12.2.1 Bipartite Systems

For bipartite systems, that is, n = 2, we denote the parties by A and B, often referred to as Alice and Bob. Any bipartite pure state |Ψ〉 can be written in its Schmidt decomposition, that is, there exist local unitary operations U_A  ⊗ U_B such that

12.2

where λ ₀ ≥ λ ₁ ≥  ⋯ ≥ λ_d are ordered Schmidt coefficients that are positive and real and sum up to 1. Since the unitary operations U_A , U_B correspond to the choice of a local basis in A, B, the entanglement properties of a pure state |Ψ〉 are completely determined by its Schmidt coefficients. A state |Ψ〉 is entangled if it has two (or more) nonzero Schmidt coefficients, while it is a product state if λ ₀ = 1. The state is called maximally entangled if all Schmidt coefficients are equal, λ ₀ = λ ₁ = ⋯ = λ _d−1 = 1/d.

In the context of quantum information processing, it is an important question whether a certain pure state |Ψ〉 can be transformed by means of local operations and classical communication (LOCC) to some other pure state |Φ〉 and vice versa. If this is possible, the two pure states can be used to perform the same tasks and can be used for the same applications. Many variants of this problem exist, reaching from restricted kinds of classical communication to entanglement‐assisted transformation leading to catalysis effects. We will consider throughout this paper only two‐way classical communication and arbitrary sequences of local operations, where in the case of pure‐state transformations it turns out that one‐way classical communication is in many cases already sufficient. For arbitrary finite‐dimensional systems, a simple necessary and sufficient criterion for the LOCC transformation of states |Ψ〉 to states |Φ〉 is known, both for deterministic and probabilistic transformations (13–15). While in the first case the transformation always succeeds, in the latter case the transformation only succeeds with some nonzero probability p. The criterion can be expressed as a simple majorization relation between the Schmidt coefficients of state |Ψ〉 and of state |Φ〉. A deterministic transformation from |Ψ〉 to |Φ〉 by means of LOCC is possible if and only if is majorized by , that is,

12.3

Similarly, the maximal success probability for the transformation can be determined from the maximum p such that is majorized by . Note that these theorems can also be applied to systems consisting of multiple copies of a certain pure state. For instance, one can answer a question whether N copies of a pure state |Ψ〉 can be transformed to M copies of a pure state |Φ〉,

12.4

and the maximal success probability for this transformation can be determined. As a special instance, this includes distillation of maximally entangled states, where target states |Φ〉 are maximally entangled states with equal Schmidt coefficients.

In the asymptotic limit of many copies, N → ∞, it turns out that a single quantity, the entropy of entanglement S_A , determines the ratio M/N for transformations between pure states. The entropy of entanglement of a pure state |Ψ〉 is given by the von Neumann entropy of the reduced density operator ρ_A  = tr _B |Ψ〉〈Ψ|, E(Ψ) = −trρ_A log₂ ρ_A , which only depends on the Schmidt coefficients, E(Ψ) =  . The transformation |Ψ〉^⊗N  → |Φ〉^⊗M by means of LOCC succeeds with vanishing error if and only if (16–18)

12.5

In particular, this implies that transformations between pure states are reversible in the asymptotic limit. In this sense, the entropy of entanglement is a unique measure of entanglement for finite‐dimensional bipartite systems. For instance, Eq. 12.5 implies that N copies of a nonmaximally entangled state |Ψ〉 can be transformed into NS_A (Ψ) copies of maximally entangled pure qubit states |Φ〉 = and vice versa. In this asymptotic sense, all bipartite entangled pure states are qualitatively equivalent, while the quantitative measure is provided by the entropy of entanglement.

12.2.2 Multipartite Systems

For multipartite systems, the situation is far more involved, mainly due to the fact that no analog of the Schmidt decomposition exists (19,20). However, the Schmidt measure (21), that is the minimum number of product terms required to represent a state, is an analog of the Schmidt number for bipartite systems (the number of terms in the Schmidt decomposition). The Schmidt measure is an entanglement measure that gives rise to a (coarse‐grained) classification of multipartite quantum states (21,22), and to necessary conditions for state transformation. However, no simple necessary and sufficient criterion for transformation by means of (probabilistic or deterministic) LOCC between single copies of multipartite pure states is known, and for more than three parties, in general, two pure states cannot be transformed into each other with nonzero probability of success (23,24). Only certain special cases, for example, the optimal transformation of an arbitrary three‐qubit state to maximally entangled Greenberger–Horne–Zeilinger (GHZ) states,

12.6

have been solved.

Also, the asymptotic transformation in the many‐copy limit – which leads to a significant simplification in the bipartite case – seems to be less tractable. In order to obtain reversible transformations between multiple copies of an arbitrary state |Ψ〉 and some set of standard states, it was shown that this set has to include several different kinds of multipartite entangled states. In particular, all kinds of maximally entangled bipartite states shared between parties A_k and A_l as well as all combinations of m‐party GHZ states have to be included in this set, as they cannot be reversibly transformed into each other (25). For instance, the three different maximally entangled bipartite states shared between A ₁ − A ₂, A ₁ − A ₃, and A ₂ − A ₃ cannot be reversibly converted into tripartite GHZ states. For special classes of multipartite pure states, it was shown that reversible transformation between states within this class and the set of m‐party GHZ states (including all possible permutations for all m ≤ n) is possible (26). However, in general the set of states has to be enlarged to ensure reversible interconvertability (27), and it is not known whether a set with finite cardinality is sufficient in general (28).

12.3.1 Distillable Entanglement and Yield

Given N copies of an arbitrary bipartite mixed state, ρ ^⊗N , the distillable entanglement E _D is defined as the fraction M/N of the number of copies M of maximally entangled pure states of two qubits, |Φ〉 =  (|00〉 + |11〉, that can be created in an asymptotic, approximate sense by means of LOCC. That is, in the limit N → ∞ one is interested in the fraction of maximally entangled EPR pairs that one can generate, where the entanglement of distillation is measured in e‐bits. Here, one allows for approximate transformations, that is, the (global) fidelity of the resulting state σ ^⊗M needs to be ε close to 1, F = 〈Φ| ^⊗M |σ^⊗M |Φ〉^⊗M  ≥ 1 − ε, ∀ε > 0 (see (28) for more details).

The sequence of LOCC that achieves the transformation ρ ^⊗N  → σ^⊗M (with σ close to maximally entangled pairs) is often called an EPP, and the fraction M/N is referred to as the yield of the procedure. In this sense, the entanglement of distillation is given by the yield of the optimal procedure.

