30
Quantum Repeater

Wolfgang Dür1, Hans‐J. Briegel1, Peter Zoller1 and Peter v Loock2

1 University of Innsbruck, Institute for Theoretical Physics, Technikerstr. 25, 6020 Innsbruck, Austria

2 Johannes Gutenberg‐Universität, Institut für Physik, Mainz, Germany

30.1 Introduction

The reliable transmission of quantum information over noisy quantum channels is one of the major problems of quantum communication and quantum information processing. One of the main obstacles for high‐fidelity transmission over large distances is the exponential scaling of channel noise and absorption with the distance. Since quantum signals can neither be cloned (1) nor amplified (2), standard techniques from classical communication technology (such as amplification of signals or the usage of repeater stations) cannot be directly applied.

In principle, methods developed in the context of quantum error correction can be used to protect a quantum signal against the influence of noise during transmission. One may for example use redundant encoding, that is, encoding of each logical qubit into a number of physical qubits, using a concatenated error correction code (3). This provides a method where the required resources (overhead) only scale polynomially with the distance. However, the requirements on measurements, local control operations, and channel noise are rather stringent. Before the influence of noise becomes too big (channel error rate must not exceed about 10 −2), error correction needs to be performed. That is, one needs to split the channel into small segments, where at intermediate local nodes error correction is applied. The small tolerable error rates limit the distance between such local nodes. In addition, the acceptable error rates for local operations (required to perform the error correction) are at the order of 10 −5 to 10 −4, far below experimentally achievable accuracies with present‐day technology.

There exists an alternative approach based on entanglement purification (4,5) and teleportation (6). The problem of transmitting arbitrary, unknown quantum states over a noisy channel is replaced by the task to establish a maximally entangled pair (or a pair with high fidelity) between two communication partners. This pair is then used for teleportation (6), thereby allowing for high‐fidelity transmission of arbitrary quantum states, or for quantum key distribution (7). In this case, the state to be prepared (a maximally entangled pair) is fixed and known, which makes this task potentially easier to be performed. By sending parts of maximally entangled states through a noisy quantum channel, one can obtain several copies of nonmaximally entangled states, which can then be postprocessed (using an entanglement purification protocol) to obtain a smaller number of entangled pairs with enhanced fidelity.

In the limit of perfect local control operations, the distillation of perfect maximally entangled pure states is possible (4,5). Hence, faithful transmission over noisy quantum channels can be achieved. However, the acceptable channel noise such that entanglement purification can be successfully applied, is limited. In particular, the channel should be such that one can still produce entangled pairs. For instance, for depolarizing channels this implies that the fidelity of output pairs has to be larger than 1/2. The exponential distance dependence of noise and losses limit the maximal length of the channel. However, substantially larger distances than in the case when relying on quantum error correction techniques are possible.

For long‐distance communication, one should split up the channel into segments of sufficiently small length (and hence sufficiently small channel noise). Then, maximally entangled pairs across each of the segments can be established. Finally, one can use entanglement swapping (8,9), that is, the teleportation of an already entangled qubit, to create maximally entangled pairs over larger distances.

However, the procedure described so far only works if maximally entangled pairs are available. When considering also imperfect local operations, as it is necessary in realistic scenarios, it is no longer possible to create maximally entangled pure states by means of entanglement purification. One can, however, still increase the fidelity and hence the amount of entanglement of the states. The maximal reachable fidelity thereby depends on the specific entanglement purification protocol and, more importantly, on the fidelity of local operations. A remarkable robustness of certain entanglement purification protocols under the influence of noisy local operations has been shown (10,11). In particular, errors of the order of several percent can be tolerated, while the fidelity of the entangled pairs can still be increased. For short‐distance quantum communication, this already provides a way to achieve the desired goal, that is high‐fidelity quantum communication.

