Key distillation from quantum channels using two-way communication protocols

We provide a general formalism to characterize the cryptographic properties of quantum channels in the realistic scenario where the two honest parties employ prepare and measure protocols and the known two-way communication reconciliation techniques. We obtain a necessary and sufficient condition to distill a secret key using this type of schemes for Pauli qubit channels and generalized Pauli channels in higher dimension. Our results can be applied to standard protocols such as BB84 or six-state, giving a critical error rate of 20% and 27.6%, respectively. We explore several possibilities to enlarge these bounds, without any improvement. These results suggest that there may exist weakly entangling channels useless for key distribution using prepare and measure schemes.


I. INTRODUCTION
Quantum Cryptography, that is, Quantum Key Distribution (QKD) followed by one-time pad, is one of the most important quantum information applications. The existing cryptographic methods using classical resources base their security on technical assumptions on the eavesdropper, often called Eve, capabilities, such as finite computational power or bounded memory [1]. Contrary to all these schemes, the security proofs of QKD protocols, e.g. the BB84 protocol [2], do not rely on any assumption on Eve's power: they are simply based on the fact that Eve's, as well as the honest parties' devices are governed by quantum theory [3]. Thus, well-established quantum features, such as the monogamy of quantum correlations (entanglement) or the impossibility of perfect cloning [4], make QKD secure. Actually, any possible quantum attack by Eve would introduce errors and modify the expected quantum correlations between the honest parties, Alice and Bob. The amount of these errors can be estimated using public discussion, so the honest parties can judge whether their quantum channel can be used for secure QKD, or abort the insecure transmission otherwise.
The monogamy of entangled quantum states (see [5]) can be simply illustrated in the scenario where two distant parties know to share a two-qubit maximally entangled state, the so-called ebit, Since the state is pure, it cannot be correlated with a third eavesdropping party. So, Alice and Bob can safely map their ebit into a secret bit by just measuring in the computational bases (see, Fig. (1)). It is meant by secret bit a random bit shared by Alice and Bob that is uncorrelated to Eve, namely P (A, B, E) = P (A, B)P (E) and P (A = 0, B = 0) = P (A = 1, B = 1) = 1/2, where P (A, B, E) denotes the probability distribution describing Alice, Bob and Eve's correlations. Then, a simple QKD protocol could consist of Alice locally preparing a state |Φ + , sending half of this state through the channel to Bob, and then measuring in the computational bases. However, any realistic channel between Alice and Bob is in general noisy, so the state sent by Alice interacts with the environment and is transformed into a mixed state, ρ AB . As a consequence of the noisy interaction with the environment, Alice and Bob measurement outcomes are no longer perfectly correlated. The honest parties then should know how to deal with errors. They should safely assume that Eve has the power to control all the environment, so all the errors are due to her interaction with the sent states: the larger the observed error rate, the larger Eve's information. Entanglement distillation protocols [6] offer a possible solution to the problem of errors or decoherence in the quantum channel. It is a technique that allows two separate parties to transform by local operations and classical communication (LOCC) many copies of a known entangled mixed state into a fewer number of pure ebits. These ebits can later be consumed to establish secret bits. However, entanglement distillation protocols are by far not feasible with present-day technology. This is because they require the use of a quantum memory, a device able to store quantum states, and controlled coherent operations. Both techniques turn out to be experimentally very challenging.
However, in order to establish secret bits, Alice and Bob do not necessarily have to go through entanglement distillation. A much more feasible family of protocols consist of the honest parties measuring their quantum states at the single-copy level and then applying classical distillation techniques to the obtained measurement outcomes. We denote these SIngle-copy Measurements plus ClAssical Processing protocols as SIMCAP [8]. Actually, it is well known that in the case of SIMCAP protocols, the honest parties do not have to use entanglement at all for the correlation distribution [9]. Indeed, Alice's preparation of the entangled two-qubit state plus measurement can be replaced by the preparation of a one-qubit state that is sent trough the noisy channel to Bob, who later measures it. That is, any SIMCAP protocol in the entanglement picture is equivalent to a prepare and measure scheme [9], which is much more feasible from an applied point of view. The BB84 and the six-state [10] protocols constitute known examples of prepare and measure QKD schemes.
Independently of the type of measurements or distilla-  1: Schematic diagram for key distillation from quantum states: a secret key can be distilled either by entanglement distillation plus measurement, which is an experimentally challenging process, or by measurement plus classical processing of the outcomes, whose implementation is much more feasible.
tion techniques employed in the protocol, a first and crucial step in any QKD scheme consists of a tomographic process by Alice and Bob to obtain information about their connecting quantum channel. By means of this process, Alice and Bob should conclude whether the secrecy properties of their channel are sufficient to run a QKD protocol. In the standard formulation, the cryptographic properties of quantum channels are referred to a specific protocol. For instance, a standard problem is to determine the critical quantum bit error rate (QBER) in the channel such that key distillation is possible using one-or two-way distillation techniques using the BB84 protocol. However, it appears meaningful to identify and quantify the cryptographic properties of a quantum channel by itself, independently of any pre-determined QKD protocol. Indeed, this is closer to what happens in reality, where the channel connecting Alice and Bob is fixed. Therefore, after the tomographic process, the two honest parties should design the protocol which is better tailored to the estimated channel parameters. In this sense, it is well known that no secure QKD can be established using entanglement-breaking channel [11,12], while the detection of entanglement already guarantees the presence of some form of secrecy [13]. Beyond these two results, little is known about which channel properties are necessary and/or sufficient for secure QKD.
In the present work, we analyze the cryptographic properties of quantum channels when Alice and Bob employ QKD schemes where (i) the correlation distribution is done using prepare and measure techniques and (ii) the key distillation process uses the standard one-way and two-way classical protocols. Indeed, these are the techniques presently used in any realistic QKD implementation. It should be clear, then, that none of the protocols considered here require the use of entangled particles. However, for the sake of simplicity, we perform our analysis in the completely equivalent entanglement picture. As it becomes clearer below, the problem then consists of identifying those quantum states that can be distilled into secret bits by SIMCAP protocols restricted to the known distillation techniques. A first step in this direc-tion has recently been given in [14]. There, a rather easily computable and powerful necessary condition for secure QKD is derived, which is shown to be sufficient against the so-called collective attacks (see below). In general, the derived necessary condition is more restrictive than the entanglement condition. In this work, we first rederive the security condition of [14], improving the security analysis. Since collective attacks have been proven to be as powerful as general attacks [15], our condition actually applies to any attack. We show how to apply this condition to the standard BB84 and six-state protocols. Next, we explore several possibilities to improve the obtained security bounds. Remarkably, all these alternatives fail, which suggests the existence of non-distillable entangled states under general SIMCAP protocols. Then, we move to higher dimensional systems, also called qudits, and extend the results to generalized Bell diagonal qudit channels. The obtained security condition turns out to be tight for the so-called (d + 1)-and 2-bases protocol of Ref. [16].
The article is organized as follows. Section II defines what we call realistic protocols. In section III, we introduce and classify several eavesdropping attacks. Exploiting the connection between QKD and the de Finetti theorem established by Renner [15], we can restrict the security analysis to the so-called collective attacks, where Eve applies the same interaction to each quantum state. Then, we briefly review some of the existing security bounds for the two most commonly used prepare and measure protocols, BB84 and six-state (section III D). In the next section, we derive the announced security condition for qubit channels and apply it to the two mentioned protocols. We then show that neither pre-processing nor coherent quantum operations by one of the parties improves the obtained security bounds. In section VII, we move to higher dimensional systems, extending the security conditions to generalized Bell diagonal channels. Then, we apply this condition to the (d + 1)-and 2-bases protocols of [16], which can be understood as the natural generalization to qudits of the BB84 and the six-state protocols, and prove the tightness for these protocols. Finally, section IX summarizes the main results and open questions discussed in this work. Most of the technical details are left for the appendices.

