## Abstract

In this paper we propose feasibility demonstration of twin-field quantum key ditribution system based on multi-mode weak coherent phase-coded states. Their utilization provides indisputable advantages described in the paper. We also provide the detailed description of nontrivial interference scheme for those states and derive detection and quantum bit error rates. Since we propose the feasibility scheme we present in this paper only asymptotic secure key estimation and show that in principle it can beat well-known fundamental limit of repeaterless quantum communications , i.e., the secret key capacity of the lossy communication channel. Also we present here the experimental setup and provide the experimental values of detection rates dependence on the modulation signals phase difference.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the last decade huge amount of experimental works in the quantum key distribution (QKD) field was performed [1–3]. Nevertheless the strict security proof for several protocols (assuming ideal case) are already exists [4–8], a lot of loopholes can be found in real QKD setup related to non-ideal equipment, e.g. detector blinding [9–12], "trojan horse" attack [13] and etc. One of the most crucial from this list is detector blinding. The best solution to overcome the problem is to make measurement node untrusted from the very beginning of the protocol development. The solution as measurement-device-independent (MDI) QKD system was proposed in [14,15] and developed in [16–18]. As alternative to single-photon approach first realization of twin-field (TF) QKD scheme with coherent states was proposed in [19] which allows to overcome the well-known fundamental limit of repeaterless quantum communications, i.e., the secret key capacity of the lossy communication channel [20] (also known as the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound) [20,21]. Moreover a bunch of new approaches for realization of TF QKD protocol were proposed in [22–24] . Despite the fact that those devices allow to increase the secure key rate and critical distance, they remain difficult to be realized experimentaly.

In this work we would like to present the feasibility demonstration of twin-field (TF) QKD system based on multimode coherent phase-coded states [25–27]. The proposed approach is quite similar to the one proposed in [19,28], except the fact that the states used here are multimode ones (due to phase modulation). Their utilization provides several useful advantages. In some sense considered realization is quite similar to well-known unbalanced Mach-Zehnder interferometer, however phase difference is transferred from commonly used time domain to frequency domain. Unbalanced Mach-Zehnder interferometer requires careful stabilization and crucially subjects to environment fluctuations, whereas we need only to stabilize phase modulators (the effective quantum model of electro-optical phase modulator can be found in [29]). Also modulation signal has radio frequency (in order of GHz) and both Alice’s and Bob’s modulation signals can be easily synchronized by electrical or optical synchronization with high precision. Besides, one can use the carrier as the reference pulse (due to frequency multiplexation) and provide the calibration and active phase compensation (as it is shown in Section 4) without artificial addition of reference signal. Also phase-modulated states overlaps are real [25,27] hence absence of imaginary part reduce the probability of unambiguous state discrimination compare to single-mode phase-coded coherent states.

However, it was shown in [28], that one cannot implement "the widely used photon number channel model [30] used in the security proof of MDI-QKD is proven to be invalid for this kind of setting". Thus we estimate the asymptotic key rate using Devetak-Winter bound [31] similarly to the analysis provided in [32].

This paper is organised as follows. In the Section 2 we describe implemented protocol in details. Section 3 gives the overview of proposed system, including description of used quantum states, their transformation, detection, estimation of the raw and sifted key rates, quantum bit error rates and asymptotic secure key rate. Section 4 part presents the obtained experimental results. Section 5 concludes the article.

## 2. Proposed protocol

The proposed protocol is as follows:

(i) **Classical pre-processing**. Alice and Bob sends the unmodulated bright light to the measuring station, to let Charlie stabilise the communication channels and perform phase tuning using phase modulator with constant voltage.

(ii) **Preparation of quantum states**. Alice and Bob each chooses a random basis $\{1,\ldots, M\}$ and random phase value $\{0,\pi \}$ related to the bit value $\{0,1\}$ respectively and prepares multi-mode weak coherent states (Eq. (1)).

(iii) **Measurement**. Incoming signals (Eq. (5)) interferes at the Charlie’s side and he records which detector clicks. When the distribution phase is over, he publicly announces either none, detector 1, detector 2 or both clicked. All the runs where none of or both of his detectors clicked are discarded. Otherwise Alice and Bob will correspondingly agree on bit values.

(iv) **Reconciliation**. The reconciliation step is as follows. Alice or Bob announces which basis was chosen in each round. In case when bases have been chosen correctly they keeps their raw bits respectively, otherwise they discard the bits. If the first detector clicks they accept the sent bit, otherwise Bob (or Alice, it should be discussed beforehand) changes the bit value on the opposite one.

