Memory-assisted measurement-device-independent quantum key distribution

A protocol with the potential of beating the existing distance records for conventional quantum key distribution (QKD) systems is proposed. It borrows ideas from quantum repeaters by using memories in the middle of the link, and that of measurement-device-independent QKD, which only requires optical source equipment at the userʼs end. For certain memories with short access times, our scheme allows a higher repetition rate than that of quantum repeaters with single-mode memories, thereby requiring lower coherence times. By accounting for various sources of nonideality, such as memory decoherence, dark counts, misalignment errors, and background noise, as well as timing issues with memories, we develop a mathematical framework within which we can compare QKD systems with and without memories. In particular, we show that with the state-of-the-art technology for quantum memories, it is potentially possible to devise memory-assisted QKD systems that, at certain distances of practical interest, outperform current QKD implementations.


Introduction
Despite all commercial [1] and experimental achievements in QKD [2,3,4,5,6,7,8,9,10], reaching arbitrarily long distances is still a remote objective.The fundamental solution to this problem, i.e., quantum repeaters, is known for over a decade.From early proposals by Briegel et al. [11] to the latest no-memory versions [12,13,14], quantum repeaters, typically, rely on highly efficient quantum gates comparable to what we may need for future quantum computers.While the progress on that ground may take some time before such systems become functional, another approach based on probabilistic gate operations was proposed by Duan and co-workers [15], which could offer a simpler way of implementing quantum repeaters for moderate distances of up to around 1000 km.The latter systems require quantum memory modules with high coupling efficiencies to light and with coherence times exceeding the transmission delays, which are yet to be achieved together.In this paper, we propose a protocol that, although is not as scalable as quantum repeaters, for certain classes of memories, relaxes, to some extent, the harsh requirements on memories' coherence times, thereby paving the way for the existing technologies to beat the highest distance records achieved for no-memory QKD links [2].The idea behind our protocol was presented in [16], and independent work has also been reported in [17].This work proposes additional practical schemes and rigorously analyses them under realistic conditions.
Our protocol relies on concepts from quantum repeaters, on the one hand, and the recently proposed measurement-device-independent QKD (MDI-QKD), on the other.The original MDI-QKD [18] relies on sending encoded photons by the users to a middle site at which a Bell-state measurement (BSM) is performed.One major practical advantage of MDI-QKD is that this BSM can be done by an untrusted party, e.g., the service provider, which makes MDI-QKD resilient to detector attacks, e.g., timeshift, remapping, and blinding attacks [19,20,21,22,23,24,25,26]. The security is then guaranteed by the reverse EPR protocol [27].Another practical advantage is that this BSM does not need to be a perfect measurement, but even a partial imperfect BSM implemented by linear optical elements can do the job.In our scheme, by using two quantum memories at the middle site, we first store the state of the transmitted photons in the memories, and perform the required BSM, only when both memories are loaded.In that sense, our memory-assisted MDI-QKD is similar to a single-node quantum repeater, except that there is no memories at the user's end.This way, similar to quantum repeaters, we achieve a rate-versus-distance improvement as compared to the MDI-QKD schemes proposed in [18,28,29,30], or other conventional QKD systems that do not use quantum memories.
There is an important distinction between our protocol and a conventional quantum repeater system that relies on single-mode memories.In such a quantum repeater link, which relies on initial entanglement distribution among neighbouring nodes, the repeat period for the protocol is mainly dictated by the transmission delay for the shortest segment of the repeater system [31,32].In our scheme, however, the repeat period is constrained by the writing time, including the time needed for the herald/verification process, into memories.This implies that using sufficiently fast memories, i.e., with short writing times, one can run our scheme at a faster rate than that of a quantum repeater, thereby achieving higher key generation rates, as compared to conventional QKD links, and at lower coherence times, as compared to probabilistic repeater systems.This increase in clock rate is what our proposal shares with the recently proposed third generation of quantum repeaters, which use quantum error correction codes to compensate for loss and errors, thus also being able to speed up the clock rate to local processing times [12].The need for long coherence times remains one of the key challenges in implementing the first generations of quantum repeaters before the latest no-memory quantum repeater proposals can be implemented.
The above two benefits would offer a midterm solution to the problem of longdistance QKD.While our scheme is not scalable the same way that quantum repeaters are, it possibly allows us to use the existing technology for quantum memories to improve the performance of QKD systems.In the absence of fully operational quantum repeater systems, our setup can fill the gap between theory and practice and will become one of the first applications of realistic quantum memories in quantum communications.
It is worth mentioning that the setups we propose here are compatible with different generations of hybrid quantum-classical (HQC) networks [33].In such systems, home users are not only able to use broadband data services, but they can also use quantum services such as QKD.MDI-QKD offers a user-friendly approach to the access part of such networks as the end users only require source equipment.Whereas, in the first generation of HQC networks, the service provider may only facilitate routing services for quantum applications, in the future generations, probabilistic, deterministic, and eventually no-memory quantum repeaters constitute the quantum core of the network.In each of these cases, our setups are extensible and compatible with forthcoming technologies for HQC networks.
The rest of the paper is structured as follows.In Section II, we describe our proposed schemes and the modelling used for each component therein.Section III presents our key rate analysis, followed by some numerical results in Section IV.Section V concludes the paper.

