Abstract
Quantum cryptography harnesses quantum light, in particular single photons, to provide security guarantees that cannot be reached by classical means. For each cryptographic task, the security feature of interest is directly related to the photons’ nonclassical properties. Quantum dotbased singlephoton sources are remarkable candidates, as they can in principle emit deterministically, with high brightness and low multiphoton contribution. Here, we show that these sources provide additional security benefits, thanks to the tunability of coherence in the emitted photonnumber states. We identify the optimal optical pumping scheme for the main quantumcryptographic primitives, and benchmark their performance with respect to Poissondistributed sources such as attenuated laser states and downconversion sources. In particular, we elaborate on the advantage of using phononassisted and twophoton excitation rather than resonant excitation for quantum key distribution and other primitives. The presented results will guide future developments in solidstate and quantum information science for photon sources that are tailored to quantum communication tasks.
Introduction
With the rise of quantum algorithms capable of breaking modern encryption schemes, there follows a global response to search for stronger security levels^{1,2,3}. While the security of most current schemes relies on the complexity of solving difficult mathematical problems, quantummechanical laws can provide security against adversaries endowed with unlimited computational power for some tasks^{4,5}. This type of security, known as informationtheoretic security, motivates research towards a quantum internet^{6}.
Modern communication networks rely on a handful of fundamental building blocks, known as cryptographic primitives^{7,8}. These can be combined with one another to provide security in various applications such as message encryption, electronic voting, digital signatures, online banking, anonymous messaging, and software licensing, to name a few. In order to reach informationtheoretic security through quantum primitives, information is typically encoded onto quantum properties of light, such as photonic path, timebin, polarization, and photon number^{5}. In the quantum realm, the uncertainty principle then ensures that any eavesdropper attempting to access quantumencoded information, while unaware of the preparation basis, will alter the quantum states in a way that is detectable by the honest parties^{4,9}.
For such quantum primitives, it is expected that quantum dotbased singlephoton sources (QDS) can excel by generating photons ondemand, with high brightness and low multiphoton contribution^{10,11}. In fact, source brightness is crucial in achieving highspeed quantum communication^{12,13}, while low multiphoton contribution minimizes information leakage to a malicious eavesdropper^{14}. In contrast to these ondemand singlephoton sources, widely used Poissondistributed sources (PDS), such as attenuated laser states^{5} and spontaneous parametric downconversion^{15}, suffer from a stringent tradeoff between high brightness and low multiphoton emission. Despite elaborate countermeasures proposed to overcome this tradeoff^{16,17}, the distance and rate of secure quantum communication can be increased using ondemand singlephoton sources.
Some pioneering works have already implemented instances of quantum key distribution (QKD) employing QDS^{18,19,20,21,22,23,24,25} or other singlephoton sources^{26,27}, comparing their performance to PDS in terms of secret key rate. In these works, brightness and purity are the sole figures of merit used to establish a comparison, while the additional tuning capabilities of QDS and their role in quantum cryptography have not yet been investigated.
In this work, we explore features of QDS to enhance the performance and security of quantumcryptographic primitives, with an emphasis on telecom wavelengths. We first optimize and compare the brightness and singlephoton purity of three main optical pumping schemes, using realistic intracavity simulations of quantum dot dynamics. We then show how photonnumber coherence generated from QDS, experimentally demonstrated in^{28}, can be erased or preserved to boost the performance of practical QKD, and match its fundamental security requirements^{17,29}. We further explain how the field of mistrustful quantum cryptography^{7,8}, not yet implemented with QDS, can significantly benefit from this feature. Our findings are designed to bridge the gap between the quantum dot and quantum cryptography communities: we optimize and benchmark QDS optical pumping schemes for four main quantum cryptographic primitives, exploiting the combined advantage of brightness, singlephoton purity, and photonnumber coherence. The studied primitives include quantum key distribution (standard BB84, decoy and twinfield)^{5,16,30,31}, unforgeable quantum tokens^{32,33,34}, quantum strong coin flipping^{35,36,37}, and quantum bit commitment^{38,39,40} under storage assumptions.