12.3.2 Criteria for Entanglement Distillation

In general, it is very difficult to calculate the entanglement of distillation, as this entanglement measure is operationally defined. That is, one has to maximize over all LOCC procedures that accomplish the transformation in question. These LOCC procedures may, for example, include (possibly infinite) sequences of (weak) measurements in A, classical communication of the results to B, measurements in B (depending on the outcome of A), communication of the results to A, measurement in A (depending on all previous outcomes), and so on. The class of LOCC transformations is very difficult to deal with, which makes the calculation of E _D a highly nontrivial task. In general, only upper and lower bounds are known. Any EPP that is capable of purifying a certain state provides us with a lower bound for the entanglement of distillation. The lower bound is given by the yield of the protocol. Upper bounds can be derived by considering the transformation of states under larger classes of operations – including the set of LOCC transformations as special instances. For example, one can consider operations that preserve the positivity of the partial transpose (see Section 12.3.2.1 for the definition of partial transposition), which are easier to handle than LOCC and derive in this way upper bounds for the efficiency of transformations. Upper bounds for the efficiency for all protocols based on positivity‐preserving operations automatically lead to upper bounds for distillable entanglement (using protocols based on LOCC) (29). There exist examples of states where upper and lower bounds coincide and hence the distillable entanglement is known. This is the case for incoherent mixtures of two maximally entangled states of two qubits,

12.7

with |Φ⁺〉 =  (|00〉 + |11〉), Ψ⁺〉 =  (|01〉 + |10〉). Here, the distillable entanglement is given by

12.8

In general, the value of E _D, even for simple mixed states such as Werner states (30) (a mixture of a maximally entangled state with a completely mixed state), is however not known. More strikingly, even the question whether a given (high‐dimensional) mixed state is distillable entangled or not can in general not be answered. As we will see below, there exist necessary criteria for distillation, and sufficient criteria for distillation. In general, these criteria are not conclusive in the sense that for many states, it is not possible to judge whether the state is distillable entangled or not. Only for low‐dimensional systems, in particular all 2 × d systems, a necessary and sufficient condition is known.

12.3.2.1 Partial Transposition as a Necessary Criterion for Distillation

The partial transposition of a density operator turns out to provide a simple, necessary criterion for distillation. The partial transposition of a density operator ρ with respect to the first subsystem, , written in the standard basis {|0〉, |1〉, …,|d − 1〉} is given by (31)

12.9

The partial transposition is basis dependent, but the eigenvalues are not. We say that ρ has positive partial transposition (PPT) if all eigenvalues of are positive, while ρ is said to be NPPT (nonpositive partial transposition) or simply NPT (negative partial transposition) if at least one of the eigenvalues of is negative. If all eigenvalues of are positive, the state ρ is said to be PPT (positive partial transpose). It turns out (32,33) that NPT of ρ is a necessary condition for distillability. This can be readily seen from the fact that any sequence of local operations does not change the positivity of the partial transposition. One uses the operator identity

12.10

where one only needs to consider the case C = A ^†, D = B ^†. That is, a density operator ρ (i.e., an operator with a positive spectrum), which is PPT by assumption, is converted by local transformation in another density operator (right‐hand side of Eq. 12.10). The partial transposition of this transformed density operator can also be obtained by applying (different) local transformations on the partial transpose of the initial density operator (left‐hand side of Eq. 12.10). As the spectrum of is positive by assumption, also the spectrum of the operator on the left‐hand side of Eq. 12.10 is positive. Hence, also the spectrum of the locally transformed operator (right‐hand side of Eq. 12.10 is positive. As the maximally entangled target state |Φ〉〈Φ| is NPT, it follows that only states that are initially also NPT can be converted to |Φ〉. The argument also holds for approximative transformations and multiple copies.

For 2 × d systems, that is, states consisting of a qubit and a d‐level system, NPT turns out to be a necessary and sufficient condition for distillability (33,34). This can be shown as follows: First, there exists a projector into a two‐dimensional subspace in B such that the resulting state is still NPT (34). Second, in 2 × 2 systems NPT implies that there exist local filtering measurements such that a state can be created from ρ that has fidelity F = 〈Φ| |Φ〉 > 1/2 (33). Finally, there exists an entanglement distillation protocol that allows one to create maximally entangled states whenever F > 1/2 (7). This protocol will be discussed in more detail in Section 12.4.

For higher‐dimensional d ₁ × d ₂ systems, there exist, however, states that have the puzzling property that they are PPT (and hence not distillable), but which are nevertheless entangled (i.e., nonseparable) (32). These states are called bound entangled, as their entanglement cannot be converted to a useful (pure‐state) form. Whether NPT is also a sufficient condition for distillability for d × d systems is presently unknown. Strong evidences for the existence of such NPT bound entangled states have been reported in (34,35) (see also (36,37)). Note that the existence of such states would imply the nonadditivity of entanglement of distillation (38), as one can distill entanglement if both certain (conjectured) NPT bound entangled states and PPT bound entangled states are available.

12.3.2.2 Sufficient Conditions for Distillation

Only few sufficient criteria for distillability are known. A criterion that is simple to check and valid for d × d systems is the reduction criterion developed by the Horodecki family (39). In particular, we have that if a state ρ violates the reduction criterion,

12.11

then the state is distillable. For mixtures of a maximally entangled state and a completely mixed state (global white noise, described by a density operator ), the criterion reads F ≥ 1/d. However, many distillable states are not detected by the reduction criterion.

A second criterion follows from the fact that NPT is a sufficient condition for distillation in 2 × 2 systems. Hence, if for a d × d system in a mixed state ρ local projections in two‐dimensional subspaces in A, B exist such that the resulting state is NPT, then ρ is distillable. This property is in fact called 1‐distillability, where k‐distillability is defined as the existence of such projectors when operating jointly on k copies of ρ, ρ ^⊗k . A state is distillable if there exists a k such that it is k‐distillable. This criterion is however not a practical one, as in general it is difficult to check due to the optimization over all two‐dimensional projections.

More practical criteria are in a certain sense provided by entanglement distillation protocols, where successful applicability of a protocol clearly implies distillability of the corresponding state. In this sense, the regime where a protocol can be successfully applied (which can often be expressed in terms of fidelity or of diagonal entries of the density matrix written in a certain basis, as we will see in Section 12.4) automatically translates into a sufficient condition for distillability. For instance, a fidelity F > 1/2 with a maximally entangled state is sufficient for the applicability of the protocols (7,8) for two‐qubit systems and is hence a sufficient condition for distillability.

12.4.1 Filtering Protocol

The most simple protocols operate on a single copy of the mixed state ρ and consist in the application of local filtering measurements (including weak measurements). A weak measurement may, for example, be realized by a joined, local operation on the system and an (high‐dimensional) ancilla, followed by a von Neumann measurement of the ancilla. Hence, (sequences of) local operations, including (weak) measurements, are applied in such a way that for specific measurement outcomes the resulting state σ is more entangled than the initial state ρ. Note that the output state σ is obtained only with a probability p < 1. Mixed states where such a filtering method can be applied include, for example, certain rank two states (40)

12.12

Application of the local operators O_A  = O_B  =  |0〉〈0| + |1〉〈1| (which correspond to a specific branch of a local positive operator valued measure, POVM) leads to a nonnormalized state of the form ρ′ = Fε|Ψ⁺〉〈Ψ⁺ | + (1 − F)ε ² |00〉〈00|. The fidelity of the resulting state is given by F′ = Fε/[Fε + (1 − F)ε ²]. Note that for small ε, F′ → 1, that is, states arbitrarily close to the maximally entangled state |Ψ⁺〉 can be created. However, the probability to obtain the desired outcome corresponding to O_A , O_B , p _suc = Fε + (1 − F)ε ², goes to zero as ε → 0. There is a tradeoff between the reachable fidelity of the output state and the probability of success of the procedure.