System Description
Our scheme relies on "loading" quantum memories (QMs) with certain, unknown, states of light.This loading process needs to be heralding, that is, by the end of it, we should learn about its success.Within our scheme, two types of memories can be employed, which we refer to by directly versus indirectly heralding QMs.Some QMs can operate in both ways, while some others are more apt to one than the other.By directly heralding memories we refer to the class of memories to which we can directly transfer the state of a photon and we can verify-without revealing or demolishing the quantum statewhether this writing process has been successful.An example of such memories is a trapped atom in an optical cavity [34].In the case of indirectly heralding memories, a direct writing-verification scheme may not exist.Instead, we assume that we can entangle a photonic state with the state of such QMs [15,35,36,37,38,39,40], and later, by doing a measurement on the photon, we can effectively achieve a heralded writing into the memory.These two approaches of writing cover most relevant practical examples to our scheme.
The scheme for directly heralding memories works as follows [16,17]; see figure 1(a).The two communicating parties, Alice and Bob, send BB84 encoded pulses [41], by either single-photon or weak laser sources, towards QM units located in the middle of the link.Each QM stores a photon in a possibly probabilistic, but heralding, way.Once both memories are loaded, we retrieve their states and perform a BSM on the corresponding photons.A successful BSM indicates some form of correlation between the transmitted bits by Alice and Bob.
We can easily extend the above idea to the case of indirectly heralding memories.An additional BSM, on each side, along with an entangling process between photons and QMs, can replace the verification process needed for directly heralding memories.In this case, see figure 1(b), a successful BSM between the transmitted photon by the users and the one entangled with the QM, would effectively herald a successful loading process, that is, the state of the QM is correlated with the quantum state sent by the users.
In order to entangle a QM with a photon, one can think of two standard ways.One approach would be to generate a pair of entangled photons, e.g., by using spontaneous parametric down-converters [42,43], and then store one of the photons in the memory and use the other one for interference with the incoming photon sent by the user.While this approach is not fully heralding (because we cannot be sure of the absorption of the locally generated photon by the memory), it is still a viable option for highly efficient writing procedures.Another approach to entangle a photon with a memory, which this paper is mainly concerned with, is to start from the memory and generate a photon entangled with the memory by driving certain transitions in the memory [40,15].With entangling times as short as 300 ps reported in the literature [44], high repetition rates are potentially achievable for indirectly heralding memories.
In either approach, it is possible to have multiple-excitation effects, which can cause errors in our setup.In this paper, for readability reasons, we make the simplifying assumption of having only single excitations in the memories, and address the multiple excitation effect in a separate publication [45].Furthermore, here we only consider the polarization entanglement.The extension to other types of entanglement is straightforward and will be dealt with in forthcoming publications.
Under all above assumptions, suppose once we entangle the memory A with a single photon P , the joint state of the two is given by 1 Figure 1: (a) MDI-QKD with directly heralding quantum memories.Alice and Bob use the efficient BB84 protocol to encode and send pulses to their respective QM in the middle of the link.At each round, each QM attempts to store the incoming pulse.Once they are both loaded, we retrieve the QMs' states and perform a BSM on the resulting photons.(b) MDI-QKD with indirectly heralding quantum memories.At each round, an entangling process is applied to each QM, which would generate a photon entangled, in polarization, with the QM.These photons interfere at the BSM modules next to the QMs with incoming pulses from the encoders.As soon as one of these BSMs succeeds, we stop the entangling process on the corresponding QM, and wait until both QMs are ready for the middle BSM operation.In this case, QMs are not required to be heralding; a trigger event is declared by the success of the BSM located between the QM and the respective encoder.(c) The original MDI-QKD protocol [18].(d) One possible energy-level configuration for a QM suitable for polarization encoding.
photons, and |s H A and |s V A are the corresponding memory states; see figure 1(d).In equation (2.1), the conditional state of the photon, knowing the memory state, has the same form as in BB84.Each leg of figure 1(b), from the user end to the respective QM, is then similar to an asymmetric setup of the original MDI-QKD scheme as depicted in figure 1(c).The working of the system in figure 1(b) will then follow that of the original MDI-QKD.We will use this similarity in our analysis of the system in figure 1(b).
The main advantage of our scheme as compared to the original MDI-QKD, in figure 1(c), is its higher resilience to channel loss and dark count.In the no-memory MDI-QKD, both pulses, sent by Alice and Bob, should survive the path loss before a BSM can be performed.The key generation rate then scales with the loss in the entire channel.In our scheme, each pulse still needs to survive the path loss over half of the link, but this can happen in different rounds for the signal sent by Alice as compared to that of Bob.We therefore achieve the quantum repeater benefit in that the key generation rate, in the symmetric case, scales with the loss over half of the total distance.Moreover, in the case of directly heralding memories, our scheme is almost immune against dark counts [17].This is because the measurement efficiency in the BSM module is typically a few orders of magnitude higher than that of dark count rates.Dark counts will then only sightly add to the error rate.In our scheme, memory decoherence errors play a major role as we will explain in this and the following sections.
In the following, we describe the protocol and its components in more detail.

