{'fraction_non_alphanumeric': 0.022636743417856795, 'fraction_numerical': 0.0069221801224132904, 'mean_word_length': 4.513994910941475, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Relay QKD Networks&Bit Transport

We show how it is possible to operate end-to-end relays on a QKD network by treating each relay as a trusted eavesdropper operating an intercept/resend strategy. It is shown that, by introducing the concept of bit transport, the key rate compared to that of single-link channels is unaffected. The technique of bit transport extends the capability of QKD networks. We also discuss techniques for reducing the level of trust required in the relays. In particular we demonstrate that it is possible to create a simple quantum key exchange scheme using secret sharing such that by the addition of a single extra relay on a multi-relay channel requires the eavesdropper to compromise all the relays on the channel. By coupling this with multi-path capability and asynchronous quantum key establishment we show that, in effect, an eavesdropper has to compromise all relays on an entire network and collect data on the entire network from its inception.


Introduction
Quantum Key Distribution (QKD) is an elegant and ingenious application of quantum mechanics to the problem of key establishment in classical cryptosystems (see [1,2] for the 2 seminal works that established the two basic methods of QKD; for some excellent reviews of QKD see [3]). Its principal limitation, in practical terms, is the distance over which it is possible to exchange the keys securely. Whilst the security of the technique is robust to loss, detector imperfections mean that even modest amounts of loss, such as that seen in optical fibres, can limit the distance of the technique over a single channel. Any widespread implementation of QKD must, of course, operate on a network, and this introduces further complications. Active processing of any quantum signal will, in general, destroy the integrity of the quantum states and render quantum key exchange impossible. QKD works beautifully on passive optical networks [4,5], but passive network switching elements can also be seen as effectively introducing further loss, thus adding to the difficulty of operating a QKD network over the distances that might be required in any realistic network implementation.
The current design of QKD network trials (see, for example, [6]) adopt an approach based on relay nodes operated in a link-by-link fashion in order to extend the distance (we use the terminology 'LL relays' to describe such linkby-link operation). Whilst this is an eminently practical solution to the problem of extending the distance of any QKD scheme it is preferable, from a security perspective, to have the capability of exchanging end-to-end keys between any two network users. Bechmann-Pasquinucci and Pasquinucci [7] show how, by considering a relay to be a trusted eavesdropper operating a standard intercept/resend strategy, it is possible to construct such an end-to-end relay (we shall use the term 'IR relays' to describe these). It is argued in [7] that such relays cannot be used, however, to extend the distance for a QKD channel. By a suitable adaptation of the operation of these IR relays we show here how, contrary to this conclusion, it is indeed possible to use such relays to extend the distance.
Both LL and IR relays must be trusted network elements and this, to some extent, reduces the attractiveness of the technology. After all, perhaps the most compelling feature of single link QKD is to allow key exchange in a provablysecure fashion. We discuss techniques for reducing the level of trust required in such relay solutions. In particular, we show that by adding in redundancy at the relay level, we can construct channels in which an eavesdropper has to compromise all of the relays. The implication of this is that our network users, Alice and Bob, only need to be able to trust at least one relay on such channels 1 .
A naive application of IR relays would suggest that each relay reduces the effective key rate by a factor of 2 from the key rate that can be achieved by single link QKD. This exponential reduction in key rate as we add more relays is catastrophic for our original goal of extending the distance. By introducing the notion of bit transport we show that, in fact, the single link QKD rate can be achieved over channels with any number of relays. Bit transport is a powerful classical post-processing technique that can be used to extend the flexibility and capability of QKD networks [8] and we discuss some applications here, including the ability of Alice and Bob to perform eavesdropper detection on a duplex QKD channel without the necessity of public bit comparison.