(v) **Classical post-processing**. After reconciliation step Alice and Bob can provide classical post-processing procedures as error correction and privacy amplification (the latter requires special consideration).

It should be mentioned, that technique of state preparation proposed here is very similar to one called "phase randomization" [19,28,33] (in this paper we use $M$ bases according to work [34], where was shown that 10 states is enough to assume weak coherent states to be phase-randomized). Moreover such scheme is clearly resembles the phase-encoding MDI-QKD protocol [14,17]. Also the reconciliation step is close to one called phase postcompensation method proposed in [17]. In [28] Alice and Bob retain their signals when their announced phases $\varphi _A$ and $\varphi _B$ are either exactly the same or with a $\pi$ difference. In our case they also retain their signals only then bases are mathched.

In this paper we do not propose the detailed security analysis of the system since this problem is beyond the scope of the paper due to its complexity and ambiguity. Despite we provide asymptotic key rate based on the Devetak-Winter bound [31].

## 3. Asymptotic security analysis

#### 3.1 Generation of sifted key

It should be noted that further we will consider symmetrical case (Alice and Bob have identical equipment) just in order to show the feasibility of generating sifted (identical) bit strings between Alice and Bob.

The main feature here is the used multimode state. So we denote prepared states as follows. The input (unmodulated) state at Alice’s or Bob’s (further denoted by $A$ or $B$) modulator can be denoted as $|\sqrt {\mu _0}\rangle _0\otimes |\mathrm {vac}\rangle _{SB}$, where $|\mathrm {vac}\rangle _{SB}$ is the vacuum state of the sidebands and $|\sqrt {\mu _0}\rangle _0$ is the coherent state of the carrier wave with the amplitude determined by the average number of photons $\mu _0$ in a transmission window provided with coherent monochromatic light beam with optical frequency $\omega$. The carrier wave phase is accepted as reference and all other phases are calculated with respect to it. Electro-optical phase modulator (with the frequency of the microwave field $\Omega$ and its encoding phase $\varphi _A$ or $\varphi _B$) rearranges the energy between the interacting modes (the field at the modulator output acquires sidebands at frequencies $\omega _k=\omega +k\Omega$, we limit our consideration to $2S$ sidebands and let the integer $k$ run between the limits $-S\le k\le S$), so that the state of the field at the modulator output is a multimode coherent state:

We introduce useful approximations that help us estimate quantum bit error rate (QBER) and sifted key rate. First of all we consider the case of large number of interaction modes ($S\rightarrow \infty$) which is true for standard optical fiber modulators. One may use approximation of d-function as follows:

where $J_k(m)$ is Bessel function of the first kind. Considering small $m$, which is also the case, and recalling classical modulation theory we use first order series approximation of Bessel function. Also according to Eq. (3) $\beta \rightarrow 0$ hence one may use first order series approximations to express Eq. (8) in terms of $m$ implying proportionality in Eq. (11). Finally we derive Eq. (9) as follows:Let us consider joint probability distribution $P_{ABCB'}(a,b,c,b')$, where $A$, $B$, $C$, and $B'$ are random variables characterized by events $a$, $b$, $c$, and $b'$ respectively. Events $a$ denote sending of bit «0» or «1» by Alice. Events $b$ denote sending of bit «0» or «1» by Bob. Events $c$ denote detection of signal in either detectors («1» or «2») or absence of clicks and double clicks (we combine them together and denote as «?»). Events $b'$ denotes including of bit «0» or «1» or nothing (when «?» in $c$ occurs) in sifted key depending on his sendings $b$ and the result of detection $c$. Thus one can estimate mutual information $I(a;b')$. It can be done by consideration of binary symmetric erasure channel, where characteristics of the channel $P_{B'|A}(b'|a)$ can be expressed in terms of $P_{ABC}(a,b,c)$ joint probability distribution as follows:

#### 3.2 Impossibility of PNS attack on phase-coded states

Let us consider weak coherent state $|\alpha e^{i \varphi }\rangle$, where $\alpha$ is the amplitude of the states and $\varphi$ is the phase, which encoded the bit. Probability of obtaining an anticipated measurement result indicating the presence of $n$ photons in a given sample or pulse can be obtained by considering the trace of the product of the density matrix of the coherent state $\rho$ and the projector on the Fock basis $|n\rangle \langle n|$:

Therefore, the state after measurement will be reduced as follows:

- 1. $m \neq n$$$\sqrt{|n\rangle\langle n|}|m\rangle = \sum_{k=0}^{\infty} \frac{({-}1)^k(2k)!}{(1-2k)(k!)^2 4^k}(|n\rangle\langle n|-\hat{I})^k|m\rangle ,$$where $\hat {I}$ is identity operator. Using the following properties we obtain
- 2. $m=n$
Then using the next properties

$$\begin{aligned}(|n\rangle\langle n|)^k |n\rangle = (|n\rangle\langle n|)^{k-1}=\\ =|n\rangle\langle n|n\rangle = (|n\rangle\langle n|)^{k-1} |n\rangle = \cdots= |n\rangle, \end{aligned}$$then$$\begin{aligned}\sqrt{|n\rangle\langle n|}|n\rangle &= \sum_{k=0}^{\infty} \frac{({-}1)^k(2k)!}{(1-2k)(k!)^2 4^k}(|n\rangle\langle n|-\hat{I})^k|n\rangle\\ &=|n\rangle + \sum_{k=1}^{\infty} \frac{({-}1)^k(2k)!}{(1-2k)(k!)^2 4^k}(|n\rangle\langle n|-\hat{I})^k|n\rangle\\ &=|n\rangle + 0 = |n\rangle. \end{aligned}$$

#### 3.3 Asymptotic secure key rate

To provide asymptotic key rates we use well-known method introduced by Devetak-Winter [31]. We already have estimated $I(a;b')$ thus estimation of information obtained by Eve (interceptor and illegitimate user) should be provided. Since we present only feasibility of the method further we will not provide thorough security estimation (we think that such complex problem requires separate consideration). However brief sketch will be discussed.

We consider several cases of how Eve may obtain information of distributed key. Eve operates in both quantum channels that connects Alice with Charlie and Bob with Charlie. Maximal amount of information that Eve may obtain from the states in one channel is Holevo bound denoted by $\chi$. Further we assume that $\chi$ denotes ratio of distinguished state to the total amount of sent ones. Thus in average probability of state to be discriminated is numerically equal to $\chi$. Hence three cases are as follows. Eve obtain information from the first channel only with probability $(1-\chi )\chi$. Probability of obtaining information form the second channel only is the same $(1-\chi )\chi$. Finally the probability of obtaining information from both channels simultaneously is $\chi ^2$. Either case Eve can recover sent bit with information of detection events from Charlie. Hence Eve may obtain the following amount of information in total:

We have estimated Holevo information for multi-mode coherent states presented in Eq. (1) according to [32]:## 4. Experimental results

Experimental setup implementing TF QKD based on multi-mode coherent phase states is shown on Fig. 2. It should be noted that such experimental realization has the similar concept as the one presented in [19], apart from the utilized states (sideband modulation).

In our feasibility demonstration to simplify an adjustment we use only one laser source and split it’s radiation in two parts with fiber optic 50:50 beam splitter $BS1$. This approach allows us to imitate the case that Alice and Bob have well-correlated light sources. As the light source we use coherent DFB laser $L1$ with a narrow linewidth (less than 100 kHz) and wide wavelength range adjustment (from 1530.0 nm to 1565.5 nm) with precise accuracy (20 pm). The initial output power is equal to 10 mW. Due to sensitivity of DFB laser’s resonator to reverse propagating light a passive optical isolator $I$ based on the Faraday effect with isolation 50 dB was installed in the optical scheme. To implement single-photon mode a variable optical attenuator $VOA$ was used with a fine adjustment of the attenuation degree.

After the light beam being emitted by a continuous-wave laser it is sent through the two arms of the interferometer via beam splitter (thus imitating Alice and Bob separate emitters) we can encode the information using phase modulators. The outputs of the first beam splitter are connected to two independent 10 GHz $LiNbO_3$ electro-optical phase modulators (with integrated polarizers) $PM1$ and $PM2$. Polarizers reduce the sensitivity of the modulators to the polarization of the input radiation and increase the visibility of the interference pattern. In terms of current experiment it was enough to set up polarization controller $PC$. For further development of the system, for instance, passive polarization distortion compensation scheme similar to one in [35] or motorized fiber polarization controllers with feedback should be implemented. Electrical inputs of $PM1$ and $PM2$ were connected with digital-to-analog converters $DAC$ outputs with modulating $sin$-wave 4.8 GHz RF-signals. The amplitudes of that signals are adjusted in a way that sidebands intensity should be equal for Alice and Bob. The modulation index that shows the fraction of energy on sidebands according to the carrier’s energy in result of modulation should be 5 %. In that case the optimal ratio of the signal on sidebands to noise of the carrier trespassing through the filter is observed. Thereby the mean optical power on sidebands is equal to 2.56 pW that corresponds to $\mu =0.2$.