It turns out that filtering protocols are of limited applicability for general mixed states, even for the simplest case of two qubits. In particular, as shown in (41,42), the fidelity of a single copy of a full rank state can in general not be increased by any local operation. This seriously restricts the applicability of filtering procedures and requires one to consider protocols that operate jointly on two (or more) copies of the state in order to increase fidelity and ultimately to obtain maximally entangled states.

12.4.2 Recurrence Protocols

In the following, we discuss a class of conceptually related protocols (7,8,10) that allow one to produce states arbitrarily close to a maximally entangled pure state by iterative application. Before we go into technical details, we describe the general concept underlying these (and more generally, almost all) EPPs. The basic idea of all EPPs is to decrease the degree of mixedness of the ensemble of mixed state. To this aim, one needs to gain information, which is done by performing suitable measurements. As the relevant information is nonlocal, one needs to use the entanglement inherent in states of the ensemble to reveal this information. In fact, by first operating on several copies of the ensemble in a local way, information about this subensemble is transferred to one of the states. This state is then measured to reveal the information, and in this way to increase the information about the remaining states. In many protocols, the remaining states are only kept if a specific measurement outcome was found. This is due to the fact that one finds for certain measurement outcomes (measurement branches) that the entanglement of the remaining states is increased, while for other outcomes it is decreased or the states are no longer entangled. In this way, it is also guaranteed that on average, entanglement cannot increase under LOCC. Recurrence protocols operate in each purification step on two identical copies of a mixed state. After local manipulation, one of the copies is measured, and depending on the outcome of the measurement the other copy is kept (we refer to this as a successful purification step) or discarded. In the case of a successful purification step, the fidelity of the remaining pair is increased. The procedure is iterated, whereby states resulting from a successful purification round are used as an input for the next purification round. Typically, these protocols converge to a fixed point which – in case the initial fidelity was sufficiently large – is given by a maximally entangled state.

We now turn to specific recurrence protocols that allow one to purify bipartite entangled states. We will not describe these protocols as they were originally presented, but provide an equivalent description that will allow us a unified treatment of bipartite and multipartite EPPs. In particular, we describe protocols that operate on states in a (locally) rotated basis and describe the corresponding states in terms of their stabilizing operators. To this aim, we start by fixing some notation. We consider two parties, A and B, each holding several copies of noisy entangled states described by a density operator ρ_AB acting on Hilbert space ℂ² ⊗ ℂ². We denote by

12.13

a maximally entangled state of two qubits, where |0〉 _z , |1〉 _z [|0〉 _x , |1〉 _x ] are eigenstates of σ_z [σ_x ] with eigenvalue (±1) respectively. That is, σ_x|1〉 _x  = −|1〉 _x , and |0〉 _x  =  (|0〉 _z  + |1〉 _z ). We also define

12.14

with k ₁, k ₂ ∈ {0, 1}. The states form a basis of orthogonal, maximally entangled states, the so‐called Bell basis. We remark that the states are joint eigenstates of correlation operators

12.15

with eigenvalues and respectively. Whenever several copies of a mixed state are involved, we will refer to the different copies by numbers. For instance, refers to the first copy of a state, while refers to the second copy. In this case, party A holds two qubits, A ₁ and A ₂.

We consider mixed states that we write in the Bell basis,

12.16

One can always depolarize the state to a standard form by a suitable sequence of (random) local operations in such a way that the fidelity of the state, F ≡ 〈Φ₀₀ |ρ_AB|Φ₀₀〉 is not altered. To be specific, by probabilistically applying one of the local operations corresponding to { , K ₁, K ₂, K ₁ K ₂}, one produces a density operator that is diagonal in the Bell basis,

12.17

and in which diagonal coefficients remain unchanged, . This can be understood as follows: Consider for instance the action of K ₁ on basis states . For k ₁ = 0, the state is left invariant while a phase of (−1) is picked up if k ₁ = 1. It follows that off‐diagonal elements of the form in 12.16 are transformed to , that is, pick up a phase if k ₁ ≠ j ₁. Consequently, when applying the local operation K ₁ with probability p = 1/2 and with probability p = 1/2 leaving the state unchanged, the resulting density operator has no off‐diagonal elements where k ₁ ≠ j ₁. In a similar way, all off‐diagonal elements are cancelled by the (random) application of , K ₁, K ₂, K ₁ K ₂. Note that all diagonal elements – in particular the fidelity of state – remain unchanged by this depolarization procedure. Using similar techniques, one can further depolarize the state by equalizing all but one of the diagonal elements. The resulting states are called Werner states (30),

12.18

where the fidelity F = (3x + 1)/4 is unchanged. This can be accomplished by randomly applying local unitary operations that leave the state |Φ₀₀〉 (up to a phase) invariant, which is the case for all operations of the form U ⊗ HU*H with H being the Hadamard gate (43) and * denoting complex conjugation. The unitaries can be chosen uniformly (according to the Haar measure), or selected from a specific finite set of operations (7). What is important in our context is that any state with fidelity F can always be brought to Werner form. It is thus sufficient to provide an entanglement purification method which works for Werner states, because such a method automatically allows one to purify all states with same fidelity. We consider such a purification procedure in the following.

12.4.2.1 BBPSSW Protocol

In 1996, Bennett et al. (7) introduced a purification protocol that allows one to create maximally entangled states with arbitrary accuracy starting from several copies of a mixed state ρ, provided that the fidelity F with some maximally entangled state fulfills F > 1/2. The protocol consists of the following steps: (i) depolarize ρ to Werner form; (ii) apply bilateral local CNOT operations (44); (iii) measure qubit A ₂ [B ₂] in eigenbasis of σ_z [σ_x ] locally with corresponding results respectively, where ξ ₁, ζ ₁ ∈ {0, 1}. The effect on other particles of this local measurement is the same as the measurement of the observable ; (iv) keep the state of A ₁ B ₁ if (ξ ₁ + ζ ₁) mod2 = 0, that is, measurement results coincide.