For long‐distance quantum communication, one may try to straightforwardly apply the scheme sketched above, that is, use entanglement swapping to create far distant entangled pairs. In this case, however, the fidelity of the resulting long‐distance pair will depend on the fidelity of the small‐distance pairs, and will in fact decrease exponentially with the number of connected pairs. In particular, it might happen that the resulting pair is no longer entangled, and hence cannot be used for faithful teleportation. The solution to this problem is the quantum repeater (10,11), which we discuss in this chapter. The basic idea is to connect only a few short‐distance pairs with fidelity F 0, thereby decreasing the fidelity only slightly, and purifying the longer‐distance pairs to the initial fidelity F 0. Using a nested scheme described below, one can generate entangled pairs over arbitrary distances with only polynomial overhead in the distance. In addition, such a protocol shows essentially the same robustness against local noise as standard entanglement purification protocols, that is, large‐distance quantum communication is feasible even for error rates of the order of percent. We emphasize that quantum repeaters cannot only deal with any kind of channel noise, but also with absorption and losses. The exponential scaling of noise and absorption with the distance can be overcome, and can be translated into only polynomial overhead in resources.

We remark that the generation of long‐distance entanglement is not only useful in the context of quantum communication and quantum key distribution, but may also find applications in distributed quantum computation, for scalable quantum computation, or even for fault‐tolerant quantum computation to improve error thresholds. In this context, high‐fidelity entangled states – created by means of entanglement purification, or more generally, by quantum repeaters – are used to implement nonlocal two‐qubit gates between distant qubits (12–14).

30.2 Concept of the Quantum Repeater

Entanglement purification and connection of entangled pairs via a process known as entanglement swapping (8,9) or teleportation (6) are the main tools required for a quantum repeater. While entanglement purification is discussed in detail in a separate chapter (see Chapter 11), we will briefly review the connection of nonmaximally entangled pairs here. Having these tools available, we proceed by introducing in detail the nested purification loop, the key ingredient of the quantum repeater. Required resources, in particular the polynomial scaling with the distance, will be discussed. We also show how to translate the polynomial overhead in spatial resources (i.e., qubits to be stored at repeater stations) into temporal resources.

30.2.1 Entanglement Purification

Entanglement purification protocols are discussed in detail in Chapter 11. What is important in the present context is that one can generate entangled pairs with fidelity F 1, starting from pairs with some initial fidelity F 0, if (i) F 1 < F max, that is, the required fidelity is smaller than the maximal reachable fidelity of the entanglement purification protocol and (ii) F 0 > F min, that is, the initial fidelity is larger than the minimal required fidelity. The purification range of the entanglement purification protocol is given by the interval (F min , F max). On average, a certain number of elementary pairs, specified by the yield images , will be required to achieve this aim. We call the inverse of this number M in the following. Typically, only a few (say 3–4) purification steps will be required, and hence M will be reasonably small (typically 20–30). In any case, M can be treated as a constant in the following.

30.2.2 Connection of Elementary Pairs

Given two maximally entangled pairs, one may connect them by means of a Bell measurement. That is, given pairs A–C 1 and images , one can teleport the particle C 1 using the pair images , where a Bell measurement on particles images is performed. The resulting state is a maximally entangled pair shared between A and B. That is, the entanglement is swapped and now shared between A and B, and hence this process is sometimes also called entanglement swapping (8,9). If A–C 1 and images are short‐distance entangled pairs shared between A–C 1 and images , where C 1 might be some intermediate location between A and B, the resulting state a long‐distance entangled pair, now shared between A and B (where we dispose the entangled pair shared between images here). In a similar way, one may connect L of these elementary pairs and obtain a maximally entangled pair of distance Ll 0, where l 0 is the distance of elementary pairs. The connection of L pairs may be done (i) sequentially or, more practically, (ii) in parallel. Regarding (i), one first connects at location C 1, then C 2 and so on, where L − 1 connections are required. In (ii), one first connects simultaneously the neighboring pairs at C 1, C 3, …, C L − 1. This leaves us with longer pairs (A–C 2), (C 2–C 4),…, (C N − 2–B). Then, one connects simultaneously these longer pairs at C 2, C 6,…, C N − 2, and so on, until we get a final pair between A and B.