Results
Comparison of pumping schemes
Solidstate singlephoton sources can be excited under different optical pumping schemes, and we aim to provide a fair comparison of their performance for quantum cryptography. Using realistic intracavity simulations for GaAsbased QDS, we calculate the emitted photonnumber occupations up to three photons for resonant excitation (RE), longitudinal phononassisted (LA) excitation and twophoton excitation (TPE). In Fig. 1, we then compare each scheme’s brightness and singlephoton purity, and estimate the fullwidthathalfmaximum excitation pulse length which maximizes both properties (marked with black symbols).
RE schemes are based on resonant excitation of a twolevel system^{41,42,43}. The spectral degeneracy of the excitation laser and the emitted photons usually imposes separation based on polarization filtering, which may cause significant collection losses of around 50%^{42,44}. Other methods, exploiting dichromatic pumping or trion recombination in asymmetric cavities however, can overcome such limitations^{45,46}. Since RE exhibits Rabi oscillations of the excitonic state populations, its brightness and singlephoton purity are susceptible to pump power fluctuations—thus presenting challenges for quantum network applications^{47}. As we show in Fig. 1a, for a fixed RE πpulse area, singlephoton purity decreases with pulse length due to reexcitation processes, while brightness decreases as the emission statistics tend to a Poisson distribution^{48,49}.
The main limitations of RE may be overcome by using LA excitation schemes. Here, the pump energy is slightly higher than the relevant excitonic transition, and the fast emission of a longitudinalacoustic (LA) phonon precedes the population of the excited state. Due to this additional incoherent step, Rabi oscillations vanish, and the purity of the emitted single photons becomes less sensitive to small pump power fluctuations^{50}. Recently, it was shown that LA excitation can reach even smaller multiphoton components than its RE counterpart^{51}, while still enabling spectral filtering of the pump^{52,53}. As regards to brightness, Fig. 1b displays an increase with pulse length, as was experimentally demonstrated in^{52}. With longer pulses however, the peak intensity decreases for a fixed pulse area, thus lowering the efficiency of the phonon excitation process.
RE and LA multiphoton contributions can be greatly reduced by addressing the excitonbiexciton cascade through TPE schemes, usually employed to generate spectrallyseparated entangled photon pairs. Here, the reexcitation probability scales quadratically with the pulse length, as opposed to linearly in the resonantlydriven RE scheme^{49}, which can reduce multiphoton emission by several orders of magnitude^{54}. Moreover, TPE offers the possibility to overcome the collection efficiency limitations of RE: the spectral separation of the generated photons allows for frequency filtering of the pump laser^{55}, avoiding the polarisation filtering losses. We show in Fig. 1c that the brightness is low for short pulse lengths, due to a remaining overlap with the exciton transition, causing the biexciton level to be only partially populated. Although the emitted exciton polarization is random, which would limit the brightness to half the values of Fig. 1c, recent works have shown the possibility of deterministically preparing the exciton polarization with nearunit brightness by adding a stimulated biexciton excitation after the original twophoton excitation^{56,57,58}.
Photonnumber coherence
We now discuss the presence of coherence in the photonnumber basis, a usually disregarded feature of interest, under each excitation scheme. For PDS such as attenuated laser states, this quantity refers to a fixed phase relationship between the various Poissondistributed number states. For QDS, this will materialize as a coherent superposition of vacuum, single and twophoton states.
In RE, it was experimentally demonstrated in^{28} that the coherentlydriven Rabi oscillations translate into emitted photonnumber coherence: values of coherence purity as high as 96% for πpulse areas were measured. On the other hand, this coherence can gradually vanish as the pump is detuned from resonance in LA schemes, along with the vanishing of Rabi oscillations^{51}. Our quantum dot dynamics simulations support these findings: for the optimal pulse lengths of Fig. 1, the normalized offdiagonal density matrix elements of LA between the vacuum and singlephoton components are around 10 times smaller than the RE ones. Accordingly, we will assume in this work that states emitted under RE are pure in photonnumber basis, while those emitted under LA present vanishing offdiagonal elements.
In TPE, our simulation results display offdiagonal elements around 20 times smaller than the RE ones. Although the biexciton level follows Rabi oscillations under resonant TPE^{54,55}, loss of coherence arises from a radiative decay between the biexciton to exciton state, which creates a timing jitter similar to the phononinduced jitter in LA schemes. We will therefore also assume that states emitted under TPE present vanishing offdiagonal elements. Note that this assumption is expected to hold for the stimulated schemes discussed in^{56,57,58}, since the remaining jitter due to the biexciton transition is significantly larger than the jitter responsible for coherence erasure.