II. REALISTIC PROTOCOL
There exist plenty of QKD protocols in the literature. Here, we restrict our considerations to what we call realistic protocols where Alice prepares and sends states from a chosen basis to Bob, who measures in another (possibly different) basis. This establishes some classical correlations between the two honest parties. Of course this process alone is clearly insecure, since Eve could apply an intercept resend strategy in the same basis as Alice's state preparation, acquiring the whole information without being detected. Therefore, from time to time, Alice and Bob should change their state preparation and measurements to monitor the channel and exclude this possibility. Alice and Bob announce these symbols to extract information about their channel, so these instances do not contribute to the final key rate. Indeed these symbols are waisted in the tomographic process previously mentioned. However, in the limit of large sequences, the fraction of cases where Alice and Bob monitor the channel can be made negligible in comparison with the key length, but still sufficient to have a faithful description of some channel parameters, such as the QBER [17]. The states sent by Alice will be transformed into a mixed state because of Eve's interaction. This decoherence will produce errors in the measurement values obtained by Bob. The security analysis aims at answering whether the observed decoherence in the channel is small enough to allow Alice and Bob distilling a secret key. We call these protocols realistic in the sense that they do not involve experimentally difficult quantum operations, such as coherent measurements, quantum memories or the generation of entangled particles. The establishment of correlations is done by just generating one-qubit states and measuring them in two or more bases. Additionally, one could think of including a filtering single-copy measurement on Bob's side. This operation is harder than a standard projective measurement, but still feasible with present-day technology [18].
The above scenario can be explained in the completely equivalent entanglement-based scenario [9], that turns out to be much more convenient for the theoretical analysis. In the entanglement-based scheme, the information encoding by Alice is replaced by generating and measuring half of a maximally entangled state. That is, Alice first locally generates a maximally entangled two-qubit state and sends half of it to Bob through the channel. A mixed state ρ AB is then shared by the two honest parties, due to the interaction with the environment controlled by Eve. Now, Alice and Bob measure in two bases to map their quantum correlations into classical correlations. For instance, if Alice and Bob measure in the computational bases, the QBER simply reads It can be imposed that Alice's local state cannot be modified by Eve, since the corresponding particle never leaves Alice's lab, which is assumed to be secure. It has to be clear that the techniques of [9] imply the equivalence between SIMCAP protocols on entangled states and prepare and measure QKD schemes: the correlation distribution is, from the secrecy point of view, identical. This equivalence, for instance, is lost if one considers entanglement distillation protocols for QKD, where the particles are measured by the honest parties after applying coherent quantum operations.

A. Classical key distillation
After the correlation distribution, either using prepare and measure or SIMCAP protocols, Alice and Bob share partially secret correlations to be distilled into the perfect key. The problem of distilling noisy and partially secret correlations into a secret key has not been completely solved. Recently, general lower bounds to the distillable secret-key rate by means of error correction and privacy amplification using one-way communication have been obtained in [19]. In case the correlations are too noisy for the direct use of one-way distillation techniques, Alice and Bob can before apply a protocol using two-way communication. The obtained correlations after this two-way process may become distillable using oneway protocols. Much less is known about key distillation using two-way communication. Here we mostly apply the standard two-way communication protocol introduced by Maurer in [20], also known as classical advantage distillation (CAD). Actually, we analyze the following two slightly different CAD protocols: • CAD1. Alice and Bob share a list of correlated bits.
Alice selects N of her bits that have the same value and publicly announces the position of these symbols. Bob checks whether his corresponding symbols are also equal. If this is the case, Bob announces to Alice that he accepts, so they use the measurement values (they are all the same) as a bit for the new list. Otherwise, they reject the N values and start again the process with another block.
• CAD2. Alice locally generates a random bit s. She takes a block of N of her bits, A, and computes the vector such that A i + X i = s. She then announces the new block X through the public and authenticated classical channel. After receiving X, Bob adds it to his corresponding block, B + X, and accepts whenever all the resulting values are the same. If not, the symbols are discarded and the process is started again, as above.
These protocols are equivalent in classical cryptography and in the completely general quantum scenario. Nevertheless, it is shown in section IV C that they are different in some particular, but still relevant, scenarios.
In what follows, we restrict the analysis to key distillation protocols consisting of CAD followed by standard oneway error correction and privacy amplification. Thus, it is important to keep in mind that any security claim is referred to this type of key-distillation protocols. Although these are the protocols commonly used when considering two-way reconciliation techniques, their optimality, at least in terms of robustness, has not been proven.

III. EAVESDROPPING STRATEGIES
After describing Alice and Bob's operations, it is now time to consider Eve's attacks. With full generality, we suppose that Eve has the power to control all the environment. That is, all the information that leaks out through the channel connecting Alice and Bob goes to Eve, so all the decoherence seen by Alice and Bob is introduced by her interaction. Following Ref. [14], eavesdropping strategies can be classified into three types: (i) individual, (ii) collective and (iii) coherent. Once more, although most of the following discussion is presented in the entanglement picture, the same conclusions apply to the corresponding prepare and measure scheme.

A. Individual attacks
In an individual attack Eve is assumed to apply the same interaction to each state, without introducing correlations among copies, and measure her state right after this interaction. In this type of attacks, all three parties immediately measure their states, since no one is supposed to have the ability to store quantum states. Therefore, they end up sharing classical-classical-classical (CCC) correlated measurement outcomes [21], described by a probability distribution P (A, B, E). In this case, standard results from Classical Information Theory can be directly applied. For instance, it is well known that the secret-key rate using one-way communication, K → , is bounded by so-called Csiszár-Körner bound [22], Here I(A : B) denotes the classical mutual information between the measurement outcomes A and B. It reads where H denotes the standard Shannon entropy. In this type of attacks, Eve's interaction can be seen as a sort of asymmetric cloning [23] producing two different approximate copies, one for Bob and one for her. This cloning transformation reads U BE : |Φ + AB |E → |Ψ ABE where ρ AB = tr E |Ψ Ψ| ABE . It has been shown that in the case of two qubits, two honest parties can distill a secret key secure against any individual attacks whenever their quantum state ρ AB is entangled [8].
It is clear that to prove security against individual attacks is not satisfactory from a purely theoretical point of view. However, we believe it is a relevant issue when dealing with realistic eavesdroppers. Assume Eve's quantum memory decoherence rate is nonzero and the honest parties are able to estimate it. Then, they can introduce a delay between the state distribution and the distillation process long enough to prevent Eve keeping her states without errors. Eve is then forced to measure her states before the reconciliation, as for an individual attack.

B. Collective Attacks
Collective attacks represent, in principle, an intermediate step between individual and the most general attack. Eve is again assumed to apply the same interaction to each quantum state, but she has a quantum memory. In other words, she is not forced to measure her state after the interaction and can arbitrarily delay her measurement. In particular, she can wait until the end of the reconciliation process and adapt her measurement to the public information exchanged by Alice and Bob. After a collective attack, the two honest parties share N independent copies of the same state, ρ ⊗N AB , where no correlation exists from copy to copy. Without losing generality, the full state of the three parties can be taken equal to |ψ ⊗N ABE , where After a collective attack, and the measurements by Alice and Bob, the three parties share classical-classicalquantum (CCQ) correlations, described by a state where a and b denote Alice and Bob's measurement outcomes associated to the measurement projectors [a] and [b]. Throughout this paper, square brackets denote onedimensional projector, e.g.
The following result, obtained in [19,24], is largely used in the next sections. After a collective attack described by a state like (6), Alice and Bob's one-way distillable key rate satisfies Here, the correlations between Alice and Bob's classical variables are again quantified by the standard mutual information, I(A : B). The correlations between Alice's classical and Eve's quantum variables, A and E, are quantified by the Holevo quantity, where S denotes the Shannon entropy, so S(E) = S(ρ E ) and S(E|A) = a p (a)S(ρ E |A = a). Actually the "same" equation (7) applies when Bob is also able to store quantum states and the three parties share classical-quantum-quantum (CQQ) correlations. In this case, both mutual information quantities between Alice's classical variable, A, and Bob's and Eve's quantum states, denoted by B and E, should be understood as Holevo quantities [19]. Notice the similarities between (3) and (7). Indeed, the obtained bounds represent a natural generalization of the CK-bound to the CCQ and CQQ correlations scenarios.

C. General Attacks and the de Finetti Theorem
Finally, one has to consider the most general attack where Eve can perform any kind of interaction. In this case, Alice and Bob cannot assume to share N copies of the same quantum state. Compared to the previous attacks, there did not exist nice bounds for the extractable key-rate under general attacks. However, very recently a dramatic simplification on the security analysis of QKD protocols under general attacks has been achieved by means of the so-called de Finetti theorem [15]. Indeed, Renner has proven that general attacks cannot be more powerful than collective attacks in any protocol that is symmetric in the use of the quantum channel. This provides a huge simplification in security proofs, since by means of the de Finetti arguments (see [15] for more details), Alice and Bob can safely assume to share N copies of a quantum state consistent with their tomographic process, and then apply the existing bounds for this scenario. Note that the de Finetti theorem should also be employed if one wants to use entanglement distillation as a key distillation technique. In what follows, then, we can restrict our analysis to collective attacks, without underestimating Eve's capabilities.