Given two copies of a state with fidelity F, it is straightforward to calculate the fidelity of the resulting state when applying (i–iv). The effect of (ii) on two Bell states is given by

12.19

The effect of (iii) and (iv) is to select states in A ₂ B ₂ that are eigenstates of with eigenvalue (+1), while eigenstates with eigenvalue (−1) are discarded. That is, only initial states with k ₂ ⊕ j ₂ = 0 will pass the measurement procedure, which implies that, when considering mixed states, only these components will contribute to the final density operator. The final state turns out to be not of Werner form; however due to step (i) the state is brought back to Werner form when iterating the procedure. Hence, the essential parameter is the fidelity F′ after successful purification. One finds

12.20

which fulfills F′ > F for F > 1/2. The success probability is given by the denominator of Eq. 12.20, p _suc = F ² + 2F (1 − F)/3 + 5[(1 − F)/3]². The iteration of the procedure, which means to take two identical copies of states with fidelity F′, resulting from a previous, successful purification round, allows us to further increase the fidelity. In fact, it is straightforward to see that the map Eq. 12.20 has F = 1 as an attractive fixed point. Hence states arbitrarily close to maximally entangled states can be produced. Although the probability of success of the purification steps tends to 1 for F → 1, the yield of the procedure goes to zero as always one pair is measured and has to be discarded. Fixing however the desired accuracy of resulting states to a value F > 1 − ε, a finite number of purification steps suffices and hence the yield will be finite. We remark that obtaining states with F = 1 seems to be a question of only theoretical relevance, since imperfections in an apparatus used in the preparation of the state and in the purification procedure limit the reachable fidelity.

12.4.2.2 DEJMPS Protocol

The DEJMPS, introduced by Deutsch et al. in (8), is conceptually very similar to the BBPSSW protocol. It operates however not on Werner states, but on states diagonal in a Bell basis (see Eq. 12.17). The main advantage of this protocol is that it has better efficiency. The protocol operates on two identical copies of a state and consists essentially of the same steps as the BBPSSW protocol. The only difference is that step (i) is replaced by depolarization of ρ to a Bell diagonal state (Eq. 12.17), and in addition applying before step (ii) something as step (i)a, an additional local basis change |0〉 _z →  (|0〉 _z − i|1〉 _z ), |1〉 _z →  (|1〉 _z − i|0〉 _z ) in A and |0〉 _x →  (|0〉 _x  + i|1〉 _x ), |1〉 _x →  (|1〉 _x  + i|0〉 _x ) in B. The action of step (i)a is (up to some irrelevant phases) to flip the diagonal components of |Φ₁₀〉 and |Φ₁₁〉, that is, λ ₁₀ ↔ λ ₁₁. The total effect of the protocol (steps (i–iv)) can be described as a nonlinear map for the diagonal components of ρ to ρ′ (written in the Bell basis), that is, a map from ℝ⁴ → ℝ⁴. To be specific, the map reads

12.21

where N = (λ ₀₀ + λ ₁₁)² + (λ ₀₁ + λ ₁₀)² is the probability of success of the protocol. Again, the protocol can be iterated, and the diagonal coefficients of the state (written in the Bell basis) after k successful purification steps can be calculated by k iterations of the map Eq. 12.21. One can show that the map has λ ₀₀ = 1, λ_ij  = 0 for ij ≠ 11 as an attracting fixed point, and in fact all states with λ ₀₀ > 1/2 (i.e., F > 1/2) can be purified (45).

12.4.2.3 (Nested) Entanglement Pumping

While both the BBPSSW and DEJMPS protocol allow one to successfully produce entangled states with arbitrary high fidelity, the requirements on local resources are rather demanding. In particular, since at every round two identical states resulting from previous successful purification rounds are required, the total number of pairs that have to be available initially increases (exponentially) with the number of steps and will typically be of the order of several hundred. In particular, these pairs have to be stored by some means. For many physical set‐ups, however, the number of particles that can be stored is limited.

The requirements in memory space can however be translated into temporal resources. The corresponding purification protocol is called (nested) entanglement pumping. The basic idea is to repeatedly produce elementary entangled pairs (e.g., resulting from sending parts of a locally generated maximally entangled state through noisy channels) and using always a fresh elementary pair to purify a second pair. If a purification step is not successful, one has to start again from the beginning, using two elementary pairs. The actual sequence of local operations is either given by the BBPSSW or DEJMPS protocol, where the pair to be purified acts as pair 1 (source pair), while the fresh, elementary pair plays the role of pair 2 (target pair) that is measured. In case the purification step was successful, the fidelity of the first pair is increased by a certain amount. It is straightforward to determine the maps corresponding to Eqs. 12.20 and 12.21 for nonidentical input states. One finds

12.22

in the case of two Werner states with fidelity F ₁, F ₂. In this map, F ₂ is to be considered as a constant since the second pair is always an elementary one. For two Bell diagonal states with coefficients λ_ik and μ_ik, we obtain

12.23

Again, the second pair is always an elementary one, and hence μ_ik is fixed. Iteration of the corresponding maps allows in both cases to improve the fidelity; however in general no maximally entangled states can be generated. That is, the fixed point of the maps, Eqs. 12.22 and 12.23 depends on the fidelity of the elementary pair (or more generally on the coefficients μ_ik ) (34).

As elementary pairs can be generated on demand, they do not need to be stored. Hence in A and B only two qubits need to be stored (corresponding to the pair to be purified and the elementary pair respectively). The reduction in spatial resources leads however to an increase of temporal resources. In protocols BBPSSW and DEJMPS, the purification of different pairs corresponding to a single purification step can be implemented in parallel (i.e., the temporal resources are given by the number of steps), while the probabilistic character of entanglement purification manifests itself in the fact that many identical pairs need to be simultaneously available. In entanglement pumping, in contrast, the probabilistic character of purification leads to increased number of required repetitions, as in the case of an unsuccessful purification step the procedure has to be started from beginning and pairs are sequentially generated.

One can improve the entanglement pumping scheme in such a way that the number of qubits that have to be locally stored remain small (≈4 for practical purposes), while it is possible to generate maximally entangled states rather than only enhancing the fidelity by a finite amount. The corresponding scheme is called nested entanglement pumping (10) and works as follows: At nesting level 1, elementary pairs created between A ₁ − B ₁ are used to purify a pair shared between A ₂ − B ₂ via entanglement pumping. The fidelity of elementary pairs at level 1 is given by F ₁. It turns out that after a few purification steps, the fidelity of the pair A ₂ − B ₂, F ₂, is already close to the reachable fixed point. The resulting pair with improved fidelity F ₂ now serves as elementary pair at nesting level 2. That is, an elementary pair at nesting level 2 shared between A ₃ − B ₃ is purified by means of entanglement pumping, where always (elementary) pairs (of nesting level 2) with fidelity F ₂ shared between A ₂ − B ₂ are used. The fidelity of the resulting pair A ₃ − B ₃ after a few purification steps is given by F ₃ with F ₃ > F ₂ > F ₁. We remark that an unsuccessful purification step at a higher nesting level requires to restart the procedure at the lowest nesting level 1. Still, the required temporal resources increase only polynomially. The overall procedure can be viewed as a stochastic process, or equivalently as a one side bounded random walk. With each nesting level, one additional particle has to be stored at each location. However, it turns out that for practical purposes (say required accuracy of ε = 10 ⁻⁷) a few nesting levels (≈3) suffice to generate states with fidelity F > 1 − ε (10). Hence, the storage requirements remain very moderate, while the required temporal resources increase.