Practical sources and security
We now discuss the role of brightness, singlephoton purity and photonnumber coherence in quantum primitives involving two parties, exchanging a sequence of classical and quantum (photonic) messages that do not rely on quantum entanglement. Each of these primitives achieves a different functionality within quantum networks, and thus also requires its own security figure of merit.
The main efficiency limitations of PDS may be understood upon inspection of the generated state \(\mathop{\sum }\nolimits_{n = 0}^{\infty }{C}_{\mu }\left(n\right)\leftn\right\rangle\), where the \({P}_{\mu }\left(n\right)={\left{C}_{\mu }\left(n\right)\right}^{2}\) coefficients follow a Poisson distribution with average photon number μ, and \(\{\leftn\right\rangle \}\) span the photonnumber basis. Increasing the source brightness (i.e., increasing μ) comes at the cost of increasing the multiphoton components n ⩾ 2, which renders the respective quantum primitive vulnerable to attacks involving photon number splitting on lossy channels^{14}. Thus, μ is typically kept very low in quantumcryptographic implementations, in the range μ ~ 0.005–0.5^{5,32,35,39}, which limits the communication rate. On the other hand, singlephoton purity in QDS can be increased without an intrinsic penalty on the multiphoton component. Achieving higher QDS brightness is then ultimately a technological challenge, limited by the collection efficiency of the source^{11,46}, and not a fundamental limitation as in the case of PDS.
In contrast to their PDS counterparts, QDS have not yet been optimized to suit the security requirements of quantum primitives. Most importantly, a main assumption behind the implementation of decoy QKD and other primitives is that the global phase of PDS must be actively scrambled, to effectively destroy the coherence in the number basis^{17,29}:
Under this assumption, the adversary’s cheating strategy is restricted to performing an attack conditioned on the photonnumber content of each pulse. Many works rely on this feature to prove the security of quantum primitive implementations^{5,16,31,32,35}.
Achieving phase randomization with active phase modulation or laser gain switching imposes practical limitations of a few GHz on repetition rates^{12,59}. These limitations, combined with the low values of μ required due to fundamental PDS source statistics, can bring effective communication rates down to a few MHz. Unwanted remnants of coherence in the number basis, furthermore, can be exploited for a large spectrum of attacks, using unambiguous state discrimination for instance^{60,61}. In contrast, as discussed in this work, QDS can be excited in such a way that this coherence is intrinsically suppressed, thus circumventing the need for active phase scrambling. With demonstrated Purcellenhanced photon lifetimes of tens of picoseconds^{46,62} and source efficiencies now beyond the 50% level^{46}, QDS have the potential to enable secure communication rates of tens of GHz^{13}, i.e. around 3 orders of magnitude higher than effective PDS communication rates. This is not only true for QKD but for many quantum primitives as shown in the following sections.
Quantum key distribution
A few decades after the birth of quantum key distribution (QKD)^{4}, experimentalists started demonstrating that QDS with low collection efficiencies can already outperform PDS in terms of secret key rate^{22,23,24,25}. We first show that, while this is true for standard QKD implemented without the decoystate countermeasure, beating PDS with decoy states^{16} requires much higher QDS collection efficiencies at an equal repetition rate. Our results, based on the optimal performance of pumping schemes in Fig. 1, are displayed in Fig. 2: without decoy states (a), QDS with collection efficiency 1% are enough to outperform PDS after 100 km, while infinite decoy schemes (b) require at least 30%. We should emphasize, however, that this benchmark must be scaled by a repetition rate factor for QDS which could achieve considerably higher repetition rates than phaserandomized PDS. Any state preparation losses, including modulator losses, can be absorbed in the QDS collection efficiency, and the resulting key rate inferred from Fig. 2. Comparisons can also be established using finite (and different) number of decoy intensities for QDS and PDS.
The optimal pumping scheme for QKD then follows from Eq. 1: the states’ global phase must be uniformly randomized^{17,29}, which implies that standard and decoystate QKD should only be implemented with LA and TPE. Any remaining photonnumber coherence will lead to a decrease in the secure key rate, as shown in^{29}.
