Research ArticlePHYSICS

Quantum key distribution with correlated sources

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Science Advances  09 Sep 2020:
Vol. 6, no. 37, eaaz4487
DOI: 10.1126/sciadv.aaz4487

Abstract

In theory, quantum key distribution (QKD) offers information-theoretic security. In practice, however, it does not due to the discrepancies between the assumptions used in the security proofs and the behavior of the real apparatuses. Recent years have witnessed a tremendous effort to fill the gap, but the treatment of correlations among pulses has remained a major elusive problem. Here, we close this gap by introducing a simple yet general method to prove the security of QKD with arbitrarily long-range pulse correlations. Our method is compatible with those security proofs that accommodate all the other typical device imperfections, thus paving the way toward achieving implementation security in QKD with arbitrary flawed devices. Moreover, we introduce a new framework for security proofs, which we call the reference technique. This framework includes existing security proofs as special cases, and it can be widely applied to a number of QKD protocols.

INTRODUCTION

Quantum key distribution (QKD) allows two distant parties, Alice and Bob, to securely exchange cryptographic keys in the presence of an eavesdropper, Eve (1). Despite notable progress made in recent years, there is still a big gap between the information-theoretic security promised by the security proofs and the actual security offered by the practical implementations of QKD. The most pressing problem is the discrepancy between the idealized device models used in the security proofs and the functioning of the real devices used in the experiments. This is so because typical security proofs rely on assumptions to describe the behavior of these devices and ignore their inherent imperfections. In practice, any deviation from these theoretical models might open security loopholes that could lead to side-channel attacks, thus compromising the security of QKD. A possible solution to this problem is to construct more realistic security proofs that can take into account device flaws. Lately, there have been notable advances in this direction. This includes, e.g., the proposal of the decoy-state method (24), allowing the use of practical light sources while maintaining a high secret key rate. In addition, measurement device–independent QKD (MDI-QKD) (5) can effectively eliminate all detector side channels and is practical with the current technology (611). The missing step toward achieving implementation security in QKD is to better characterize and secure the parties’ sources.

Security loopholes in the source could emerge from three main causes: from state preparation flaws (SPFs) due to the finite precision of the modulation devices, from information leakage either due to side channels arising from mode dependencies or due to Trojan horse attacks (THAs) (1216), or they could be caused by undesired classical correlations between the generated pulses. Mode dependencies of the emitted signals occur when the optical mode of a pulse depends on Alice’s setting choices. That is, Alice’s setting choices might be encoded in various degrees of freedom of the generated signals, not only on the desired one. Moreover, Eve can perform a THA by sending bright light into the source and then observing the back-reflected light to obtain partial information about Alice’s internal settings. Last, pulse correlations imply that the state of each pulse depends on the previous setting choices, such as bit and basis choices.

SPFs can be efficiently treated with the original loss-tolerant (LT) protocol (17). This is so because in this scheme, the resulting secret key rate is almost independent of the source’s flaws. Its main drawback is the requirement that the states of the pulses are described by qubit states, which is hard to guarantee in practice because of unavoidable potential side channels. To address this limitation, a generalization of the LT protocol was put forward recently (18). This latter protocol encompasses SPFs, mode dependencies, and THAs without requiring detailed information about the state of the side channels, which simplifies their experimental characterization. There are also other techniques that can deal with mode dependencies and THAs, such as the Gottesman-Lo-Lütkenhaus-Preskill (GLLP) type security proofs involving the quantum coin idea (1921) (from now onward, we shall refer to them as GLLP type security proofs) or the numerical approaches introduced in (2224).

The final piece toward guaranteeing implementation security is, thus, to consider pulse correlations among the emitted signals. These pulse correlations are purely classical, and they arise from the limitations of practical modulators. In general, due to memory effects of these modulation devices, the state of a pulse depends not only on the current modulation setting but also on the previous ones, meaning that the secret key information, i.e., the bit and the basis choices, is encoded not only into a single pulse but also between subsequent pulses. Theoretically, it is believed that this correlation is very hard to model because the dimensionality of the state space becomes very large. All existing security proofs circumvent this imperfection by simply neglecting it, which means that they cannot guarantee the security of practical implementations. We remark that a few recent works (2527) have incorporated in their analysis certain pulse correlations between the emitted signals. However, all these works only consider restricted scenarios. In particular, the results in (25, 26) and in (27) only consider setting choice–independent pulse correlations and intensity correlations between neighboring pulses, respectively. Therefore, none of them can deal with pulse correlations in terms of the secret key information nor with long-range correlations. Another reason why these correlations have been ignored so far is because one expects that, in practice, they are small. However, a small imperfection does not necessarily mean a small impact on the secret key rate, as Eve could, in principle, enhance such imperfection by exploiting, say, channel loss, resulting in a poor secret key rate (1921). Therefore, we note that pulse correlations could be a serious threat to the security of QKD.

Here, we present a general and simple framework to guarantee the security of QKD in the presence of arbitrary classical pulse correlations. The key idea is very easy yet very useful, that is, we regard the leaked information encoded into the correlations of subsequent pulses as a side channel for each of the pulses. The key features of our method include the following: (i) When combined with the generalized LT (GLT) protocol (18) or with the reference technique (RT) introduced in this work, it can analytically guarantee the security of QKD with practical devices that suffer from typical source imperfections, i.e., SPFs and side channels (including mode dependencies, THAs, and pulse correlations), even if the state of the side channels is totally unknown; (ii) due to its simplicity, our method is compatible with many other security proofs including those based on the inner product structure of the emitted pulses such as, for instance, the GLLP type security proofs (1921) and the numerical techniques in (2224); and (iii) our method can be applied to many QKD protocols such as, e.g., the BB84 scheme (28), the six-state protocol (29), the SARG04 protocol (30), distributed-phase-reference protocols (3133), and MDI-QKD (5). Our results indicate the feasibility of secure QKD with arbitrary flawed devices, and therefore, they constitute an essential step toward closing the big gap between theory and practice in QKD.

In addition, a second contribution of this work is a new framework for security proofs, the RT, that can provide high performance in the presence of source imperfections. More precisely, this is a parameter estimation technique that includes existing security proofs as special cases (see the Supplementary Materials). The RT incorporates the original LT protocol and can reproduce the GLT protocol and the GLLP type security proofs. The key idea is to consider some reference states, which are close to the actual states prepared by the protocol of interest, and use them to simplify the estimation of the parameters needed to guarantee the security of the protocol. More precisely, by bounding the maximum deviation between the probabilities associated with the reference states and those associated with the actual states, one can obtain a relationship for the probabilities involving the actual states based on those of the reference states. In doing so, one can estimate the parameters needed to guarantee the security of the actual protocol from the estimation that uses the reference states. We remark that the freedom to choose the reference states is very useful when dealing with source imperfections. In particular, this freedom allows us to analytically prove the security of a QKD protocol without any information on the side-channel states. This is important for achieving implementation security since a full characterization of the side-channel states, which, in principle, could live in unknown physical modes, is certainly very challenging in practice. In this work, we consider three special cases of the RT and evaluate their secret key rate in the presence of pulse correlations and SPFs.

RESULTS

Pulse correlations occur, for instance, when the emitted signals depend on the previous values of the encoding device (e.g., a phase modulator). In other words, subsequent pulses leak information about Alice’s former encoding choices. The key idea of our work to evaluate this complex scenario is to interpret these correlations as a side channel. By realistically modeling the source, we can bound this passive leakage of information and ensure secure QKD after performing enough privacy amplification. In what follows, we first outline the assumptions used in our security analysis, which is presented afterwards.

Assumptions on Alice’s and Bob’s devices

For simplicity, we consider a three-state protocol in which modulation devices are used to encode the bit and the basis choices. We do not explicitly consider the use of the decoy-state method (24); however, we remark that our framework could be combined with that method and also incorporate the effect of correlated intensity modulators and other imperfections of the intensity modulators (15). Furthermore, we assume an asymptotic scenario where Alice sends Bob an infinite number of pulses. We note, however, that the work presented here also applies to other protocols that use more than three states, as discussed in the next section.

Additional assumptions might be required depending on the particular security proof technique that is combined with our method. For instance, if the RT based on the GLT protocol (18) or the RT based on the original LT protocol (17), which we will present below, are used, then one also needs to assume that certain information about the states prepared by Alice is known. To be precise, for a setting choice j ∈ {0Z,1Z,0X}, the state of the kth pulse is in general purified into systems CkBkE and expressed asψjCkBkE=ajϕjCkBkλE+1aj2ϕjCkBkE(1)

Here, we take aj as a non-negative number satisfying 0 ≤ aj ≤ 1, which is possible by appropriately choosing the global phase of the states. The subscript CkBkE stands for all the systems, which include not only the kth qubit (system Bk) that Alice sends to Bob over the quantum channel but also the system Ck, which is needed for purifying the state of system Bk, and E is a system that includes Eve’s system. System E includes the systems sent by Alice over the quantum channel, such as the back-reflected light from a possible THA and the ancilla systems kept in Eve’s laboratory. As we will discuss further later, in general, this system also includes Alice’s ancilla systems used in the virtual entanglement-based protocol, which is equivalent to the actual protocol. Some of the latter systems store the setting information for all the pulses sent before the kth pulse. This means, in particular, that ∣λ〉E could depend on the setting choices for all the previous pulses. If it is not possible to find such a state, then aj becomes simply zero. From construction, Eq. 1 is the most general state that can be prepared in a QKD protocol. In other words, Eq. 1 simply decomposes a state ∣ψjCkBkE in a given Hilbert space into two states, each of which belongs to an orthogonal space. Precisely, one of them is the qubit state ∣ϕjCkBk∣λ〉E (as the set of states {∣ϕjCkBk∣λ〉E}j constitutes a qubit space), with ∣λ〉E being a state independent of the kth setting choice, and the other is the setting-dependent side-channel state ϕjCkBkE that corresponds to unwanted and possibly unknown modes. This decomposition can always be done for an appropriate choice of aj with 0 ≤ aj ≤ 1. The characterization of ϕjCkBkE is not required for the RT, and, in particular, no relationships between the states ϕjCkBkE and ϕj˜CkBkE and between ∣ϕjCkBkE and ϕj˜CkBkE for jj˜ are required, where j represents a different setting choice to j. To use the RT, we only need to know a lower bound on the coefficient aj in Eq. 1 and a full characterization of the density operator of the qubit Bk. The main contribution of our work is to show that one can accommodate the effect of pulse correlations through the parameter aj in Eq. 1.