In this experimental feasibility demonstration we used only two encoding phases at both Alice and Bob ($\varphi _{A,B}\in \{0,\pi \}$, here compressed subscript notation $\varphi _{A,B}$ is used to indicate both $\varphi _A$ and $\varphi _B$, referring the first label to Alice and the second to Bob). The phase difference ($\Delta \varphi$) between Alice and Bob was produced by changing the IQ table of DACs using self-developed FPGAs and software. In case of measuring dependence of detection rates on phase difference ($\Delta \varphi$) the phase was shifted sequentially with a $\varphi _{step}\ = 10^{\circ }$ step. The obtained states at the output of $PM1$ and $PM2$ can be described in according to the Eq. (1).

After the states were prepared we send them into the quantum channel. To compensate the phase mismatch (to adjust the relative phase of optical signals) in the two arms of the interferometer we use phase modulators $PM3$ and $PM4$. This adjustment was implemented by controlling the DC voltage by two independent electrical outputs of waveform generators $GEN1$ and $GEN2$. The second phase modulator’s pair outputs were connected to inputs pair of fiberoptical 2x2 50:50 beam splitter $BS2$.

The measurements were carried out in the photon counting mode on a SNSPD (Scontel) single photon detector with two independent receivers $D1$, $D4$. The quantum efficiency for each was equal to 10 % and the value of own dark counts less then 50 Hz for each. The measured value of illuminations and detectors dark counts $\gamma _{dark}=1.5$ kHz. The detection station was organized as follows. After the relative phase of optical signals is adjusted two states enter at the second interfering beam splitter ($BS2$). The description of the states at the output of the beamsplitter are presented in Eq. (5). The following discussion assuming relative optical phase $\phi _0\approx 0$. If the the phase difference between Alice’s and Bob’s modulation signals is equal to zero ($\Delta \varphi =0$) the whole spectrum goes to the one arm, otherwise even modes of the spectrum (including the central carrier) go to the same arm and odd modes go to the second arm. In case of $\Delta \varphi =0$ we need to split our spectrum. Optical spectral filtering in Charlie’s module aims at removing the relatively strong carrier wave since we assume that the sideband modes encodes quantum states. Unfortunately, in a practical SCW QKD system this wave can only be attenuated by a factor $\vartheta \ll 1$, as it shown in Eq. (6). Thus the mean photon number which arrives to the detector $D1$ can be calculated according to Eq. (7). It should be mentioned that since we use low mean photon number only the first pair of the sidebands is significant. Case of $\Delta \varphi =\pi$ demonstates non-trivial result. The multimode state splits at the beam splitter and the central mode (and all odd sidebands) goes in the first arm and all even sidebands (we assume that only the first pair of sidebands is significant) goes to the second arm. Thus optical phase justification may replace optical filtering in the one arm of the detection scheme. The indicator of successful optical phase adjustment is that the carrier is always staying in the first arm and is being detected by $D2$.

Experimentally measured values (four rounds) of detection rates ($F\mathcal {P}_{1,2}^{+}$) dependent on phase difference ($\Delta \varphi$) are shown in Fig. 3. Experimental dots are accompanied by numerical simulations (according to Eq. (12)) lines in order to compare them. Parameters were estimated as follows: $\mu =0.2$, $\mu _0=4$, $F=100$ MHz, $\gamma _{dark}=1.5$ kHz, $\vartheta =10^{-2.5}$, $\varphi _0\approx 0$, $\eta =10^{-1.65}$, $\varphi _m\approx 3^{\circ }$.

## 5. Conclusion

In this paper we present the feasibility demonstration of twin-field quantum key distribution system based on multi-mode weak coherent states. We propose the analytical description of the optical part of the protocol which includes preparation of the multi-mode states, their transformation and detection. Also we provide expression for raw key rates, quantum bit error rates, and asymptotic secure key rates. We also demonstrate that the protocol in principle can overcome the PLOB bound. Moreover we provide the possible realization of experimental setup, compare the experimental results with derived theoretical model and found them match.

## Funding

Government of Russian Federation (Grant 08-08).

## Acknowledgments

A.K. and An.G. provided theoretical description of the protocol, V.C. conducted the experiment, Ar.G. and G.M. supervised experiment and theoretical work respectively. Moreover, we would like to thank referees for useful advises contributing the improvement of the manuscript.

## Disclosures

The authors declare no conflicts of interest.

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