3 x x'}. \end{equation} \tag{ 2.8 }$$

The resulting state is closer to the target state |Φ⁺〉 if both ρ and ρ' are entangled (that is, if x, x' > 1/3) and if x > x' > 2x/(1 + 4x − 3x²). It is important to note that this operation is not deterministic since it succeeds only with probability

$$\begin{equation} {\rm prob}(\rho\otimes\rho'\mapsto\sigma) = \frac{1+x x'}{2}. \end{equation} \tag{ 2.9 }$$

This procedure can be iterated, with x increasing after each step, until it is arbitrarily close to 1 (considering perfect operations); this asymptotic technique is sometimes referred as distillation. However, one is often interested in the yield, which is defined as the asymptotic ratio of the number of input states to the number of output states. The above protocol requires a diverging number of states to produce one arbitrarily pure state and thus has a vanishing yield. In order to get a positive yield, the previous method is applied until states of sufficiently large x are generated; then one switches to purification techniques using one-way communication. Many improvements and variants over this construction exist; see [15, 45] and references therein.

The basic operation to propagate the entanglement in a quantum network is the so-called entanglement swapping [16, 48–50] depicted in figure 3: by performing a Bell measurement on qubits b and b' at station B, one creates a quantum link between the previously unconnected stations A and C. Then, a local unitary that depends on the outcome m of the measurement is applied on qubit c, so that the resulting entangled pair has the standard form given in (2.3) or in (2.4). This operation is equivalent to teleporting b to c.

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For two mixed states ρ = ρ_W(x) and ρ' = ρ_W(x'), it is easy to see that the entanglement swapping produces the quantum state σ_m = ρ_W(x x') for all measurement outcomes, that is,

$$\begin{equation} C(\sigma_m) = \max\left\{0,\frac{3xx'-1}{2}\right\}\tqs\forall \ m. \end{equation} \tag{ 2.10 }$$

In the case of pure states |ρ〉 and |ρ'〉, however, the result depends on the outcome and one gets either a Bell pair or a state that is weakly entangled. Remarkably, the average entanglement of the resulting states |σ_m〉 is not less than that of the initial states:

$$\begin{equation*} \bar{E}(\sigma_m) = \frac{1}{4} \sum_{m=1}^{4} E(\sigma_m) = 2 \min\{\rho_1,\rho_1'\}, \end{equation*}$$

which is, for |ρ〉 = |ρ'〉, the 'conserved entanglement' described in [51]. This property of pure states will be used in various protocols of the entanglement propagation described in section 3.

Maximizing the average entanglement of the outcomes is of prime importance for random or statistical processes. However, one may also desire that every possible outcome results in a state with a reasonable amount of entanglement. In this scenario, one can use the rotated Bell basis $\mathcal{B}_X \equiv (X \otimes {\rm id}_2)\,\mathcal{B}$ to perform the entanglement swapping; see section 1.1 in [52]. In this case, one gets four outcomes |σ_m〉 satisfying

$$\begin{equation*} E(\sigma_m) = 1 - \sqrt{1-16 \rho_0\rho_1\rho_0'\rho_1'} \tqs\forall m, \end{equation*}$$

or, equivalently,

$$\begin{equation*} C(\sigma_m) = C(\rho)\,C(\rho') \tqs \forall m. \end{equation*}$$

The entanglement swapping procedure can be iterated, creating a quantum connection between qubits that are more and more distant. However, the resulting long-distance entanglement that is generated in this way decreases exponentially with the number of swappings. This is clear in the case of mixed states and for pure-state Bell measurements in the $\mathcal{B}_X$ basis; the general proof can be found in section 1.3 in [52]. Because of this important loss of entanglement, new schemes have to be designed to efficiently entangle any two stations of a quantum network. The various protocols proposed so far that achieve this task are described in the following chapters.

Noisy operations. Thus far, we have assumed that all quantum operations are ideal or perfect. However, in practice, every operation or measurement introduces some noise. Here we consider two sources of error: those arising from applying an imperfect gate (unitary operation) and those arising from an imperfect measurement. In the first case, we model the error by including a small depolarizing term in a channel, which replaces a fraction of the density operator with a completely decoherent (i.e. unknown) state [5, 52]. For a multi-qubit state, a gate $O_{S}^{\rm{ideal}}$ acting on a subset S of qubits with an error probability ε is replaced by the map

$$\begin{equation} \rho \mapsto O_{S}[\rho] = (1-\varepsilon)\ O_{S}^{\rm{ideal}}[\rho] + \varepsilon'\ {\rm id}_{S} \otimes \tr_{S}[{\rho}], \end{equation} \tag{ 2.11 }$$

where tr_S denotes the partial trace over the subsystem S and ε' is such that the resulting state has trace one. It can be shown that this model correctly models isotropic errors for single qubit rotations [6]. On the other hand, suppose we model a measurement error on a single qubit by assuming that we have a small fixed probability ε of reading 1 when the qubit was actually measured into the 0 state. In this case, the measurement operators read:

$$\begin{eqnarray} M_0^\eta & = \sqrt{1-\varepsilon}{\,|{0} \rangle\!\langle {0}|\,} + \sqrt{\varepsilon} {\,|{1} \rangle\!\langle {1}|\,}, \\ M_1^\eta & = \sqrt{1-\varepsilon}{\,|{1} \rangle\!\langle {1}|\,} + \sqrt{\varepsilon} {\,|{0} \rangle\!\langle {0}|\,}. \end{eqnarray} \tag{ 2.12 }$$

Propagating the errors according to this simple model allows us to estimate the error resulting from a particular protocol.