However, if the elementary pairs are nonmaximally entangled, but have some fidelity F < 1, the resulting state after the connection procedure will not be maximally entangled either. This is already clear from the teleportation picture, as a nonmaximally entangled state used for teleportation corresponds to imperfect transmission. For instance, if the elementary pairs are Werner states (15),

30.1 equation

One obtains that the resulting state after L connections (and subsequent depolarization), is again a Werner state ρW (x L) with reduced fidelity x L .01 One finds

30.2 equation

and one may derive a similar formula when taking into account also noisy operations. Let us illustrate the influence of errors by considering the simple error model used in Chapter 11. Imperfect two‐qubit operations are in this case modeled by first applying local white noise (depolarizing channels M) to the individual qubits, followed by the perfect operation, images with images . The action of such a single‐qubit depolarizing channel on one qubit of a Werner state ρW (x) leads again to a Werner state, ρW (px), with reduced fidelity. It follows that the imperfect connection of L Werner states leads, after depolarization, again to a Werner state with reduced fidelity

30.3 equation

where the exponent 2(L − 1) of p can be understood from the fact that L − 1 connection processes (Bell measurements) are required. Similar expressions can be obtained taking into account more general errors (correlated noise, errors in measurement and depolarization) (11), leading essentially to the same behavior.

30.2.3 Nested Purification Loops

We are now in a position to introduce (nested) entanglement purification, the basic notion of a quantum repeater (10,11). Our aim is to create an entangled pair between two distant locations A–B, which are connected by a noisy quantum channel. Due to exponential scaling of channel noise and absorption losses with the distance l, any quantum signal sent through the channel will be absorbed with large probability, and even if it finds its way through the channel it will be completely corrupted. To overcome this limitation, we divide the long channel into N smaller segments of length l 0 = l/N, where l 0 is chosen in such a way that entangled pairs with sufficiently high fidelity F > F min can be created. Several of these pairs are then purified to constitute elementary pairs of length l 0 with some working fidelity F.

Given several copies of such elementary pairs of length l 0 and fidelity F, one creates by (i) the connection of L such pairs and (ii) the repurification to the working fidelity F new pairs of length Ll 0, again with fidelity larger or equal F. The connection of L pairs reduces the fidelity, while entanglement purification restores the fidelity to the initial value. In order that such a process can work, one needs that the fidelity after the connection of L pairs is still larger than F min, the minimum required fidelity for entanglement purification, and that the working fidelity F is smaller than F max, the maximum reachable fidelity of entanglement distillation. Such an elementary purification loop is illustrated in Figure 30.1.

Graphical illustration of Purification loop: Connection of L elementary pairs and repurification to initial fidelity F.

Figure 30.1 Purification loop: Connection of L elementary pairs and repurification to initial fidelity F.

(Briegel et al. (1998) (10). Copyright 2014, American Physical Society.)

The requirement that one always needs to stay within the purification regime of the entanglement purification protocol limits the number L of pairs that can be connected before repurification. Hence, one needs a nested procedure to generate entanglement over a large distance. After one such purification loop, one has pairs of length Ll 0, again with fidelity F. That is, one has an equivalent situation as at the beginning, but now the length of elementary pairs (at nesting level 1) is Ll 0. Performing again a purification loop with these elementary pairs at nesting level 1, one ends up with pairs of distance L 2 l 0 and fidelity F, which now serve as elementary pairs at nesting level 2. Proceeding in the same way, we have that after n nesting levels, the distance of the pairs is Ln , that is, only a logarithmic number n = logL N of nesting levels is required to cover the distance l = l 0 N.

30.2.4 Resources

The logarithmic number of required nesting levels translates into a polynomial number of total resources (see Figure 30.2).

Scheme for Nested purification with an array of elementary EPR pairs.

Figure 30.2 Nested purification with an array of elementary EPR pairs.

(Briegel et al. (1998) (10). Copyright 2014, American Physical Society.)