The assumptions on Bob’s devices also depend on the security proof. For example, in the case of the RT based on the GLT protocol or based on the original LT protocol, one assumes that Bob measures the incoming pulses in the Z or the X basis. More precisely, Bob’s measurements are represented by the positive operator–valued measures (POVMs) {m̂0Z,m̂1Z,m̂f} and {m̂0X,m̂1X,m̂f}, respectively. Here, m̂αβ corresponds to Bob obtaining the bit value α ∈ {0,1} when selecting the basis β ∈ {Z, X}, and m̂f is associated with an inconclusive outcome. That is, we assume that these measurements satisfy the basis-independent efficiency condition, i.e., we impose that the operator m̂f is the same for both basis. Note that this condition is usually used in security proofs to remove detector side-channel attacks exploiting channel loss (34, 35); however, it is not necessary in MDI-QKD, to which our framework also applies. Furthermore, we emphasize that our method to deal with pulse correlations could be used as well with security proofs where the basis-independent efficiency condition is not guaranteed, such as in (36).

Security analysis in the presence of pulse correlations

In this section, we present the security analysis of QKD with pulse correlations. For this, we consider a security proof with the following properties. It uses an entanglement-based virtual protocol where Alice prepares pulses in an entangled state, and she (Bob) measures the local (incoming) systems to distill a secret key. In addition, it considers a particular detected pulse to estimate the phase error rate (or the phase error rate as a bound of the min-entropy). For simplicity, in what follows, we shall explicitly mention only the phase error rate, but it applies to both cases. Security against coherent attacks can then be guaranteed with the help of Azuma’s inequality (37), Kato’s inequality (38), or by applying the techniques in (39, 40). Moreover, we assume that the security proof can be generalized such that it applies to a particular pulse with a side channel. That is, it can be used to prove the security of QKD in the presence of active and/or passive information leakage. Thanks to the reduction technique presented below, a particular pulse affected by correlations can be regarded as a pulse with a side channel, and therefore, the security of QKD with pulse correlations is guaranteed. As an example, we now demonstrate that running a three-state protocol in the presence of nearest-neighbor pulse correlations can be regarded as a three-state protocol in which each of the pulses entails side channels. We emphasize, however, that it is straightforward to generalize this reduction technique to an m-state protocol, as discussed below, and to arbitrarily long-range correlations (see Materials and Methods for more details).

Nearest-neighbor pulse correlations. Let {ψjB}j=0Z,1Z,0X be the set of three quantum states used in the three-state protocol. We assume that Alice chooses ∣ψjB with probability pj and sends the pulse prepared in the chosen state to Bob over the quantum channel. As for Bob’s measurements, as already mentioned above, the assumptions vary according to the selected security proof. In an entanglement-based picture with nearest-neighbor pulse correlations, the transmission of n pulses by Alice can be described by first preparing n ancilla systems A and n pulses in the stateΨAB=j1j1A1ψj1B1j2j2A2ψj2j1B2jnjnAnψjnjn1Bn(2)and then by sending system B to Bob. In Eq. 2, A = A1, A2, …, An (B = B1, B2, …, Bn) refers to the composite system of Alice’s ancilla systems (Bob’s pulses), where Ak (Bk) for k ∈ {1,2, …, n} denotes Alice’s kth ancilla system (Bob’s kth pulse), the index jk ∈ {0Z,1Z,0X}, and {∣jkAk}jk ∈ {0Z,1Z,0X} is a set of unnormalized orthogonal states in a three-dimensional Hilbert space with jkAk=pjk, e.g., 0ZAk=p0Z. Importantly, ∣ψjkjk − 1Bk represents any nearest-neighbor classical pulse correlation, namely, this is the state of the kth emitted pulse when Alice selects the setting jk, given that her previous setting choice was jk − 1.

Now, suppose that after Alice sends Bob system B, Bob obtains click events for some of the received signals. Then, Alice and Bob perform fictitious measurements on their systems to generate the raw data in the experiment in order. We are interested in the state of their kth systems only before the fictitious measurements, which resulted in a click at Bob’s detectors. To obtain this state, recall that any operations and measurements on system B, including the detection measurements on the pulses received by Bob, commute with Alice’s measurements. Hence, we can assume that Alice has already measured her first k − 1 ancillas before sending system B. Then, we have the resulting state asj1A1ψj1B1jk1Ak1ψjk1jk2Bk1jkjkAkψjkjk1Bkjk+1jk+1Ak+1ψjk+1jkBk+1jnjnAnψjnjn1Bn(3)where j1′, ⋯, jk − 1 represent the outcomes of Alice’s measurement on her first k − 1 ancillas. To simplify this state, we introduce the following definitionjk+1Ak+1,,An,Bk+2,,Bnjk+1Ak+1jk+2jk+2Ak+2ψjk+2jk+1Bk+2jnjnAnψjnjn1Bn(4)which forms a set of orthogonal bases as {∣jk + 1Ak + 1, ⋯, An, Bk + 2, ⋯, Bn}jk + 1 = 0Z,1Z,0X. In addition, we define the stateλjkAk+1,,An,Bk+1,,Bnjk+1jk+1Ak+1,,An,Bk+2,,Bnψjk+1jkBk+1(5)

By using the above two states, we can rewrite Eq. 3 asj1A1ψj1B1jk1Ak1ψjk1jk2Bk1jkjkAkψjkjk1BkλjkAk+1,,An,Bk+1,,Bn(6)

As a reference, recall that if there were no pulse correlations in the three-state protocol, then the resulting state, instead of being in the form given by Eq. 6, would becomej1A1ψj1B1jk1Ak1ψjk1Bk1jkjkAkψjkBkλAk+1,,An,Bk+1,,Bn(7)where the state ∣λ〉Ak + 1, ⋯, An, Bk + 1, ⋯, Bn is independent of Alice’s setting choice jk and can be expressed asλAk+1,,An,Bk+1,,Bn=jk+1jk+1Ak+1ψjk+1Bk+1jnjnAnψjnBn(8)

In the security proof for the three-state protocol without pulse correlations, one typically obtains the phase error rate by considering any attack on system Bk in jkjkAkψjkBk in Eq. 7. On the other hand, when there are nearest-neighbor pulse correlations, one can see from Eq. 6 that Alice’s information jk is encoded not only on system Bk but also on the systems Bk + 1, ⋯, Bn, and the state ∣λjkAk + 1, ⋯, An, Bk + 1, ⋯, Bn, serves as side-channel information about the state ψjkjk1Bk. This suggests that if we obtain the phase error rate for the composite systems Bk and Bk + 1, ⋯, Bn in jkjkAkψjkjk1BkλjkAk+1,,An,Bk+1,,Bn, then the security follows. In other words, the three-state protocol with pulse correlations can be simply regarded as a three-state protocol where Alice prepares the states {ψjkjk1BkλjkAk+1,,An,Bk+1,,Bn}jk{0Z,1Z,0X} for any k and sends systems Bk, Bk + 1, ⋯, Bn to Bob.

Note that our framework is also valid for the case where Alice emits mixed states instead of pure states. The emission of mixed states might happen because of imperfections in Alice’s devices or when the prepared pure states are entangled with Eve’s systems because of, say, a THA. To treat this latter scenario, the mixed states can be purified by introducing an ancilla system Ck, with k ∈ {1,2, ⋯, n}, which contains Alice’s and Eve’s systems. As a result, Eq. 6 becomesj1A1ψj1C1B1jk1Ak1ψjk1jk2Ck1Bk1jkjkAkψjkjk1CkBkλjkAk+1,,An,Ck+1Bk+1,,CnBn(9)

Again, if a security proof for the three-state protocol without pulse correlations shows that one can estimate the phase error rate for ΣjkjkAk∣ψjkCkBk, then it follows that jkjkAkψjkjk1CkBkλjkAk+1,,An,Ck+1Bk+1,,CnBn is also secure if one can obtain the parameters needed for the security proof given these latter states. Furthermore, we remark that only for the purpose of estimating the phase error rate, in some cases, it may make the mathematical analysis simpler to fictitiously consider an arbitrary attack on the systems Ak + 1, ⋯, An (which, in reality, are inaccessible by Eve) besides the composite systems Bk and Bk + 1, ⋯, Bn. Note that the number of systems that we include as side channels does not matter, but what matters is how much the state ∣λjkAk + 1, ⋯, An, Bk + 1, ⋯, Bn depends on Alice’s information jk. Therefore, this fictitious attack on Ak + 1, ⋯, An should not result, in general, in a lower key rate because these ancillas do not directly entail information about jk.

Last, we remark that all the discussions in this section and also in the next one do not require jk to be chosen from only three possibilities, i.e., {0Z,1Z,0X}. That is, by only considering jk ∈ {1,2,3, ⋯, m}, our method applies for an m-state protocol.