All building blocks needed to construct a quantum network have been demonstrated; in fact, small-scale quantum networks are now a reality. For instance, while the first demonstrations of teleportation were made on laboratory scales [48, 55, 56], presently, entangled photons distributed in free space can be used for teleportation over 150 km [57, 58] (see also the recent improvements in the direction of telecommunication [59]). Moreover, atom-photon interfaces have also been used in the demonstration of teleportation [60, 61] and entanglement swapping was also shown in different scenarios [62, 63].

One of the first implementations of a quantum network, the DARPA Quantum Network, consists of several nodes and supports both fibre-optic and free-space links [64]. It is capable of distributing quantum keys between sites separated by a few tens of kilometres. Another example is the SECOCQ network in Vienna, which consists of six nodes connected by links ranging from 6 to 85 km [65].

Hierarchical graphs iterate certain geometric structures, so that at each level of iteration either the number of neighbours or the length of the connections increases. Various such graphs were considered in [66]; let us give here yet another example in which both pure and mixed state entanglements can be generated between nodes that lie at any level of the hierarchy. In this example, two infinite ternary trees, in which each link is an entangled state ρ, are connected at their leaves by a state σ; see figure 5(a). The protocol runs as follows: first, two entanglement swappings are performed on the central links ρ, σ and ρ, yielding a state $\tilde{\sigma}$. Second, the three states $\tilde{\sigma}$ that connect the new leaves of the ternary trees are purified, leading to a single connection σ'. The procedure can be iterated as long as E(σ') ⩾ E(σ). In this way, nodes lying at higher and higher levels of the hierarchical graphs become entangled, that is, larger and larger quantum connections are created. In the following, we determine the minimum amount of pure or mixed-state entanglements of ρ and σ for this strategy to be successful.

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Pure states. In section 2.2.4, we saw that the result of the entanglement swappings depends on the basis that is chosen for performing the two-qubit measurement. In order to facilitate the comprehension of the mechanism, we choose the $\mathcal{B}_X$ basis, so that all outcomes are equally entangled: $C(\tilde{\sigma}) = C^2(\rho)\,C(\sigma)$ ; see (2.10). The optimal purification of $\tilde{\sigma}^{\otimes 3}$ into σ', as described in section 2.2.3, requires working with the Schmidt coefficient $\tilde{\sigma}_0 = (1 + \sqrt{1-C^2(\tilde{\sigma})})/2$. One then finds $\sigma_0' = \max\{\frac{1}{2},\tilde{\sigma}_0^3\}$ ; the recursion relation for the entanglement of the central connections reads:

$$\begin{equation} E' = \min\left\{1,\,2-\case{1}{4}\Big( 1+\sqrt{1-\mu\, E(2-E)}\Big)^3\right\}, \end{equation} \tag{ 3.1 }$$

with E' = E(σ') and E = E(σ). In this equation, the parameter μ = C⁴(ρ) lies in the interval [0, 1]. It is an easy calculation to show that E' < E for any value of E if $\mu<\mu_{\rm c}=\frac{1}{3}$, that is, if E(ρ) < E_c ≈ 0.35. This means that the entanglement cannot be propagated in the hierarchical graph if the links ρ are too weakly entangled. In contrast, if μ > μ_c, one stable and non-trivial fixed point appears in (3.1); see figure 6. In this case, nodes lying at any level of the hierarchy can be connected by an entangled state of two qubits; one further shows that this state is a perfect Bell pair if μ ⩾ μ^* ≈ 0.655.

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Mixed states. The scenario of a mixed-state hierarchical graph, where the connections are Werner states ρ = ρ_W(x) and σ = ρ_W(y), is quite similar to that of pure states. First, two consecutive entanglement swappings are performed on the central states ρ, σ and ρ, leading to a state $\tilde{\sigma} = \rho_{{\rm W}}(\tilde{y})$ with $\tilde{y} = x^2y$. Then, we try to concentrate the entanglement of the three states $\tilde{\sigma}$ into one connection σ'. As for the pure states, we would like this operation to be deterministic, but the purification of mixed states is intrinsically probabilistic. In order to get a result in a predictable fashion, we purify two connections only and take the third one if the purification failed. From (2.8) and (2.9), this succeeds with probability $p = (1+\tilde{y}^2)/2$ and the average entanglement of σ' ≡ ρ_W(y') is

$$\begin{equation*} y'(x,y) = \frac{x^2y}{6}(5+4x^2y-3x^4y^2). \end{equation*}$$

If the links of the graph satisfy $x>x_c=\sqrt{18/19}$ and $y>y_c(x)= (2x-\sqrt{19x^2-18})/(3x^3)$, then a stable fixed point y^* appears. In this case, iterating the entanglement swappings and the purifications generates some long-distance pairs of qubits whose entanglement approaches $y^* = (2x+\sqrt{19x^2-18})/(3x^3)$.

We have just shown that entangled pairs of qubits can be generated over a large distance in graphs with a hierarchical structure, under the condition that the entanglement of the bonds is larger than a critical value E_c. The self-similarity of these graphs allows one to design natural sequences of entanglement swappings and purifications but suffers a physical limitation: either the length of the bonds or the number of qubits per node grows exponentially with the iteration depth. We now consider regular two-dimensional lattices, that is, periodic configurations of links throughout the plane in which the nodes have a fixed number Z of neighbours.