Intercept-Resend Relays
An IR relay as envisioned in [7] acts precisely as an eavesdropper performing an intercept/resend strategy on each transmitted quantum state. The relay chooses at random one of the coding bases used by Alice for each timeslot and simply retransmits the state corresponding to the measurement result. The difference between an eavesdropper and the relay is that now we consider the relay to be a cooperative entity on the network. We can also view the action of the relay as another network user, such as Bob, whose function is to re-transmit the measured result to the next network user. One of the nice features of using relays in IR mode is that we do not have to consider a single dedicated path between Alice and Bob, but can consider the establishment of the shared key on multiple distinct paths between them on the network. This feature alone allows us to put less trust in any individual path and the relays on that path. When this is combined with the technique of secret-sharing we can reduce the level of trust required in any individual relay still further [9,10]. We shall discuss the issue of trust as it applies to the relay nodes of a quantum network later.
We shall assume the coding bases used are represented by the operatorsX andŶ which have eigenstates |± X and |± Y , respectively, and we adopt the coding interpretation |+ ≡ 1 and |− ≡ 0. Thus, the quantum key is to be established using the BB84 protocol [1]. We shall label the relays on a channel by R j and the two users who wish to establish a key, Alice and Bob, by A and B, respectively. The distance between A and B is such that a quantum key cannot be successfully established by single link QKD. A channel between Alice and Bob with a single relay is of the form The possible transmissions on the channel, when Alice chooses to use the coding basis represented byX, can be split into the 4 cases shown in the table below with a similar table for the situations in which Alice makes the choice of coding basis represented byŶ . It is clear that a quantum key can be established on the first of these entries in the table in which we have all 3 parties using the same coding basis. At first sight this suggests that an IR relay can be used to extend the distance for key establishment, but at the expense of sacrificing 3/4 of the transmissions, instead of the loss of 1/2 of the transmissions on filtering that we would obtain for single link QKD. However, if the channel between Alice and the relay is already operating at the limit for single link QKD then simple re-transmission of the measured results by the relay will lead to a quantum signal between the relay and Bob that is insufficient to overcome the signal-tonoise limitations at Bob's detector. Bechmann-Pasquinucci and Pasquinucci [7] prove that a relay channel operated in this fashion cannot be used to extend the distance for QKD. The argument presented in [7] is applicable to the situtation when we think of the overall channel A → R → B as a single channel. The problem is that the final quantum signal that gets to Bob is too weak to be successfully distinguished from the detector noise. However, we do not need to operate this as a single channel, but can effectively split the communication into 2 separate channels. In order to overcome the signal-to-noise problem at Bob's detector we can do one of two things: 1. The relay delays retransmission until sufficient data has been gathered so that the signal sent on to Bob is of sufficient strength to overcome the SNR limitations at Bob's detector 2. The relay pads the retransmission with a separate QKD communication between the relay and Bob. The relay can keep track of which timeslots are from Alice and which are padding qubits. The padding qubits can be used to establish a separate quantum key between the relay and Bob, if desired.
Using either of the techniques (1) and (2) we see that we can use IR relays to successfully extend the distance for quantum key establishment. The apparent problem with this is that we have halved the key rate between Alice and Bob. It is clear that the addition of another relay further exacerbates the situation and each additional relay will introduce a further loss factor of 2 into the final key rate. However, this reasoning is based on the assumption that only the channels where all entities have used the same coding basis can be used to establish an end-to-end key between Alice and Bob. As we shall see in the next section, the notion of bit transport reveals that this is an unduly restrictive assumption and that the introduction of relays does not affect the final key rate.

Bit Transport
In order to illustrate the technique of bit transport we shall consider a channel that requires two relays, R 1 and R 2 , to span the distance between A and B. Each relay is operated in IR mode as described above. The transmission sequence is schematically described by In the timeslots where Alice chooses the coding basis X we have the following possible situations 2 : We can think of the entire transmission (in which Alice chooses the coding basis X) as being made up of 8 distinct channels, in which each channel occurs at random in the sequence of timeslots. There is, as discussed previously in the case of the single-relay channel, an equivalent table describing the situations where Alice chooses the coding basis Y . Channel 3 in the above table, for example, represents those instances where relay 1 chooses the coding basis X, relay 2 the coding basis Y , and Bob chooses the coding basis X.