Protocol
In our protocol, Alice and Bob, at a rate R S , send BB84 encoded pulses to the middle station (dashed boxes in figure 1).At the QMs, for each incoming pulse, we either apply a loading process by which we can store the state of the photons into memories and verify it, or use the indirectly heralding scheme of figure 1(b).Once successful for a particular QM, we stop the loading procedure on that QM, and wait until both memories are loaded, at which point, a BSM is performed on the QMs.The BSM results are sent back to Alice and Bob, and the above procedure is being repeated until a sufficient number of raw key bits is obtained.The rest of the protocol is the same as that of MDI-QKD.Sifting and postprocessing will be performed on the raw key to obtain a secret key.In this paper, we neglect the finite-size-key effects in our analysis [29].

Component modeling
In this section, we model each component of figure (1) including sources and encoders, the channel, QMs, and the BSM module.

Sources and encoders
We consider two types of sources: ideal single-photon sources and phase-randomized weak laser pulses.The latter will be used in the decoystate [46] version of the protocol.Each source, at both Alice's and Bob's sides, generates pulses at a rate R S .Each pulse is polarization encoded in either the rectilinear (Z) or diagonal (X) basis.In the case of ideal single photons, we, correspondingly, send states |H and |V in the Z basis, and in the X basis.In each basis, the two employed states, respectively, represent bits 1 and 0. In the case of the decoy-state protocol, the single-photon states are replaced with weak phaserandomized coherent states of the same polarization.Here, we use the efficient version of BB84 encoding, where the Z basis is used much more frequently than the X basis [47].The pulse duration is denoted by τ p and it is chosen in accordance with the requirements of the memory system in use.
There are several sources of nonideality one may be concerned with at the encoder box.For instance, in [48], one major source of error is in not generating fully orthogonal states in each basis.Note that secure exchange of keys may still be possible, although at a possibly reduced rate, by using even uncharacterised sources [49].Another possible issue would be in having multiple-photon components if one uses parametric downconverters to generate single photons [50,51].Although all these issues, among others, are important in the overall performance of the system, here we would rather focus on the memory side of the system, which is newly introduced, and deal with the details of source imperfections, and their effects on the secret key generation rate in a separate publication [45].

Channels
The distance between Alice (Bob) and the respective QM is denoted by L A (L B ).The total distance between Alice and Bob is denoted by The transmission coefficient for a channel with length l is given by where L att is the attenuation length of the channel (roughly, 22 km for 0.2 dB/km of loss).The channel is considered to have a background rate of γ BG per polarization mode, which results in an average p BG = 2γ BG τ p background photons per pulse.This can stem from stray light or crosstalk from other channels, especially if classical signals are multiplexed with quantum ones in a network setup [6,5,52,53,54].
We also consider setup misalignment in our analysis.We assume certain polarization maintenance schemes are in place for the Alice's and Bob's channels, so that the reference frames at the sources and memories are, on average, the same.We, nevertheless, consider a setup misalignment error probability e dK , for K = A, B, to represent misalignment errors in each channel.