A deterministic strategy to entangle two infinitely distant nodes in lattices in which each node has a number of nearest neighbours Z ⩾ 4 was proposed in [66]. It is very similar to the recursive method developed in the previous example, but it works only with pure states. In the case of a square lattice of links |φ〉, the idea is to sequentially shorten the legs of a 'centipede', so that the entanglement of the links is gradually concentrated; see figure 7. This eventually yields a perfect Bell pair at the spine of the centipede, on which infinitely many entanglement swappings can be applied and therefore, long-distance entangled pairs of qubits are generated. More precisely, one starts by applying two entanglement swappings in the $\mathcal{B}_X$ basis on the states |σ〉 (dotted line in figure 7) and its neighbour states |φ〉 at the extremity of each leg. This results in a state ${|{\tilde{\sigma}} \rangle}$ and, as in the previous example, we have $C(\tilde{\sigma}) = C^2(\varphi)\ C(\sigma)$. The difference is that the purification is now performed on ${|{\tilde{\sigma}} \rangle} \otimes {\,|{\varphi} \rangle}$ rather than on ${|{\tilde{\sigma}} \rangle}^{\otimes 3}$. Very similarly to (3.1), one finds the following recursion relation:

$$\begin{equation} E' = \min\left\{1,\,2-\varphi_0\Big( 1-\sqrt{1-\mu\, E(2-E)}\Big)\right\}, \end{equation} \tag{ 3.2 }$$

where μ = C⁴(φ) is a function of φ₀. One can show that there always exists a non-trivial stable fixed point for this equation. However, the fixed point of (3.2) is strictly smaller than unity when E(φ) < E_c ≈ 0.649. In this case, although we do concentrate some entanglement along the spine of the centipede, we still face the problem that the spine is a one-dimensional system, which therefore exhibits an exponential decrease of the entanglement with its length. On the other hand, if E(φ) > E_c, then the fixed point is reached in a finite number of steps and a maximally entangled state is generated. Since the spine now consists of perfect connections, any two nodes lying on it can share a Bell pair, regardless of their distance.

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Classical percolation is perhaps the fundamental example of critical phenomena, since it is a purely statistical one [70]. At the same time, it is quite universal because it describes a variety of processes, with applications in physics, biology, ecology, engineering, etc [71]. Two types of models are typically considered: site- and bond percolation. In bond percolation, the neighbouring nodes of a lattice are connected by an open bond with probability p, whereas they are left unconnected with probability 1 − p; see figure 8. In site percolation, the sites (i.e. nodes) rather than bonds are occupied with probability p. In either case, for an infinite lattice, one would like to know whether an infinite open cluster exists, that is, whether there is an infinitely long path of connected nodes. It turns out that a unique infinite cluster $\mathcal{C}$ appears if, and only if, the connection probability p is larger than a critical value p_c that depends on the lattice. Few lattices have a threshold that is exactly known. Among them, we find the important honeycomb, square and triangular lattices, with , $p_{\rm c}^{\square} = 1/2$ and $p_{\rm c}^{\opentriangle} = 2\sin(\pi/18)$, respectively [72].

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Suppose now that we want to generate some entanglement between two distant stations A and B in a quantum lattice, where each connection denotes a partially entangled pure state |φ〉. CEP runs as follows [23]: every pair of neighbouring nodes tries to convert its two-qubit state into a Bell pair, which succeeds with an optimal probability p = E(φ); see section 2.2.3. If this value is larger than the threshold p_c of the lattice, that is, if the entanglement of the links is large enough, then an infinite cluster $\mathcal{C}$ appears. The probability that both A and B belong to $\mathcal{C}$ is strictly positive; in this case a path of Bell pairs between these two nodes can be found. Then, exactly as described in the previous section, one performs the required entanglement swappings along this path, such that A and B become entangled. Note that the path of Bell pairs is randomly generated by the measurement outcomes at the nodes, which contrasts with the deterministic location of the 'backbone' generated by the purification method.

A quantity of primary interest when studying the efficiency of CEP is the percolation probability

$$\begin{equation*} \theta(p)\equiv P(A\in\mathcal{C}), \end{equation*}$$

which is the probability that a randomly chosen node A belongs to the infinite cluster. This value is closely related to the percolation threshold: in fact, in an infinite lattice, we have θ(p) = 0 for p < p_c, whereas θ(p) > 0 for p > p_c. In our case, we are interested in the probability P(A ↔ B) of creating a Bell pair between two nodes A and B separated by a distance L. For p < p_c, this probability decays exponentially with L/ξ(p) [73], where the correlation length ξ(p) describes the typical radius of an open cluster. Above the critical point, the two nodes are connected only if they are both in $\mathcal{C}$. In the limit of large L, the events $\{A\in\mathcal{C}\}$ and $\{B\in\mathcal{C}\}$ are independent, so that

$$\begin{equation*} P(A\leftrightarrow B) = \theta^2(p), \end{equation*}$$

and therefore the problem is reduced to studying θ(p).

A natural question is whether the entanglement thresholds E_c = p_c defined by the classical percolation theory are optimal. In fact, percolation of entanglement represents a related but different theoretical problem, where new bounds may be obtained. This is of course equivalent to determining if the measurement strategy based on local Bell pair conversions is optimal in the asymptotic regime. Several examples that go beyond the classical picture were developed in [23, 66, 69], proving that CEP is not optimal; such results are referred to as QEP strategies. All these examples are based on the average conservation of the entanglement after one swapping, as described in section 2.2.4, but they do not provide a general construction to surpass the classical percolation strategy. In the following paragraph we review the original example of [23], since it gives some insights into the way a quantum lattice can be transformed by local measurements; we let the reader consult the articles [66, 69] for the other examples. Finally, an improved strategy that makes use of multipartite entanglement will be described in section 3.2.3.