Particular device model

Having stated the framework for the security proof in the presence of pulse correlations, we now consider a particular device model with only nearest-neighbor pulse correlations. The purpose of this section is to show how to obtain the parameters needed in Eq. 1 for a particular example of device model. Once this is achieved, one can directly apply the RT to guarantee the security of practical QKD implementations. We remark that for simplicity, below, we do not consider THAs or mode dependencies. However, they could readily be included by using the method in (18). In addition, we assume that a single-photon source is available, and as a concrete example for modeling pulse correlations, we select the following instance of nearest-neighbor pulse correlationψjkjk1Bk=1ϵϕjkBk+eiθjkjk1ϵϕjkBk(10)for the three states. Here, ψjkjk1Bk is a single-photon state living in a qubit space with jk ∈ {0Z,1Z,0X}, ∣ϕjkBk is a qubit state, the parameter ϵ intuitively quantifies the strength of the correlation, θjkjk − 1 represents how the kth state depends on the previous information jk − 1, and ϕjkBk is a state, in the same qubit space, that is orthogonal to ∣ϕjkBk. Note that, when there are no pulse correlations, i.e., ϵ = 0, the state ψjkjk1Bk becomes the perfect state ∣ϕjkBk, which does not depend on the previous setting jk1. However, in the presence of pulse correlations, i.e., when ϵ > 0, the overall state ψjkjk1Bk diverges from the ideal state ∣ϕjk〉, since it becomes dependent on the previous setting choice. The physical intuition of this model derives from the functioning of a phase modulator. To be precise, the state of an emitted pulse is typically affected by the modulation of the previous pulses such that there is a deviation depending on its preselected phase, which is quantified in the example given in Eq. 10 by θjkjk1.

Below, we show how to derive the state in the form of Eq. 1 for this particular example starting from Eq. 10. For this, we follow the idea introduced in the previous section and obtain the states ψjkjk1BkλjkAk+1,,An,Bk+1,,Bn given by Eq. 6. By using Eq. 10, we have thatψjkjk1Bkjk+1jk+1Ak+1,,An,Bk+2,,Bnψjk+1jkBk+1=(1ϵϕjkBk+eiθjkjk1ϵϕjkBk)jk+1jk+1Ak+1,,An,Bk+2,,Bn(1ϵϕjk+1Bk+1+eiθjk+1jkϵϕjk+1Bk+1)(1ϵ)ϕjkAk+1,,An,Bk,Bk+1,,Bn+1(1ϵ)2ϕjkjk1Ak+1,,An,Bk,Bk+1,,Bn(11)whereϕjkAk+1,,An,Bk,Bk+1,,Bn=ϕjkBkjk+1jk+1Ak+1,,An,Bk+2,,Bnϕjk+1Bk+1(12)is a qubit state (note that the set {ϕjkAk+1,,An,Bk,Bk+1,,Bn}jk{0Z,1Z,0X} spans a two-dimensional space) since ∑jk + 1jk + 1Ak + 1, ⋯, An, Bk + 2, ⋯, Bn∣ϕjk + 1Bk+1 is a normalized state independent of the information jk, and ϕjkjk1Ak+1,,An,Bk,Bk+1,,Bn is a state orthogonal to this qubit state. The explicit form of ϕjkjk1Ak+1,,An,Bk,Bk+1,,Bn is omitted here for simplicity, but it could be straightforwardly obtained from Eq. 11. We can regard our protocol as a protocol that uses the states in Eq. 11 rather than the ideal states ∣ϕjkBk for any k. We emphasize once again that the parameter ϵ and the state ϕjkjk1Ak+1,,An,Bk,Bk+1,,Bn in Eq. 11 represent most of the source imperfections (i.e., SPFs, mode dependencies, and THAs could be incorporated in a state of the form given by Eq. 11) (18), not only pulse correlations. This comes from the generality of Eq. 1.

Now, our formalism to deal with pulse correlations can be used directly with the RT since the states in Eq. 11 are in the form of Eq. 1. For the RT (described in the next section), we only require to know a lower bound on the coefficient 1 − ϵ and a full characterization of the state ∣ϕjkBk. We remark, however, that this framework can also be applied to the numerical techniques in (2224) if, in addition, the form of the state ϕjkjk1Ak+1,,An,Bk,Bk+1,,Bn is known or if bounds involving the inner products Ak+1,,An,Bk,Bk+1,,Bnϕjkjk1ϕj˜kjk1Ak+1,,An,Bk,Bk+1,,Bn and Ak+1,,An,Bk,Bk+1,,Bnϕjkjk1ϕj˜kjk1Ak+1,,An,Bk,Bk+1,,Bn for jkj˜k can be estimated, where j˜k represents a different setting choice to jk.

Here, we restricted the discussion to the case of nearest-neighbor pulse correlations, but our analysis also applies to arbitrarily long-range correlations. For instance, these correlations could be characterized byBkψjkjk1,,jw+1,j˜w,jw1,,j1ψjkjk1,,jw+1,jw,jw1,,j1Bk21ϵkw(13)for any w and k with w < k. That is, the correlation could be characterized through the response according to the change of the wth index. In other words, we can quantify the correlation represented by ϵkw, where k–w is the range of the correlation, by looking at the distinguishability of the states. Here, k–w can be any non-negative number, meaning that our method can incorporate arbitrary long-range correlations. One can show that from this model, it is straightforward to obtain the three states in the form given by Eq. 1 (see Materials and Methods) and consequently apply the RT.

RT based on the original LT protocol

In this section, we introduce a new framework for security proofs, the RT, which results in a high secret key rate in the presence of source imperfections. In what follows, we outline the intuition behind the key idea of the RT by applying it to the original LT protocol (17). A full description of the RT, including the detailed security proof, is presented in Materials and Methods. To simplify the discussion, here, we shall assume collective attacks; however, our analysis can be generalized to coherent attacks (see Materials and Methods for more details). Only as an example, we consider a protocol with a single-photon source in the presence of side-channel information, such as pulse correlations, in which Alice prepares the following three states for each pulse emissionψjkjk1B=(1ϵ)ϕjkB+1(1ϵ)2ϕjkjk1B(14)where B denotes the system to be sent to Bob. We remark that this subscript B could be replaced with Ak + 1, ⋯, An, Bk, Bk + 1, ⋯, Bn and then we would recover Eq. 11. However, in this section, we prefer to use Eq. 14 rather than Eq. 11 for simplicity of notation. Note that, here, we analyze the case of nearest-neighbor pulse correlations, but the RT is also applicable to arbitrary long-range pulse correlations. In Eq. 14, ∣ϕjkB is a qubit state while ϕjkjk1B corresponds to the side-channel state for jk ∈ {0Z,1Z,0X} that lives in any dimensional Hilbert space and is orthogonal to ∣ϕjkB for each setting choice jk. However, we do not assume any relationship between ϕjkjk1B and ϕj˜kjk1B for jkj˜k. For instance, ∣ϕjkB can be defined as in (18) such thatϕ0ZB=0ZB,ϕ1ZB=sin (δ2)0ZB+cos (δ2)1ZB,ϕ0XB=cos (π4+δ4)0ZB+sin (π4+δ4)1ZB(15)where {∣0Z〉, ∣1Z〉} is a qubit basis and δ( ≥ 0) is the deviation of the phase modulation from the intended value due to SPFs (18). That is, when there is no side-channel information, the states of the single photons sent by Alice have the form given by Eq. 15, but in the presence of side-channel information, however, these states are defined by Eqs. 14 and 15.

To prove the security of this protocol, we need to evaluate its phase error rate. The key idea of the RT is to consider the phase error rate estimation that we would obtain if we replace the actual set of states of the protocol, {ψ0Zjk1B,ψ1Zjk1B,ψ0Xjk1B}, with another set of states, which we call the reference states. Being the intuition that since the actual and the reference states are close to each other, one should be able to obtain a relationship between the events associated with the actual states by slightly modifying the relationship for the reference states. Note that the choice of reference states is, in principle, infinite; however, for higher secret key rates, they should be linearly dependent states such that unambiguous state discrimination (41, 42) is not possible. This allows us to use directly the original LT protocol (17) to estimate precisely some quantities associated with the reference states and their relationship as an intermediate step toward obtaining the phase error rate associated with the actual states.

