Quantum key distribution (QKD) [1] is a pioneering application of quantum information theory enabled by fundamental particle and wave quantum features of light. Advantageously, in experiments at optical wavelengths, QKD can exploit complementary photon counting and homodyne detection methods of quantum optics. Naturally, both methods have advantages and disadvantages, fundamental as well as technical. Therefore, optimal implementation of a quantum-secure network will be likely hybrid in the future, combining advantages and suppressing the weaknesses of different protocols respectively to the requirements and conditions [2]. Currently, homodyne detection is fast, efficient, with extremely low-noise, and tolerant to background noise in the channel [3]. This hardware already opened space for a high-speed secret key generation. For a long time, the homodyne detection stimulated a large set of theoretical proposals [4–6] and experimental protocols with coherent states of light [7–12]. With nonclassical squeezed and entangled states, the continuous-variable (CV) protocols [13–15] become more robust and potentially applicable at distances up to 300 km [16] in optical fibers with attenuation 0.2 dB per kilometer, and tolerant to data processing inefficiency [17]. Such protocols can be advantageously implemented in both optical fiber links [16] and free-space atmospheric channels with realistic turbulence [18]. Moreover, higher security can be offered by relaxing the assumption about trusted devices for both coherent-state and entanglement-based protocols, as it was demonstrated by implementation of one-sided device-independent protocols [19,20].

A rate of secret key can be increased in CV QKD by frequency multiplexing of transmission channels [21]. Frequency (wavelength) multiplexing is a well-known technique from classical optical communications [22], also with the homodyne detection [23]. It can be similarly considered to increase the secret key rate of CV QKD protocols and has recently been studied using Gaussian modulation of frequency comb states [24], as well as using independent lasers for each mode for subsequent discrete [21] or Gaussian modulation [25]. In this paper we test the entanglement-based CV QKD protocol that uses multiple frequency modes to multiplex the signal. However, in practice generated multiple entangled modes can become mutually coupled to each other, resulting in cross correlations as well as excess noise in the modes that can be destructive for CV QKD. Importantly, homodyne detection of field quadratures gives sufficient information about states of light in the individual modes in order to compensate for the cross talk. Based on these advantages, a cross-talk elimination based on state or data manipulations has been addressed in Refs. [26,27] demonstrating that such a limiting factor for multiplexed QKD can be in principle deterministically eliminated by optimized data manipulation, using the whole multimode structure (contrary to modes selection, e.g.,  used to improve quantum steering in Ref. [28]). Different to protocols with single-photon detectors, it is therefore not required to implement an active strategy of optical decoupling which is very challenging for a large number of transmitted modes.

Nowadays, CV QKD reaches a new level at which a substantial increase of secret key rate of mid-range protocols is a relevant target of the ongoing development and requires, in particular, development of cross-talk elimination methods. The previously suggested methods either required heterodyne detection with optimal engineering of auxiliary input states and were not applied to CV QKD security analysis [26] or considered only cross-talk interaction between the neighboring modes of the otherwise perfect entangled states [27]. In the current paper we suggest the multimode cross-talk compensation method based on data manipulation, equivalent to linear state manipulations, experimentally test it on the real multiplexed entangled states measured with the mode-discriminating homodyne detection suitable for CV QKD, and evaluate security of the resulting CV QKD protocol. Without such a test on the multimode state with real cross talk and errors it is impossible to estimate applicability of the cross-talk compensation for a large number of modes. A positive result demonstrating that secret key rate can be enhanced by channel multiplexing with high efficiency, despite cross talk substantially reducing the achievable key rate already in the source, is necessary to open a pathway for further implementations and applications of frequency-multiplexed CV QKD. For the test we use simultaneously frequency-multiplexed source of entanglement with eight pairs of modes and mode-discriminating homodyne detection. For them, the cross talk is a very natural phenomenon pronouncedly reducing the key rate to a tiny number of 0.015 bits. We suggest and apply optimized data manipulation, which allows decoupling of modes under cross talk and brings large improvement to an achievable secret key rate. The secret key rate can be enhanced by almost a factor of 15. Reducing the channel noise in all frequency-multiplexed channels by this data manipulation, alternatively, can extend the secure distance (channel range at which generation of the secret key is still possible) by approximately 100 km. Moreover, as the source emits femtosecond pulses, allowing for high system clock rates, the performance of the system can be also further increased by time multiplexing. Our result solves the major problem of mode cross talk in the source; however, it can be equally applicable to cross talk in the link and detection (although the mode coupling inside an optical fiber is weak, if present at all, hence one can expect the mode interaction in the source to be the dominating cause of cross talk). Therefore, it opens the possibility for high-speed and high-capacity entanglement-based CV QKD with femtosecond frequency-multiplexed states.