Honeycomb lattice with double bonds. Let us consider a honeycomb lattice where each pair of neighbouring nodes is connected by two copies of the same state |φ〉; see figure 9(a). The CEP protocol converts all bonds shared by two parties into a single connection and from majorization theory we know that a double bond can be optimally purified into one pair of qubits with entanglement $E' = 2(1-\varphi_0^2)$. Setting this value to be equal to the percolation threshold for the honeycomb lattice, one finds that the entanglement can be propagated if E(φ) > E_c, with

However, there exists another measurement pattern yielding a better percolation threshold (figure 9): half the nodes perform on their qubits three entanglement swappings, which maps the honeycomb lattice into a triangular one. Since the entanglement of the connections is not altered, on average, it follows that a lower threshold is found:

$$\begin{equation*} \hat{E}_{\rm c}=p_{\rm c}^{\vartriangle} \approx 0.347. \end{equation*}$$

This proves that CEP is not optimal since it cannot generate long-distance entanglement in the range $E(\varphi)\in (\hat{E}_{\rm c},E_{\rm c})$, whereas QEP achieves it with a strictly positive probability.

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It is interesting to note that this example has also been used to show that the close analogy between quantum entanglement and classical secret correlations can be applied to entanglement percolation [74]. In this analogy, secret key bits, rather than entanglement are shared between neighbouring nodes.

We have seen that CEP can be enhanced by first applying some quantum operations at the nodes [23, 66, 69]. All examples proposed in these articles consist of transforming the quantum lattices by a sequence of entanglement swappings, thus conserving the average entanglement of the bonds. They are, however, restricted to purely geometrical transformations and they apply to specific lattices only. In this respect, it was not clear whether the CEP strategy and particularly the corresponding threshold, could be improved in general. A positive answer to this question was given in [75], in which a class of percolation strategies exploiting multi-partite entanglement was introduced. To that end, one performs the entanglement swappings in a more refined way, which we describe below.

Generalized entanglement swapping. The key ingredient of the multi-partite method is to consider a generalized entanglement swapping at the nodes. First, an incomplete measurement (i.e., not a complete projection) is performed at the central node by applying the operators

$$\begin{eqnarray} &&M_1 = {|{0} \rangle\!\langle {00}|\,} + {\,|{1} \rangle\!\langle {11}|},\\ &&M_2 = {|{0} \rangle\!\langle {01}|\,} + {\,|{1} \rangle\!\langle {10}|}, \end{eqnarray} \tag{ 3.3 }$$

for which the completeness relation $\sum_{i=1}^2 M_i^{\dagger}M_i^{\vphantom{\dagger}} = {\rm id}_4$ is satisfied. This measurement leaves a two-dimensional subspace at the central station entangled with the two outer nodes. Thus, the central station still plays a role in the propagation of entanglement through the lattice. Second, not only two but n ⩾ 2 links are 'swapped' at the same time; then the Procrustean method of entanglement concentration is performed on the resulting state; see (2.7). These operations succeed with a finite probability that depends on the number of links and on their entanglement [75], generating the Greenberger–Horne–Zeilinger (GHZ) state

$$\begin{equation*} {|{{\rm GHZ}_n} \rangle} \equiv \frac{{\,|0 \rangle}^{\otimes n}+{\,|{1} \rangle}^{\otimes n}}{\sqrt{2}}, \end{equation*}$$

which is the generalization of the Bell pair |Φ⁺〉 to n qubits.

Entanglement thresholds: from bond to site percolation. In the multi-partite strategy, one creates from the links |φ〉 of a quantum lattice $\mathcal{L}$ a new lattice $\hat{\mathcal{L}}$, where the nodes represent the GHZ states obtained with probability p by the generalized entanglement swappings. Two vertices in $\hat{\mathcal{L}}$ are connected by a bond if the corresponding GHZ states share a common node in the original lattice. This defines a site percolation process with occupation probability p and, above the site percolation threshold of the new lattice, the entanglement is propagated over a large distance as follows. Consider the situation in which two GHZ states of size n and m sharing one node have been created. One builds a larger GHZ state on n + m − 1 particles with unit probability by performing a generalized entanglement swapping on the two qubits of the common node. This operation is iterated, eventually yielding a giant GHZ state spanning the lattice. Then, given a GHZ state of any size, a perfect Bell pair is created between any two of its qubits by measuring all other qubits in the X basis.

An example of multi-partite entanglement percolation is given in figure 10. In this case, since the probability to create a GHZ state of three qubits is equal to that of creating a Bell pair (only two links are required), the minimum amount of entanglement of the bonds for generating long-distance entanglement is $\hat{E}_{\rm c} \approx 0.650$, which is equal to the site percolation threshold in $\hat{\mathcal{L}}$ [76]. Since the critical value for CEP is E_c ≈ 0.677 (the threshold for bond percolation in $\mathcal{L}$ [77]), it follows that multi-partite entanglement percolation surpasses CEP in the range $E(\varphi)\in(\hat{E}_{\rm c},E_{\rm c})$.

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The previous example appeared in [75] together with many other lattices for which the critical values using the multi-partite strategy are lower than the thresholds for CEP. In particular, an improvement is found for all Archimedean lattices, which are tilings of the plane by regular polygons (such as the square or the triangular lattice). Moreover, it is shown that not only the thresholds but also the probabilities P(A ↔ B) are better for any amount of entanglement $\hat{E}_{\rm c}<E(\varphi)<1$. This clearly indicates that the interplay of geometrical lattice transformations and multi-partite entanglement manipulations is a key ingredient for propagating entanglement in a quantum network.

