Self-coherent phase reference sharing for continuous-variable quantum key distribution
SSelf-coherent phase reference sharing for continuous-variable quantum key distribution
Adrien Marie ∗ and Romain Alléaume Télécom ParisTech, LTCI, CNRS, 46 Rue Barrault, 75013 Paris, France (Dated: August 31, 2016)We develop a comprehensive framework to model and optimize the performance of CV-QKD with a locallocal oscillator (LLO), when phase reference sharing and QKD are conjointly implemented with the samehardware. We first analyze the limitations of the only existing approach, called LLO-sequential, and showthat it requires high modulation dynamics and can only tolerate small phase noise, leading to expensivehardware requirements. Our main contribution is to introduce two original designs to perform LLO CV-QKD with shared hardware„ respectively called LLO-delayline and LLO-displacement, and to study theirperformance. Both designs rely on a self-coherent approach, in which phase reference information andquantum information are coherently obtained from a single optical wavefront.We show that these designs can lift some important limitations of the existing LLO-sequential approach.The LLO-delayline design can in particular tolerate much stronger phase noise and thus appears as anappealing alternative to LLO-sequential that can moreover be deployed with affordable hardware. We alsoinvestigate, with the LLO-displacement design, how phase reference information and quantum informationcan be multiplexed in a single optical pulse. By studying the trade-off between phase reference recoveryand phase noise induced by displacement we however demonstrate that this design can only tolerate lowphase noise. On the other hand, the LLO-displacement design has the advantage of minimal hardware re-quirements and can be applied to multiplex classical and quantum communications, opening practical pathtowards the development of coherent quantum communications systems compatible with next-generationnetworks requirements.
I. INTRODUCTION
Quantum key distribution (QKD) [ ] is a promising technology that has reached the commercialization stepsince the last decade [
4, 5 ] . Targeting deployment over large-scale networks, next-generation QKD should relyon affordable optical components. It will in particular consist in highly integrated systems able to operate athigh rate and to be deployed over modern optical networks. Relying on standard telecommunication equipment,Continuous-Variable (CV) QKD is an attractive approach towards this new step of QKD development [
6, 7 ] . Whilefirst results towards CV-QKD practical photonics chip integration have been pursued [
8, 9 ] , the possibility to ef-fectively deploy CV-QKD in coexistence with intense wavelength-division multiplexing classical channels has beendemonstrated [ ] . Furthermore, high repetition rates (up to the order of hundreds of MHz) [
11, 12 ] CV-QKDsystemps have also been demonstrated recently. More sensitive to optical losses than discrete-variable based QKD,long distance CV-QKD has however been demonstrated by controlling excess noise [ ] and developping high ef-ficiency error correction codes [ ] . These important steps in the recent development of CV-QKD are, in addition,likely to benefit from the rise of classical coherent communications [ ] , with the prospect of d convergence ofclassical and quantum communication techniques and simplified photonic integration. This positions CV-QKD andmore generally quantum coherent communications as an appealing technology for the the development of modernquantum communications.If we compare with discrete-variable quantum communications, quantum coherent communications howeverhave to address one specific challenge, namely phase reference sharing. As a matter of fact the receiver mustperform a phase-sensitive detection, using an optical beam usually called “local oscillator” whose phase drift withrespect to the emitter must be controlled, or estimated and corrected. We will review in Sec. II the differentmethods that have been considered to perform phase reference sharing in CV-QKD, and explain why generating"locally" the local oscillator is a fundamental requirement for continuous-variable quantum key distribution(CV-QKD), both for performance and security reasons. We will first analyze the question of phase reference sharingwithin to the broader body of work on reference frame agreement and then focus more specifically on the issue ofphase reference sharing in coherent optical communications. Sharing a reference frame.