12.4.3 N → M Protocols, Hashing, and Breeding

The protocols discussed in the previous section operate on two copies of a given mixed state, and produce one copy as output if they are successful. More general protocols are conceivable that operate on N input copies of the state and produce M copies as output. We will refer to such protocols as N → M protocols, and discuss them in this subsection. A protocol of this kind of particular importance is the so‐called hashing protocol, which operates in the limit N, M → ∞. The general idea behind N → M protocols is very similar as in the case of standard recurrence protocols operating on two copies: To obtain information about a subensemble – in this case consisting of M copies of the state – the remaining N − −M copies are measured after applying suitable local operations.

12.5.1 n‐Party Distillability

In the following, we will consider distillability of mixed states in multipartite systems. We denote by A ₁, A ₂, …, A_n n (possibly spatially separated) parties, and by ρ a n‐qubit density operator they share. We will be interested in the entanglement properties of ρ, that is, in its nonlocality properties. As in the case of bipartite systems, one can consider distillation of pure state entanglement, that is, the question whether one can create from many copies of the state ρ some entangled pure states by means of local operations assisted by classical communication. We will assume that two‐way classical communication between any pair of parties is available.

In contrast to the bipartite case, many variations of the problem are conceivable. The most natural one is the n‐party distillation of some genuine n‐party entangled pure state. In this case, all operations are n‐local, where locality is understood with respect to the parties. That is, each of the parties is allowed to operate on their qubits (belonging to different copies of ρ), where the action may depend on results of previous measurements and operations performed by other parties, and arbitrary sequences of this kind of operations can be performed. We remark that any genuine multiparty entangled pure state can be used in the definition of n‐party distillability. This is due to the fact that any pair of genuine multipartite entangled pure states |ψ ₁〉, |ψ ₂〉 can be interconverted if many copies are available. That is, Bell pairs between pairs of parties can be generated from many copies of |ψ ₁〉, which can then be connected or used for teleportation to create any other desired state. To be more precise, from the results of (54) follows that from a genuine multipartite entangled pure state, one can generate Bell pairs shared between pairs of parties in such a way that these Bell pairs form a connected graph. This already implies that each pair of parties can be connected by Bell pairs, and hence teleportation can be applied. Note that this qualitative equivalence of all kinds of multipartite entangled pure states no longer holds when considering a single copy of the state, or some restricted kind of classical communication.

We emphasize that the possibility of distilling a n‐party entangled pure state is equivalent to the possibility of distilling Bell pairs between all pairs of parties, where the remaining parties can assist the distillation process. This provides a convenient tool to prove n‐party distillability of states.

12.5.2 m‐Party Distillability with Respect to Coarser Partitions

One may also consider different partitions of the system into m < n groups of parties, and determine the distillability properties of ρ with respect to a given partition. That is, local operations are understood with respect to the m groups of parties (i.e., the partition), and one attempts to distill a m‐party entangled state shared among the n groups of parties. When considering bipartitions of the system, that is, partitions into two groups of particles, one recovers the situation discussed in Section 12.3. In particular, the criteria for distillation discussed in this section can be applied. Recall for instance that a necessary condition for distillability of a state with respect to a given partition is that its partial transpose is nonpositive (NPT) with respect to this bipartition.

Similarly, one can obtain necessary conditions for distillation with respect to arbitrary partitions (55). For instance, one finds that all bipartitions that include a given m‐partition need to be NPT in order that the state can be m‐party distillable (i.e., the bipartition can be obtained from the m‐partition by joining some of the groups of parties). This follows again from the fact that local operations cannot change the state from PPT to NPT for a given bipartition. Since this is not possible by operations that are local with respect to a given bipartition, this implies that also no local operations with respect to the finer m‐partition can achieve this. However, the desired m‐party entangled pure state is NPT with respect to all such bipartitions. Hence, the necessity of NPT with respect to all bipartitions including the n‐partition of the initial state follows.

12.5.3 Bound Entanglement in Multipartite Systems

The strong requirement that a state needs to be NPT with respect to a large number of bipartitions in order to be distillable leads to various kinds of multipartite bound entangled states with rather puzzling properties. Examples of such states have been discussed in (55). For instance, one can construct states where one can choose for each bipartition independently whether the state should be distillable with respect to this partition or not. This allows one to find states where entanglement can only be distilled if certain groups of parties join. That is, the entanglement is bound when considering the n party system (i.e., the corresponding n‐partition), but can be activated by allowing some parties to join (or, equivalently, by allowing these parties to share entanglement) (see also (34,56)). For instance, states where entanglement can be distilled only if two groups of macroscopic size (i.e., each including, say, more than 40% of the particles) are formed can be constructed.

Even more surprisingly, by classically mixing different states, all of which are nondistillable with respect to the finest n‐partition, one can obtain a distillable state (57). Given the close connection of distillability properties of multipartite states and the quantum capacity of multipartite quantum channels (54), binding entanglement channels (corresponding to the bound entangled states) can be constructed in such a way that their channel capacity is not additive, but in fact superadditive (54).

12.6.1 Graph States

We start by defining graph states. A graph G is given by a set of n vertices {1, 2, …, n} connected in a specific way by edges E. To every such graph, there corresponds a basis of n‐qubit states {|Φ _μ 〉 _G }, where each of the basis states |Φ _μ 〉 _G is the common eigenstate of n commuting correlation operators K_j ^G with eigenvalues (−1) ^µj , μ  = μ ₁ μ ₂…μ_n . To relax notation, we will omit the index G and assume that an arbitrary but fixed graph G is considered. Graph states fulfill the set of eigenvalue equations

12.24

j = 1, …, n. The correlation operators are uniquely determined by the graph G and are given by

12.25

A graph is called two‐colorable if there exists two groups of vertices A,B such that there are no edges inside either of the groups, that is, {k, l} ∉ E if k, l ∈ A or k, l ∈ B. For graph states associated with two‐colorable graphs, which we call two‐colorable graph states, we will split the multiindex μ into two parts, μ  =  μ _A , μ _B , belonging to subsets A and B, respectively.

Graph states have first been introduced in (61), generalizing the notion of cluster states as introduced in (4). A detailed investigation of their entanglement properties has recently been given in the paper by Hein et al. (22). Graph states occur in various contexts in quantum information theory, in which multiparty quantum correlations play a central role. Examples are multiparty quantum communication, measurement‐based quantum computation, and quantum error correction. Prominent examples of two‐colorable graph states are GHZ states, cluster states (4), and codewords of error correction codes (62) (see, for example, (60)). In fact, as has been shown recently (63), two colorable graph states are equivalent to codewords of the CSS codes. We also remark that the correlation operators {K_j } are the generators of a group that is often called stabilizer of the state |Φ ₀ 〉 _G , and the corresponding description in terms of the stabilizers is also referred to as the stabilizer formalism.