Twinfield QKD, on the other hand, requires two states, generated by Alice and Bob, to interfere on an untrusted party’s beamsplitter^{30,31}. This forces Alice and Bob to scramble the global phase of their states in an active manner (using a modulator for instance), such that they can record their original fixed phase encoding. We therefore argue that twinfield schemes must be implemented with RE QDS, in order to provide the two parties with a shared phase reference before the scrambling. We simulate the protocol performance of^{31} under these conditions in Fig. 2c, assuming the implementation of decoy states. We note that performing TFQKD with a perfect single photon source (RE πpulse and unit singlephoton purity) is impossible, since there is no accessible phase to encode the key information. However, by decreasing the pulse area a little below π, in the same manner as^{28}, it is possible to create a coherent superposition of vacuum and singlephoton, thus providing an accessible phase to perform the protocol.
Our results are focused on QKD implemented with discrete degrees of freedom, such as polarization or timebin. For completeness, we note the existence of QKD protocols optimized for continuous degrees of freedom of PDS, such as phase and amplitude^{63}. These can actually compete with discretevariable QKD over metropolitan distances, using simpler technology, but are sensitive to larger attenuations due to heavier postselection and noise estimation techniques^{64,65}.
Quantum primitives beyond QKD
Quantum cryptography presents a broad spectrum of other primitives, many of which belong to the branch of mistrustful cryptography^{7,8}: unlike in QKD, Alice and Bob are not collaborators, but adversaries wishing to compute a common function. Decoystate methods are then more challenging to apply (although not impossible for all protocols), since Alice and Bob do not trust each other.
Remarkably, the desired security properties for such primitives can be very sensitive to photonnumber coherence. In this instance, substituting PDS by appropriately optimized QDS can yield even more benefits than in QKD. To show this, we extend the practical security analyses of three mistrustful quantum primitives^{32,35,38} to the QDS framework, and estimate the QDS collection efficiencies required to outperform PDS for the relevant security figures of merit. Our performance results are summarized in Table 1 for all primitives.
We display the performance of QDS and PDS for one example primitive in Fig. 3: unforgeable quantum tokens. This primitive allows a central authority to issue tokens, comprised of quantum states, whose unforgeability is intrinsically guaranteed by the nocloning theorem, thus requiring no hardware assumptions. One famous application is quantum money, which, in its privatekey form, can prevent banknote forgery^{9}, doublespending with credit cards^{32,66} and guarantee features such as user privacy^{34}.
In Fig. 3a, we compare the noise tolerance of the quantum token scheme from^{66} for PDS and QDS as a function of source efficiency. Noise tolerance indicates how much experimental error rate can be tolerated such that the unforgeability property holds, while source efficiency is the probability that a threshold singlephoton detector will click in a lossless setting. Naturally, PDS reach a maximal noise tolerance for source efficiencies around 63%, corresponding to μ ≈1, before dropping again when the multiphoton contribution becomes too significant. For QDS, we notice a striking difference between schemes with coherence (RE) and those without (LA and TPE): the latters give an overhead of almost 2% on the noise tolerance with respect to RE at high source efficiencies. This difference is crucial in making implementations feasible, since boosting the fidelity of quantum state preparation and quantum storage by a few percent can be extremely challenging. These differences are also reflected in Fig. 3b, which identifies the collection efficiencies at which QDS can outperform the best PDS performance: while LA and TPE require 44% and 38%, respectively, RE must be pushed to 47% to beat PDS. For information purposes, we also select three stateofthe art experimental QDS, and show how they would perform in such a beyondQKD protocol with their reported values of brightness and singlephoton purity.
Figure 3c finally compares the performance of each source as a function of distance. Once again, the difference between LA/TPE and RE is significant due to the coherence feature. We notice here that the maximal distance for all sources is much shorter than in QKD schemes, since our selected quantum token scheme bears a maximal loss tolerance of 50%: above this limit, an adversary can clone the quantum token without introducing any errors^{66}.
Our work showcases the importance of engineering optical pumping schemes towards specific primitives. Table 1 displays nontrivial requirements for quantum strong coin flipping for instance: unlike with QKD and quantum tokens, QDS perform better at lower collection efficiencies, thus rendering stateofthe art QDS already capable of providing quantum advantage in such mistrustful primitives. Furthermore, the absence of photon number coherence in LA and TPE allow these schemes to reach quantum advantage over significantly longer distances than RE schemes: for the selected protocol from^{35}, these values read 86 km and 36 km for TPE and LA, respectively, vs. 25 km for RE.