A first hint of the usefulness of multi-partite quantum states was given in section 3.2.3. In that setting, a giant GHZ state is created on the lattice by extracting perfect entanglement from the bonds adjoining each node. Then, measurements in the X basis of all but two qubits transform the giant state into a Bell pair between the two remaining qubits. Finally, a local basis rotation depending on the measurement outcomes further converts the Bell pair into, say, the maximally entangled state |Φ⁺〉. The procedure is rather similar here, but the links of the networks are used to create a GHZ state (section 4.3.1), a surface code (section 4.3.2) or a cluster state (section 4.3.3). The key point of the construction is that while these states are simple enough to be created by local operations on the nodes of the quantum network, they also are tolerant of a certain amount noise, as described in the following sections.

In the protocols involving pure states, it is known exactly whether a conversion of partially entangled states into Bell pairs succeeds or fails, since this information is given by the outcome of a measurement. In contrast, in mixed-state quantum networks, some noise enters the system randomly and there is no way, a priori, to know where this happens. In fact, because every connection is a Werner state with non-unit fidelity, the generation of the multi-partite state based on such connections leads to a quantum state that contains errors (for instance, bit-flip and phase errors on some of its qubits). Hence, if one decides to measure all but the two target qubits right after the generation of the multi-partite state, then the choice of the final basis rotation will be correct with a probability of approximately only one fourth. This means that we have no knowledge at all about which one of the four Bell pairs we are dealing with, or in other words, the qubits are in a separable quantum state.

One strength of the network-based error correction is that one can gain some information about the errors without damaging the long-distance quantum correlations. In fact, the multi-partite entangled states created in the quantum network are highly symmetrical and satisfy a set of eigenvalue equations that can be checked by local measurements: if the outcomes do not match the symmetry of the target state at the corresponding nodes, which is called a syndrome, then one immediately knows that at least one adjacent link inserted an error into the system; see section 4.3. The question of determining which link is responsible for the syndrome is treated in what follows.

Syndromes are defined on the nodes of the dual lattice of the quantum network, which is either a square lattice (first two schemes) or a cubic lattice (third scheme). The generation of the multi-partite entangled state is such that the noise entering the system corrupts every link of the dual lattice with probability p. This creates chains of errors, which are consecutive corrupted links of the dual lattice. Syndromes correspond to the endpoints of these chains and thus come in pairs, as depicted in figure 12(a).

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If one knew the location of all chains of errors, then it would be possible to perfectly restore the target multi-partite state. The difficulty of the error recovery is that while the positions of the syndromes are known, no other information about the chains is available. Since different chains of errors can lead to a similar syndrome pattern, the recovery is ambiguous and may lead to a wrong correction of the errors. However, Dennis et al showed that, in an infinite square lattice, a (partial) recovery is possible if the error rate p does not exceed a critical value p_c [83]. This threshold is found via a mapping to the random-bond Ising model and is approximately equal to 10.94%, which is numerically calculated in [84]. In order to obtain long-distance quantum correlations for p < p_c, one should be able to compute all patterns of errors that lead to the measured syndromes and then to choose the one that most likely occurred. This is infeasible in practice, but for small error rates p, a good approximation of the optimal solution is the pattern in which the total number of errors is a minimum. In fact, such a pattern may be efficiently found by a classical algorithm, known as the minimum-weight perfect matching algorithm [85, 86]. Illustrations of these ideas are given in figures 12(b) and (c).

Any entangled state of two qubits can be transformed by LOCC to the Werner state defined in (2.4), but to understand the scheme of [26] it is more appropriate to consider a slightly different parameterization of a two-qubit entangled mixed state:

$$\begin{eqnarray} &&\fl \rho(\varepsilon_b,\varepsilon_p) \equiv \Big((1-\varepsilon_b)(1-\varepsilon_p),\, \varepsilon_b(1-\varepsilon_p),\nonumber\\ &&\tqs\quad \varepsilon_p(1-\varepsilon_b),\,\varepsilon_b\varepsilon_p\Big)_{\mathcal{B}}. \end{eqnarray} \tag{ 4.1 }$$

In this equation, ε_b and ε_p stand for the probability that the second qubit of the ideal connection |Φ⁺〉 has been affected by a bit-flip and a phase error, respectively. This state is as general as a Werner state in the sense that any quantum state of two qubits can be brought to this form using LOCC only.

Protocol in the case of bit-flip errors only. The links of a N × N square lattice are used to create a giant GHZ state of N² qubits; see figure 13. For each direction, the stations perform the generalized entanglement swapping described in (3.3). If one temporarily assumes that phase errors are not present in the links of the network, then the resulting multi-partite state is a mixture of GHZ states whose qubits are flipped with probability p = ε_b. If the error rate does not exceed the critical value p_c ≈ 10.9%, then most bit-flip errors can be corrected, as depicted in figure 14.

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The bit-flip error correction presented above can be applied to arbitrary planar networks, as shown by Broadfoot et al in [79]. They also prove that it can be generalized to entangled mixed states of rank three, but another method has to be used in the case of full rank mixed states, which is the topic of the following paragraph.