Quantum memories
We use the following assumptions and terminologies for the employed QMs.This list covers most relevant parameters in an experimental setup relying on polarization encoding, whether the QM is operated in the directly or indirectly heralding mode.
• In the case of a successful loading, each QM in figure 1 ideally stores a polarization qubit corresponding to the polarization of the incoming pulse.We assume that such a squashing operation occurs [55,56] even if at the input of the QM there is a non-qubit state, e.g., a phase-randomized coherent state.That is, if, for instance, two photons with horizontal polarizations are at the input of the memory, the QM would only store the polarization information, and ignores the photon-number information.In practice, the loading efficiency would be a function of input photon numbers, but, for simplicity, here we neglect this dependence.This is in line with our single-excitation assumption we have adopted in this paper.One suitable energy level structure for such a memory is the double-Λ configuration in figure 1(d), with a common ground state and two other metastable states corresponding to two orthogonal polarizations.The excited states can then facilitate Raman transitions from the ground state to each of the metastable states, using known optical transition techniques [57,58], in response to the input polarization state.We assume that each QM only stores one spatio-temporal mode of light.Our protocol can be extended to incorporate multimode QMs [59,60,61,62] or multiple QMs [31], in which case a linear improvement in the rate is expected.In this work, we focus on the case of a single logical memory per user and leave extensions to future work.
• For directly heralding memories, we denote the QM's writing efficiency by η w .The writing efficiency is the probability to store a qubit and herald success conditioned on having a single-photon at the QM's input.Note that η w also includes the chance of failure for our verification process.For indirectly heralding memories, we introduce an entangling efficiency, η ent , which is the probability of success for entangling a photon with our QM.
• We denote the QM's reading efficiency by η r .That is the probability to retrieve a single photon out of the QM conditioned on a successful loading in the past.The reading efficiency is expected to decay over a time period t as η r (t where T 1 is the memory amplitude decay time and η r0 is the reading efficiency right after loading.In our example of a double-Λ-level memory of figure 1(d), such a decay corresponds to the transition form one of the metastable states |s H or |s V to the ground state |g , in which case, no photon will be retrieved from the memory.
• We denote the QM's writing time by τ w .For directly heralding memories, it is the time difference between the time that a pulse arrives (beginning of the pulse) at the QM and the time that a successful/unsuccessful loading is declared.This is practically the fastest repeat period one can run our protocol.In the case of indirectly heralding memories, τ w includes the time for the entangling process as well as that of the side BSM operation.Accounting for such timing parameters is essential in enabling us to have a fair comparison between memory-assisted and no-memory QKD systems.One must note that in a practical setup there will be time periods, e.g., for synchronization purposes or memory refreshing, over which no raw key is exchanged.The total number of key bits exchanged over a period of time must therefore exclude such periods once the total key generation rate is calculated.In our work, we neglect all these overhead times, with the understanding that one can easily modify our final result by considering the percentage of the time spent on such processes within a specific practical setup.
• We denote the QM's reading time by τ r .It is the time difference between the time that the retrieval process is applied until a pulse (end of the pulse) is out.
• We denote the QM's coherence (dephasing) time by T 2 .For an initial state ρ(0) of the QM at time zero, its state at a later time t is given by [31] where p(t) = [1 + exp(−t/T 2 )]/2.Note that dephasing would only occur if we are in a superposition of Z eigenstates, e.g., the eigenstates of X.The above model of decoherence is expected to have more relevance in some practical cases of interest [34,40,31] than the model used in [17], in which the memory state switches suddenly from an intact one to a fully randomized version after a certain time.We discuss the implications of each model in our numerical result section.It is, however, beyond the scope of this paper to fully model every possible decoherence mechanisms in QMs.Specific adjustments are needed if one uses a memory that is not properly modelled by our T 1 and T 2 time constants.

BSM module
Figure 2 shows the schematic of the BSM module used in our analysis.This module enables an incomplete BSM over photonic states.In order to use this module, in our scheme, we first need to read out the QMs and convert their qubit states into polarization-encoded photons.The BSM will then be successful if exactly two detectors click, one H-labelled and one V -labelled.Depending on which detectors have clicked and what basis is in use, Alice and Bob can identify what bits they ideally share [28].
We assume the BSM module is symmetric.We lump detector quantum efficiencies with other possible sources of loss in the BSM module and denote it by η d for each detector.We also assume that each detector has a dark count rate of γ dc , which results in a probability p dc = γ dc τ p of having a dark count per pulse.The implicit assumption here is that the retrieved and the writing photons have the same pulse width.Finally, we assume that there is no additional misalignment error in the BSM module.

Key Rate Analysis
In this section, we find the secret key generation rate for our proposed schemes in figures 1(a) and 1(b).The common assumption in our predicting the relevant observed parameters in a QKD experiment is that we work under the normal mode of operation, where there is no eavesdropper present, and we are only affected by the imperfections of the system, behind which an eavesdropper can in principle hide.We later compare our results with two conventional QKD schemes, namely, BB84, summarized in Appendix A, and the original MDI-QKD in figure 1(c), summarized in Appendix B, that use no memories.In all cases, we consider both single-photon and decoy-state sources.In all forthcoming sections, f denotes the inefficiency of the error correction scheme, i.e., the ratio between the actual cost of error correction and its minimum value obtained by the Shannon's theorem, assumed to be constant, and we denote the binary entropy function as h(p) = −p log 2 (p) − (1 − p) log 2 (1 − p), for 0 ≤ p ≤ 1.