At nesting level 1, one needs in total LM elementary pairs, as L pairs are connected, and M copies are required on average for repurification. At nesting level 2, blocks of the size LM now play the role of elementary pairs at nesting level 1. In total, L such blocks are connected and again M copies are required for repurification. Hence, at nesting level 2, the total number of resources is given by (LM)LM = (LM)2. Hence, the total number R of elementary pairs will be (LM) n . This result can be re‐expressed as

30.4 equation

which shows that the resources grow polynomially with the distance N.

The number of parallel channels required between the repeater stations is given by images when using recurrence protocols of Refs. (4,5), where all pairs are purified simultaneously. As shown in Refs. (10,11), one can translate the polynomial overhead in spatial resources (i.e., number of required parallel channels, or, equivalently, the number of particles to be stored at each local node) into a logarithmical overhead in spatial resources and a polynomial overhead in temporal resources. That is, the required number of particles that need to be stored at local nodes is at most n + 1, while the temporal resources (i.e., the time required to obtain a long‐distant entangled pair with high fidelity) grows polynomially. This translation of the vertical axes in Figure 30.2 to a temporal axis is achieved using entanglement pumping rather than recurrence schemes of Refs. (4,5) (see Chapter 11 for details). In the case of entanglement pumping, only two particles need to be stored at each site. Elementary pairs need to be sequentially generated, and are used to purify a second pair. This leads to the polynomial overhead in temporal resources. One additional particle needs to be stored at each nesting level, as one pair corresponding to this nesting level needs to be stored, while all other particles are already involved in the generation of elementary pairs at this nesting level. This results into a total of n + 1 = log L N + 1 number of particles that need to be stored at certain repeater stations (the end points). In all other repeater stations – which are used at lower nesting levels – the required spatial resources are smaller.

A further improvement in the required spatial resources has been achieved in (16). In this scheme, only a constant number of qubits (namely two) need to be stored at each site. The basic idea is to make use of entanglement pumping, however once a pair over distance Ll 0 between sites C 1 and C L is generated, one attempts to generate a new elementary pair of distance Ll 0 by generating and purifying a pair of distance (L − 2)l 0 between the two neighboring sites C 2 and C L − 1. This is possible because all intermediate repeater stations are not occupied with storage of another qubit, only sites C 1 and C 2 are. Finally, short‐distance pairs between C 1 − C 2 and C L − 1 − CL are generated and connected with the pair C 2 − C L − 1 to form a new elementary pair of distance Ll 0, which is used to purify the initial pair. This scheme avoids the logarithmical increase of spatial resources with the distance, while leading to slightly more stringent error thresholds.

In the schemes described above, it is assumed that memory errors can be neglected at timescales required for the generation of long‐distance entangled pairs. That is, entangled pairs need to be reliably stored until additional pairs required for entanglement purification are available. For the schemes with reduced spatial but increased temporal resources, this becomes challenging for larger distances as entangled pairs are created sequentially. The times to generate long‐distance entangled pairs (and hence the required storage times) on an intercontinental scale have been estimated to be of the order of seconds. This implies that a reliable quantum memory with sufficiently long decoherence times is a necessary ingredient of a quantum repeater.

It is also worth mentioning that arbitrary channel errors, including absorption and losses, can be handled and overcome by the quantum repeater. In the case of absorption, one can devise schemes to detect the absence of a traveling qubit (e.g., a photon) (17). The detection of such absorption errors is sufficient to guarantee that the standard quantum repeater scheme – with respective polynomial resources – can be applied. This scheme can at the same time overcome arbitrary additional channel noise, provided that absorption and error probability are not too big (which can always be achieved by choosing channel segments sufficiently short).