We will also consider mixed states ρ, which for a given graph G, can be written in the corresponding graph state basis {|Φ _μ 〉 _G }, ρ = ∑ _μ,ν λ _μν |Φ _μ 〉〈Φ _ν |. We will often be interested in fidelity of the mixed state, that is, the overlap with some desired pure state, say |Φ ₀ 〉 _G , F = 〈Φ ₀ |ρ|Φ ₀ 〉. We remark that depolarization of ρ to a standard form ρ_G ,

12.26

can be achieved by randomly applying correlation operators K_j (59,60). The diagonal elements, in particular the fidelity, are left unchanged by this depolarization procedure. Note that both the notation and the description of the depolarization procedure are similar to those used for Bell states in this chapter. Bell states – as used in this chapter – are in fact graph states with two vertices, connected by a single edge.

12.6.2 Recurrence Protocol

In the following, we will discuss a family of EPPs that allow one to purify an arbitrary two‐colorable graph state. To be precise, for each two‐colorable graph there exists a purification protocol that allows one to obtain the pure state |Φ ₀ 〉 _G as output state, provided the initial fidelity is sufficiently large. The recurrence scheme (59,60) to purify a two‐colorable graph state is very similar to the BBPSSW and DEJMPS protocol to purify Bell pairs. We consider two subprotocols, P1 and P2, each of which acts on two identical copies ρ ₁ = ρ ₂ = ρ, ρ ₁₂ ≡ ρ ₁ ⊗ ρ ₂. The basic idea consists again in transferring (nonlocal) information about the first pair to the second, and revealing this information by measurements.

In subprotocol P1, all parties who belong to the set A apply local CNOT operations (44) to their particles, with the particle belonging to ρ ₂ as source, and ρ ₁ as target. Similarly, all parties belonging to set B apply local CNOT operations to their particles, but with the particle belonging to ρ ₁ as source, and ρ ₂ as target. The action of such a multilateral CNOT operation is given by (59)

12.27

where μ _A  ⊕  ν _A denotes bitwise addition modulo 2.

The second step of subprotocol P1 consists of a measurement of all particles of ρ ₂, where the particles belonging to set A [B] are measured in the eigenbasis {|0〉 _x , |1〉 _x } of σ_x [{|0〉 _z , |1〉 _z } of σ_z ] respectively. The measurements in sets A [B] yield results , with ξ_j , ζ_k  ∈ {0, 1}. Only if the measurement outcomes fulfill mod2 = 0 ∀j – which implies μ _A  ⊕  ν _A  = 0 – the first state is kept. In this case, one finds that the remaining state is again diagonal in the graph‐state basis, with new coefficients

12.28

where K is a normalization constant such that tr( ) = 1, indicating the probability of success of the protocol.

In subprotocol P2, the roles of sets A and B are exchanged. The action of the multilateral CNOT operation is in this case given by

12.29

which leads to new coefficients

12.30

for the case in which the protocol P2 was successful.

The total purification protocol consists of a sequential application of subprotocols P1 and P2. While subprotocol P1 serves to gain information about μ _A , subprotocol P2 reveals information about μ _B . Typically, subprotocol P1 increases the weight of all coefficients , while P2 amplifies coefficients . In total, this leads to the desired amplification of λ _0,0 .

The regime of purification in which these recurrence protocols can be successfully applied is rather difficult to determine analytically, due to the nontrivial structure of the nonlinear maps describing the protocol. Numerical investigation has been performed in (60), and we refer the interested reader to this article for details. We remark here that the fidelity does not provide a suitable measure to compare purification regimes for different number of particles n, as typically the required fidelity will decrease exponentially for all states. This is related to the exponential growth of the dimension of the Hilbert space with the number of particles n. One can alternatively consider the maximum acceptable amount of local noise per particle such that the state remains distillable by means of the recurrence protocol. That is, one assumes that each of the particles belonging to a given graph state is sent through a noisy quantum channel (e.g., a depolarizing channel) to its final location. One then finds for linear cluster states (or, more generally, all graph states with a constant degree) that the maximum acceptable amount of noise per particle is essentially independent of the particle number. For GHZ states, however, the acceptable amount of noise per particle decreases with increasing particle number. That is, GHZ states of large number of particles become more and more difficult to purify as the number of particles increases.

12.6.2.1 Example: Binary‐Type Like Mixture

It is illustrative to consider the purification of a special family of states in some detail. We consider the example of mixed states of the form

12.31

These states arise, for example, in a (hypothetical) scenario where all particles within set A are only subjected to phase flip errors (described by σ_z ), while all particles within set B are subjected to bit flip errors (σ_x ). The iterative application of protocol P1 is sufficient to purify states of the form 12.31, as only information about μ _A has to be extracted. A single application of protocol P1 leads again to a state of the form ρ_A , with new coefficients

12.32

where is a normalization constant indicating the probability of success of the protocol. That is, the largest coefficient is amplified with respect to the other ones. Iteration of the protocol P1 thus allows one to produce pure graph states |Φ _0,0 〉 with arbitrary high accuracy, given the coefficient λ _0,0 is larger than all other coefficients . The family of states ρ_A includes states up to rank , where n_A denotes the number of particles in group A. Depending on the corresponding graph, n_A can be as high as n − 1 and hence the rank can be as high as 2 ⁿ⁻¹.

As a concrete example, consider the one‐parameter family ρ_A (F) with λ _0,0  = F,  = (1 − F)/ for μ _A  ≠ 0, where F is the fidelity of the desired state. Application of protocol P1 keeps the structure of those states and leads to

12.33

This map has as an attracting fixed point for F ≥  . The probability of success for a single step is given by p = F ² + (1 − F)² / .

12.6.3 Hashing Protocol

In a similar way, one can design a hashing protocol for any two‐colorable graph state. The first protocol of this type, capable of purifying GHZ states with nonzero yield, was introduced in (47). Hashing protocols for arbitrary two‐colorable graph states were presented in (60,63). The central tool in these protocols is already evident from Eqs. 12.27,12.29. These equations state how information about indices are transferred from one state to another. To be more precise, information about all indices belonging to set A is transferred from copy one to copy two by the multilateral CNOT operations as specified in the first step of protocol P1, while information transfer occurs for all indices corresponding to set B when the direction of CNOT operations is reversed (as it is done in P2). Again, by determining the parity of the bit values for random subsets – which is done in a similar way as for Bell pairs, but here all bits belonging to set A or B can be determined simultaneously – one can learn the required amount of information in such a way that the remaining ensemble is in a tensor product of pure graph states. To be precise, one needs to learn the classical information of which nonlocal state is at hand.