Protocol including both bit-flip and phase errors. The global error correction works exactly as described above, but each qubit is replaced by a logical qubit which is an encoded block of n qubits. Furthermore, all quantum operations on the logical qubit are implemented by an appropriate protocol at the encoded level. Phase errors are then suppressed by the redundancy of the following code:

$$\begin{equation*} {|0 \rangle} \mapsto {\,|{\tilde{0}} \rangle}=\frac{1}{\sqrt{2}} \left({\,|{+} \rangle}^{\otimes n}+{\,|{-} \rangle}^{\otimes n}\right), \end{equation*}$$

$$\begin{equation*} {|{1} \rangle} \mapsto {\,|{\tilde{1}} \rangle}=\frac{1}{\sqrt{2}} \left({\,|{+} \rangle}^{\otimes n}-{\,|{-} \rangle}^{\otimes n}\right), \end{equation*}$$

where |±〉 are the eigenvectors of the X basis. Using majority vote, up to t phase errors can be corrected for each block of n = 2t + 1 qubits. A detailed discussion of a fault-tolerant implementation of this encoding is done in [26], but here it is sufficient to state the final result: long-distance quantum correlations can be obtained with the encoded version of the global bit-flip correction, but the number of links between neighbouring stations has to increase logarithmically with the distance. The main difference with 1D quantum repeaters is that all operations can be applied simultaneously, so that no quantum memory is needed.

Surface codes have been used to implement error correction in quantum computation and communication [87–92]. Here, we review three ways in which the surface code is used to generate entanglement over large distances. The protocol described in [27] differs from the previous protocols in that the data to be transmitted is encoded across the network as the multi-partite state is created. In [29], the authors propose to use the entanglement existing in the links of the network to apply stabilizer measurements and to create the surface code across the network. In both proposals, the number of required entangled pairs scales with the distance at which the entanglement is to be generated. We describe these two protocols in what follows, whereas the third one, which requires a constant number of quantum connections only, is described in section 4.3.3.

Planar surface code. For the planar surface code, half of the qubits encode the data and half are only used to measure syndromes. The code is implemented via stabilizers. A stabilizer of |ψ〉 is an operator A satisfying A |ψ〉 = |ψ〉. Data is encoded by preparing the array in a simultaneous eigenstate of a set of commuting stabilizers, which are defined as follows. A lattice is formed by associating an edge with each data qubit. Each face represents a stabilizer formed by the tensor product of four Pauli Z operators tensored with the identity of all other data qubits. Vertex stabilizers are defined similarly, substituting the X operator for the Z operator. The exceptions are on the edges of the array, where the stabilizers are products of only three Pauli operators. The dimensions of the lattice are chosen to give two boundaries with partial vertex stabilizers and two with partial face stabilizers, which ensures that the dimension of the space satisfying the stabilizer conditions is 2. That is, the array encodes one logical qubit. Details of surface codes and error correction can be found in [83].

Surface code communication protocol. The planar surface code is implemented on a 2D rectangular array of entangled qubits and encodes a single logical qubit; see figure 15. We begin with a long rectangular array that supports this code. Square sections on the left and right ends also each support a surface code. The left end is prepared in a logical state ${|{\tilde\Psi} \rangle}$ and the right end in the logical state ${|{\tilde 0} \rangle}$. A sequence of operations is performed in such a manner that it spreads the state ${|{\tilde\Psi} \rangle}$ until the entire array encodes ${|{\tilde\Psi} \rangle}$. Finally, we perform some measurements that leave the array on the right end in the state ${|{\tilde\Psi} \rangle}$. Errors are measured during the entire procedure and passed classically to the right end, where they are processed to correct the result.

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Implementing the surface code communication protocol. The array is initialized with ${|{\tilde\Psi} \rangle}$ on the left and ${|{\tilde 0} \rangle}$ on the right, while all the data qubits in the middle are measured in the Z basis. The state ${|{\tilde\Psi} \rangle}$ is then spread across the array by measuring each stabilizer a number of times equal to code depth d, which is the same as the height of the lattice. In general, increasing d allows better correction for gate errors. The sequence of six gates shown in figure 15 is designed so that all stabilizers may be measured simultaneously. As a result of the repeated measurements, the syndromes mark chains of errors that extend in time (for d discrete steps) as well as space. A change in a stabilizer measurement signals a syndrome marking the beginning or end of a chain. These error chains can be corrected in the same way as described in the previous subsection.

As it stands, this protocol creates and transports entanglement only through local interactions. Thus, it is not sufficient for communication between distant parties. For long distance communication, each syndrome qubit is replace by two syndrome qubits, each of which is coupled to one party of a Bell pair, as shown in figure 16. The Bell pairs must be generated for each of the d steps in spreading the surface code.

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Assuming no loss in transmission and that all gates within the repeater nodes are perfect, the average time to failure of a single link only grows non negligibly with the code depth d if the fidelity of the entangled pairs satisfies F ≳ 0.92. Furthermore, the authors find that by modelling loss in transmission as measurement in an unknown basis, loss rates less than 0.45 can be handled efficiently. They find that realistic gate error rates do not affect these results significantly. The number of qubits per repeater grows with the code depth d, which scales as log(N/p_c), where N is the number of links and p_c is the desired communication error rate. Reference [27] includes a more thorough discussion of the error rate and transmission rate.

Network-based surface code protocol. The authors of [29] suggest a different scheme to generate long-distance entanglement based on the surface code. The main idea is to use the entanglement present in the links of the network to perform stabilizer measurements on qubits sitting in different nodes. More specifically, the authors consider a square lattice where, besides the four qubits composing the network, each node has one extra qubit (called a processing qubit); see figure 17. The nodes are divided in three categories, black, red and blue, in such a way that each red or blue node is surrounded by four black nodes. After initializing the processing qubits of black nodes in the state |0〉, the entanglement shared between neighbouring nodes is used to perform stabilizer measurements. According to the circuit depicted in figure 18, red nodes perform the stabilizer measurements $\bigotimes_{\mathcal{N}_r} X_{\mathcal{N}_r}$, while blue nodes measure $\bigotimes_{\mathcal{N}_{\rm b}} Z_{\mathcal{N}_{\rm b}}$ ( $\mathcal{N}_{r,{\rm b}}$ denotes the neighbours of red or blue nodes). If the entanglement shared in the network is perfect, that is, if the links are given by Bell states, the state of the black processing qubits after the stabilizer measurements is transformed into an eigenstate of the surface code. The last step of the protocol consists of measuring all black processing qubits except the ones that are held by Alice and Bob. The basis for these measurements is chosen to create a maximally entangled state between Alice and Bob.