In full, therefore, we have a complete transmission consisting of N timeslots. On average, in N/2 of these Alice will choose the coding basis X and the coding basis Y in the remaining timeslots. Thus the entire transmission can be partitioned into 16 channels, distributed at random in the timeslot sequence, which represent all of the possible coding choices of the participants. In each timeslot the participants record a tuple (t, c, b) where t is the index for the timeslot (which we can simply think of as an integer so that t = 1, 2, 3, . . . , N ), c is the bit representing the choice of coding basis (we adopt the convention that X ≡ 0 and Y ≡ 1) and b is the actual received/transmitted bit value. Thus in a timeslot t we have the participants recording the following tuples: The partitioned channels we have described above are generated by the possible values of the bit string c A c 1 c 2 c B . The 8 partitioned channels in the above table are given by the possible values of the bit string 0c 1 c 2 c B . In ideal operation we therefore have: which simply states that if adjacent participants in the channel choose the same coding basis they will record the same bit value. In other words, if adjacent users choose the same coding basis then they could establish a quantum key on that link, if they desired. We describe these links as 'open' and links in which adjacent users have chosen a different coding basis as 'closed'. Using the symbol to denote an open link and the symbol to denote a closed link the 8 channels in the above table can be rewritten as Consider channels 2 and 5. We can see that a quantum key can only propagate between A, R 1 and R 2 in timeslots described by channel 2, and only between R 2 and B in timeslots described by channel 5. However, Alice and Bob are only interested in establishing a shared bit at the end of the process and so R 2 can 'repair' the broken link by correlating a timeslot from channel 2 configurations with another timeslot from channel 5 configurations such that the same bit value is propagated. For example, suppose in timeslots 5 and 12 we have these configurations, respectively, then we might have the recorded tuples Alice R 1 R 2 Bob Channel 2 (5, 0, 1) (5, 0, 1) (5, 0, 1) (5, 1, 0) Channel 5 (12, 0, 1) (12, 1, 0) (12, 0, 1) (12, 0, 1) The relay R 2 simply needs to announce to Alice and Bob that Alice should use the bit value from timeslot 5 and Bob should use the bit value from timeslot 12. Thus 2 partially closed channels are effectively combined into one open channel and Alice and Bob can establish a shared bit value by combining their data in timeslots 5 and 12. Alice ignores her recorded data in timeslot 12 and Bob ignores his recorded data in timeslot 5.
For each partially closed channel in the above there is a dual channel which can be used to correlate the data so that a shared bit value can be established with the assistance of the intermediate relays. There are 6 partially closed channels, one fully open channel (X X X X) and one fully closed channel (X Y X Y ). Thus by transporting bits across broken links in this fashion we can see that Alice and Bob can establish N/2 shared bits, on average, just as they would for single link QKD.
If a partitioned channel is described by the bit string c A c 1 c 2 c B then the bit string for the dual channel is given by where ⊕ is the bitwise exclusive-or of the bit strings andc is the bit complement of c. This can be extended to the situation with n relays: where we can partition the data into 2 n channels distributed randomly across the sequence of timeslots. If one of these partitions is described by the bit string c A c 1 c 2 c 3 . . . c n−2 c n−1 c n c B then its dual partition is described by It is clear that to each partially closed channel there is a single unique dual channel (also partially closed). The fully-open and fully closed partitions are, of course, duals of one another. Hence we can see that if there are N timeslots, then, even with n relays, we can use this bit transport technique over dual partitions to establish a shared key between Alice and Bob that is, on average, N/2 bits in length.
Bit transport, as outlined above, is a classical post-processing technique (although it can be performed during the quantum key transmission). When seen in this perspective we note that we can think of the entire quantum transmission as a series of N sequential experiments. It is the post-processing of the data that allows us to establish a shared secret key from this data. This is true of single link QKD as it is for relay-based schemes. This change of perspective allows us to consider new network operations such as asynchronous quantum key exchange [8]. Indeed, by considering the data collected on the quantum transmissions simply as 'data' we might also wish to use the network relay nodes as a sensitive network monitoring tool where problems with particular paths or links can be detected and communications re-routed. There is no reason why we have to use the recorded quantum data only in security applications.