We consider the use of an entanglement source in the multimode CV QKD testbed based on the frequency-multiplexed femtosecond pulses of light, consisting of 16 modes and with mode-discriminating homodyne detection, as described in Fig. 1. In our proof-of-principle experiment, all the 16 modes are generated in a single beam, and to test the applicability of the source for QKD purpose, we assume that the lower half of the frequency modes are distributed to Alice, and the other half are to Bob.

The experimental setup is shown in Fig. 2. The main laser is a Ti-sapphire pulse laser, having a duration of 120 fs centered at λ0 (=795nm) with a repetition rate of 76 MHz. The beam from the laser splits into two beams, where one is used for generating frequency-multiplexed entangled light, and the other serves as a local oscillator (LO) for mode-discriminating homodyne detection. To generate the entangled light, we employ a synchronously pumped optical parametric oscillator (SPOPO) including a 2 mm thick BiB3O6 (BiBO) crystal, which operates below the threshold [29,30]. The pump laser for the SPOPO (centered at λ0/2) is prepared by second-harmonic generation of the main laser in a 0.2 mm thick BiBO crystal. As a result, an entangled state of femtosecond pulses of light in multiple frequency modes (centered at λ0) is generated in a single beam, and the efficiency of the process is enhanced by the cavity constituting the SPOPO. For our purpose, we consider 16 frequency-band modes of the generated multimode light and assume that the lower half (eight) frequency modes are measured by the trusted sender Alice, while the other half frequency modes are measured by the trusted receiver Bob after a multimode channel. We stress that even if in practice this separation is not performed in the current experiment, spectrally splitting a beam in two halves can be readily done experimentally with a simple dispersive element, such as a grating, a dichroic mirror, or a prism. It is possible to use a high-efficiency grating or prism, or fiber-based wavelength-division multiplexing [31]. These dispersive elements would introduce only small additional losses, leading to the excess noise in the generated multimode states. Such noise can be however considered trusted and will only have a limited negative effect on the key distribution [32].

To measure the generated multimode state, we use homodyne detection which can discriminate different frequency modes. As the LO of homodyne detection determines the frequency mode, we control the LO based on a pulse shaping technique. For this purpose, a pulse shaper in the 4-f configuration is employed: an input beam is diffracted by an optical grating (1200 grooves/mm), which is subsequently focused by a cylindrical lens (190 mm focal length). On the Fourier plane of the lens, a reflection-type spatial light modulator having 512×512 pixels controls the amplitude and the phase of frequency modes. The reflected beam comes back to the lens and the grating. The overall wavelength resolution is found to be 0.1 nm. Using the pulse shaper, a covariance matrix associated with the 16 frequency modes was obtained by measuring quadrature outcomes in a sequential way from the mode-discriminating homodyne detection; in the homodyne detection, the two photodiodes have a quantum efficiency of 99%, fringe visibility is 93%–95%, and demodulation frequency is 1 MHz [33]. The obtained covariance matrix is presented in Appendix A.

Given the resolution of the pulse shaper, we consider that all the measured modes are realistically matched to the local oscillator. It also does not limit the applicability of the method which can be applied to the cross talk in the multimode detector equally well. If the unmatched modes are present, they will contribute to noise [34] and may act as a detection side channel [35] but can be compensated for by increase of the brightness of the local oscillator [36].