As an example, we select the reference states to be {ϕ0ZB,ϕ1ZB,ϕ0XB}, which are defined in Eq. 15 and that correspond to the qubit part of the actual states in Eq. 14. In addition, we fictitiously consider that Alice chooses the reference states with the same probabilities as the actual states. Now, we can apply the RT in the following way. The first step is to find an expression for the probability of a phase error in terms of the reference states, which is a key parameter to be estimated in the security proof. For this, we consider an entanglement-based virtual protocol (see Materials and Methods for further details) using the reference states, where Alice prepares the virtual statesϕαXvirB=ϕ0ZB+(1)αϕ1ZB2(1+(1)αBϕ0Zϕ1ZB)(16)with α ∈ {0,1} and where, for simplicity, we assumed that the selection probabilities in the Z basis satisfy p0Z=p1Z. We can then define the probability of a phase error conditional on the reference states asP(phRef)p1XvirpZBTr[ϕ1Xvirϕ1XvirBM̂0X]+p0XvirpZBTr[ϕ0Xvirϕ0XvirBM̂1X](17)where pαXvir=12pZA(1+(1)αBϕ0Zϕ1ZB) is the probability that Alice sends the virtual states defined in Eq. 16, pZAp0Z+p1Z(pZB) is the probability that Alice (Bob) selects the Z basis, and M̂αX is Bob’s POVM element after any attack by Eve in the actual protocol. That is, M̂αXeK̂em̂αXK̂e, where K̂e is the Kraus operator representing Eve’s action in the actual protocol, e corresponds to her measurement outcome, and m̂αX is Bob’s POVM element for detecting αX in the actual protocol. The probabilities Tr[ϕ1Xvirϕ1XvirBM̂0X] and Tr[ϕ0Xvirϕ0XvirBM̂1X] in Eq. 17 cannot be directly obtained since they involve reference and virtual states, which are never sent in reality. However, by exploiting the fact that the reference states are all qubit states, one can follow the idea of the original LT protocol (17) and get a simple relationship between these probabilities and the probabilities associated with the reference states. To see this, first, note that in a qubit space, the following expressions holdϕ1Xvirϕ1XvirB=aϕ0Zϕ0ZB+bϕ1Zϕ1ZBcϕ0Xϕ0XB,ϕ0Xvirϕ0XvirB=ϕ0Xϕ0XB(18)where the coefficients a, b, and c are defined in Materials and Methods. We remark that if there are no SPFs, then the coefficients become a = b = c = 1. Then, by substituting Eq. 18 into Eq. 17, we obtain an expression for the probability of a phase error in terms of the reference states0=p1XvirpZBaTr[ϕ0Zϕ0ZBM̂0X]+p1XvirpZBbTr[ϕ1Zϕ1ZBM̂0X]+p0XvirpZBTr[ϕ0Xϕ0XBM̂1X][p1XvirpZBcTr[ϕ0Xϕ0XBM̂0X]+P(phRef)] (19)

In the RT, we call Eq. 19 the reference formula since it is used as a reference to obtain a similar expression in terms of the actual states. Note that we cannot use the reference formula directly in the security proof because it entails probabilities associated with the reference states, rather than the actual states.

Fortunately, by evaluating the deviation between the reference and the actual states, we can obtain bounds on the probabilities associated with the actual states and, consequently, the phase error rate of the actual protocol. This part of the RT corresponds to the deviation evaluation part (see Materials and Methods for further details). By following the analysis in the Supplementary Materials, we have that this deviation is quantified by usinggL(Tr[AAM̂],AR)Tr[RRM̂]gU(Tr[AAM̂],AR)(20)where M̂ is any non-negative bounded operator such that 0M̂1 and A and R are any normalized states associated with the actual and reference states, respectively. Here, the functions gL(x, y) and gU(x, y) are defined asgL(x,y)={0x<1y2x+(1y2)(12x)2y(1y2)x(1x)x1y2(21)andgU(x,y)={x+(1y2)(12x)+2y(1y2)x(1x)xy21x>y2(22)

No measurement, including any measurement performed by Eve, can induce a larger deviation between the probabilities because Eq. 20 holds for any M̂. We remark that, here, one could also use the trace distance argument (15); however, for the problem at hand, that bound is loose, and therefore, we use a tighter bound. That is, we use the knowledge of the probability associated with the observable events in the actual protocol, i.e., Tr[AAM̂], while the trace distance does not.

Now, we apply Eq. 20 to the first three terms and the last line of Eq. 19 separately, thus converting Eq. 19 into an expression for the probability of a phase error in terms of the actual states. For instance, note that the last line can be expressed by pZBSTr[AACBM̂] with ACBp1Xvirc/S0x,A,XBCψ0Xjk1B+pZA/2S0z,A,ZBCψ0Zjk1B+pZA/2S1z,A,ZBCψ1Zjk1B and M̂P̂(0x,A,XBC)M̂0X+P̂([0z,A,ZBC1z,A,ZBC]/2)M̂0X+P̂([0z,A,ZBC+1z,A,ZBC]/2)M̂1X where S=p1Xvirc+pZA, system C is an ancilla that stores the classical information associated with Alice’s and Bob’s setting choices, and P̂(·)=·· (see Materials and Methods). Here, we have mathematically represented the summed probabilities using the trace. By obtaining a similar expression for the first three terms of Eq. 19, we find that this equation becomes (see Materials and Methods)0S+gU(p1XvirpZBaS+Tr[ψ0Zjk1ψ0Zjk1BM̂0X]+p1XvirpZBbS+Tr[ψ1Zjk1ψ1Zjk1BM̂0X]+p0XvirpZBS+Tr[ψ0xjk1ψ0xjk1BM̂1X],1ϵ)SgL(p1XvirpZBcSTr[ψ0xjk1ψ0xjk1BM̂0X]+P(phAct)S,1ϵ)(23)where S+=p1Xvira+p1Xvirb+p0Xvir andP(phAct)p˜1XvirpZBTr[ψ1Xjk1virψ1Xjk1virBMˆ0X]+p˜0XvirpZBTr[ψ0Xjk1virψ0Xjk1virBMˆ1X](24)is the probability of a phase error conditional on the actual states. In Eq. 24, ψαXjk1virB are the virtual states associated with the actual states, and pαXvir are their respective probabilities. The explicit form of ψαXjk1virB is omitted here for simplicity, but similar to Eq. 16, ψαXjk1virBψ0Zjk1B+(1)αψ1Zjk1B. Equation 23 is valid for any eavesdropping strategy, i.e., any Kraus operator K̂e, that is included in the operators M̂αX (see discussion just after Eq. 17), and it can be directly used for the phase error estimation in the actual protocol. To clearly see how Eq. 23 is related with quantities observed in an actual experiment, we rewrite it as0S+gU(p1XvirpZBaS+p0ZpXBP(q0z,0xAct)+p1XvirpZBbS+p1ZpXBP(q1z,0xAct)+p0XvirpZBS+p0XpXBP(q0x,1xAct),1ϵ)SgL(p1XvirpZBcSp0XpXBP(q0x,0xAct)+P(phAct)S,1ϵ)(25)where, e.g., P(q0z,0xAct)p0ZpXBTr[ψ0Zjk1ψ0Zjk1BM̂0X] is the joint probability (i.e., the yield) that Alice selects the setting 0Z and prepares the state ψ0Zjk1B and Bob’s measurement outcome is 0X. Last, by solving Eq. 25 with respect to P(ph∣Act), we obtain the probability of a phase error of the actual protocol. The phase error rate is then defined as eX = P(ph∣Act)/YZ, where YZP(q0z,0z ∣Act) + P(q0z,1z∣Act) + P(q1z,0z∣Act) + P(q1z,1z∣Act) is the yield in the Z basis, i.e., the joint probability that Alice and Bob choose the Z basis and Bob obtains a detection event.

Simulation of the secret key rate

To show the performance of QKD in the presence of pulse correlations, we now present the simulation results. For simplicity of discussion, here, we apply our framework to two different cases of the RT: the RT based on the GLT protocol (18) and the RT described in the previous section. We remark that the GLLP type security proofs (1921) are also regarded as a special case of the RT where we select the actual states as the reference states and skip the reference formula part (see the Supplementary Materials for the proof of this claim). However, they involve four states, rather than three states, and analytical or numerical optimization is required. The comparison between the RT based on the GLLP type security proofs and the RT based on the original LT protocol is presented in the Supplementary Materials.

The main difference between the RT based on the GLT protocol and the RT based on the original LT protocol is that, in the former, a different bound is used to estimate the probabilities associated with the actual states. More precisely, the RT based on the GLT protocol essentially uses an inequality involving eigenvalues, instead of Eq. 20, which has the formTr[ϕjkϕjkBM̂αX]+λjkminTr[ψ0Zjk1ψ0Zjk1BM̂αX]Tr[ϕjkϕjkBM̂αX]+λjkmax(26)

Here, λjkmin and λjkmax are the eigenvalues of a matrix in the form [CjkBjk*Bjk0], where Bjk=(1ϵ)1(1ϵ)2 and Cjk = 1 − (1 − ϵ)2. The inequality in Eq. 26 is valid for any M̂αX with α ∈ {0,1}, and therefore, we can use it to consider the deviation between the probabilities associated with the reference states and the ones associated with the actual states [see (18) for more details]. We emphasize that pulse correlations are not taken into account in (18); however, we can apply our method to deal with pulse correlations to this security analysis. In doing so, we simply consider a QKD protocol with the states in Eqs. 14 and 15 and apply the RT based on the GLT protocol. That is, besides pulse correlations, we also include the effect of SPFs by assuming δ > 0 in Eq. 15. Furthermore, recall that system B in Eqs. 14, 15, and 26 can include more systems, not only those sent to Bob. In these equations, the subscript B could be replaced by Ak + 1, ⋯, An, Bk, Bk + 1, ⋯, Bn, allowing us to consider pulse correlations as the side channel. Note that to simplify the mathematical analysis, we do not trace out Alice’s subsequent systems Ak + 1, ⋯, An. Since these systems are independent of the setting jk, they do not provide any relevant information to Eve, and therefore, they do not affect our estimation of the phase error rate.

For the simulations, we assume the asymptotic regime where the secret key rate formula for a single-photon source can be expressed asRYZ(1h(eX)fh(eZ))(27)where, as defined before, YZ is the yield in the Z basis and eX is the phase error rate. The term eZ is the bit error rate, h(x)=xlog2(x)(1x)log2(1x) is the binary entropy function, and f is the error correction efficiency. Note that YZ and eZ are directly observed in a practical implementation of the protocol, but in the simulations, a channel model [see (18) for more details] is used instead. The experimental parameters used are as follows: dark count rate of Bob’s detectors pd = 10−7, f = 1.16, and the probabilities for Alice and Bob to select the Z basis are, for simplicity, pZA=23 and pZB=12. Unfortunately, there are no quantitative works characterizing pulse correlations (i.e., the value of the parameter ϵ); therefore, for illustration purposes, we select the values 10−3 and 10−6 to evaluate this imperfection. In addition, to investigate how the length of the pulse correlations affects the secret key rate, we consider the nearest-neighbor correlation ϵ1, as well as correlations among two subsequent pulses, ϵ2, and among 10 subsequent pulses, ϵ10 (see Eq. 13 for the definitions of these ϵ parameters). Regarding SPFs, we choose δ = 0 and δ = 0.063 according to the experimental results reported in (4345). The results for the RT based on the GLT protocol and for the RT based on the original LT protocol are illustrated in Fig. 1.