The yield of the hashing protocol approaches unity for any state diagonal in the graph state basis with λ ₀  → 1, independent of the specific form of the state. This implies that a given mixed state of sufficiently high fidelity F can be purified with nonzero yield using the hashing protocol (combined with the depolarization procedure).

12.6.4 Entanglement Purification of Nonstabilizer States

While all bipartite and multipartite EPPs we have described so far purify stabilizer states, that is, states that are eigenstates of local stabilizer operators, very recently a multipartite EPP was obtained (64) that allows one to purify a nonstabilizer state, in particular a W state,

12.34

This protocol is a 3 → 1 protocol and, among other interesting features, it has not only the 3‐particle W state but also maximally entangled states shared between two of the parties as attracting fixed points (64).

12.7.1 Distillable Entanglement and Yield

Using the standard definition of distillability and yield is clearly inappropriate in the case of imperfect local operations. In particular, no maximally entangled pure states can be created in this case. This implies that no state will be distillable, and that the yield is zero. We therefore have to adopt the definition of distillability and yield to account for these facts.

Rather than demanding that maximally entangled pure states can be created (fidelity F = 1), we will consider the creation of states with certain fidelity. Distillability refers in this case to the possibility of approximating a given target state ψ with fidelity F ≥ F_c . Clearly, such a definition of distillability depends on both the required target state ψ and the desired fidelity F_c . To be more precise, we say that a given mixed state ρ is distillable with respect to a target state ψ and fidelity F_c if one can generate from many possible copies of ρ by means of LOCC a state σ such that the fidelity of σ with respect to ψ is larger than or equal to F_c , 〈ψ|σ|ψ〉 ≥ F_c .

One may also consider the yield of purification procedures corresponding to this notion of distillability, . In this case, however one needs to specify the exact structure of target states. In particular, when considering general distillation procedures (e.g., N → M protocols), one obtains as output a mixed state Γ of a large number of particles. Here, we will demand that the output state Γ is a tensor product of states σ_k , Γ = ⊗σ_k , where each of the σ_k fulfills 〈ψ|σ_k|ψ〉 ≥ F_c . That is, we require that after the purification procedure one possesses independent copies of the state with desired fidelity. One may also use the weaker criterion that all reduced density operators (corresponding to different output “copies” of the output state) have fidelity F ≥ F_c , where are obtained from Γ by tracing out all particles but those corresponding to state k. In this case, however, it is not clear whether the different output states can be independently used for all applications. While their fidelities certainly fulfill F ≥ F_c , there might be classical correlations among the output states that are limiting their applicability, for example, for security applications such as key distribution.

In this context, it would be interesting to see whether the definition of yield with respect to fidelities of reduced density operators is equivalent to those we use here. To this aim, one would need to show that one can produce from an ensemble of states where all reduced density operators have a sufficiently high fidelity an ensemble that consists of a tensor product of copies, where the size of the ensembles might be diminished by a sublinear amount, or the fidelity be reduced by some (arbitrarily small) δ _F. Such a “purification of classical correlations” has, however, not been reported so far.

12.7.2 Error Model

To analyze the influence of noisy local operations, we will consider a simple error model where only local two‐qubit operations are noisy, and the noise is of a simple, local form. More general error models, including correlated noise and also errors in measurements, have been analyzed, leading essentially to the same qualitative behavior of EPPs (6,10,60,65).

We model a noisy two‐qubit operation U by first applying local noise to each of the qubits, followed by the perfect unitary operation U,

12.35

We will mainly assume that local, completely positive maps ℳ _k , ℳ _l are described by white noise (depolarizing channels),

12.36

where σ_j denote Pauli operators with σ ₀ =  . In some cases, we will consider even more restricted noise models, namely local dephasing channels (or phase‐flip channels), and local bit‐flip channels, .

12.7.3 Bipartite Recurrence Protocols

We start by analyzing the BBPSSW protocol, where we assume local white noise channels as described by Eq. 12.36, but for simplicity, perfect local measurements. Given two copies of a Werner state Eq. 12.18, the influence of noisy local control operations – in this case noisy CNOT operations – can be readily obtained. The action of noisy bilateral CNOT operations is the same as applying noiseless bilateral CNOT operations to two copies of Werner states with reduced fidelity. In particular, one finds that the parameter x is reduced to xp ² due to the local depolarizing noise. That is, one applies the original protocol to two copies of Werner states ρ_W (xp ²). Rewriting Eq. 12.20, that is, the fidelity of output state as a function of input state, in terms of parameter x = (4F − 1)/3, one obtains x′ = (4x ² + 2x)/(3x ² + 3). Taking into account the effect of noisy local operations, that is, the reduction of x, we obtain that the output state after applying one purification step is again a Werner state ρ_W (x′) with

12.37

That is, the purification curve (the fidelity of the output state plotted against the fidelity of the input state) is shifted down (see Figure 12.1).

It is now straightforward to determine the maximal reachable fidelity as well as the minimal required fidelity such that entanglement purification can be successfully applied. These quantities are given by the fixed points of the map, Eq. 12.37. One finds

12.38

where the maximum reachable fidelity F _max = (3x ₊ + 1)/4 and the minimum required fidelity F _min = (3x₋  + 1)/4. The threshold value for p such that a finite purification regime remains (i.e., x ₊ > x₋ ) is given by p _min = 0 .9628. This implies that errors of the order of 4% are tolerable.

One can perform a similar analysis for the DEJMPS and (nested) entanglement pumping protocol. There, the fixed points of the corresponding nonlinear maps are more difficult to obtain analytically. One can, however, perform the analysis numerically and obtain (6,10) that (i) the maximum reachable fidelity F _max for the DEJMPS protocol is significantly higher than for the BBPSSW protocol; (ii) the minimal required fidelity F _min for the DEJMPS is significantly smaller than for the BBPSSW; (iii) the threshold for noisy operations described by p _min is smaller for the DEJMPS protocol; and (iv) reachable fidelity, minimum required fidelity and threshold for noisy operations seem to be the same for nested entanglement pumping and for the original DEJMPS protocol (10).

When assuming correlated white noise errors for local operations and errors in measurements of same order of magnitude (34), one finds tolerable errors of about 3% for the BBPSSW protocol, and 5% in the case of the DEJMPS protocol.

12.7.4 Multipartite Recurrence Protocols

A similar analysis can be performed for multipartite EPPs (60). Numerical results for the purification range (minimal required and maximal reachable fidelity) as well as error threshold for linear cluster states of different sizes are given in Figure 12.2. Again, errors of the order of several percent are tolerable.

An important observation is that the threshold value p _min is for linear cluster states independent of the number of particles n. That is, multipartite states of large number of particles also can be successfully purified, and the requirements on local control operations are independent of the system size. This is not true when attempting to purify GHZ states (60), where one finds that the required fidelity of local control operations depends on the particle number.