To extend and verify the method of decoupling following the preliminary theoretical studies [26,27] for the source depicted in Fig. 1, we assume a typical QKD scenario, where we suppose that Alice’s preparation is trusted (being fully out of control by an eavesdropper Eve) and Alice is measuring her modes locally by a multimode homodyne detector, while the Bob’s modes travel directly toward his detection. Bob is measuring his modes using mode-discriminating homodyne detectors, also assumed to be trusted (including the efficiency and the electronic noise of the detectors). The ability to address the individual local modes in the homodyne detection is crucial for channel multiplexing in CV QKD and the multimode structure of entangled states can be harmful for the protocols otherwise [37]. To controllably investigate the impact of lossy channel, we applied attenuation to Bob’s measured results. It emulates an untrusted channel, characterized by the transmittance T, which is assumed to be fully controlled by an eavesdropper, Eve, capable of collective attacks. We assume a purely lossy (attenuating) channel as the background noise is already very small in real optical fiber channels. Such an approach allows modeling fiber as well as free-space channels, where fluctuations due to atmospheric turbulence are typically slow compared to the signal repetition rate [38]. To comply with the experimental testbed, where the multimode source was characterized, we assume that the cross talk appears in the source, but our methodology can also be directly applied also to cross talk in the channel and detectors.

Security of CV QKD is evaluated as the positivity of the lower bound on the key rate, which, in the case of collective attacks and reverse reconciliation [5], reads K=max{0,βIAB−χBE},(1)where β≤1 is the post-processing efficiency (further we realistically take β=96%, which complies with the achieved post-processing efficiency [39]), IAB is the mutual classical (Shannon) information between Alice and Bob, and χBE is the Holevo bound, which upper limits the information accessible to a potential eavesdropper Eve on Bob’s measurement results. We address security against collective attacks for the Gaussian entanglement-based CV QKD [40,41], which can be directly extended to the finite-size regime [42,43] and implies security against general attacks [44,45]. The reverse reconciliation is used to test secret key distribution for mid-range distance with channel attenuation below −3dB. The positivity of the lower bound [Eq. (1)] implies that the trusted parties are able to distill the secret key with at least the rate K by using classical post-processing (error correction and privacy amplification) [46]. We therefore analyze the security of frequency-multiplexed CV QKD by evaluating the lower bound on the key rate per multimode channel use K (further also referred to as the key rate). We follow the Gaussian security proofs and respective security analysis methods, as described in Appendix A.

We demonstrate the power of the multimode states in CV QKD by confirming the gradual increase of the overall key rate for increasing the number of pairs of modes measured by Alice and Bob, as shown in Fig. 3 (left), assuming channel transmittance T=0.2. We first rank the pairs by the key rate between the individual pairs and then add pair by pair, thereby obtaining larger key rate, as seen in Fig. 3 (left, circles and the blue solid line).

For the multimode states as shown in Fig. 1, we optimize the data processing to achieve the maximum key rate Eq. (1). The applied data processing is equivalent to local passive symplectic transformations when both the sender and the receiver separately act on their respective modes by optimized beam splitter networks [47]; see Appendix A for details. The results of the optimization are given in Fig. 3 (squares and solid yellow line) and it is evident that optimized local data manipulations lead to an almost 15-fold increase of the overall key rate for the multimode states. The optimization process can, therefore, efficiently retrieve multiple pairs of entanglement from the multimode entanglement resource, providing significant improvement for CV QKD. Moreover, the key rate becomes much more robust against statistical error in the finite data ensemble, as can be seen from the respective plots in Fig. 3 (squares and dashed yellow lines).

The improvement of multimode CV QKD by the local data processing is concerned with the increase of the multimode mutual information, as can be seen from Fig. 3 (right, top), while Holevo bound is not affected, as seen from Fig. 3 (right, bottom). Indeed, the local data processing (equivalent to symplectic transformations) does not affect the quantum entropies contributing to the Holevo bound by not changing the symplectic spectrum (i.e., thermal-state decomposition) of the multimode Gaussian state. On the other hand, redistribution of modes occupation by the local symplectic transformations increases the additive classical mutual information due to increased correlations and can be optimized to achieve the best performance. This also substantially simplifies the optimization of the method by reaching the maximum mutual information, leading to the maximum key rate. Our method is focused on maximizing mutual information (not on eliminating cross correlations) and leads to optimal improvement of the key rate. Note that our method can be advantageously combined with the protocol, in which the Holevo information, maximally accessible to Eve, is minimized [48]. In this scenario by proper multiplexing and elimination of cross talk a higher key rate can be achieved at low post-processing efficiencies, while not increasing the information leakage. This can be particularly promising for high-speed CV QKD, where fast but less efficient error correction can be otherwise a very limiting bottleneck [8,17].