Fig. 1 Secret key rate R against the overall system loss measured in dB in the presence of correlations between the emitted pulses when our method is applied to the RT based on the GLT (RT-GLT) protocol and to the RT based on the original LT (RT-LT) protocol.

In all graphs, the blue and red lines are associated with the RT-GLT and the RT-LT, respectively. The solid lines correspond to the nearest-neighbor pulse correlations ϵ1, while the dashed (dashed-dotted) lines correspond to second ϵ2 (tenth ϵ10) neighbor pulse correlations, as indicated in the legend. (A) When there are no SPFs and the parameter ϵ is high, the RT-GLT and the RT-LT provide similar secret key rates. (B) As the parameter ϵ decreases, both security proofs provide higher secret key rates, but the RT-LT clearly outperforms the RT-GLT. (C) In the presence of SPFs, the secret key rate is only slightly worse for all cases since the security proofs are based on the LT protocol. (D) For high SPFs and low ϵ, the RT-LT is still superior to the RT-GLT.

As expected, this figure shows that when the magnitude of pulse correlations characterized by ϵi increases, the secret key rate decreases. In addition, as the length of the correlations, taken into account, increases, the secret key rate drops. We note, however, that even when long-range correlations are considered, a secret key can still be obtained. Namely, Fig. 1 shows that for ϵ = 10−6, one can generate a secret key even when there are correlations between 10 subsequent pulses. For a smaller value of the parameter ϵi, longer correlations can be included. If ϵi is small enough, then one can consider a very long range of pulse correlations while guaranteeing the security of QKD.

We emphasize that the security proof selected highly affects the results obtained, and this is also illustrated in Fig. 1, where we apply our technique to two different cases of the RT. To compare the RT based on the GLT protocol and the RT based on the original LT protocol as a function of pulse correlations, one can examine panels (A) and (B) or (C) and (D) of Fig. 1. Noticeably, as the magnitude of the pulse correlation ϵi increases, the secret key rate deteriorates for both of them. However, the RT based on the LT protocol outperforms the RT based on the GLT protocol in all the parameter regimes investigated. In addition, by comparing panels (A) and (C) or (B) and (D) of Fig. 1, one can see the effect of SPFs. As expected, the RT based on the GLT protocol and the RT based on the LT protocol are barely affected by this imperfection since they inherit, from the GLT protocol and the original LT protocol, respectively, high tolerance against SPFs with channel loss. The big difference observed in Fig. 1 between these two cases of the RT arises because of the following reason. Recall that we need to evaluate the deviation between the probabilities associated with the reference states and those associated with the actual states. For this, the bound used in the RT based on the GLT protocol is obtained by calculating certain eigenvalues, and thus, they entail square root terms, which deteriorate the secret key rate. Note that in the trace distance argument (15), square root terms are also present, resulting in loose bounds. On the other hand, the RT based on the original LT provides a tighter estimation of the phase error rate thanks to the bound in Eq. 20. More precisely, the square root terms in Eq. 20 include detection probabilities, which decrease as the channel loss increases, while for the other two bounds, the square root terms are constant, and thus, the high performance is maintained by using the bound in Eq. 20. Last, we remark again that the RT framework is general and can be applied to other QKD protocols as well, as shown in the Supplementary Materials.

DISCUSSION

Security proofs of QKD have to consider source imperfections in the theoretical models. Fortunately, SPFs, THAs (1216), and mode dependencies have been considered together very recently in (18). In this work, we have introduced a general framework to deal with pulse correlations, which are the last pieces required for securing the source. Our framework is compatible with those security proofs that incorporate other source imperfections, and therefore, it can be used to guarantee implementation security with flawed devices by combining it with MDI-QKD (5) and the results in (18). We remark that the decoy-state method (24) has not been considered in this work, and therefore, the imperfections of the intensity modulator have not been addressed. However, these imperfections could be straightforwardly included in our framework. The key idea for dealing with pulse correlations is interpreting the information encoded in the subsequent pulses as side-channel information. By doing so, we have shown that, as long as the magnitude of the correlations is small, a secret key can still be obtained even when there are correlations over a long range of pulses. Moreover, our framework can be directly applied in combination with existing security proofs such as the GLT protocol (18), the GLLP type security proofs involving the quantum coin idea (1921), and the numerical techniques recently introduced in (2224).

Furthermore, we have proposed a new framework for security proofs, which we call the RT. It uses reference states that are similar to the states sent in the actual protocol, thus allowing us to determine the parameters needed to prove the security of the latter. The RT is very general, and it can be applied to many QKD protocols. Moreover, it already includes the LT protocol, the GLT protocol, and the GLLP type security proofs as special cases. That is, we are able to reconstruct these security proofs by applying the RT, as shown in the Supplementary Materials. We have demonstrated that most of the source imperfections can be incorporated simultaneously into the RT, and therefore, this technique has been proven to be very useful for guaranteeing the security of practical QKD protocols. In particular, we have shown that for the RT based on the original LT, no information about the side-channel states is required, yet it is an analytical security proof, resulting in a much simpler characterization of the source. In addition, we emphasize that the RT can be applied together with analytical or numerical optimization to estimate an upper bound on the phase error rate, which could result in a higher performance. In this work, we have rigorously proven the security of the RT, and we have provided the sufficient conditions to apply this technique to other QKD protocols (see the Supplementary Materials). We remark that, for the security proof, we have not considered the probabilities to be conditional on the detection events, which is usually important for high performance in the finite-key scenario. Fortunately, thanks to the recently developed Kato’s inequality (38), this is not a problem anymore, and it does not affect the performance of the secret key rate even in the finite-key size regime.

In addition, in the Supplementary Materials, we have compared the RT based on the original LT protocol with the RT based on the GLLP type security proofs. We remark, however, that this comparison might be considered unfair because the RT based on the GLLP type security proofs requires four states and analytical or numerical optimization. Last, we note that if a better inequality to evaluate the deviation between the probabilities associated with the reference states and those associated with the actual states is available, then it could replace the inequality in Eq. 20, resulting in even higher secret key rates for the RT. In addition, our method could be applied to other problems in quantum information theory where one needs to estimate summed probabilities. In this sense, our work not only proves the security of practical QKD systems but also has a potential to contribute to quantum information theory in general.

MATERIALS AND METHODS

Reference technique

The RT is a new framework to prove the security of QKD protocols. It is general and can reproduce the GLLP type security proofs involving the quantum coin idea (1921) and the original LT protocol (17). Moreover, it can be applied to many different protocols. To see this, we refer the reader to the Supplementary Materials where we demonstrate that the GLLP type security proofs can be reconstructed from the RT. In addition, we outline the sufficient conditions to use the RT and prove the security of an m-state protocol. In this section, however, we present the key idea of the RT and show that it can be seen as a generalization of the LT protocol. For concreteness of the explanation, we concentrate on a particular example, the three-state protocol considered in Results.

Usually, to prove the security of QKD protocols, a relationship among the probabilities associated with the actual states needs to be established. Quite often, it is not straightforward to construct such a relationship, and the RT could be very useful to overcome this difficulty. The key idea is to consider a set of states, which we call the reference states, instead of the actual states. These reference states can be chosen freely, but they should be selected such that it is easy to derive a relationship among the probabilities associated with them. For this, it may be convenient to select the reference states in a structured space, such as a qubit space, and importantly, it is preferential that the resulting relationship is resilient against some imperfections in the space, such as the SPFs. Note that this relationship is associated with the reference states, and, therefore, it cannot be used directly in the security proof. However, since the reference states are chosen to be similar to the actual states, we can obtain a relationship associated with the actual states by slightly modifying the relationship for the reference states. In summary, the RT consists mainly of two parts:

1) Reference formula part: Here, we construct a relationship among the probabilities associated with the reference states.

2) Deviation evaluation part: Here, we transform the relationship for the reference states into a relationship for the actual states by evaluating the deviation between the probabilities associated with the reference states and those associated with the actual states.

We emphasize that the reference states are purely a mathematical tool to construct the reference formula, and we do not need to consider or imagine their practical implementation. Below, we show how to apply the RT in practice by presenting a rigorous security proof against coherent attacks for the three-state protocol.

Security proof of the three-state protocol with side channels. Let us assume a three-state protocol where Alice chooses a normalized state ∣ψjB from the set {∣ψjB}j = 0Z,1Z,0X with probability pj for each pulse emission. For simplicity of discussion, we assume that p0Z=p1Z. The assumptions on Bob’s side have been described in Results. Namely, he measures the incoming pulses in the Z or in the X basis with probabilities pZB and pXB, respectively. More precisely, Bob’s Z-basis (X-basis) measurement is represented by the POVM {m̂0Z,m̂1Z,m̂f} ({m̂0X,m̂1X,m̂f}), and it satisfies the basis-independent detection efficiency condition. Note that, in this protocol, the key is generated from a subset of the states indexed by j = 0Z,1Z, i.e., the Z basis and the bit values obtained by Bob’s Z-basis measurement.