Key rate for single-photon sources
With ideal single-photon sources, the secret key generation rate in the setups of figures 1(a) and 1(b) is lower bounded by [63] where efficient BB84 encoding is employed [47].In the above equation, e QM 11;X and e QM 11;Z , respectively, represent the quantum bit error rate (QBER) between Alice and Bob in the X and Z basis, when single photons are used, and Y QM 11 represents the probability that both memories are loaded with single photons of the same basis and the middle BSM is successful.
To obtain the individual terms in equation ((3.1)), we can decompose the protocol into two parts: the memory loading step and the measurement step, once both memories are loaded.The first step is a probabilistic problem with two geometric random variables, N A and N B , corresponding, respectively, to the number of attempts until we load Alice and Bob's memories with single photons.The number of rounds that it takes to load both memories is then max{N A , N B }. Once both memories are loaded, the rest of the protocol is similar to that of original MDI-QKD in terms of rate analysis: the QMs replace the sources in figure 1(c) and the total transmission-detection efficiency is replaced by the reading-measurement efficiency in the BSM module.We can therefore use many of the relationships obtained for the original MDI-QKD, summarized in Appendix B, for the memory-assisted versions of figure 1.
For finite values of T 1 , the reading efficiency for the Alice's QM could be different from that of Bob.In fact, we can assume that, once both memories are loaded, one of the memories (late) will be read immediately, while the other (early) |N A − N B | rounds after its successful loading.The effective measurement efficiency for the leg K, K = A, B, corresponding to the path originating from memory K in the BSM module will then be given by With the above setting, and considering the required time for reading from the QMs, we obtain where Y 11 is the corresponding yield term, given by equation (B.4), for the MDI-QKD protocol and N L = E{max(N A , N B )} is given by equation (C.3).Here, E{•} represents the expectation value operator with respect to N A and N B , and η ′ m = η d η r , where η r = η r0 E{exp(−|N A − N B |T /T 1 )} can be obtained from equation (C.4).In equation (3.3), N r , represents the extra rounds lost due to the nonzero reading times of QMs, once they are both loaded, and is given by where T = 1/R S is the repetition period.The condition τ w ≤ T is a matter of practicality as sending photons faster than they can be stored is of no benefit.The fastest possible rate is then obtained at T = τ w .
In the case of directly heralding memories of figure 1(a), we have as the probabilities of successful loading of Alice and Bob's QMs with single-photon sources (or background noise).In the case of indirectly heralding memories of figure 1(b), following our discussion in Section (2) about the equivalence of each leg of figure 1(b) to an asymmetric MDI-QKD system, we have where the above terms must be calculated at an effective dark count rate of γ dc + γ BG η d /2.We remark that, although obtained from different methods, the analysis in [17] also finds similar expressions for the yield term.In [17], the analysis is only concerned with the symmetric setup, and some of the parameters considered in our work take their ideal values.It can be verified, however, that in the special case of τ w = T , τ r = 0, γ BG = 0, L A = L B , η w = 1, and T 1 → ∞, for directly heralding memories, equation (3.3) reduces to the same result obtained in [17].By accounting for additional relevant parameters, our analysis offers a better match to realistic experimental scenarios.
Similarly, the error terms are given by where, e 11;Z and e 11;X , given by equation (B.4), are the corresponding error terms for the original MDI-QKD.In addition to the typical sources of error, such as loss and dark count, the above expressions are functions of misalignment parameters.This misalignment could be a statistical error in the polarization stability of our setup, modelled by e dA and e dB , or an effective misalignment because of memory dephasing [64] and/or background photons.Putting all these effects together, as we have done in Appendix D, we obtain where e dS and e dS , respectively, represent the misalignment probabilities for Alice's and Bob's memories, for basis S = X, Z, at loading probabilities η A and η B and are given by equations (D.2) and (D.5).The above equation accounts for the fact that if the state of both memories are flipped, Alice and Bob will still share identical key bits.We assume that the BSM module is balanced and does not have any setup misalignment.
Note that in equation (3.8), because of no dephasing errors for the Z eigenstates, e QM dZ is independent of N A and N B , whereas e QM dX is a function of them.The approximation in equation (3.7) assumes E{e QM dX η mA η mB } ≈ E{e QM dX }E{η mA η mB }, which is valid when T 1 ≫ T 2 , to give a more readable final result.
Equation (3.8) can also be used in the case of indirectly heralding QMs as explained in Appendix D. The main idea is to use the analogy of each leg in figure 1(b) with the original MDI-QKD in figure 1(c).