D. Review of the existing Security Bounds
Finally, we would like to summarize the existing security bounds for the two most known QKD protocols, BB84 and six-state. These bounds are usually stated in terms of the critical QBER such that key distillation is possible. Of course, these bounds depend on the type of key distillation techniques employed by the honest parties. Since the first general security proof of BB84 by Mayers [25], security bounds have been constantly improved. Using a quantum error-correction (of

BB84
Six-state 11.0% 12.4% 12.7% 14.1% General security without pre-processing General security improved by pre-processing QBER FIG. 3: Security bounds for QKD protocols using key distillation techniques with one-way communication: based on the analogy between these techniques and quantum error correction, the security bounds for the BB84 and the six-state protocols are 11% and 12.7% respectively. These bounds have later been improved by information-theoretic considerations up to 12.4% and 14.1%. The improvement is achieved using some classical pre-processing by one of the parties.
bit-flip and phase-inversion) description of classical oneway error-correction and privacy amplification, Shor and Preskill showed the general security of BB84 whenever QBER < 11% [26]. Later, Lo adapted their proof to 6-state protocol obtaining a critical QBER of 12.7% [27]. More recently, Kraus, Renner, and Gisin have improved these values by introducing some classical pre-processing by the two honest parties, obtaining critical QBER's of 12.4% for the BB84 and 14.1% for the six-state protocol [24]. More recently, the bound for BB84 has been improved up to 12.9% in Ref. [28]. On the other hand, the known upper bounds on the critical QBER are slightly higher than these lower bounds, so the exact value for the critical QBER remains as an open question.
The honest parties however can apply CAD to their outcomes before using one-way key-distillation techniques and improve these bounds. The whole process can now be mapped into a two-way entanglement distillation protocol. Based on this analogy, Gottesman and Lo have obtained that secure QKD is possible whenever the QBER is smaller than 18.9% and 26.4% for the BB84 and six-state protocol, respectively [29]. Chau has improved these bounds up to 20.0% and 27.6% respectively [30]. The generalization of the formalism [24] to two-way communication has also been done by Kraus, Branciard and Renner [31]. We show in the next sections (see also [14]) that, for larger QBER, no protocol consisting of CAD followed by one-way distillation techniques works. So, contrary to what happens in the case of one-way communication, there is no gap between the lower and upper bounds for secure key distribution using the BB84 and six-state schemes, under the considered reconciliation techniques. FIG. 4: Security bounds for QKD protocols using two-way followed by one-way communication techniques: based on the analogy between the two-way plus one-way communication and two-way entanglement distillation protocol, general security bounds of the BB84 and the six-state protocols are given by 18.9% and 26.4% respectively [29]. Later, Chau improved the error correction method and the bounds are moved to 20.0% and 27.6% [30]. In sections IV and V, we show that those bounds are tight. Note that the key distillability condition is stronger than the entanglement condition, which is 25.0% and 33.3% for the BB84 and the six-state protocols.

IV. SECRECY PROPERTIES OF QUBIT CHANNELS
After reviewing the main ideas and previous results used in what follows, we are in position of deriving our results. Consider the situation where Alice and Bob are connected by a qubit channel. Alice locally prepares a maximally entangled state of two qubits and sends half of it through the channel. Then, both parties measure the state. By repetition of this process, they can obtain a complete, or partial, characterization of their channel, up to some precision. Indeed, there exists a correspondence between a channel, Υ, and the state Now, the parties agree on a pair of bases, that will later be used for the raw key distribution. They repeat the same process but now measure almost always in these bases. However, with small probability, they have to change their measurement to the previous tomographic process in order to check the channel. After public communication, they discard the asymptotically negligible fraction of symbols where any of them did not use the right basis and proceed with the key distillation. In what follows, we provide a security analysis of this type of schemes. Two important points should be mentioned again: (i) as said, these schemes can be easily transformed into a prepare and measure protocol, without entanglement and (ii) using de Finetti theorem, Alice and Bob can restrict Eve to collective attacks. In other words, they can assume to share N independent copies of the same state, ρ ⊗N AB , that is, the channel does not introduce correlation between the states. The goal, then, consists of finding the optimal SIMCAP protocol for the state ρ AB , or equivalently, the best prepare and measure scheme for the channel Υ.
Generically, ρ AB can be any two-qubit state. However, no key distillation is possible from separable states, so Alice and Bob abort their protocol if their measured data are consistent with a separable state [13]. We can assume, if the state preparation is done by Alice, that her local state, ρ A , cannot be modified by Eve. In our type of schemes, this state is equal to the identity. Although our techniques can be used in the general situation, we mostly restrict our analysis to the case where Bob's state is also equal to the identity. This is likely to be the case in any realistic situation, where the channel affects with some symmetry the flying qubits. This symmetry is reflected by the local state on reception, i.e. ρ B = 1 1. In the qubit case, the fact that the two local states are completely random simply means that the global state ρ AB is Bell diagonal, where j λ j = 1, λ j > 0, and define the so-called Bell basis. Or in other words, Υ is a Pauli channel. Pauli channels are very useful, as it will become clearer below, in the analysis of the BB84 and six-state protocols. It is also worth mentioning here that Alice and Bob can always transform their generic state ρ AB into a Bell diagonal state by single-copy filtering operations. Actually, this operation is optimal in terms of entanglement concentration. Indeed, it maximizes the entanglement of formation of any state ) obtained after LOCC operations of a single copy of ρ AB [32]. This filtering operation succeeds with probability If ρ AB is already in a Belldiagonal form, it remains invariant under the filtering operation. Alternatively, Alice and Bob can also map their state into a Bell diagonal state by a depolarization protocol, where they apply randomly correlated change of basis, but some entanglement may be lost in this process. In view of all these facts, in what follows we mainly consider Bell diagonal states.
It is possible to identify a canonical form for these states. This follows from the fact that Alice and Bob can apply local unitary transformation such that Indeed, they can permute the Bell basis elements by per-forming the following operations Once the state has been casted in this canonical form, Alice and Bob measure it in the computational basis. The choice of the computational bases by Alice and Bob will be justified by our analysis. Indeed, once a Bell-diagonal state has been written in the previous canonical form, the choice of the computational bases seems to maximize the secret correlations between Alice and Bob, although, in general, they may not maximize the total correlations. Before Alice and Bob' measurements, the global state including Eve is a pure state that purifies Alice and Bob's Bell diagonal state, that is, where |j E define an orthonormal basis on Eve's space.
All the purifications of Alice-Bob state are equivalent from Eve's point of view, since they only differ from a unitary operation in her space. After the measurements, Alice, Bob and Eve share CCQ correlations. In the next sections we study when these correlations can be distilled into a secure key using the standard CAD followed by one-way distillation protocols. We first obtain a sufficient condition for securtiy, using the lower bounds on the secret-key rate given above, c.f. (7). Then, we compute a necessary condition that follows from a specific eavesdropping attack. It is then shown that the two conditions coincide, so the resulting security condition is necessary and sufficient, under the mentioned distillation techniques. Next, we apply this condition to two known examples, the BB84 and the six-state protocols. We finally discuss several ways of improving the derived condition, by changing the distillation techniques, including classical pre-processing by the parties or one-party's coherent quantum operations.