QKD with Bit Revelation
As an illustration of the application of this perspective of a QKD channel as a series of sequential experiments coupled with a post-processing technique, we consider the possibility of key establishment over a quantum channel in which the bit values are publicly revealed. In the standard operation of QKD the basis information is revealed and the bit values are kept secret. We show that it is also possible to establish a shared secret key by revelation of the bit values, but keeping the basis information secret.
Assuming a lossless channel and ideal detections Alice and Bob will each possess a set of N tuples {(t, c A , b A )} and {(t, c B , b B )}, respectively. Each of Alice's tuples contains 2 secret bits of information; the basis bit c A and the coded bit value b A . In the standard BB84 protocol [1] the basis information is publicly revealed and a sifting process employed to select only those tuples, for the same t, where c A = c B . Public examination of a small random sample of these sifted tuples reveals (under our assumption of ideal conditions) the presence of an eavesdropper. The possible transmissions and outcomes on a QKD channel operating the BB84 protocol are shown in the table below in terms of the recorded tuples (t, c, b) for a given timeslot labelled by t

Alice
Bob Probability where the probability refers to the probability that, for the transmitted state, Bob records that tuple given his measurement basis. At this point everything is per the usual BB84 protocol and neither basis nor bit information has been revealed publicly. Let us consider the case where Alice chooses to reveal, for a given timeslot, the bit value b A , whilst keeping the basis bit c A secret. We shall suppose that in this timeslot she has recorded the tuple (t, 0, 0) A so that her revealed bit value is b A = 0. If, in this timeslot, Bob has chosen a basis value of c B = 0, then he is using the same basis as Alice and he will have recorded the triple (t, 0, 0) B with unit probability. With Alice's announcement he knows that he has measured the correct bit value, but he cannot tell whether Alice's tuple is (t, 0, 0) A or (t, 1, 0) A . He knows that with his chosen basis and the measurement result his tuple is more likey (with a probability of 2/3) to have been a result of Alice's tuple (t, 0, 0) A , but he cannot be certain. In other words, he cannot unambiguously decode his measurement to yield Alice's secret basis value if his recorded tuple is (t, 0, 0) B . If he has chosen a basis value of c B = 1 then his possible tuples resulting from this choice and Alice's input tuple are (t, 1, 0) B and (t, 1, 1) B . If he records the tuple (t, 1, 0) B then his bit value is in agreement with Alice's revealed bit but he cannot tell (with equal probability) whether this has arisen from an input tuple of (t, 0, 0) A or (t, 1, 0) A . However, if his measurement result yields the tuple (t, 1, 1) B then he knows with certainty that, with Alice's revealed bit value of b A = 0 this could only have arisen from an input tuple of (t, 0, 0) A and so he knows that Alice's basis bit (c A = 0) is the complement of his basis bit (c B = 1). Similar considerations apply for each of Alice's possible input tuples so that we can establish the following adapted protocol for quantum key establishment using the BB84 protocol with bit revelation : This protocol variant of BB84 discards (on average) 5/8 of the transmitted timeslots and is worse than the standard protocol in this regard. However, it offers a slight advantage over the standard BB84 protocol in that an eavesdropper employing a basic intercept/resend strategy causes a higher error rate on the key data.
Whilst we would not suggest adopting this protocol over the standard operation of BB84 we note that if the eavesdropper employs a strategy that is optimised for the BB84 protocol with basis revelation then it may not be optimsed for this adapted BB84 protocol with bit revelation. As both of these protocols rely on post-processing techniques Alice and Bob can select, at random, which protocol to operate for a given recorded timeslot. Thus an eavesdropper needs to optimise any strategy over both protocol variants. Indeed, in general, an eavesdropper needs to optimise over all possible post-processing protocols that Alice and Bob can employ.