Key rate for decoy states
Suppose Alice and Bob use a decoy-state scheme with average photon numbers µ and ν, respectively, for the two main signal intensities, and infinitely many auxiliary decoy states.The secret key generation rate, in the limit of infinitely long key, is then given by where is the rate at which both memories are loaded, by Alice (Bob) sending a coherent state in the Z basis with µ (ν) average number of photons, and a successful BSM is achieved.
In the case of directly heralding memories, are the probabilities for successful loading of Alice and Bob's QMs with coherent-state sources.Similarly, is the QBER in the Z basis, and is the contribution of single-photon states in the gain term of equation (3.10).
Similar to the treatment in the previous subsection, one can find or approximate the above terms in the case of indirectly heralding memories as well.For the sake of brevity, we leave this extension to the reader.
Apart from all additional parameters considered in our model as compared to [17], our treatment of the decoy-state QKD is different from that of [17] in the way that QMs are modelled.In our work, we assume QMs store qubits, which while is not necessarily an exact model, it often serves a good first-order approximation to the reality.In [17], however, QMs are assumed to be able to store number states.This assumption seems more restrictive as many QMs, such as single trapped atoms or ions, can only store one photon.

Storage time
To get some insight into the working of our system, in this section, we simulate the achievable rates assuming L A = L B = L/2.The average number of trials to load both memories from equation(C.3) is then given by [65] where η is the probability of successfully loading a QM at distance L/2, approximately, given by η QM exp(−(L/2)/L att ), where η QM = η w for directly heralding memories, and η QM = η ent η 2 d for indirectly heralding QMs.Similarly, the average required storage time, from equation (C.5), is given by which is similar to the result reported in [17].
The secret key generation rate in equations (3.1) and (3.9) is proportional to the pulse generation rate R S = 1/T at the encoder.To maximize R S , we choose T = τ w , throughout this section and next, resulting in T st ≈ τ w /η. Figure 3 compares T st with the required storage time in multi-memory probabilistic quantum repeaters [31], L/c, where c is the speed of light in the channel.It can be seen that our scheme offers lower required coherence times until a certain distance.With fast memories of shorter than 10 ns of access time, this crossover distance could be longer than 500 km.With such memories, the required coherence time at 300 km is roughly 1 µs, or lower, as compared to over 1 ms for probabilistic quantum repeaters.
It is worth mentioning that the possible advantage of requiring low coherence times is only achievable for systems with nesting level one, i.e., with one stage of entanglement swapping.Unlike quantum repeaters, our protocol, in terms of its timing, is not scalable to higher nesting levels.Nevertheless, even with only one entanglement swapping stage, our protocol can outperform conventional QKD schemes in terms of rate-versus-distance behaviour, and, more importantly, this can possibly be achieved with existing technology for quantum memories.We explore this and other aspects of our scheme in the next section.Distance, L (km) Figure 3: Average required storage time, T st , versus distance, in our scheme, for different repetition rates 1/τ w .As compared to that of a probabilistic quantum repeater, labelled by L/c, where c = 2 × 10 8 m/s is the speed of light in optical fibre, our scheme requires lower coherence times up to a certain distance.The crossover distance at τ w = 1 µs is over 300 km and at τ w = 1 ns is nearly 700 km.In all curves, η w = η d = η ent = 1 and p BG = 0.

Numerical Results
In this section, we study the impact of various parameters on the secret key generation rate of our scheme.All results have been obtained assuming the symmetric setup described in Section (3.3), τ w = T , f = 1.16, c = 2 × 10 8 m/s, and 0.2 dB/km of loss in the channel.We also compare our scheme with the efficient BB84 and MDI-QKD protocols, whose secret key generation rates are, respectively, summarized in Appendices A and B.