A. Sufficient condition
In this section we will derive the announced sufficient condition for security using the lower bound on the secret-key rate of Eq. (7). Just before the measurements, the honest parties share a Bell diagonal state (10). This state is entangled if and only if 4 j=2 λ j < λ 1 , which follows from the fact that the positivity of the partial transposition is a necessary and sufficient condition for separability in 2 × 2 systems [33]. When Alice and Bob measure in their computational bases, they are left with classical data [i, j ] AB (i, j ∈ {0, 1}) whereas Eve still holds a quantum correlated system |e i,j E . The CCQ correlations they share are described by the state (up to normalization) where Eve's states are and the corresponding states without tilde denote the normalized vectors. So, after the measurements, Alice and Bob map ρ ⊗N AB , into a list of measurement outcomes, whose correlations are given by P AB (i, j), where This probability distribution reads as follows: Here, ǫ AB denotes the QBER, that is, Alice and Bob now apply CAD to a block of N symbols. Eve listens to the public communication that the two honest parties exchange. In particular, she has the position of the N symbols used by Alice in (2), in case the honest parties use CAD1 or the N -bit string X for CAD2. In the second case, Eve applies to each of her symbols the unitary transformation This unitary operation transforms |e i,j E into |e s,j E where s is the secret bit generated by Alice. If Alice and Bob apply CAD1, Eve does nothing. In both cases, the resulting state is where ǫ N is Alice-Bob error probability after CAD, and the last inequality tends to an equality when N → ∞. That is, whatever the advantage distillation protocol is, i.e. either CAD1 or CAD2, all the correlations among the three parties before the one-way key extraction step are described by the state (20).
We can now apply Eq. (7) to this CQQ state. The probability distribution between Alice and Bob has changed to where it can be seen that Alice and Bob have improved their correlation. The CAD protocol has changed the initial probability distribution P (A, B), with error rate ǫ AB , into P ′ (A, B), with error rate ǫ N . The mutual information between Alice and Bob I(A : B) is easily computed from the above table. I(A : E) can be derived from (20), so, after some algebra, the following equality is obtained where is the binary entropy, and the subscript 'eq' ('dif') refers to the resulting value of Alice being equal to (different from) that of Bob. Let's compute this quantity in the limit of a large number of copies, N ≫ 1, where ǫ N , Λ eq , Λ dif ≪ 1. It can be seen that in this limit The security condition follows from having positive value of the Eq. (22), which holds if More precisely, if this condition is satisfied, Alice and Bob can always establish a large but finite N such that Eq. (22) becomes positive. Eq. (25) can be rewritten as Therefore, whenever the state of Alice and Bob satisfies the security condition (25) above, they can extract from ρ AB a secret key with our SIMCAP protocol. This gives the searched sufficient condition for security for two twoqubit Bell diagonal states or, equivalently, Pauli channels. Later, it is proven that whenever condition (25) does not hold, there exists an attack by Eve such that no standard key-distillation protocol works. Condition (25) has a clear physical meaning. The r.h.s of (22) quantifies how fast Alice and Bob's error probability goes to zero when N increases. In the same limit, and since there are almost no errors in the symbols filtered by the CAD process, Eve has to distinguish between N copies of |e 0,0 and |e 1,1 . The trace distance between these two states provides a measure of this distinguishability. It is easy to see that for large N Thus, the l.h.s. of (22) quantifies how the distinguishability of the two quantum states on Eve's side after CAD increases with N . This intuitive idea is indeed behind the attack described in the next section.
Once this sufficient condition has been obtained, we can justify the choice of the computational bases for the measurements by Alice and Bob when sharing a state (10). Note that the same reasoning as above can be applied to any choice of bases. The derived security condition simply quantifies how Alice-Bob error probability goes to zero with N compared to Eve's distinguishability of the N copies of the states |e 0,0 and |e 1,1 , corresponding to the cases a = b = 0 and a = b = 1. The obtained conditions are not as simple as for measurements in the computational bases, but they can be easily computed using numerical means. One can, then, perform a numerical optimization over all choice of bases by Alice and Bob. An exhaustive search shows that computational bases are optimal for this type of security condition. It is interesting to mention that the bases that maximize the classical correlations, or minimize the error probability, between Alice and Bob do not correspond to the computational bases for all Bell diagonal states (10). Thus, these bases optimize the secret correlations between the two honest parties, according to our security condition, although they may be not optimal for classical correlations.

B. Necessary condition
After presenting the security condition (25), we now give an eavesdropping attack that breaks our SIMCAP protocol whenever this condition does not hold. This attack is very similar to that in Ref. [34].
Without loss of generality, we assume that all the communication in the one-way reconciliation part of the protocol goes from Alice to Bob. In this attack, Eve delays her measurement until Alice and Bob complete the CAD part of the distillation protocol. Then, she applies on each of her systems the two-outcome measurement defined by the projectors According to (20), all N measurements give the same outcome. If Eve obtains the outcome corresponding to F eq , the tripartite state becomes (up to normalization) In order to learn s A , Alice's bit, she has to discriminate between the two pure states |e 0,0 ⊗N and |e 1,1 ⊗N . The  (26): the security region is defined by the intersection of the entanglement condition λ1 > 1/2, the normalization condition λ1 + λ2 < 1, and the security condition (26). minimum error probability in such discrimination is [35] The corresponding error probability ǫ dif is the same as in Eq. (30), with the replacement | e 0,0 |e 1,1 | → | e 0,1 |e 1,0 |. Note that | e 0,0 |e 1,1 | ≥ | e 0,1 |e 1,0 |. Eve's information now consists of s E , as well as the outcome of the measurement (28), r E = {eq, dif}. It is shown in what follows that the corresponding probability distribution P (s A , s B , (s E , r E )) cannot be distilled using one-way communication. In order to do that, we show that Eve can always map P into a new probability distribution, Q, which is not one-way distillable. Therefore, the nondistillability of P is implied. Eve's mapping from P to Q works as follows: she increases her error until ǫ dif = ǫ eq . She achieves this by changing with some probability the value of s E when r E = dif. After this, Eve forgets r E . The resulting tripartite probability distribution Q satisfies Q(s B , s E |s A ) = Q(s B |s A ) Q(s E |s A ). Additionally, we know that Q(s B |s A ) and Q(s E |s A ) are binary symmetric channels with error probability ǫ B (= ǫ N in (21)) and ǫ eq in (30), respectively. It is proven in [20] that in such situation the one-way key rate is which is non-positive if Let us finally show that this inequality is satisfied for all values of N whenever the condition (25) does not hold. Writing z = λ 1 + λ 2 , we have 1/2 ≤ z ≤ 1, since the state of Alice and Bob is assumed entangled. Using the following inequality which holds for any positive N , the right-hand side of (34) is equal to ǫ B , whereas the left-hand side is an upper bound for ǫ eq . This bound follows from the inequality (λ 1 − λ 2 ) 2 /z 2 ≤ (1 − z)/z, which is the negation of (25). That is, if condition (25) is violated, no secret key can be distilled with our SIMCAP protocol. More precisely, there exists no N such that CAD followed by one-way distillation allows to establish a secret key. Since (25) is sufficient for security, the attack we have considered is in some sense optimal and the security bound (25) is tight for our SIMCAP protocol. It is worth analyzing the resources that this optimal eavesdropping attack requires. First of all, note that Eve does not need to perform any coherent quantum operation, but she only requires single-copy level (individual) measurements. This is because when discriminating N copies of two states, there exists an adaptative sequence of individual measurements which achieves the optimal error probability (30) [36]. However, what Eve really needs is the ability to store her quantum states after listening to the (public) communication exchanged by Alice and Bob during the CAD part of the protocol.

C. Inequivalence of CAD1 and CAD2 for individual attacks
As we have seen, the two CAD protocols lead to the same security condition. This follows from the fact that Eve is not assumed to measure her state before the CAD takes place. Then, she can effectively map one CAD protocol into the other by means of the reversible operation U E . This is no longer true in the case of individual attacks. Interestingly, in this scenario, the two two-way distillation methods do not give the same security condition. As mentioned, although the study of individual attacks gives a weaker security, it is relevant in the case of realistic eavesdroppers. Moreover, we believe the present example has some interest as a kind of toy model illustrating the importance of the reconciliation part for security. Recall that in the case of individual attacks, where Eve can neither perform coherent operations nor have a quantum memory, the security condition using CAD2 is the entanglement condition λ 1 > 1/2 [8]. However, when the honest parties apply CAD1 plus one-way communication, the security condition is (25). This holds true for two-qubit protocols, and remains open for the two-qudit protocols studied in the next sections [37].
Let us suppose that Alice and Bob apply CAD1 and consider the following individual attack. Eve knows that for all the instances passing the CAD protocol, Alice and Bob's symbols are equal with very high probability. Moreover, she knows that in all the position announced by Alice, Alice's symbol is the same. Therefore, from her point of view, the problem reduces to the discrimination of N copies of the two states |e i,i . Thus, she has to apply the measurement that optimally discriminates between these two states. As mentioned, the optimal two-state discrimination [36] can be achieved by an adaptive individual measurement strategy. Therefore, Eve can apply this adaptive strategy to her states right after her individual interaction. Her error probability is again given by (30). That is, although the attack is individual, the corresponding security condition is the same as for collective attacks.
This N -copy situation on Eve's space does not happen when Alice and Bob apply CAD2. Indeed, Eve maps CAD2 into CAD1 by applying the correcting unitary operation U i after knowing the vector X used in CAD2. This is the key point that allowed her to map one situation into the other above. This is however not possible in the case of individual attacks, where Eve is assumed to measure before the reconciliation part takes place. Under individual attacks, the security condition for CAD2 is equivalent to the entanglement condition for Bell diagonal states, as shown in [8]. Therefore, the two CAD protocols, which have proven to be equivalent in terms of robustness against general quantum attacks, become inequivalent in the restricted case of individual attacks.