Here, of course, we have not undertaken a full security analysis of this alternative protocol. If such a protocol were ever required to be used in practice then a full security analysis is necessary, but it would unduly distract us from the main point we are making. Here we are using this as an illustration of the flexibility of considering a QKD channel as a quantum 'experiment' which simply collects data. It is the post-processing of this data that gives us the possibility of bit transport and of randomizing over different choices of post-processing protocol.

QKD Without Public Bit Comparison
In order to detect errors without any public bit comparison Alice and Bob must operate a duplex QKD channel. That is, Alice transmits photons to Bob, according to the BB84 protocol, and Bob transmits photons to Alice according to the BB84 protocol. For convenience we shall imagine these to be interleaved so that for odd timeslots a photon is transmitted by Alice whereas Bob transmits in even timeslots. These can, in fact, be two entirely separate transmissions; all that is required is that we can uniquely correlate a particular transmission with a particular measurement, which we can achieve using the technique of bit transport. We can view the duplex channel as being of the form A → R → B folded back on itself. As above, a full security analysis of this protocol would unduly detract from the point we are making here; the duplex channel provides us with an example of another capability unlocked by applying post-processing techniques such as bit-transport.
It is best to illustrate the technique with an example. The table below gives an example duplex transmission over 18 timeslots. We assume that each timeslot is occupied and that each photon reaches its destination. This is, of course, not true in practice, but it is easy to accommodate timelots where nothing is transmitted or received. We use a tilde to denote values in even timeslots (transmissions from Bob to Alice) so thatb B , for example, would represent Bob's transmitted bit value in one of these even timeslots. We also use the symbol to denote the bit in situations where Alice and Bob choose a different coding basis. For clarity we have also reverted to the description of the coding basis using X or Y rather than using the bit value to denote this choice.
Alice informs Bob of her basis choices for both transmission and measurement. Bob filters this data into 3 sets. The first is the data for which they expect no agreement because they have chosen different bases. In the table this first set consists of timeslots t = 1, 4, 7, 10, 12, 13, 17. Bob informs Alice of these timeslots and they are discarded. The second and third sets consist of the remaining odd and even timeslots, respectively. In a perfect world and in the absence of an eavesdropper, Alice and Bob should have the same recorded tuples for sets 2 and 3.
In order to check this agreement Bob chooses a timeslot from set 2 and reads the measured bit value b B from the recorded tuple. He then looks for an element of set 3 in which his transmitted bit valueb B = b B . He sends Alice the t value for these timeslots. Alice compares her bit valuesb A and b A from these two timeslots. They should be equal. If there are errors on the channel, caused by an eavesdropper or practical imperfections, then there is a finite probability that Alice's comparison will fail. If Bob transmits a sufficient number of these timeslot pairs from sets 2 and 3 then the probability of an error remaining undetected can be made negligibly small.
Of course, by revelation of which timeslots have equal values, Bob has leaked information to any eavesdropper. If the eavesdropper has measured both channels using a standard intercept/resend strategy in the coding bases, then if only one of her basis guesses were correct she would know the bit value for both channels A → B and B → A. Instead of searching for an identical bit value from sets 2 and 3, Bob could simply look at sequential tuples and transmit an extra bit of information that tells Alice whether or not to perform a bit flip on her recorded bit values from set 3. Using the example data from the table we can see that these sets lead to the following tuples recorded by Bob The actual bit values recorded by Alice and Bob are never publicly revealed, although 1 bit of information, the parity bit f , is revealed. The fundamental difference between this protocol and BB84 operated in single link QKD mode, is that the bit transport mechanism allows Alice and Bob to use their entire filtered transmission to check for errors, rather than just a random sample which is then discarded. Alice and Bob, therefore have access to their entire data set, without compromising their final key, to gather information about what is happening on the channel between them. Of course they could, if they choose, also perform a standard random sampling on the data from sets 2 and 3 (discarding timeslots in which bit values are publicly revealed).