Coherence time
In this section, we discuss the effects of memory dephasing on the secret key generation rate.As mentioned before, while our scheme in figure 1(a) is particularly resilient to dark count errors, it still suffers from memory errors.Figure 4 demonstrates the secret key generation rate per pulse at different coherence times for the scheme of figure 1(a).
A finite coherence time is the only source of nonideality considered in this figure.Since, in our model, the dephasing process only affects the diagonal basis, e QM 11;Z = 0 at all distances; hence R QM ∝ 1 − h(e QM 11;X ) remains always positive.The rate is initially proportional to exp(−(L/2)/L att ), and with low values of e QM 11;X for short distances, our scheme beats the BB84 case depicted by the dashed line.Note that, because of the partial BSM in figure 2, the initial key rate at L = 0 for our scheme is lower than that of BB84.At large distances, however, the dephasing process becomes significant and results in e QM 11;X approaching 1/2; see the inset.Subsequently, R QM decays with a faster slope and at some point becomes lower than what one can achieve with an ideal BB84 system.The window between the two crossing points on each curve is the range where our scheme, can, in principle, beat a noise-free BB84 system.This window is larger for QMs with longer coherence times.In [17], authors look at the minimum required coherence time to achieve nonzero key rates, assuming e QM 11;X = e QM 11;Z within their model of decoherence.Although the models used for decoherence in our work and [17] are different, e QM 11;X has a similar behaviour in both cases.In our case, however, the transition from 0 to 1/2 is smoother than that of [17].This is expected as the model in [17] is an abrupt good-bad model for the memory.A consequence of this difference is that the minimum required coherence time is then higher in our case, which highlights the importance of the more accurate model we have used for decoherence.
The comparison in figure 4 assumes that the source rate R S is the same for both the BB84 protocol and our scheme.In our scheme, however, R S depends on the writing time of the memories.Figure 5 shows the secret key generation rate for the scheme of figure 1(a) at a fixed value of T 2 /T = 1000, but for several values of T = τ w = 1/R S .The BB84 system is run at a fixed rate of 1 GHz.Again, we assume that the only source of nonideality is memory dephasing.It can be seen that slow memories with writing times of 100 ns, or higher, can hardly compete with an ideal BB84 system.The two orders of magnitude lost because of the lower repetition rate cannot be compensated within the first 300 km.It is still possible to beat the BB84 case, at long distances, if memories have higher coherence times.

Realistic examples
It is interesting to see if any of the existing technologies for quantum devices can be employed in our scheme to beat conventional QKD systems.Figure 6 makes such a comparison between BB84, MDI-QKD, and memory-assisted MDI-QKD for particular experimental parameters.We have chosen our QM parameters based on the two lessons learned from figures 4 and 5: the QM needs to have a high bandwidth-storage product (T 2 /τ w ) on the order of 1000 or higher, and, it also needs to be fast, with writing times on the order of nanoseconds.Both these criteria are met for the QM used in [44], which particularly offers fast reading and writing with 300-ps-long pulses at a storage time of around 4 µs.The employed memory in this experiment is an atomic ensemble, which fits our indirectly heralding scheme of figure 1(b).We should, however, be careful with multiple excitations in this case, which are not considered in our model.We therefore assume that, by driving this memory with short pulses, one can ideally generate the jointly entangled state in equation (2.1) between the memory and a photon [39], where, in this case, |s H and |s V are, respectively, the corresponding symmetric collective excited states to horizontal and vertical polarizations [15,37].By keeping the entangling efficiency low at η ent = 0.05, here, we try to keep the effect of multiple excitations in such memories low [35,66,64]; further analysis is, however, required to fully account for such effects [45].We also assume that T 2 = T 1 and use the state-ofthe-art single-photon detectors with η d = 0.93 at γ dc = 1 count per second and 150 ps of time resolution [67] for all systems.
We consider two sets of parameter values for our employed QM in figure 6.In the first set, corresponding to the curve labelled A on the figure, we use the same numerical values as reported in [44], that is, η r0 = 0.3, T 1 = 4 µs, and τ w = τ r = τ p = 300 ps.We, however, assume that R S = 1/τ w , which is much faster than the repetition rate used in [44].In the curve labeled B, we improve the performance by assuming η r0 = 0.73, Figure 6: Secret key generation rate for single-photon BB84 (dotted), MDI-QKD (dashed), and our indirectly heralding scheme of figure 1(b) (solid) at practical parameter values.In all curves, η d = 0.93, γ dc = 1/s, γ BG = 0, and e dA = e dB = 0.005.For BB84 and MDI-QKD, R S = 3.3 G pulse/s, similar to R S = 1/τ w , in our scheme.For our scheme, we have used some of the experimental parameters reported in [44].For the curve labelled A, η ent = 0.05, η r0 = 0.3, T 1 = T 2 = 4 µs, and τ w = τ r = τ p = 300 ps.It is assumed that there is no multiple excitations in the QMs.For the curve labelled B, everything is the same except that η r0 = 0.73 and which is what another group has obtained for a similar type of memory [39], and T 1 = T 2 = 100 µs, which is attainable by improving magnetic shielding [68].It can be seen that, whereas the current QM employed in [44] is short of beating either of no-memory systems, our slightly boosted system, in curve B, outperforms both systems at over roughly 200 km.The cut-off distance in curve B is about 400 km, which is mainly because of memory decoherence, and it can be improved by using memories with longer coherence times.This implies that with slightly improving some experimental parameters, we would be able to employ realistic QMs to improve the performance of practical quantum communication setups.We remark that the example QM chosen in figure 6 is not necessarily the only option, and improved versions of other types of memories can potentially offer the same performance [40,61,69,70,71,72,73].
What we have proposed here is an initial step toward improving the performance of QKD systems by using quantum memories.In particular, we have shown how technologically close we are to beating a direct, no-memory, QKD link in terms of the achievable rate at certain long distances.Our scheme is not, however, scalable to arbitrarily long distances.For that matter, full quantum repeaters would eventually be needed.A possible roadmap for the development of such systems would pass through probabilistic, and then deterministic, and eventually no-memory versions of quantum repeaters [12,13,14].It is hard to make a fair comparison between all these and our scheme, as the required resources in each case are different.Some studies have nevertheless compared different repeater schemes under certain assumptions [74,64].It is only the future, in the end, that proves which system, and at what price, can be implemented over the course of time.