V. BB84 AND SIX-STATE PROTOCOLS
The goal of the previous study has been to provide a general formalism for determining the security of qubit channels under a class of realistic QKD protocols. Relevant prepare and measure schemes, such as the BB84 and six-state protocol, constitute a particular case of our analysis. Indeed, the process of correlation distribution and channel tomography in these protocols is done by Alice preparing states from and Bob measuring in two (BB84) or three (six-state) bases. In this section, we apply the derived security condition to these protocols and compare the obtained results with previous security bounds. As explained in III D, a standard figure of merit in the security analysis of a given QKD protocol is given by the maximum error rate such that key distillation is still possible. For instance, in the case of one-way communication, the values of the critical error rates keep improving (see [28] for the latest result in this sense) since the first general security proof by Mayers [25]. In the case of reconciliation using two-way communication, the best known results were obtained by Chau in [30]. It is then important to know whether these bound can be further improved. In what follows, it is shown that our necessary condition for security implies that Chau's bounds cannot be improved. In order to do that, then, one has to employ other reconciliation techniques, different from advantage distillation plus one-way standard techniques. Some of these possibilities are discussed in the next sections.

A. BB84 protocol
In the BB84 protocol [2], bits are encoded into two sets of mutually unbiased bases {|0 , |+ } and {|1 , |− } respectively, where |± = (|0 ± |1 )/ √ 2. One can easily see that in the entanglement-based scheme, a family of attacks by Eve producing a QBER Q is given by the Bell-diagonal states (see also [38]) since the QBER is and 0 ≤ x ≤ Q. When Alice and Bob apply one-way communication distillation, the attack that minimizes (7) is x = Q 2 , and leads to the well-known value of QBER = 11%, first obtained by Shor and Preskill in [26]. The corresponding unitary interaction by Eve is equal to the phase-covariant cloning machine, that optimally clones qubits in an equator (in this case, in the xz plane). When one considers the two-way distillation techniques studied in this work, condition (25), or (26), applies. Then, one can see that the optimal attack, for fixed QBER, consists of taking x = 0. Therefore, Eve's attack is, not surprisingly, strongly dependent on the type of reconciliation employed. In the case of two-way communication, Eve's optimal interaction can also be seen as a generalized phase-covariant cloning transformation, which is shown in the Appendix I. Using this attack, the derived necessary condition for security is violated when QBER = 20%. This is precisely the same value obtained by Chau in his general security proof of BB84 [30]. So, the considered collective attack turns out to be tight, in terms of robustness. Recall that the security bound against individual attacks is at the entanglement limit, in this case giving QBER = 25.0% [8,33]. The full comparison is depicted in the Fig. (4).
Note also that the state (35) with x = 0, associated to the optimal attack, does not fit into our canonical form for Bell diagonal states, since λ 2 is not the minimal Bell coefficient. This simply means that key distillation from this state using a SIMCAP protocol is still possible. Alice and Bob only have to measure in a different basis, namely in the y basis. That is, if Alice and Bob knew to share this state, or channel, and could prepare and measure states in the y basis, not used in the considered version of BB84, they would be able to establish a secure key. This channel is still useful for QKD using a prepare and measure scheme, although not using the considered version of BB84. In our opinion, this illustrates why the present approach, that aims at identifying secrecy properties of channels without referring to a given protocol, is more general.

B. Six-state protocol
If a third mutually unbiased basis, in the y direction, is added to BB84, one obtains the so-called six-state protocol. The information encoding is as follows: bit 0 is encoded on states {|0 , |+ , | + i }, and 1 in {|1 , |− , | − i }, where | ± i = (|0 ± i|1 )/ √ 2 [10]. It is easy to see that an attack by Eve producing a QBER equal to Q is given by the Bell diagonal state This attack actually corresponds to Eve applying the universal cloning transformation. Contrary to what happened for BB84, this attack is optimal for both types of reconciliation protocols, using one-or two-way communication.
Applying the security condition (25), the security bound gives a critical QBER of Q = 27.6%. This value again coincides with the one obtained by Chau in his general security proof of [30] for the six-state protocol. The present attack, then, is again tight. In the case of indidual attacks, the security bound [8] is the entanglement limit Q = 33.3%.

VI. CAN THESE BOUNDS BE IMPROVED?
The previous section has applied the obtained security condition to two well-known QKD protocols. In the corresponding attack, Eve is forced to interact individually and in the same way with the sent qubits. As discussed, the de Finetti results by Renner imply that this does not pose any restriction on Eve's attack. However, Eve is also assumed to measure her states right after CAD, while she could have delayed her measurement, for instance until the end of the entire reconciliation. In spite of this apparent limitation, the condition is shown to be tight, under the considered distillation techniques, for the two protocols. As it has been mentioned, the obtained bounds do not coincide with the entanglement limit. This raises the question whether prepare and measure schemes, in general, do attain this limit. Or in other words, it suggests the existence of channels that, although can be used to distribute distillable entanglement, are useless for QKD using prepare and measure techniques. Recall that a channel that allows to establish distillable entanglement is secure: this just follows from combining the de Finetti argument with standard entanglement distillation. So, in this sense the channel indeed contains distillable secrecy. However, our results suggest that this secrecy is non-distillable, or bound, using single-copy measurements. That is, this secrecy is distillable only if both parties are able to perform coherent quantum operations. Perhaps, the simplest example of this channel is given by (37) with Q > 27.6%, i.e. by a weakly entangling depolarizing channel.
The aim of this section is to explore two possibilities to improve the previous security bounds. We first consider the classical pre-processing introduced in [24]. In this work, previous security bounds using one-way communication protocols for BB84 and six-state protocols have been improved by allowing one of the honest parties to introduce some local noise. This noise worsens the correlations between Alice and Bob, but it deteriorates in a stronger way the correlations between Alice and Eve. Here, we study whether a similar effect can be obtained in the case of the considered two-way communication protocols. In a similar way as in Ref. [24], we allow one of the two parties to introduce some noise, given by a binary symmetric channel (BSC). In our case, however, this form of pre-processing does not give any improvement on the security bounds. Later, we study whether the use of coherent quantum operations by one of the parties helps. We analyze a protocol that can be understood as a hybrid between classical and entanglement distillation protocol. Remarkably, this protocol does not provide any improvement either. In our opinion, these results strengthen the conjectured bound secrecy of these weakly entangled states when using SIMCAP protocols [39].