In practice, the duplex technique would not be used to establish a key since single link QKD can already be operated in such a fashion as to give unconditional security [11][12][13][14]. However, one could imagine situations where this duplex technique may be useful to obtain extra information about the channel. Indeed, given that we would envision that in any practical network both Alice and Bob will have both detection and transmission capability then it costs little extra to perform duplex transmissions anyway. In the next section we consider a relay using a different technique employing GHZ-type states which performs a kind of automatic bit transport.

Relay Key Distribution with GHZ-type States
In the preceding discussion we have assumed a key distribution scheme employing a BB84 protocol and the transmission of spin-1/2 states. This combination is one of the most robust practically and can easily be achieved with quasi-single photon sources and optical fibres. Indeed, commercial QKD systems employing these mechanisms have been available for a number of years. It could be argued that the current explosion of interest in quantum information processing was kick-started by the development of QKD. It is clear that whilst QKD based on quasi-single photon sources is a robust and practical technology, it is only the tip of the iceberg as far as the possibilities for the exploitation of quantum mechanics in information processing. With increasing advances in the control and manipulation of entangled sources we expect to see many new and fascinating technologies emerge in the not too distant future. With this in mind, therefore, we consider one application of intermediate nodes on a channel that is currently practically infeasible.
In this application the intermediate node is not really acting as a relay, as such, but it is interesting because the GHZ-type correlation automatically achieves some of the features of bit transport. Let us consider an intermediate node between Alice and Bob who is able to prepare spin-1/2 states in the correlated form (and we use a binary notation for the state here, rather than our previous |± ) where |B 1 and |B 3 are Bell states given by One of the particles in the Bell states is sent to Alice and its partner to Bob whilst the intermediary holds on to the remaining particle. A number of such particles are sent to Alice and Bob in well-defined timeslots. All 3 particles are stored (technology permitting, of course) for future use. When Alice and Bob wish to exchange a secret key they inform the intermediary who then makes a measurement of the spin variable of his stored particle. The intermediary publishes this list, which is just a random binary bit string, to Alice and Bob over a public channel. Alice and Bob, for each particle, then make a spin measurement. The measurement of the intermediary will have projected their particles into either a correlated state or an anti-correlated state. Accordingly, with the random binary bit string sent by the intermediary, Alice and Bob can now establish a shared secret random bit string from their measurements. For example, if Alice measures a bit value 1 and this is associated with a bit value of 1 from the bit string sent by the intermediary, then she knows that Bob will have measured the bit value 0. They can establish a key from the inferred parity of their measurements. The interesting feature of this scheme is that Alice and Bob need not communicate directly, nor need they perform their measurements synchronously (assuming secure storage). Furthermore, they can check whether their particles are indeed correlated (after the measurement of the intermediary) by choosing a random sample and checking to see whether the Bell inequality is violated, although this does require some public communication. Another useful feature of this scheme is that all of the transmitted particles are used to establish the key.
As it stands, this scheme is vulnerable to a meet-in-the-middle attack by an eavesdropper who need only intercept both particles sent to Alice and Bob, perform a Bell measurement to determine whether the state is correlated or anti-correlated, and re-transmit the particles in a corresponding Bell state to Alice and Bob. The Bell measurement of the eavesdropper will project the intermediary's spin into the state |0 or |1 at random. In order to frustrate this attack, the intermediary randomizes the transmitted particles over the timeslots. Thus, if the intermediary prepares 5 states for transmission to Alice and Bob, then Alice's particle from state 1 could be sent in timeslot 1, with the particle sent to Bob in timeslot 1 coming from, say, state 4. When Alice and Bob wish to establish a key the intermediary publishes the random binary string representing the spin measurements made, and also a list of tuples containing the information about which timeslots must be associated together. Again we see the key idea underlying bit transport, that of an intermediary correlating disparate data sets, finding an application here.