Discussion
We have estimated threshold collection efficiencies for which GaAsbased quantumdot photon sources can outperform Poissondistributedbased implementations in four main quantumcryptographic primitives. The estimations include the combined effect of brightness, singlephoton purity, and photonnumber coherence. We have shown in particular that resonant excitation should be used for twinfield QKD, but not for decoy QKD and other quantum primitives due to the unwanted presence of photonnumber coherence that violates practical security assumptions.
We believe that these results will provide a benchmark for future achievements in quantum dot cavity structures, especially at telecom wavelengths^{13,67,68,69}. Although stateoftheart dotcavity simulation frameworks cannot account for all characteristics of quantum dots emitting in the telecom range, current telecom performance^{13,70,71} shows good agreement with our 900nm framework: the brightness and purities are not strongly dependent on the emission wavelength. Furthermore, frequency conversion of 900nm photons to telecom photons can currently reach efficiencies up to 57%^{72}. These extra losses can be absorbed in our collection efficiency quantity.
We wish to encourage future quantum key distribution experiments with optimal pumping schemes, taking into account the security assumptions provided by the quantum cryptography community. Finally, we hope to stimulate experiments that explore the full potential of quantum dotbased singlephoton sources for other quantum network primitives like unforgeable tokens^{32,33,34}, coin flipping^{35,36} and bit commitment^{38,39,40}. We believe our analysis can be extended in future works to multipartite entanglementbased quantum network primitives, such as secret sharing^{73} and anonymous messaging^{74}.
Methods
Quantum dot dynamics
Our GaAsbased quantum dot, driven by a pulsed pump laser, is modelled either as a two or a threelevel system coupled to a singlemode microcavity in the JaynesCummings manner. The interaction of the quantum dot with phonons is treated by the standard puredephasing Hamiltonian^{75,76,77,78}. In this way, we solve for the dynamics of the dotcavity system by employing a numerically exact pathintegral formalism^{79,80,81}. The detailed intracavity simulations, along with the derivation of the photon number populations, are presented in Supplementary Note 1.
Practical security analyses
Supplementary Notes 2 and 3 show how the collection efficiencies and state encodings are modelled, both in the presence and absence of photon number coherence, for PDS and QDS, respectively. Supplementary Note 4 provides some highlevel descriptions of the four main quantum primitives, and displays all results justifying our claims. Supplementary Note 5 briefly introduces mathematical tools required to understand the security analyses, namely semidefinite programs and Choi’s theorem on completely positive maps. Supplementary Notes 6 to 10 provide the practical security analyses of all protocols, and the extensions to account for the presence of coherence in the QDS framework.
Data availability
The numerical data generated and analysed during the current study is available from the corresponding author upon reasonable request.
Code availability
The code used during this study is available from the corresponding author upon reasonable request.
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Acknowledgements
M.B., M.V, J.C.L. and P.W. acknowledge support from the European Commission through UNIQORN (no. 820474), the AFOSR via QTRUST (FA95502110355), the Austrian Science Fund (FWF) through BeyondC (F7113) and Reseach Group (FG5), and from the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association. M.C., T.S. and V.M.A. are grateful for funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project No. 419036043. C.N., S.L.P. and P.M. gratefully acknowledge the funding by the German Federal Ministry of Education and Research (BMBF) via the project QR.X (16KISQ013) and the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 899814 (Qurope). The work reported in this paper was partially funded by Project No. EMPIR 20FUN05 SEQUME.
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M.B., M.V., C.N., S.L.P., P.M. and P.W. conceived the project, and V.M.A, P.M and P.W supervised the project. M.B., M.V., M.C. and T.S. performed the theoretical calculations, security analyses, and numerical simulations. M.B., M.V. and M.C. wrote the manuscript, with input from C.N., T.S., J.C.L., S.L.P., V.M.A., P.M. and P.W.
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Bozzio, M., Vyvlecka, M., Cosacchi, M. et al. Enhancing quantum cryptography with quantum dot singlephoton sources. npj Quantum Inf 8, 104 (2022). https://doi.org/10.1038/s4153402200626z
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DOI: https://doi.org/10.1038/s4153402200626z
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