A. Pre-processing by one party
Recently, it has been observed that local classical preprocessing by the honest parties of their measurement outcomes can improve the security bounds of some QKD protocols [24]. For instance, Alice can map her measurement values X into another random variable U , and this transforms the mutual information from I(X : B) into I(U : B). At the same time, I(X : E) changes to I(U : E). In general, this mapping makes the mutual information of Alice and Bob decrease, but bounds on the secret key rate may improve, e.g. I(U : B) − I(U : E) > I(X : B) − I(X : E). Actually, by applying a simple BSC of probability q, where the input value is kept unchanged with probability 1−q or flipped with probability q, Alice may be able to improve the one-way secret-key rate [24]. Using this technique, the security bounds have been moved from 11% to 12.4% for the BB84 protocol and from 12.7% to 14.1% in the six-state protocol [24]. Here, we analyze whether a similar effect happens in the case of protocols consisting of two-way communication.
Note that pre-processing is useless if applied after CAD. Indeed, recall that the situation after CAD for the attack of Section IV B is simply given by two independent BSC channels between Alice and Bob and Alice and Eve, where pre-processing is known to be useless. The only possibility left is that Alice and/or Bob apply this preprocessing before the whole reconciliation protocol takes place.
As mentioned, Alice's pre-processing consists of a BSC channel, where her measurement value j is mapped into j and j + 1 with probabilities 1 − q and q, respectively. After this classical pre-processing, the state of the three parties is and ǫ AB denotes the QBER of the original measurement data, i.e. the error rate before applying pre-processing. Again, the states with tilde are not normalized, so Next, Alice and Bob apply two-way CAD to σ ⊗N ABE . A new error rate is obtained after CAD. The rest of the distillation part, then, follows the same steps as in section V-A.
We now compute the mutual information between the honest parties after CAD. The new error rate of Alice and Bob is introduced by the BSC above, and is expressed as In the same limit, Eve's state can be very well approximated by After some patient algebra, one can see that the Holevo information of Alice and Eve channel is (see also Appendix II): and u + v = 1. The case of q = 0 (or equivalently, u = 1) recovers the initial mutual information I(A : E). Therefore, the security condition of this protocol is u| e 0,0 |e 1,1 | 2 + v| e 0,1 |e 1,0 More precisely, whenever this condition is satisfied, there exists a finite N such that I P (A : B) − I P (A : E) > 0. The derived bound looks again intuitive. The r.h.s quantifies how Alice and Bob's error probability for the accepted symbols converges to zero when N is large. If one computes the trace distance between ρ 0,0 and ρ 1,1 , as defined in Eq. (38), one can see that which gives the l.h.s. of (39). This result suggests that the derived condition may again be tight. That is, it is likely there exists an attack by Eve breaking the security of the protocol whenever (39) is not satisfied. This attack would basically be the same as above, where Eve simply has to measure after the CAD part of the protocol.
Our goal is to see whether there exist situations where pre-processing is useful. Assume this is the case, that is, there exists a state for which (39) holds, for some value of q, while (25) does not. Then, − v| e 01 |e 10 | 2 ). (41) After some simple algebra, one gets the inequality: The r.h.s. of this equation is smaller than 2, and this implies that ǫ AB > 1/2. However, this contradicts 0 ≤ ǫ AB < 1/2, so we conclude that one-party pre-processing does not improve the obtained security bound. Notice that since the reconciliation part uses communication in both directions, it seems natural to consider pre-processing by the two honest parties, where Alice and Bob introduce some noise, described by the probabilities q A and q B . In this case, however, the analytical derivation is much more involved, even in the case of symmetric pre-processing. Our preliminary numerical calculations suggest that two-parties pre-processing may be useless as well. However, these calculations should be interpreted in a very careful way. Indeed, they become too demanding already for a moderate N , since one has to compute the von Neumann entropies for states in a large Hilbert space, namely ρ ⊗N 0,0 and ρ ⊗N 1,1 . Therefore, the detailed analysis of pre-processing by the two honest parties remains to be done.
Before concluding, we would like to mention that preprocessing, before or after CAD, may help in improving the distillable secret-key rate if the initial rate without pre-processing is already positive (see for instance [15]). However, this improvement vanishes for large blocks and the obtained security bounds do not change.

B. Bob's coherent operations do not improve the security bound
In order to improve the security bound, we also consider the scenario where Bob performs some coherent quantum operations before his measurement. Thus, he is assumed to be able to store quantum states and manipulate them in a coherent way, see Fig. 8. This is very unrealistic, but it gives the ultimate limit for positive key-rate using the corresponding prepare and measure protocol. We do not solve the problem in full generality. Here we consider the rather natural protocol where Bob applies the recurrence protocol used in entanglement distillation. That is, he applies CNOT operations to N of his qubits and measures all but one. He accepts only when the results of these N − 1 measurements are zero and keeps the remaining qubit. Later Bob applies a collective measurement on all the accepted qubits. Alice's part of the protocol remains unchanged.
After Alice has measured her states and announced the position of N symbols having the same value, Alice-Bob-Eve state reads where | be i = i|ψ ABE . Note that Alice, Bob and Eve now share CQQ correlations. Bob applies his part of the protocol and accepts. The resulting state turns out to be equal to, up to normalization, Since Bob is allowed to apply any coherent operation, the extractable key rate satisfies (7), where now both information quantities, I(A : B) and I(A : E), are equal to the corresponding Holevo bound. Of course I(A : E) has not changed. It is straightforward to see that one obtains the same bound for the key rate as for the state (15). This follows from the fact that e i,i |e i,j = 0, where i = j. Then, this hybrid protocol does not provide any advantage with respect to SIMCAP protocols. Recall that if the two parties apply coherent quantum operations, they can run entanglement distillation and distill from any entangled two-qubit state. Actually a slightly different protocol where (i) both parties perform the coherent recurrence protocol previously applied only by Bob, (ii) measure in the computational bases and (iii) apply standard one-way reconciliation techniques is secure for any entangled state. As shown, if one of the parties applies the "incoherent" version of this distillation protocol, consisting of first measurement and later CAD, followed by classical one-way distillation, the critical QBER decreases.

VII. GENERALIZATION TO ARBITRARY DIMENSION
In the previous sections we have provided a general formalism for the study of key distribution through quantum channels using prepare and measure schemes and two-way key distillation. In the important case of Pauli channels, we have derived a simple necessary and sufficient condition for security, for the considered protocols. In the next sections, we move to higher dimension, where the two honest parties employ d−dimensional quantum systems, or qudits. The generalization of the previous qubit scenario to arbitrary dimension is straightforward. Alice locally generates a d−dimensional maximally entangled state, |k |k (44) measures the first particle of the pair, and sends the other one to Bob. Since the channel between Alice and Bob is noisy, the shared state will change into a mixed state ρ AB . As usual, all the noise in the channel is due to Eve's interaction.
In what follows, we consider generalized Pauli channels. For these channels, Eve introduces flip and phase errors, generalizing the standard bit-flip σ x and phaseflip σ z operators of qubits. This generalization is given by the unitary operators Thus, a quantum system in state ρ propagating through a generalized Pauli channel is affected by a U m,n flip with probability p m,n , that is When applied to half of a maximally entangled state |Φ , the resulting state is Bell-diagonal, where the states |B m,n define the generalized Bell basis The global state including Eve reads where c 2 m,n = p m,n and {|m, n } defines a basis. In the next lines, we derive a security conditions for these channels when the two honest parties measure in the computational bases. We restrict to the computational bases for the sake of simplicity, although the main ideas of the formalism can be applied to any bases, and then numerically optimized. We then generalize the previous eavesdropping attack. Contrary to what happened in the qubit case, we are unable to prove the tightness of our condition in full generality using this attack.
We then apply the derived security condition to the known protocols in d-dimensional systems, such as the 2and (d + 1)-bases protocols. These protocols can be seen as the natural generalization of the BB84 and the sixstate protocols to higher dimension [16]. Exploiting the symmetries of these schemes, we can prove the tightness of our security condition for these protocols. In the case of the (d + 1)-bases protocol, some security bounds using two-way communication have been obtained by Chau in [40]. Here, we obtain the same values, therefore proving that they cannot be improved unless another reconciliation protocol is employed. Moreover, in the case of 2-bases protocol, we derive the same security bound as in [41]. Thus, again, another reconciliation protocol is necessary if the bound is to be improved.