In a functional sense, this scheme is a kind of quantum version of the Needham-Schroeder protocol [15] for the establishment of keys. The Needham-Schroeder protocol is at the heart of the Kerberos protocol for the exchange of symmetric keys. Kerberos, and other classical key management protocols, will assume a central importance in the advent of a working quantum computer able to tackle problems of significant input size. We can see that this quantum protocol is much neater in principle than the classical protocols which require several challenge-response communications between the parties. Of course such quantum protocols are beyond the reach of existing technology, but the technology advances required to construct a quantum computer of significant capability may also be instrumental in allowing us to practically implement the more speculative protocols involving GHZ-type states such as the one we have outlined here.
In this application, the intermediary must be trusted. This is also true of the IR relays described above when acting to extend the distance for QKD. Indeed, in the QKD distance extension application, as described above, if Eve manages to compromise any one relay, then she will compromise the entire communication. An ingenious scheme [9] using distinct physical paths on a network, together with secret sharing, can alleviate some of this trust burden. As we now discuss, this technique can also be adapted to work on single dedicated paths containing relays.

Trusting the Relays
One of the most attractive features of QKD is its promise of unconditional security, when correctly implemented to overcome any side-channel attacks (see, for example, [16]). If we need to add extra trusted relays in order to operate the technique over the required network distances, then from a security perspective, the benefit of QKD over conventional key distribution systems becomes much harder to argue. The protocol operation we have so far discussed means that an attacker only has to compromise one relay in a channel in order to access the final key. Using multi-path networks, together with secret-sharing, can alleviate this problem somewhat [9]. In essence, the distinct network paths between Alice and Bob can be thought of as shares and the final key obtained by the binary addition of the keys obtained on the different paths. An attacker, therefore, has to compromise all paths in order to obtain the final key.
We can achieve the same functionality over a single path containing relays by creating distinct logical channels on that path. This is best illustrated by an example. We suppose that our two end users, Alice and Bob, require 3 relays to effectively span the distance between them and establish a quantum key. The path is, therefore, of the form Let us add an additional relay spaced such that any 3 of the 4 relays are sufficient to span the distance for the purposes of quantum key establishment. With a suitable re-labelling of the relays the channel is now of the form Now let us suppose that we operate the relays such that for any transmission timeslot they 'drop out' at random. By drop out here we mean that they allow the quantum signal to pass through unmeasured and unaffected. If 2 or more drop out in any one timeslot a quantum key cannot be established on the path because the quantum signal will not span the distance. If the probability that any one relay drops out is p, and we assume independent relays, then the probability that any timeslot will result in a successful transmission is where we describe such a path as 'open'. For p = 1/2 we see that 5/16 of the timeslots, on average, will lead to an open path. However, what Alice and Bob require is to use only timeslots in which precisely 3 out of the 4 relays are operational. This is, as we shall see, because they are going to establish their final key by binary addition of the keys established over the separate logical channels in which precisely 3 out of the 4 relays are operational. Thus the fraction of useful timeslots on average, for Alice and Bob, is given by which for p = 1/2 is 1/4. The useful timeslots for Alice and Bob will be those in which the transmission has been effected by one of the following relay configurations If we denote the keys established for each of these logical channels as QK ijk then Alice and Bob will establish their final quantum key QK AB by the simple expedient of performing the binary addition for these separate quantum keys so that Each key is a share of the final key and only those participants with access to all shares, in this case Alice and Bob, can recover the final secret key. An eavesdropper with access to the information of only one relay cannot establish any information about the final key. An eavesdropper with full control of one relay might decide to disable the drop out feature so that it is always operational. However, this can easily be detected by the legitimate network participants by examination of the quantum data since the relay is then effectively acting as an eavesdropper performing a standard intercept/resend strategy in each timeslot. In order to compromise the entire channel the eavesdropper needs to compromise every relay on the channel. In other words, in order to establish a quantum key, Alice and Bob need only trust one relay on the channel. If we couple this technique with the use of distinct physical paths on the network, then we can see that in order to compromise the key QK AB an eavesdropper needs to compromise all of the relays on the network that could be involved in the transmission between Alice and Bob. Whilst this is theoretically possible, it is a considerable practical challenge for any eavesdropper. If we also employ bit transport to enable asynchronous quantum key establishment [8] then not only must an eavesdropper compromise all the relays on the channel but must compromise those relays and collect data on all possible paths between Alice and Bob since the inception of the network. A considerable practical challenge indeed.