Conclusions
By combining ideas from quantum repeaters and MDI-QKD, we proposed a QKD scheme that relied on quantum memories.While offering the same rate-versus-distance improvement that quantum repeaters promise, the coherence-time requirements for the quantum memories employed in our scheme could be less stringent than that of a general single-mode probabilistic quantum repeater system.That would provide a window of opportunity for building realistic QKD systems that beat conventional no-memory QKD schemes by only relying on existing technologies for quantum memories.In our work, we showed that how close some experimental setups would be in achieving this objective.Our protocol acts as a middle step on the roadmap to long-distance quantum communication systems. in the (infinitely many) decoy-state case, where µ is the average number of photons for signal states, which is dominantly used.In equation (A.1), Y 1 is the yield of single photons, or the probability that Bob gets a click on his measurement devices assuming that Alice has sent exactly one photon, and is given by where η = η ch (L)η d , and correspond, respectively, to the terms that, in the absence of misalignment, result in identical (Correct) versus non-identical (Error) bits shared by Alice and Bob.The QBER, e 1 , which is the same for both bases, is given by where e 0 = 1/2 and e d = e dA + e dB is the total misalignment probability for the channel.Similarly, in equation (A.2), are the corresponding gain terms [46], and gives the QBER.

Appendix B. MDI-QKD key rate analysis
The secret key generation rate for the MDI-QKD scheme of figure 1(c) is lower bounded by [28] in the single-photon case, and in the decoy-state case, where µ (ν) is the average number of photons for signal states sent by Alice (Bob).Here, Q 11 is the probability of a successful BSM if Alice and Bob, respectively, send pulses with µ and ν average number of photons and is given by with e d being the total misalignment probability.In the scheme of figure 1(c), e d = e dA (1 − e dB ) + e dB (1 − e dA ).Similarly, using the results obtained in [28], we have where The loading process in the setups of figures 1(a) and 1(b) are probabilistic ones, with two geometric random variables N A and N B playing the major role.Suppose the success probability for each loading attempt corresponding to these random variables is, respectively, given by η A and η B .Then, we obtain the following probability distribution for |N A − N B |: where Using the above expressions, we then obtain where we used the fact that e In the case of indirectly heralding QMs, we assume that each erroneous click on the side BSMs will effectively result in a flip to the corresponding QM state, and can also be modeled as misalignment.This assumption is valid at low distances where majority of errors are caused by the setup misalignment.We then obtain

Figure 2 :
Figure 2: Bell-state measurement module for polarization states.

Figure 4 :
Figure 4: Secret key generation rate per pulse for the heralded scheme of figure 1(a) for different values of T 2 /T using single-photon sources.The dashed line represents the ideal efficient BB84 case.Unless explicitly mentioned, all other parameters assume their ideal values: T 1 → ∞, η w = η r0 = η d = 1, γ BG = γ dc = 0, e dA = e dB = 0, and τ r = 0.
dZ = e 11;Z (η d η ch (L K ), η d η ent , e dK ), K = A, B, (D.10) for indirectly heralding QMs, where e 11;Z can be calculated from equation (B.4) at an equivalent dark count rate of γ dc +η d γ BG /2.At long distances, most errors originate from dark counts or background photons, whose effective misalignment effect will approach half of e 11;Z in the above equation.As a conservative assumption, we use the expression in equation (D.10) for all distances.All other terms in equations (3.7) and (3.8) can be obtained following the same expressions in eqations (D.5)-(D.9)at β K = 1 − 2e (K) dZ , for K = A, B, and using equation (D.10) for e (K) dZ .
)where |H P and |V P , respectively, represent horizontally and vertically polarized single