A. Sufficient condition
After sending half of a maximally entangled state through the Pauli channel, Alice and Bob share the state where the probabilities p m,n characterize the generalized Pauli channel. After measuring in the computational bases, the two honest parties obtain correlated results. We denote by F , fidelity, the probability that Alice and Bob get the same measurement outcome. It reads In a similar way as for the qubit case, we introduce a measure of disturbance for the d − 1 possible errors. Denote Alice's measurement result by α. Then, Bob obtains α + j, with probability The total disturbance is defined as Of course, D 0 = F . Notice that all the D j can be taken smaller than F , without loss of generality. Indeed, if this was not the case, the two honest parties could apply local operations U m,n to make the fidelity F larger than any other D j . Note also that the errors have different probabilities D j . We now include Eve in the picture, the resulting global state being (47). As for the qubit case, Eve's interaction by means of the Pauli operators can be formulated as an asymmetric 1 → 1 + 1 cloning transformation [23]. In what follows, and again invoking the de Finetti argument, it is assumed that Alice, Bob and Eve share many copies of the state (47). After the measurements by Alice and Bob, the quantum state describing the CCQ correlations between the three parties is Eve's states are where the algebra is modulo d and β = α. As above, the states with tilde are not normalized, Note that e α,β |e x,y = 0 whenever β − α = y − x, so Eve can know in a deterministic way which error (if any) occurred between Alice and Bob. After the measurements, Alice and Bob have a list of correlated measurement outcomes. They now apply CAD. First, Alice locally generates a random variable, s A , that can take any value between 0 and d − 1 with uniform probability. She then takes N of her symbols (α 1 , · · · , α N ) and announces the vector X = (X 1 , · · · , X N ) such that X j = s − α j . Bob sums this vectors to his corresponding symbols (β 1 , · · · , β N ). If the N results are equal, and we denote by s B the corresponding result, he accepts s B . It is simple to see that Bob accepts a symbol with probability p ok = F N + d−1 j=1 D N j . After listening to the public communication used in CAD, Eve knows (X 1 , · · · , X N ). As in the previous qubit case, she applies the unitary operation: This unitary operation transforms Eve's states as follows, As above, this operation makes Alice, Bob and Eve's state independent of the specific vector used for CAD. The resulting state reads up to normalization. As above, the goal is to see when it is possible to find a finite N such that the CCQ correlations of state (52) provide a positive key-rate, according to the bound of Eq. (7).
The new disturbances D ′ j , j = 1, . . . , d − 1, after the CAD protocol are equal to where, again, the last inequality tends to an equality sign for large N . The mutual information between Alice and Bob is For large N , this quantity tends to Let us now compute Eve's information. Again, since Alice and Eve share a CQ channel, Eve's information is measured by the Holevo bound. For very large N , as in the case of qubits, we can restrict the computation of χ(A : E) to the cases where there are no errors between Alice and Bob after CAD. So, Eve has to distinguish between N copies of states |e k,k . Thus, in this limit, Denote by A η , with η = 0, . . . , d − 1, the eigenvalues of ρ E . As shown in Appendix III, one has Decomposing the eigenvalue A η into the term with k = k ′ and with k = k ′ , we can write A η = (1 + X (N ) Note that X Moreover, X (N ) η goes to zero when N increases. Using the approximation log(1 + x) ≈ x/ ln 2 valid when x ≪ 1, we have As above, the security condition follows from the comparison of the exponential terms in the asymptotic expressions I(A : B) and χ(A : E), having This formula constitutes the searched security condition for generalized Bell diagonal states. Whenever (57) is satisfied, there exists a finite N such that the secret-key rate is positive. In the next section, we analyze the generalization of the previous attack for qubits to arbitrary dimension.

B. Eavesdropping attack
We consider here the generalization of the previous qubit attack to arbitrary dimension. Unfortunately, we are unable to use this attack to prove the tightness of the previously derived condition, namely Eq. (57), in full generality. However, the techniques developed in this section can be applied to standard protocols, such as the 2-and d + 1-bases protocol. There, thanks to the symmetries of the problem, we can prove the tightness of the security condition.
The idea of the attack is the same as for the case of qubits. As above, Eve measures after the CAD part of the protocol. She first performs the d-outcome measurement defined by the projectors where j = 0. The outcomes of these measurement are denoted by r E . Using this measurement Eve can know in a deterministic way the difference between Alice and Bob's measurement outcomes, s A and s B . If Eve obtains the outcome corresponding to M eq , she knows the tripartite state is (up to normalization) Now, in order to learn s A , she must discriminate between the d pure states |e xx ⊗N . Due to the symmetry of these states, the so-called square-root measurement(SRM) [45,46] is optimal, in the sense that it minimizes the error probability (see Appendix IV for more details). She then guesses the right value of s A with probability where being real. Note that Y (N ) η tends to zero for large N . The error probability reads ǫ eq = 1 − P success eq . If Eve obtains the outcome corresponding to M j after the first measurement, she knows that the three parties are in the state (up to normalization) Eve again applies the SRM strategy, obtaining where the associated error probability being ǫ j = 1 − P success j . As a result of this measurement, Alice, Bob and Eve share the tripartite probability distribution P (s A , s B , (s E , r E )), where (s E , r E ) represents Eve's random variables, r E (s E ) being the result of the first (second) measurement. For each value of r E , Eve knows the difference between Alice and Bob's symbol and the error in her guess for Alice's symbol. It would be nice to relate the distillation properties of this tripartite probability distribution to the derived security condition (57), as we did in the qubit case. Unfortunately, we are at present unable to establish this connection in full generality. Actually, we cannot exclude that there exists a gap for some Bell diagonal states. However, as shown in the next section, the considered attack turns out to be tight when applied to standard protocols, such as the 2-and d + 1-bases protocols.
Let us conclude with a remark on the resources Eve needs for this attack. After applying the same unitary operation on each qudit, Eve stores her quantum states in a quantum memory. After CAD, she measures her corresponding block of N quantum states. Recall that in the qubit case, Eve does not need any collective measurement, since an adaptative individual measurement strategy achieves the fidelity of the optimal collective measurement [36]. In the case of arbitrary dimension, it is unknown whether there exists an adaptative measurement strategy achieving the optimal error probability, at least asymptotically, when N copies of d symmetrically distributed states are given [37]. We now apply the previous security condition to specific protocols with qudits, namely the so-called 2-and (d + 1)-bases protocols [16], which are the generalization of the BB84 and the six-state protocols to higher dimension. In the first case, Alice and Bob measure in two mutually unbiased bases, say computational and Fourier transform, while in the second, the honest parties measure in the d + 1 mutually unbiased bases [42].
The optimal cloning attack for these protocols gives a Bell diagonal state (45). However, due to the symmetries of the protocols, the coefficients c m,n , or p m,n , are such that where the normalization condition implies v 2 + 2(d − 1)x 2 + (d − 1) 2 y 2 = 1. For the d + 1-bases protocol, which is more symmetric, one also has x = y.
The fidelity, that is, the probability that Alice and Bob obtain the same outcome, is for m, n = 0.
In the 2-bases protocol, y is a free parameter that can be optimized for each value of the error rate, D, and depending on the reconciliation protocol. For instance, if Eve's goal is to optimize her classical mutual information, the optimal interaction (1 → 1+1 cloning machine) gives (see [16] for more details) In a similar way as in the qubit case, this choice of coefficients is not optimal when considering two-way reconciliation protocols, as shown in the next lines.

A. Security bounds
Having introduced the details of the protocols for arbitrary d, we only have to substitute the expression of the coefficients into the derived security condition. Because of the symmetries of the problem, all disturbances D j and overlaps e m,m |e 0,0 are equal, which means that the security condition simply reads After patient algebra, one obtains the following security bounds: 1. For (d + 1)-bases protocol, positive key rate is possible if The critical QBER for the 6-state protocol, 27.6%, is easily recovered by taking d = 2. Recently, Chau has derived a general security proof for the same protocols in Ref. [40]. Our critical values are the same as in his work.
2. For the 2-bases protocol, the critical disturbances D are The optimal attack, in the sense of minimizing the critical error rate, is always obtained for y = 0, see (65). The critical QBER for the BB84 protocol is recovered when d = 2. These values coincide with those obtained in [41] for 2-bases protocols.
Once again, there exists a gap between this security condition and the entanglement limit. For instance, in the case of d + 1-bases protocols, the entanglement limit coincides with the security condition against individual attacks [43] | e k,k |e l,l | > D (d − 1)F , which looks very similar to (68). Thus, there exists again weakly entangling channel where we are unable to establish a secure key using a prepare and measure scheme.

B. Proof of tightness
Finally, for these protocols, and because of the symmetries, we are able to prove the tightness of the derived security condition, under the considered reconciliation techniques. The goal is to show that the probability distribution P (s A , s B , (s A , s E )), resulting from the attack  9: Comparison of the security bounds and the entanglement condition. The security condition against collective attacks requires stronger correlation than the entanglement limit. Again, there may exist some entangled states that are useless for key distillation with the considered techniques. described in section VII B, cannot be distilled using oneway communication from Alice to Bob (the same can be proven if the communication goes from Bob to Alice by reversing the role of these parties).
In order to do that, we proceed as in the case of qubits. Alice-Bob's probability distribution is very simple: with probability F their symbols agree, with probability D j = D/(d − 1) they differ by j. After CAD on blocks of N symbols, the new fidelity between Alice and Bob is One can see that, again, Eve's error probability in guessing Alice's symbol is larger when there are no errors between the honest parties. As in the qubit case, Eve worsens her guesses by adding randomness in all these cases and forgets r E . After this process, she guesses correctly Alice's symbol with probability, see Eq. (60), independently of Bob's symbols. Here we used the fact that e m,m |e 0,0 = (v−x)/F when m = 0 for the analyzed protocols. After Eve's transformation, the one-way distillability properties of the final tripartite probability distribution are simply governed by the errors, as in the qubit case. Thus, we want to prove that at the point where the security condition is no longer satisfied, i.e. when ( for any block size N . Define t 2 = D/((d − 1)F ), where 0 ≤ t ≤ 1 because F > 1/D. What we want to prove can also be written as, see Eqs. (71) and (72), for all N and all d, where 0 ≤ t ≤ 1. Actually, using that 0 ≤ t ≤ 1, it suffices to prove the case N = 1, since all the remaining cases will follow by replacing t N → t and using the condition for N = 1. After patient algebra, one can show that (74) is satisfied for N = 1, which finishes the proof. Therefore, for the considered protocols, the attack introduced above breaks the security whenever our security condition does not hold. Therefore, this condition is tight for the considered reconciliation techniques.