The technique outlined above can clearly be extended to any number of relays. The important feature is that if we have just enough relays to span the distance then the entire channel is vulnerable to the compromise of a single relay.
If we add just one extra relay, superfluous to our distance spanning requirement, then we can operate the relays in such a way as to ensure that the channel can only be compromised if every relay on the channel is compromised. There is a price, however. If we need n − 1 relays to span the distance then with the addition of an extra relay and the drop out operation described above, we would only have a fraction of useful channels for Alice and Bob given by Thus if we require 9 relays to span the distance, the addition of a single relay operated as above with p = 1/2 means that f ≈ 0.01 so that, on average, approximately only 1 out every 100 timeslots is useful to Alice and Bob in forming the final key, and this is on the assumption that every timeslot contains transmissions that reach their destination.
The operation of the relay channel described above is not, of course, the only way to operate such channels, nor are we limited to adding in only one extra relay. All that is required for Alice and Bob to be able to establish a final key is that (i) Alice and Bob participate in all logical channels (ii) They ensure that every relay is absent from at least one of the channels used to form the key The relays thus have no knowledge of the final key established between Alice and Bob. Indeed, from a security perspective we might wish to add a relay on a channel, even when it is not strictly necessary, in order to access this feature. We might, for example, know that the relays are very secure (consider a relay placed on a satellite, for example, then practical limitations might make us more confident that it cannot be compromised). Alice and Bob, for very good reasons, may still not wish another party to have access to their key. This feature is particularly important in cases where the provision of a QKD network might be offered as a managed service by a network operator.

Conclusion
QKD is a mature and established practical technology [17] that has been implemented in several large-scale trials (see, for example [18]) as well as commercially. The current principal limitation on any network implementation is the effective distance over which the technique is feasible. Overcoming this limitation requires the use of intermediate relays which are conventionally operated in a link-by-link mode. This reduces the attractiveness of a network solution that we might wish to claim as providing unconditional security.
Here, we have shown how, with a suitable adaptation, IR relays can be operated on a QKD network to extend the distance over which successful quantum key exchange can be performed. By introducing the technique of bit transport we have shown that the size of final key established between Alice and Bob is unaffected by the introduction of such relays. Bit transport is a powerful technique for extending the capability and flexibility of QKD networks and moves the perspective from a sychronous QKD network to that of a network-based collection of quantum experiments in which the collected data can be used to establish a key both synchronously or asynchronously. Furthermore, the potential establishment of quantum keys between any two network nodes also gives us a pool of data for monitoring purposes, quite independent of any security application.
The possibility of duplex channels for establishing keys without public revelation is another application of bit transport. This technique also gives us the capability to use the entire filtered transmission to monitor the channel without compromising the security of the final key. Some of the features of bit transport can automatically be achieved by using more complex entangled quantum states such as the GHZ-type state we have considered. Whilst such schemes are currently impractical they suggest that our exploitation of the capabilities of quantum networks is still in its infancy, although significant progress is being made in this direction [19]. Indeed, it is fair to say that a radical revision of our understanding of information processing has been engendered by the exploitation of quantum mechanics (see, for example, Shor's seminal paper on quantum computing [20] which has revolutionised our perspective on computation. Other examples of the implications of this shift to quantum information can be found in [21]). It seems to us likely that further significant advances are still to be made.
Any network node, whether used for routing or to increase the distance over which a signal can be transmitted, must be trusted. We have discussed here how the creation of distinct logical channels using a random drop-out technique can radically alter the trust requirements on the intermediate nodes. This technique, when coupled with others such as multi-path QKD secret sharing and asynchronous quantum key establishment, gives us a very powerful methodology for the operation of a quantum key exchange network. Whilst the commercial arguments for single link QKD may not be compelling, it is clear that networkbased QKD with the techniques outlined above becomes a commercially more attractive proposition.