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Due to imperfections, however, errors in the entanglement shared in the network and in the operations result in incorrect stabilizer-measurement outcomes. In order to detect these errors, the stabilizer measurements are repeated N times. For each measurement run, a new entangled pair of qubits must be generated between neighbouring sites. Since the desired state is an eigenstate of the stabilizer measurements, consecutive measurements with differing outcomes indicate an error. Finally, the errors are corrected by pairing the error syndromes through the network-based error correction.

The described protocol tolerates an error rate of approximately 1.67% in the quantum channels composing the network, which, in turn, corresponds to links composed by Werner states with x ≈ 0.98. One of the advantages of the scheme presented in [29] is that the same black processing qubit can be used to measure its four neighbouring qubits.

In the error-correction based protocols described above, the number of entangled pairs shared by neighbouring nodes has to increase with the size of the network in order to generate long-distance entanglement. On the contrary, it is shown in [28, 30] that a constant number of Werner states between neighbours is sufficient to achieve this task. In the following, we discuss the proposal of [28] which uses cluster states in a 3D lattice. We then briefly describe the main idea of [30], which works both for 2D and 3D lattices.

Three-dimensional cluster states. The first protocol achieving long-distance entanglement with a constant number of connections was proposed in [28]. In that article, it was shown that if the fidelity of the connections is larger than a critical value F_c, then entanglement can be generated between two qubits A and B lying on opposite faces of a simple cubic lattice of size N³, with N → ∞. In this case, however, it must be stressed that the local quantum operations are assumed to be perfect, which is not necessary for other strategies that are fault-tolerant. In what follows, we describe in detail how this protocol works, but let us first define the cluster state, which lies at the heart of this proposal.

A cluster state |C〉 is an instance of graph states [93] and it is usually constructed by inserting a qubit |+〉 at each vertex of the graph and by applying a controlled-phase⁷ between all neighbours. This state obeys the eigenvalue equation K_u |C〉 = |C〉 for all vertices u, where K_u is the stabilizer

$$\begin{equation} K_u \equiv X_u \prod_{v\in \mathcal{N}(u)} Z_v, \end{equation} \tag{ 4.2 }$$

and where $\mathcal{N}(u)$ stands for the neighbourhood of u in the graph. In [28], the desired controlled-phases are non-local quantum operations and therefore they have to be performed indirectly through the use of the entangled connections and by applying some generalized measurements at the nodes. If one uses the noisy connections defined in (4.1), some errors are introduced into the system so the cluster state is not perfect anymore. Setting the bit-flip and phase error rates to ε, it can be shown that this results in local Z errors occurring independently at the nodes with a probability p ≈ 6 ε.3D cluster states are known to have an intrinsic capability of error correction. In fact, long-range entanglement was shown to be possible between two faces of an infinite noisy cubic cluster state [94]. The difference between [28] and [94] is that local quantum operations are assumed at every node in the latter case, in particular in the two faces. The error correction runs as follows. First, all qubits but A and B are measured in either the X or the Z basis. The measurement pattern is such that, in the ideal case, the qubits A and B are maximally entangled:

$$\begin{equation*} X_A X_B{\,|{\psi_{AB}} \rangle} = \lambda_X{\,|{\psi_{AB}} \rangle}, \end{equation*}$$

$$\begin{equation*} Z_A Z_B{\,|{\psi_{AB}} \rangle} = \lambda_Z{\,|{\psi_{AB}} \rangle}, \end{equation*}$$

where the eigenvalues λ_X, λ_Z ∈ {−1, +1} depend on the measurement outcomes x and z and are calculated from the stabilizer equations (4.2). The effect of the local Z errors is to change the sign of these eigenvalues, thus ruining the quantum correlations if no error recovery is performed. Then, the lattice is virtually divided into two interlocked cubic sublattices (one for each correlation λ_X and λ_Z) and a parity syndrome is assigned to nearly all vertices u:

$$\begin{equation*} s(u) = \prod_{v\in\mathcal{N}(u)} K_v = \prod_{v\in\mathcal{N}(u)} X_v \prod_{w\in\mathcal{N}'(u)} Z_w, \end{equation*}$$

where $\mathcal{N}'$ designates the neighbourhood in the corresponding sublattice. Since this equation arises from a product of stabilizers, we have that s = 1 if no noise is present in the system. However, Z errors do not commute with X measurements; the construction is such that an error changes the sign of two syndromes that are neighbours in the other sublattice.

We are thus back to the error recovery described in section 4.2.3, with two differences nonetheless. First, of course, the lattice is 3D and not planar. Second, some syndromes lying on the opposite faces cannot be assigned a value because of the specific measurement pattern, whereas all syndromes are known in the usual error correction. This results in imperfect quantum correlations, but the Monte Carlo simulations performed in [28] indicate that the distant qubits A and B are entangled as long as the error rate p is smaller than the threshold p_c ≈ 2.3%.