Now, we write the states sent by Alice in the form of Eq. 1. That is, we expand the states ∣ψjB by using an orthonormal basis, and in doing so, we select a qubit space that is common over the three states. This suggests that ∣ψjB can be, most generally, decomposed intoψ0ZB=(1ϵ0Z)ϕ0ZB+1(1ϵ0Z)2ϕ0ZB,ψ1ZB=(1ϵ1Z)ϕ1ZB+1(1ϵ1Z)2ϕ1ZB,ψ0XB=(1ϵ0X)ϕ0XB+1(1ϵ0X)2ϕ0XB(28)where the state ∣ϕjB represents the qubit part of the state ∣ψjB and the state ϕjB is a (possibly) unknown side-channel state that lives in any Hilbert space and is orthogonal to ∣ϕjB. We stress that this orthogonality is needed only for each setting choice j but not between different choices of j. Examples of these states were presented in Eq. 14; however, for generality, we do not restrict ourselves only to that scenario. In the security proof, we assume that the qubit parts {∣ϕjB}j = 0Z,1Z,0X, which are to be adopted as the reference states, are perfectly characterized and stable in time, but we do not require any knowledge about the side-channel states {ϕjB}j=0Z,1Z,0X. From an experimental viewpoint, the unnecessity of characterizing the side-channel state ϕjB in Eq. 28 is a great advantage, as in practice, it is very challenging to perform measurements on arbitrary physical degrees of freedom. In Eq. 28, the coefficient ϵj satisfying 0 ≤ ϵj ≤ 1 quantifies the deviation of the state ∣ψjB(j ∈ {0Z,1Z,0X}) from the qubit space. That is, the states ∣ψjB are ideally qubit states; however, due to the presence of side channels, such as THA or pulse correlations, they deviate from this perfect scenario. Note that if ϵj = 1, then it means that it is impossible to find such a qubit space for particular j. In our example, we assume that we know an upper bound ϵ on ϵj, i.e., ϵj ≤ ϵ for all j. Furthermore, we shall assume that the qubit states ∣ϕjB are those defined in Eq. 15. To summarize, even if there is no information about the side-channel state ϕjB, our security proof works as long as we adopt ∣ϕjB as the reference states and they are perfectly characterized and stable in time, and we know ϵ (or more generally, ϵj). In particular, this means that the state ϕjB can vary in time and can be dependent on the previous pulses, and therefore, the states ∣ψjB emitted by Alice’s source do not need to be regarded as independently and identically distributed. This point will become clearer after Eq. 55. We remark, however, that if we select the reference states containing side-channel states, then they will no longer be perfectly known or stable in time. In this case, to make the mathematical analysis simpler, one could use analytical or numerical optimization to consider the worst-case scenario for the side-channel states, i.e., the case that maximizes the phase error rate. This maximization removes the potential dependence on the previous pulses and thus effectively provides pulses that are independent and stable in time. This is a purely mathematical step, and it does not require any extra assumptions on Alice’s source, e.g., the states ∣ψjB do not need to be regarded as independently and identically distributed.

Having finished the description of the states, we move on to the security proof using the RT. We are interested in proving the security of the bit values generated from the Z-basis events. From Eve’s perspective, this instance is equivalent to the one in which Alice selects the Z basis, prepares systems A and B in the state12(0ZAψ0ZB+1ZAψ1ZB)(29)and sends system B to Bob while keeping system A in her laboratory, and then both Alice and Bob perform their measurements in the Z basis. To prove the security of the Z-basis events, we need to estimate the phase errors (17, 46), which are defined in the X basis. That is, we consider the errors that Alice and Bob would have obtained if Alice had performed the X-basis measurement {∣0XA, ∣1XA} (with 0XA(0ZA+1ZA)/2 and 1XA(0ZA1ZA/2) and Bob had used a basis complementary to the Z basis (a suitable choice under the basis independent efficiency condition may be the X basis used in the actual protocol) for the measurement on the joint state defined in Eq. 29. This leads us to consider a virtual protocol in which Alice sends the virtual states ψ0XvirBψ0ZB+ψ1ZB and ψ1XvirBψ0ZBψ1ZB (17) to Bob with probabilitiesp˜αXvir=12pZA[1+(1)αRe(Bψ0Zψ1ZB)](30)where pZA is the probability that Alice selects the Z basis. Here, p0Xvir(p1Xvir) is the joint probability that Alice selects the Z basis and prepares the normalized virtual state ψ0XvirB (ψ1XvirB) through the X-basis measurement.

In the security proof, it is convenient to represent the actual protocol in terms of a virtual entanglement-based protocol. As explained above, in this virtual protocol, we consider replacing Alice’s and Bob’s bases with the X basis when both of them select the Z basis. From Eve’s viewpoint, the actual protocol with this replacement can be equivalently described by Alice and Bob fictitiously preparing the following entangled stateψCBp0ZpXB0z,A,XBCψ0ZB+  p1ZpXB1z,A,XBCψ1ZB+p0XpXB0x,A,XBCψ0XB+    p0XpZB0x,A,ZBCψ0XB+p˜0XvirpZB0x,Avir,XBCψ0XvirB+      p˜1XvirpZB1x,Avir,XBCψ1XvirB(31)and then performing a measurement on system C, which is associated with Alice’s and Bob’s setting choices. In particular, ∣0z, A, XBC (1x,Avir,XBC) represents the events when Alice selects the actual state for 0Z (the virtual state for 1X) and Bob chooses the X basis. Note that there are six states of system C that store different classical information related with Alice’s and Bob’s setting choices, and they are all normalized and orthogonal to each other. After Alice prepares the entangled state in Eq. 31, we imagine that she performs an orthogonal measurement that projects system C onto one of these six states, and Bob performs the measurement according to the basis directed by the measurement outcome. We remark that the first four terms in Eq. 31 correspond to the actual events that occur in the actual protocol, while the last two terms correspond to the virtual events. That is, the last two terms represent the events in which Alice and Bob select the Z basis in the actual protocol; however, their basis choice is replaced by the X basis for the security proof. The virtual events and the actual events are clearly defined in system C, and they correspond to disjoint events. The actual protocol can then be regarded as, repeatedly, say, N times, preparing systems B and C in the state ∣ψ〉CB followed by the measurements by Alice and Bob. Now, following the steps of the RT introduced above, we can estimate the phase errors associated with this protocol.

Reference formula part. As an example, we choose the reference states to be the qubit part of the actual states. For the actual states defined in Eq. 28, this corresponds to selecting the set {ϕ0ZB,ϕ1ZB,ϕ0XB} (see Eq. 15 for their explicit form). Now, we need to construct a relationship associated with these reference states. First, we consider the virtual entangled state12(0ZAϕ0ZB+1ZAϕ1ZB)(32)

Note that Eq. 32 is analogous to Eq. 29, but the actual states have been replaced with their respective reference states. Then, we may imagine that Alice measures system A in the X basis and sends Bob the virtual states ϕ0XvirBϕ0ZB+ϕ1ZB and ϕ1XvirBϕ0ZBϕ1ZB with probabilitiespαXvir=12pZA[1+(1)αRe(Bϕ0Zϕ1ZB)](33)

Here, p0Xvir(p1Xvir) could be interpreted as the joint probability that Alice selects the Z basis and prepares the normalized virtual state ϕ0XvirB(ϕ1XvirB) through the X-basis measurement. Now, we mathematically replace all the actual and virtual states in Eq. 31 with their respective reference statesp0ZpXB0z,A,XBCϕ0ZB+p1ZpXB1z,A,XBCϕ1ZB+p0XpXB0x,A,XBCϕ0XB+p0XpZB0x,A,ZBCϕ0XB+p0XvirpZB0x,Avir,XBCϕ0XvirB+p1XvirpZB1x,Avir,XBCϕ1XvirB(34)

Again, we emphasize that this entanglement-based protocol with the reference states is purely a mathematical tool for the security proof, and we do not need to consider or imagine its practical implementation. The reason why we have selected {ϕ0ZB,ϕ1ZB,ϕ0XB} as the reference states is twofold. First, these states are close to their respective actual states {ψ0ZB,ψ1ZB,ψ0XB}. Therefore, we expect that the probabilities associated with the reference states should be similar to those associated with the actual states. Second, by directly using the idea of the LT protocol for a qubit-based protocol (17), we can obtain a relationship between the reference states and the virtual states, which is expected to be LT against SPFs. More concretely, below, we consider that the reference states used are the ones defined in Eq. 15, and, in this case, we can express the virtual states for the reference states as in Eq. 18. We rewrite it here for convenienceϕ1Xvirϕ1XvirB=aϕ0Zϕ0ZB+bϕ1Zϕ1ZBcϕ0Xϕ0XB,ϕ0Xvirϕ0XvirB=ϕ0Xϕ0XB(35)where from Eq. 15, the coefficients a, b, c ≥ 0 are given bya2 sin (π4+δ4)cos (π4+3δ4)3 sin (π4+δ4),b2 sin (π4+δ4)cos (π4+3δ4)3 sin (π4+δ4),csin (δ2)+1sin (δ2)+1(36)

We remark that, in Eq. 35, we have highly exploited the properties of a qubit space, i.e., even with a negative sign in front of the coefficient c, ϕ1Xvirϕ1XvirB is still a density operator, which would not be the case in general for a density operator in a Hilbert space with a higher dimension.