IX. CONCLUSION
This works provides a general formalism for the security analysis of prepare and measure schemes, using standard advantage distillation followed by one-way communication techniques. The main tools used in this formalism are the de Finetti argument introduced by Renner and known bounds on the key rate. We derive a simple sufficient condition for general security in the important case of qubit Pauli channels. By providing a specific attack, we prove that the derived condition is tight. When applied to standard protocols, such as BB84 and six-state, our condition gives the critical error rates previously obtained by Chau. Since our condition is tight, these critical error rates cannot be improved unless another reconciliation technique is employed. Here, most of our analysis focus on conditions for security. However, the same techniques can be used to compute key rates. Actually, our results imply that the critical error rates of 20% ad 27.6% for the BB84 and six-state protocols can be reached without any pre-processing by Alice, contrary to previous derivations by Chau [30] or Renner [15]. The rates we obtain, then, are significantly larger. We then extend the analysis to arbitrary dimension and generalized Bell diagonal states. The corresponding security condition can be applied to obtain critical error rates for the 2-and d + 1-bases protocols. For these protocols, we can also prove the tightness of the condition.
We explore several possibilities to improve the obtained security bounds. As shown here, pre-processing by Alice or a coherent version of distillation by Bob do not provide any improvement. This is of course far from being an exhaustive analysis of all possibilities, but it suggests that it may be hard, if not impossible, to get the entanglement limit by a prepare and measure scheme. In our opinion, this is the main open question that naturally follows from our analysis. The easiest way of illustrating this problem is by considering the simple qubit depolarizing channel of depolarizing probability 1 − p. This is a channel where the input state is unchanged with probability p and map into completely depolarized noise with probability 1 − p. The corresponding state is a two-qubit Werner state. When p = 1/3, the channel is entanglement breaking, that is, it does not allow to distribute entanglement, so it is useless for any form of QKD. As shown here, the same channel can be used to QKD using a prepare and measure scheme when p > 1/ √ 5. Triv-ially, the entanglement limit can be reached if one allows coherent protocols by the two parties, such as entanglement distillation. However, is there a prepare and measure scheme with positive key rate for 1/3 ≤ p < 1/ √ 5?

Appendix I. Cloning Based Attacks
Asymmetric cloning machines have been proven to be a useful tool in the study of optimal eavesdropping attacks. In a cryptographic scenario, the input state to the cloning machine is the one sent by Alice, while one of the outputs is forwarded by Eve to Bob, keeping the rest of the output state. For instance, in the BB84 case, where Alice uses states from the x and z bases, the optimal eavesdropping attack is done by a 1 → 1 + 1 phase-covariant cloning machine [44] that clones the xz equator. The output states for Bob and Eve are ρ B = 1 2 (I + η B xz (n B x σ x + n B z σ z ) + η B y n B y σ y ) ρ E = 1 2 (I + η E xz (n E x σ x + n E z σ z ) + η E y n E y σ y ), where η i are usually called the shrinking factors.
Appendix II. Eve's information in the case of pre-processing In this appendix, we show how to compute Eve's information in the case Alice applies pre-processing before the CAD protocol, for large blocks. In this limit, Eve is faced with two possibilities, ρ ⊗N 0,0 and ρ ⊗N 1,1 , that read Indeed, if N ≫ 1, there are almost no errors in the symbols accepted by Alice and Bob. Eve's Holevo bound then reads where we used the fact that S(ρ ⊗N 0,0 ) = S(ρ ⊗N 1,1 ) = N h(u). The main problem, then, consists of the diagonalization of σ E . Note however that the states ρ 0,0 and ρ 1,1 have rank two and their eigenvectors belong to different two-dimensional subspaces. This implies that σ E decomposes into two-dimensional subspaces that can be easily diagonalized. The corresponding eigenvalues are For large N and nonzero u, the only relevant terms in the previous sum are such that | e 0,0 |e 1,1 | r | e 0,1 |e 1,0 | N −r ≪ 1. One can then approximate h((1 + x)/2) ≈ 1 − x 2 / ln 4, having S(σ E ) ≈ N h(u) + 1 − (u| e 0,0 |e 1,1 | 2 + v| e 0,1 |e 1,0 | 2 ) N ln 4 , Appendix III. Properties of geometrically uniform states A set of d quantum states {|ψ 0 , ..., |ψ d−1 } is said to be geometrically uniform if there is a unitary operator U that transforms |ψ j into |ψ j+1 for all j, where the indices read modulo d. All sets of geometrically uniform states, if the cardinality is the same, are isomorphic. Therefore, we do not lose any generality when assuming that those states are of the form: • Given a set of geometrically uniform states {|ψ 0 , ..., |ψ d−1 }, an orthonormal basis spanning the support of those states can explicitly obtained as follows: • The uniform mixture of geometrically uniform states gives the orthogonal decomposition in the basis defined above {|x n }: Therefore, the eigenvalues of the equal mixture of geometrically uniform state are c 2 n . Using (81), these eigenvalues can be written as: In our case, we are interested in the eigenvalues of the state ρ = 1 d α |e α e α | ⊗N , which approximates Eve's state after CAD in the limit of large N . The states |e α ⊗N are geometrically uniform, so the searched eigenvalues are: We describe the so-called square-root measurement along the lines given in Ref. [46]. Suppose that Alice encodes a classical random variable i that can take l different values into a quantum state |φ i ∈ C d , with l ≤ d, and sends the state to Bob. Suppose the l states are non-orthogonal and span an m dimensional subspace of C d . Denote by Π m the projection into this subspace, i.e. Π m |φ i = |φ i for all i. Bob has to read out the encoded value from the quantum state in an "optimal" way. There exist several "optimal" measurements depending on the figure of merit to be optimized. Here, following [46], we consider that Bob applies a measurement consisting of l rank-one operators [m i ], satisfying i [m i ] = Π m . The figure of merit to be optimized is the squared error E = l−1 i=0 E i |E i , where |E i = |φ i −|m i are the error vectors. As shown in [46], the measurement strategy minimizing E is the so-called SRM, also known as pretty-good measurement. The construction of this optimal measurement works as follows.
Denoted by Φ the matrix whose columns are |φ i . The SRM is constructed from the structure of the matrix Φ. Applying singular value decomposition to Φ = U DV † , the optimal measurement matrix is [46] where |u i and |v i are the column vectors of the two unitary matrices U and V , respectively. Here the column vectors of M define the optimal choice of measurement projectors |m i .
Moving to our cryptography problem, the states Eve has to discriminate are the geometrically uniform states β n e 2πi(γn/d) |x n where |x n is an orthonormal basis in a d-dimensional Hilbert space, and γ runs from 0 to d − 1. Each |e γ is normalized. In our problem, Eve aims at minimizing her error probability. Interestingly, in the case of geometrically uniform state, the previous measurement strategy turns out to minimize the error probability as well [46]. So, we only have to derive the optimal measurement matrix from Φ = γ |e γ x γ |. Using relations Φ † Φ = V DV † , the unitary V is the d-dimensional Fourier transform F |x u = 1 √ d w exp(− 2πi d wu)|x w , and the diagonal matrix is D = diag( √ d|β n |). Therefore, the optimal measurement matrix is Using this measurement, the probability of guessing correctly a given state |e j is | m j |e j | 2 . Then, the average success probability is The last equality is obtained taking into account that all |e j are equally probable, p(j) = 1/d. In particular, for the d + 1-or 2-bases protocols, the success probability reads, in terms of v and z, P success = (v + (d − 1)z) 2 /dF . When N copies of the states are given, |e j ⊗N , we can apply a collective measurement strategy. The SRM is constructed in the same way as above, and the success probability, assuming that all states are equi-probable, is