We also analyze the robustness of CV QKD against channel attenuation (which is equivalent to channel distance) with the full set of eight pairs of modes before and after the optimized data manipulation as shown in Fig. 4 (the eight-pair covariance matrix after optimized data processing is illustrated in Fig. 5 in Appendix A). By eliminating the cross talk, we reduce the state preparation noise in the individual channels and respectively increase the maximum tolerable channel noise. It is evident from the plot that optimized local data manipulation increases the maximum tolerable channel attenuation of the protocol from approximately 8 dB to 28 dB of loss, thus demonstrating potentially more than threefold increase of the secure distance of the multimode CV QKD protocol (assuming 0.2 dB/km loss). We extrapolate the obtained results for the cases of 25 and 50 modes, which are expected to further improve the efficiency and robustness of the CV QKD protocol up to approximately 34 and 36 dB; see Appendix A for details. We also address the efficiency of our method by comparing the achieved results to the bounds set by 8 times maximum performance of one best pair of modes (as the total number of pairs in our experiment is eight). In the way similar to maximization of the total key rate, we now run optimization to have as much as possible key in this particular pair. In terms of mutual information, the maximum in one pair is 0.28 bit per channel use, the total maximum mutual information achieved by our method is 0.517 bit per channel use, and the bound (8 times the maximum value for one pair) is 2.24 bit per channel use, which is 4.3 times larger than we achieve. For the key rate the maximum for one pair is 0.163 bit per channel use, the total achieved key rate is 0.212 bit per channel use, and the bound is 1.304 bit per channel use, which is 6.15 time larger than we achieve. We define the decoupling efficiency as the ratio between the secret key rate achieved and the secret key rate that could be achieved in a perfect setting with all eight pairs having maximal mutual Shannon information. The efficiency of our method for the secret key rate therefore reaches 0.16. This removable limitation is caused by source imperfections beyond the linear cross talk and would require development of additional advanced experimental and data processing methods to further improve practical frequency-multiplexed CV QKD. Our results show that despite drastic improvement achieved with the suggested method for cross-talk elimination, even higher performance can be achieved by larger number of frequency channels with faster data processing and by further developed experimental techniques aimed at reduction of cross talk.

By optimally applying data manipulations we were able to compensate the cross talk in the frequency-multiplexed CV QKD with femtosecond-pulsed entangled states and substantially increase the mutual information between the sets of modes, measured by the trusted parties, while the leaked information, upper bounded by a function of Gaussian quantum entropies, did not change. Thus, we can increase the achievable key rate for continuous-variable quantum key distribution or, equivalently, extend the secure distance of the protocols. The results of the optimized local data manipulations show the possibility to increase the overall key rate by almost the factor of 15 and extend the secure distance for the multiplexed entanglement-based protocol by the factor of 3. Note that while higher key rates can be as well obtained by increasing the system repetition rate, our method does not affect the information leakage and only increases the mutual information, hence drastically increasing the key. Nevertheless, our method can be further combined with the increase of the repetition rate to achieve even higher key rates. In the present demonstration, the calculations were performed on a covariance matrix obtained by mode-discriminating homodyne measurement and can be easily extended to a large number of modes [49]. Furthermore, the cross talk between the modes can also be further reduced or adapted to the measurement system manipulating the source through spectral shaping of the pump [50]. While we applied data manipulations to compensate cross talk in the multimode CV QKD source, the method can be also used to eliminate cross talk that appears in the multimode quantum channel. Our method is therefore very promising for improving key rates of continuous-variable quantum key distribution and can also be combined with the protocol based on minimization of the information leakage [48], especially with elimination of channel noise and efficient channel estimation techniques, in order to overcome the limitations imposed by realistic fast post-processing. Moreover, we can combine our method with the existing tools to eliminate correlated noise [51] and side channels [35]. We therefore open the pathway to very high-speed practical realization of quantum key distribution using continuous variables. It should be followed by a test of complete multiplexed protocol together with secret key generation and can be extended to networking entanglement-based communication settings. Furthermore, the suggested cross-talk compensation technique can be useful in other applications of continuous-variable quantum information, such as quantum imaging [52] or quantum illumination [53].