30.3 Proposals for Experimental Realization

A quantum repeater requires two main ingredients: (i) the possibility to generate entanglement over relatively short distances and (ii) the possibility to store and manipulate a few qubits at each repeater station to perform entanglement swapping and entanglement purification. The requirements on physical qubits for (i) and (ii) differ. While (i) is achieved by transmission of entangled qubits, and hence photons are ideal candidates to perform this task, (ii) is based on controlled manipulation and storage of qubits, where long coherence times and strong interactions between qubits are required. Since photons are in general difficult to store and interact only weakly, atomic qubits or solid‐state‐based qubits seem to be more suitable in the case of (ii). This implies that interfaces between flying qubits (e.g., photons) and qubits required for storage and manipulation (e.g., trapped atoms or ions) are desirable. In fact, theoretical proposals for such interfaces have been put forward (18), for example, based on atoms surrounded by a cavity. In the following, we will briefly discuss theoretical proposals for the implementation of quantum repeaters.

30.3.1 Photons and Cavities

An implementation of a quantum repeater, based on atomic qubits for storage and manipulation, and photonic qubits for transportation was proposed in (17). In this scheme, atoms are embedded in high‐finesse optical cavities that are connected by optical fibers. Atomic and photonic states are mapped onto each other using the interface proposed in (18). The usage of auxiliary atoms in each of the cavities allows one to design a scheme that can detect and correct photon losses (absorption) that may occur during transmission. That is, the usage of a “back‐up atom” allows one to check whether the transmission of the photon was successful or not, while maintaining the coherence (and possible entanglement) of the transmitted quantum information. In case of nonsuccessful transmission (absorption of the photon), the process can simply be repeated. Also, additional errors arising due to nonstationary environment – which leads to phase noise – can be corrected using a purification protocol (17). This finally allows for the design of a quantum repeater that can generate entangled states over large distances.

30.3.2 Atomic Ensembles

A scheme for a quantum repeater based on atomic ensembles interacting with light was proposed in (19). Details of this scheme can be found in Chapter 27.

30.3.3 Quantum Dots

The implementation of a quantum repeater in a solid‐state architecture has been proposed recently (20). In this case, the primary goal is to establish high‐fidelity entangled pairs within a single solid‐state device. That is, the quantum repeater is there not a tool to achieve high‐fidelity quantum communication, but rather a source for distant entangled pairs within in the device. These entangled pairs can, for example, be used to implement two‐qubit gates between distant qubits in a quantum processor.

The solid‐state architecture in question consists of quantum dots, where spin degrees of freedom of trapped electrons are used for quantum processing. Rather than using single electron spins directly, each (logical) qubit consists of two spins, where images with images . That is, a (dynamical) decoherence free subspace is used, thereby suppressing most dominant noise sources and increasing coherence times by several orders of magnitude. Entanglement is generated between logical qubits, and hence both entanglement purification and connection have to be adopted accordingly. Locally generated entangled states are distributed by moving electrons – which is achieved by charge manipulation of trapping potentials – and eventually purified and connected following the standard repeater scheme. Based on directly available operations in such a set‐up (partial Bell measurement, exchange interaction), a novel entanglement purification scheme and connection scheme for entangled states of logical qubits was designed (20). The resulting purification map within the logical subspace is exactly the same as that for the recurrence protocol of (5) – also discussed in Chapter 11. In addition, all leakage errors, that is, errors leading outside the logical subspace, are also corrected. The proposed scheme provides a valuable tool to generate distant entanglement in such quantum dot devices, which may, for example, be used as a basic resource in scalable quantum computation architectures.

30.4 Summary and Conclusions

A quantum repeater is a fundamental tool for long‐distance quantum communication, with potential applications also in scalable quantum computation design. While an experimental realization of a fully operating quantum repeater has not been reported so far, important parts required for a quantum repeater have already been experimentally demonstrated. These demonstration experiments include the generation of entangled pairs over a few tens of kilometers, entanglement swapping (8,9) and entanglement purification (21). These experiments have been performed with entangled photons. Given the moderate error thresholds of the order of a percent, reliable creation of long‐distance entanglement on demand seems feasible.

Acknowledgments

This work has been supported by the Austrian Science Foundation (FWF), the European Union (IST‐2001‐38877, ‐39227, OLAQUI, SCALA), the Österreichische Akademie der Wissenschaften through project APART (W.D.), and the Deutsche Forschungsgemeinschaft (DFG).

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