Teleportation-based protocol. In [26] it is shown that the problem of transmitting entanglement in a 2D network is equivalent to the problem of fault-tolerant quantum computation in a 1D array of qubits restricted to next-neighbour gates. Let us discuss how it works. First, consider a square lattice in which we would like to create long-distance entanglement and let us suppose for the moment that all links of the network are perfect. Consider one of the diagonal array of qubits of this lattice as a1D quantum computer at the initial time t = 0; see figure 19(a). By teleporting each qubit of the computer twice, first right and then above, the state of the computer is mapped to an upper-right diagonal array of qubits. The computer is now at time t = 1. If between t = 0 to t = 1 one has to implement a two-qubit gate between two neighbouring qubits, one first teleports one of the qubits up and the other right; see figure 19(b). In this way, they end up at the same location, where now the gate can be locally applied. Finally, each qubit is further teleported to the diagonal defining t = 1.

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In the case that each link corresponds to a Werner state, the teleportation scheme can be seen as a quantum computation where errors occur. In this way, a fault-tolerant quantum computation scheme should be used. In fact, such a scheme exists, where two qubits per site are used [90]. Note that in fault-tolerant error-correction schemes, one starts by encoding a known state. However, in practice, in the very first stage the qubits are exposed to noise and thus decohere. Furthermore, in order to create an entangled state between two distant nodes, one would need a decoding scheme that is also fault-tolerant.

In reference [30], the authors develop a fault-tolerant encoding–decoding scheme into a 1D concatenated code [90] and into a 2D planar code [83] that works for unknown states. Both the encoding and decoding protocols can be done in a one-shot manner by measuring syndrome operators (products of X and Z operators) and using error correction schemes based on syndrome patterns. By combining the encoding–decoding scheme in 1D or 2D, with the teleportation method shown in figure 19, one can establish long distance entanglement in the 2D and 3D square lattice network, respectively.

A simpler scheme of entanglement distribution in a 3D lattice was also discussed in [30], where the authors use a 2D topological code.

One of the main goals of random graph theory is to determine the probability p at which a specific property of a graph G_{N, p} typically arises, as N tends to infinity. For fixed p, the typical node-degree diverges with N, leading to a highly connected, boring graph. Instead, we let p = p(N), with details of the dependence allowing us to examine different interesting phenomena.

Many properties of interest appear suddenly, i.e. a critical probability p_c(N) exists such that almost every graph has this property if p ⩾ p_c(N) and fails to have it otherwise. Such graphs are said to be typical. For instance, it was shown that G_{N, p} is fragmented into small isolated clusters if p ≲ N⁻¹, whereas percolation occurs above this threshold, that is, one single giant component forms in the network.

A subgraph F = (V, E) of G_{N, p} is defined as a collection of n ⩽ N vertices connected by l edges. Like the giant cluster, the subgraphs have distinct thresholds at which they typically form. It is proven in [106] that the critical probability p_c for the emergence of F is

$$\begin{equation} p_{\rm c}(N)=c\,N^{-n/l}, \end{equation} \tag{ 5.1 }$$

where c is independent of N. It is instructive to look at the appearance of subgraphs assuming that p(N) scales as N^z, with z ∈ (−∞, 0] a tunable parameter: as z increases, more and more complex subgraphs appear; see table 1. In particular, only node-to-node connections appear in the regime z = −2, whereas complete subgraphs (of order four or more) emerge above the percolation threshold z = −1.

Table 1. Some critical probabilities, according to (5.1), at which a subgraph F appears in random graphs of N nodes connected with probability p ∼ N^z. For instance, simple subgraphs appear at a small connection probability, whereas cycles and trees of all orders emerge at the critical value z = −1. After Albert and Barabási [9].

The main result in [31] states that, in an infinitely large quantum random graph and in the regime p ∼ N⁻², the state

$$\begin{equation*} {\,|{F} \rangle} \equiv \bigotimes_{i=1}^l {\,|{\Phi^+} \rangle}_{E_i*}, \end{equation*}$$

which corresponds to any subgraph F = (V, E) composed of n vertices and l edges, can be generated with a strictly positive probability using LOCC only. All critical exponents of table 1 thus collapse onto the smallest non-trivial value z = −2. This clearly indicates that the properties of disordered graphs change completely when they are governed by the laws of quantum physics. It is not the purpose of this Review to prove this result, but let us describe the main quantum operation that lies behind its proof. In fact, it is another example of a generalized measurement on many qubits that enhances the distribution of entanglement in quantum networks.

A joint measurement at the nodes. As for multipartite entanglement percolation (section 3.2.3), allowing strategies that entangle the qubits within the nodes yields better results than a simple conversion of the links into Bell pairs. In the current case, the construction of a quantum subgraph |F〉 is based on an incomplete measurement of all qubits at each node. More precisely, the measurement operators are projectors onto the subspaces that consist of exactly m states |1〉 out of M = N − 1 qubits:

$$\begin{eqnarray*}&&P_m \equiv \sum_{\pi_m}\pi_m| \underset{\scriptstyle M-m}{\underbrace{0\ldots0}} \underset{\scriptstyle m}{\underbrace{1\ldots1}} \rangle\langle {0\ldots 01\ldots 1}| \pi_m^{\dagger}, \end{eqnarray*}$$

where π_m denotes a permutation of the qubits. Applied on all nodes of the quantum random graph, this measurement generates a highly entangled state shared by a random subset of nodes. The quantum correlations corresponding to |F〉 are then extracted by a suitable series of local operations at these nodes [31]. Note that this measurement allows not only the creation of any quantum subgraph, but also the generation of some important multipartite states, such as the GHZ states [52].