We now consider the following quantityP(k)(phRef)p1XvirpZBTr[ϕ1Xvirϕ1XvirBM̂0X(k)]+p0XvirpZBTr[ϕ0Xvirϕ0XvirBM̂1X(k)](37)which, as described in Results, could be interpreted as the probability of a phase error for the kth pulse when using the reference states and where M̂αX(k) with α ∈ {0,1} is Bob’s POVM element for the kth pulse after a coherent attack in the actual protocol, that is, M̂αX(k)eK̂e(k)m̂αXK̂e(k). Here, K̂e(k) is the Kraus operator representing the action that the kth pulse is subjected to. This operator is obtained by Eve’s coherent attack that acts, in general, on all the pulses sent by Alice simultaneously and by considering all the k − 1 previous measurements by Alice and Bob. Here, e represents a particular outcome of the measurement conducted by Alice, Bob, and Eve. Note that, due to the entanglement caused by Eve’s coherent attack, K̂e(k) depends on all the k − 1 previous measurement outcomes obtained by Alice and Bob. The goal now is to transform the quantities associated with the reference states in Eq. 37 into those associated with the actual states for the kth pulse P(k)(phAct)p˜1XvirpZBTr[ψ1Xvirψ1XvirMˆ0X(k)]+p˜0XvirpZBTr[ψ0Xvirψ0XvirBMˆ1X(k)](38)

Using Eq. 35, we can express Eq. 37 asP(k)(phRef)=p1XvirpZBaTr[ϕ0Zϕ0ZBM̂0X(k)]+p1XvirpZBbTr[ϕ1Zϕ1ZBM̂0X(k)]p1XvirpZBcTr[ϕ0Xϕ0XBM̂0X(k)]+p0XvirpZBTr[ϕ0Xϕ0XBM̂1X(k)](39)which is equivalent to0=p1XvirpZBaTr[ϕ0Zϕ0ZBM̂0X(k)]+p1XvirpZBbTr[ϕ1Zϕ1ZM̂0X(k)]+p0XvirpZBTr[ϕ0Xϕ0XBM̂1X(k)](p1XvirpZBcTr[ϕ0Xϕ0XBM̂0X(k)]+p1XvirpZBTr[ϕ1Xvirϕ1XvirBM̂0X(k)]+p0XvirpZBTr[ϕ0Xvirϕ0XvirM̂1X(k)])(40)

Here, we emphasize that Eq. 40 is derived on the basis of the idea of the LT protocol, and therefore, it entails the robustness against the SPFs in the qubit space. That is, if there are no side channels, i.e., ϵ = 0, then Eq. 40, which is exactly the expression that is used in the original LT protocol (17), results in a secret key rate that is LT against SPFs. Therefore, this shows that the RT includes the LT protocol in the reference formula part. Next, we transform the relationship for the reference states in Eq. 40 into a relationship for the actual states. That is, we enter the deviation evaluation part of the RT.

Deviation evaluation part. For the transformation of Eq. 40, we use the bound in Eq. 20. We rewrite it here for conveniencegL(Tr[AAM̂],AR)Tr[RRM̂]gU(Tr[AAM̂],AR)(41)where ∣R⟩ (∣A⟩) is any normalized state associated with the reference (actual) states andgL(x,y)={0x<1y2x+(1y2)(12x)2y(1y2)x(1x)x1y2(42)gU(x,y)={x+(1y2)(12x)+2y(1y2)x(1x)xy21x>y2(43)

Note that −gL(x, y) and gU(x, y) are concave with respect to 0 ≤ x ≤ 1 for any fixed 0 ≤ y ≤ 1, and ∂y gL(x, y) ≥ 0 and ∂y gU(x, y) ≤ 0 hold. For more details on the derivation of Eq. 41, see the Supplementary Materials. Now, we consider the first three terms in Eq. 40, which are reexpressed aspZBS+Tr[p1XviraS+ϕ0Zϕ0ZBM̂0X(k)+p1XvirbS+ϕ1Zϕ1ZBM̂0X(k)+p0XvirS+ϕ0Xϕ0XBM̂1X(k)](44)where S+p1Xvira+p1Xvirb+p0Xvir is a normalization factor. Next, we rewrite the term Tr[ · ] in Eq. 44 asTr[P̂(p1XviraS+0z,A,XBCϕ0ZB+p1XvirbS+1z,A,XBCϕ1ZB+p0XvirS+0x,A,XBCϕ0XB)M̂+(k)]Tr[P̂(R+CB)M̂+(k)](45)withR+CBp1XviraS+0z,A,XBCϕ0ZB+p1XvirbS+1z,A,XBCϕ1ZB+p0XvirS+0x,A,XBCϕ0XB,M̂+(k)P̂(0z,A,XBC)M̂0X(k)+P̂(1z,A,XBC)M̂0X(k)+P̂(0x,A,XBC)M̂1X(k)where P̂(·)=··. Note that this is purely a mathematical reinterpretation of the summed probabilities. We are interested in mathematically replacing ϕ0ZB, ϕ1ZB, and ϕ0XB in Eq. 45 with ψ0ZB, ψ1ZB, and ψ0XB, respectively, by using Eq. 41. For this, we may selectA+CBp1XviraS+0z,A,XBCψ0ZB+p1XvirbS+1z,A,XBCψ1ZB+p0XvirS+0x,A,XBCψ0XB(46)and as a result, we have transformed the first three terms of Eq. 40 intop1XvirpZBaTr[ϕ0Zϕ0ZBM̂0X(k)]+p1XvirpZBbTr[ϕ1Zϕ1ZM̂0X(k)]+p0XvirpZBTr[ϕ0Xϕ0XBM̂1X(k)]pZBS+gU(Tr[P̂(A+CB)M̂+(k)],1ϵ)(47)where we have selected an upper bound on Tr [P̂(R+CB)M̂+(k)] to obtain an upper bound on the phase error probability and used ∣CBA+R+CB ∣ = 1 − ϵ. Here, note that to calculate this inner product, we need to calculate the terms B〈ψj∣ϕjB rather than ψjϕj˜BB with jj˜, which shows the aforementioned simplicity of the state characterization needed in our proof. Now, to clearly see how the term Tr[P̂(A+CB)M̂+(k)] is related with the quantities obtained from an experimental implementation of the actual protocol, we writeTr[P̂(A+CB)M̂+(k)]=Tr[p1XviraS+ψ0Zψ0ZM̂0X(k)+p1XvirbS+ψ1Zψ1ZM̂0X(k)+p0XvirS+ψ0Xψ0XM̂1X(k)]=p1XviraS+p0zpXBP(k)(q0z,0xAct)+p1XvirbS+p1zpXBP(k)(q1z,0xAct)+p0XvirS+p0xpXBP(k)(q0x,1xAct)(48)

Here, P(k)(q0z,0x∣Act) is the joint probability that Alice selects the setting 0Z, and Bob’s measurement outcome is 0X at the kth instance, conditional on the first k − 1 measurements by Alice and Bob in the entanglement-based protocol for the actual protocol. The other probabilities are defined in a similar manner. This finishes the transformation of the first three terms with respect to the probabilities associated with the actual protocol.

Next, we consider the last three terms in Eq. 40, which are reexpressed aspZBSTr[p1XvircSϕ0Xϕ0XBM̂0X(k)+p1XvirSϕ1Xvirϕ1XvirBM̂0X(k)+p0XvirSϕ0Xvirϕ0XvirBM̂1X(k)](49)where Sp1Xvirc+p1Xvir+p0Xvir=p1Xvirc+pZA, with pZAp0Z + p1Z is the normalization factor. The term Tr[ · ] in Eq. 49 can be expressed asTr[P̂(p1XvircS0x,A,XBCϕ0XB+pZA2S0z,A,ZBCϕ0ZB+pZA2S1z,A,ZBCϕ1ZB)M̂(k)]Tr[P̂(RCB)M̂(k)](50)withRCBp1XvircS0x,A,XBCϕ0XB+pZA2S0z,A,ZBCϕ0ZB+pZA2S1z,A,ZBCϕ1ZB,M̂(k)P̂(0x,A,XBC)M̂0X(k)+P̂(0z,A,ZBC1z,A,ZBC2)M̂0X(k) +P̂(0z,A,ZBC+1z,A,ZBC2)M̂1X(k)(51)

Here, we have used the fact that the state ϕαXvirB=12(ϕ0ZB+(1)αϕ1ZB)/pαXvir/pZA and that P̂(0z,A,ZBC1z,A,ZBC/2) and P̂(0z,A,ZBC+1z,A,ZBC/2) correspond to the events associated with the states 1x,Avir,XB and 0x,Avir,XB in the entanglement-based virtual protocol, respectively. Again, we remark that this is purely a mathematical reinterpretation of the summed probability. To mathematically replace the states involving the reference states with those involving the actual states, we may selectACBp1XvircS0x,A,XBCψ0XB+pZA2S0z,A,ZBCψ0ZB+pZA2S1z,A,ZBCψ1ZB(52)

As a result, we have transformed the last three terms in Eq. 40 intop1XvirpZBcTr[ϕ0Xϕ0XBM̂0X(k)]+p1XvirpZBTr[ϕ1Xvirϕ1XvirBM̂0X(k)]+p0XvirpZBTr[ϕ0Xvirϕ0XvirBM̂1X(k)]pZBSgL(Tr[P̂(ACB)M̂(k)],1ϵ)(53)where we have selected a lower bound on Tr[P̂(RCB)M̂(k)] to obtain an upper bound on the phase error probability and used ∣CBARCB ∣ = 1 − ϵ. As before, to calculate this inner product, we need to calculate the terms B〈ψj ∣ ϕjB rather than ψjϕj˜BB with jj˜. Now, we look at Tr[P̂(ACB)M̂(k)], which is expressed and interpreted byTr[P̂(ACB)M̂(k)]=Tr[P̂(p1XvircS0x,A,XBCψ0XB+   pZA2S0z,A,ZBCψ0ZB+pZA2S1z,A,ZBCψ1ZB)M̂(k)]     =Tr[p1XvircSψ0Xψ0XBM̂0X(k)+p˜1XvirSψ1Xvirψ1XvirBM̂0X(k)+       p˜0XvirSψ0Xvirψ0XvirBM̂1X(k)]     =p1XvircSp0XpXBP(k)(q0x,0xAct)+1SpZBP(k)(phAct)(54)where we have used Eq. 38, namely, the definition of P(k)(ph∣Act) and the fact that the states ψαXvirB=12(ψ0ZB+(1)αψ1ZB)/pαXvir/pZA.