The qualitative difference of cluster and GHZ states can already be understood from an analytically solvable toy model (60), where one considers mixtures of GHZ states |Φ _0,0 〉 and |Φ_1,0 〉 and a restricted error model of only bit flip errors in set B that keep the structure of such states. Using that bit flip errors in B act as phase flip errors in A, and the fact that subprotocol P1 is sufficient to purify such states, one obtains a lower bound on the threshold value p _min given by p _min =  This follows from arguments along the same lines as used in the derivation of purification curve for the bipartite BBPSSW protocol. Performing a similar analysis for binary‐like mixtures of linear cluster states under this restricted noise model, one observes that the threshold value p _min is essentially independent of the number of particles n, in agreement with the numerical observations for systems of up to size n = 10 for a more general noise model.

12.7.5 Hashing Protocols

While for perfect local operations recurrence protocols have zero yield, only hashing protocols, operating simultaneously on an asymptotic number of copies, have a nonzero yield. For imperfect local operations, the situation changes drastically. When requiring output states to have only a sufficiently high fidelity F ≥ F_c , one finds that recurrence protocols may have a nonzero yield as long as F_c ≤ F _max, that is, as long as the required fidelity is smaller than the fidelity reachable by the protocol. At the same time, the hashing protocol fails completely in the case of imperfect local operations. The reason for this is that one operates on a asymptotic amount of states m → ∞ to reveal one bit of information. That is, one performs m bilateral CNOT operations with a given copy always serving as target state. As each of the CNOT operations is noisy, noise is accumulated in the target state. Assuming that the target state was initially in a maximally entangled pure state, the target state ends up in a Werner state ρ_W (p ^2m ). Clearly, if the amount of noise is too big (as is the case for sufficiently large m, in particular for m → ∞, even if p is close to 1), no information about the remaining ensemble can be extracted. In other words, the information loss due to imperfect local operations exceeds the possible information gain per measurement (maximum one bit). This implies that hashing in its original form cannot be applied in the case of imperfect local operations.

It would be interesting to perform a detailed analysis of the performance of general N → M protocols for finite N in the presence of noisy operations. First steps in this direction have been reported in (49).

12.8.1 Quantum Communication and Cryptography

In a (multiparty) quantum communication scenario, two (or more) parties attempt to communicate and exchange quantum information. They might, for example, want to establish a secret key – to ensure secure classical communication, or to perform distributed quantum computation. When dealing with realistic scenarios, both the quantum channels and local control operations are noisy. This limits the possibility to faithfully transmit quantum information in a direct way, and additional effort is required to overcome the influence of noise.

While classical information can be transmitted over basically arbitrary distances using repeaters, the situation is more complicated in the case of quantum information. Here, the no‐cloning theorem does not permit to copy or amplify a quantum signal. However, one may use techniques from quantum error correction, and encode each qubit of the transmitted signal into several qubits. This technique, known as redundant coding, allows one in principle to faithfully transmit quantum information over noisy channels. One has, however, a substantial overhead, and the requirements of intermediate error detection and correction procedures are rather stringent (same as for fault‐tolerant quantum computation).

An alternative approach is given by entanglement purification. It is sufficient to generate a known maximally entangled state shared between two parties to ensure perfect quantum communication. This is due to the fact that such states (together with classical communications) provide the necessary resource to perform teleportation. Thus, the problem of transmitting arbitrary, unknown quantum states over noisy channels reduces to the generation of a specific, known maximally entangled state as long as classical communication is available. Such a task seems to be much easier to achieve. In fact, when assuming perfect local control operations, EPPs for bipartite systems allow one to faithfully transmit quantum information if the channel noise is not too big. To be precise, a sufficient condition that entanglement purification can be applied is, when sending part of a maximally entangled state through the noisy channel, that the output state has fidelity F > 1/2. If this is not the case – as might for example happen if the distance between parties is large – one may use quantum repeaters, described in detail in Chapter 30.

In the case where not only the channels but also the local operations are imperfect, entanglement purification can still be applied. As we have seen in the previous section, one can increase the fidelity of entangled states – and hence the quality of the channel when using the purified entangled states for teleportation. More importantly, the entanglement produced by entanglement purification, although not perfect, is private (66). That is, although no maximally entangled states can be produced, any eavesdropper will be factored out. This implies that a secret key can be established between two parties, even in the presence of noisy channels and imperfect apparatus (66). This provides an alternative proof of unconditional security of quantum key distribution, and is an important application of entanglement purification for quantum cryptography.

12.8.2 Secure State Distribution

The secure and secret distribution of an unknown multipartite state with high fidelity provides a basic quantum primitive, as multipartite entangled states can serve as a resource to perform certain quantum information processing tasks. The specific type of entanglement determines the tasks that can be performed. Hence, it easy to imagine scenarios where the involved parties do not want any third party to learn which secret state they possess, and they wish at the same time their entanglement to be private. While in an idealized scenario where one assumes perfect local operations, this task can be achieved rather easily, under nonidealized conditions (as one typically faces) the problem becomes nontrivial. (Multipartite) entanglement purification is the main tool to achieve the secure and secret distribution of high‐fidelity multipartite entanglement. However, standard EPPs need to be adopted to take care of additional secrecy and security requirements. In particular, even parties involved in the purification process may not be allowed to learn which state they are purifying.

In (67), three different solutions to the secure‐state distribution problem were put forward. The first solution is based on bipartite entanglement purification, which serves to purify channels. Together with teleportation, this enables one to generate arbitrary multipartite entangled states. The second solution makes use of direct multipartite EPPs, which is combined with basis randomization and adopted accordingly to ensure security. Security in the third solution, again based on direct multipartite purification, is ensured by purifying enlarged states. Each of the solutions offers its own advantages, and there in fact exist parameter regimes (for local noise, channel noise, desired target fidelity) such that one of the three schemes can be applied, while the other two fail.

12.8.3 Quantum Error Correction

Since certain two‐colorable graph states constitute codewords of error correction codes, one may use the purification of these graph states to achieve high‐fidelity encoding without making use of complicated encoding networks (60). In particular, a certain 7‐qubit code (a Calderbank–Shore–Steane (7, 1, 3) code) can be obtained by using a two‐colorable graph state of eight vertices (a cube) as resource, and teleportation. Concatenated codes of this kind can be obtained by appending to each vertex of the cube another cube. Encoding into the graph state can be achieved by a single Bell measurement (60), where the qubit to be encoded is coupled by the Bell measurement to the eighth vertex of the cube. A similar procedure is considered for the (5, 1, 3) code in (62), where the notion of graph codes was introduced. The fidelity of the encoding mainly depends on the fidelity of the two‐colorable graph state used in the procedure described above. Hence, multipartite entanglement purification can be applied to generate high‐fidelity entangled states which are then used to achieve high‐fidelity encoding.

12.8.4 Quantum Computation