Now, we combine Eqs. 40, 47, 53, and 54 to obtain a relationship for the kth pulse associated with the actual states0pZBS+gU(Tr[P̂(A+CB)M̂+(k)],1ϵ)pZBSgL(p1XvircSp0XpXBP(k)(q0x,0xAct)+1SpZBP(k)(phAct),1ϵ)(55)with Tr[P̂(A+CB)M̂+(k)] given by Eq. 48. We stress that Eq. 55 does not depend on pαXvir, and the inner products ∣CBA+R+CB∣ and ∣CBARCB∣ have the value of 1 − ϵ. Therefore, Eq. 55 does not depend on the inner products of the side-channel states or on the inner products between the side-channel states and the qubit states. In particular, this means that our security proof works even if we do not know anything about the side-channel states, and, thus, they can vary in time and depend on the previous pulses, as discussed above. Note that Eq. 55 is the required relationship for the actual states. This finishes the deviation evaluation part.

Last, we have to convert Eq. 55 into a relationship in terms of numbers rather than probabilities. The procedure for this step is quite standard (17, 18, 26, 47). For this, first, note that gU(x, y) and −gL(x, y) are concave functions with respect to 0 ≤ x ≤ 1 for any fixed 0 ≤ y ≤ 1. In addition, recall that the use of Azuma’s inequality (37) or Kato’s inequality (38) converts the summed probabilities into the corresponding number in the asymptotic limit of a large number of pulses sent. That is, for N → ∞, kNP(k)(qj,jBAct)N(qj,jBAct), where N(qj,jB ∣ Act) is the number of events with Alice’s setting choice equal to j and Bob’s outcome equal to jB in the experiment after N runs of the quantum communication protocol. Here, we emphasize that the use of Azuma’s or Kato’s inequality can deal with any correlations between Alice and Bob’s measurement outcomes, making our proof valid against coherent attacks. Now, we take a summation over k ∈ {1,2, ⋯, N} in Eq. 55, and together with the two ingredients mentioned above, we find the final expression as0S+gU(p1XviraS+p0ZpXBN(q0z,0xAct)Np1XvirbS+p1ZpXBN(q1z,0xAct)N+p0XvirS+p0XpXBN(q0x,1xAct)N,1ϵ)SgL(p1XvircSp0XpXBN(q0x,0xAct)N+1SpZBN(phAct)N,1ϵ)(56)

This inequality involves only the number of events defined in the actual protocol, and by solving this with respect to N(ph∣Act), the security proof is done. We emphasize that our proof is valid for any coherent attack because Eqs. 55 and 56 hold for any K̂e(k).

Arbitrarily long-range pulse correlations

In this section, we show how to extend our analysis to accommodate arbitrarily long-range correlations between the pulses. To simplify the discussion, we consider the three-state protocol, but this formalism can be easily extended to any number of states. Our starting point is the assumption in Eq. 13. We rewrite it here for convenienceBkψjkjk1,,jw+1,j˜w,jw1,,j1ψjkjk1,,jw+1,jw,jw1,,j1Bk21ϵkw(57)where k ∈ {1,2, ⋯, n} and 1 ≤ wk − 1. Note that the difference between both states is in the jwth index. Also, the right-hand side of Eq. 57 does not depend on the indices jk, jk − 1, ⋯, j1 and ~jw, and the term kw is associated with the correlation under consideration. For example, when kw = 1, it refers to the nearest-neighbor pulse correlation considered in Results. Furthermore, without loss of generality, we can assume the relationBkψjkjk1,,jw+1,jw,jw1,,j1ψjkjk1,,jw+1,0X,jw1,,j1Bk0(58)after appropriately choosing the phase of the state ∣jwAw. Using these assumptions, an extension of our framework is now presented. That is, we show how to obtain a lower bound on the parameter aj in Eq. 1 starting from Eq. 57. More generally, the entangled state prepared by Alice, shown in Eq. 2, can now be written asΨABjnj1ζ=1njζAζψjζjζ1,,j1Bζ(59)where jζ ∈ {0Z,1Z,0X} and ζ ∈ {1,2, ⋯, n}. Note that j0 represents having no condition, and the state ∣ψjζjζ − 1, ⋯, j1Bζ represents the long-range pulse correlations, that is, the state of the ζth pulse depends on all the previous setting choices. As before, we suppose that Alice measures her ancilla systems up to the kth pulse. More precisely, she measures the first k − 1 systems of {Aζ}ζ = {1,2, ⋯, n} by using the computational basis. The whole (unnormalized) state can then be expressed asΨjk1,,j1AB(ζ˜=1k1jζ˜Aζ˜ψjζ˜jζ˜1,,j1Bζ˜)jkjkAkψjkjk1,,j1Bk(jnjk+1ζ=k+1njζAζψjζjζ1,,jk+1,jk,jk1,,j1Bζ)(60)

To clarify, after Alice’s measurement, the state ∣Ψ〉AB in Eq. 59 becomes the state Ψjk1,,j1AB in Eq. 60, where the subscripts indicate its dependence on the previous measurement results jk1,,j1. Note that Eq. 60 corresponds to Eq. 6 in Results.

Now, similar to our analysis for the nearest-neighbor pulse correlations, to see how the information jk is encoded in the state Ψjk1,,j1AB, defined in Eq. 60, we rewrite it asΨjk1,,j1AB=(ζ˜=1k1jζ˜Aζ˜ψjζ˜jζ˜1,,j1Bζ˜)jkjkAkψjkjk1,,j1Bk(ajk,jk1,,j1Φjk1,,j1Ak+1,,An,Bk+1,,Bn+bjk,jk1,,j1Φjk,jk1,,j1Ak+1,,An,Bk+1,,Bn)(61)where Φjk1,,j1Ak+1,,An,Bk+1,,Bn and Φjk,jk1,,j1Ak+1,,An,Bk+1,,Bn are normalized states, and Φjk1,,j1Ak+1,,An,Bk+1,,Bn is orthogonal to Φjk,jk1,,j1Ak+1,,An,Bk+1,,Bn. Recall that the subscripts in the variables, e.g., ajk,jk1,,j1, bjk,jk1,,j1, or in the state ψjk1,,j1AB, indicate their dependence on previous results. Importantly, the state Φjk1,,j1Ak+1,,An,Bk+1,,Bn does not depend on jk but Φjk,jk1,,j1Ak+1,,An,Bk+1,,Bn does. In other words, Φjk,jk1,,j1Ak+1,,An,Bk+1,,Bn is the side-channel information for jk. Furthermore, note that ψjkjk1,,j1BkΦjkjk1,,j1Ak+1,,An,Bk+1,,Bn in Eq. 61 corresponds to ϕjkjk1Ak+1,,An,Bk,Bk+1,,Bn in Eq. 11.

Next, we obtain a lower bound on the coefficient ajk,jk1,,j1. For Φjk1,,j1Ak+1,,An,Bk+1,,Bn, one may choose a state such that it becomes independent of jk. One of these choices could beΦjk1,,j1Ak+1,,An,Bk+1,,Bnjnjk+1ζ=k+1njζAζψjζjζ1,,jk+1,0X,jk1,,j1Bζ(62)which is the state of the last (n − k) systems in Eq. 60 with only the kth index of ψjζjζ1,,jk+1,jk,jk1,,j1Bζ being fixed to 0X. This state is independent of jk. Since ajk,jk1,,j1 is equal to the inner product between the state given by Eq. 62 and the vectorjnjk+1ζ=k+1njζAζψjζjζ1,,jk+1,jk,jk1,,j1Bζ(63)which is the expression in the last parenthesis of Eq. 60, we can evaluate a lower bound for ajk,jk1,,j1 asajk,jk1,,j1=jnjk+1ζ=k+1npjζBζψjζjζ1,,jk+1,0X,jk1,,j1ψjζjζ1,,jk+1,jk,jk1,,j1Bζ=jnjk+1ζ=k+1npjζBζψjζjζ1,,jk+1,0X,jk1,,j1ψjζjζ1,,jk+1,jk,jk1,,j1Bζjnjk+1ζ=k+1npjζ(1εζk)1/2=ζ=1nk(1εζ)1/2(64)

In the second equality, we use the result given by Eq. 58, and the inequality comes from Eq. 57.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/37/eaaz4487/DC1

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REFERENCES AND NOTES

Acknowledgments: We thank G. Currás Lorenzo and Á. Navarrete for very valuable discussions. We also thank an anonymous referee for finding a mistake in a previous version of this manuscript. Funding: This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO), the Fondo Europeo de Desarrollo Regional (FEDER) through the grant TEC2017-88243-R, and the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 675662 (project QCALL). K.T. acknowledges support from JSPS KAKENHI grant nos. JP18H05237 18H05237 and JST-CREST JPMJCR 1671. Author contributions: M.P. and K.T. conceived the initial idea of how to deal with pulse correlations. K.T. constructed the RT through discussions with all the other authors. Then, G.K., A.M., and M.C. contributed to the development and discussion of the initial idea and the RT with M.P. and K.T. M.P. performed the numerical simulations. All authors contributed to writing the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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