A reference frame, shared or partially shared, between an emitter and a receiveris a typical requirement in communication protocols, even though this requirement is often implicit. Information ∗ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] A ug n the reference frame allows the receiver to more faithfully translate the received physical signals into logicalinformation. It can for example consist in the knowledge of the relative angle between spatial two-dimensionalcartesian reference frames [ ] , in the synchronization of spatially separated clocks [ ] , or information about therelative phase between two lasers, respectively at emitter and receiver side, when coherent optical communicationis performed [ ] . This latter problem, phase reference frame sharing considered in the context of CV-QKD, willbe the main focus of this article .The problem of sharing a reference frame is specific in the sense that reference frame information constitutes unspeakable information , that can only be shared through physical carriers exchanged between emitter and receiver [ ] . On the other hand, it is important to emphasize that although quantum mechanics gives a precise frameworkto formulate the question of reference frame sharing, in relation with quantum metrology [ ] , this question canbe solved “classically”, using macroscopic signals to exchange reference frame information. The type of questionsrelated to phase reference sharing is not whether it is possible, but whether it can be achieved given resourceconstraints, dictated by the hardware resources and by the characteristics of the channel, such as losses and noise.In line with the recent work on LLO CV-QKD [ ] , we will focus on in this article on the issue of jointlyperforming, with the same hardware, phase reference sharing and CV-QKD.Another question related to reference frame sharing in quantum communications consist in performing“referenceless” quantum communication in which quantum information is encoded so that it can be recovered“without reference frame” at the receiver, under some assumptions about the channel such as collective noise.Such approach can for example be used with polarization encoding, when emitter and receiver spatial referenceframe are slowly rotating, by encoding quantum information over noiseless subspaces [ ] . Referencelessquantum communications can be seen as a specific approach to perform reference frame sharing. This approachhowever requires to encode information over entangled quantum states, and cannot be easily used to designpractical optical encodings for CV-QKD. We will therefore not consider the referenceless protocols in this article. Phase reference sharing in classical and quantum coherent communication.
Coherent communication sys-tems have the advantage of offering higher sensitivity (information per photon) than systems based on directdetections (for example On-Off-Keying modulation, where the information is encoded solely in intensity), and clas-sical coherent systems are gradually becoming more and more used in modern classical optical networks, especiallyin core networks, over long-distance segments. Phase reference sharing is an important requirement in coherentcommunication systems, in order to correct the phase drift between the phase of the emitter laser and the phase ofthe local oscillator laser, placed at the receiver side. The generic objective is essentially to solve this phase referencesharing problem with minimal resource overhead and minimum penalty on the associated communication proto-col. An essential point is to notice that the constraints and thus the solutions that can be adopted in the classicaland in the quantum cases to solve the phase reference sharing issue significantly differ.Classical coherent detectors are designed to detect intense light pulses, typically coherent states containinga very large number of photons, while coherent detectors used in quantum communications must typically beoperated in the shot-noise regime, i.e with electronic noise significantly below the signal variance associated withthe detection of one photon. This limits the intensity that can be handled by shot-noise limited coherent detectorsbefore saturation [ ] . While analogic phase lock loops where used until the 70’s in classical coherent opticalsystems to solve the issue of phase locking, suffering however from phase lock loop bandwidth limitations, theadvent of GHz-clocked electronics and fast digital signal processing now allows to recover both signal informationand phase reference information from discrete modulation, such as binary phase shift keying (BPSK) or higherorder modulations [ ] . Such phase recovering techniques, used for classical coherent communication systems,that require a high number of photons at reception and high-speed modulations / detections, cannot be directlyapplied to perform quantum coherent communications. This makes the problem of phase reference sharing inquantum communications more constrained, and requires specific approaches.Since the problem of phase reference sharing can be solved by sending classical reference pulses, one simpleapproach to the problem, in the context of quantum coherent communication is indeed to use an external classical(intense) phase reference sharing scheme. Such “classical” method is always possible but will typically requires theuse of two separate detectors, one shot-noise limited detector for (weak) quantum signals and a second detector,with a large linearity range, to detect classical phase reference signals. This classical method hence implies notonly techniques for multiplexing and demultiplexing reference and quantum signals, but basically to deploy twoseparate detection hardwares.It is natural to seek how one can lift the extra hardware requirement of the “classical” method in order to jointlyperform phase reference sharing and quantum communication with the same hardware. This question has beenaddressed in recent works aiming at demonstrating CV-QKD operation with a local local oscillator (LLO) [ ] ,however with performance limitations and constraints on the hardware.2 ontributions of this work. We identify and discuss the existing approaches to the phase reference sharingproblem for LLO CV-QKD. Recent works [ ] all rely on time-multiplexed quantum signals pulses with referencepulses in order to jointly perform phase recovery and quantum communication. In this work, we introduce newelements in the standard noise model of CV-QKD analysis, considering new practical constraints imposed by thesimultaneous quantum signal and phase reference transmission of LLO-based CV-QKD. In particular, the amplitudemodulator (AM) dynamics and the linearity range of Bob’s detector are studied and we show that the AM dynamicsis a key parameter in order to compare performance of realistic implementations of LLO-based CV-QKD. As acontribution, our resulting noise model is a refined framework for realistic CV-QKD analysis, including LLO regimes.Based on this comprehensive model, we show that there exist fundamental and practical limitations in the phasenoise tolerance of the designs introduced in [ ] , that we designate as LLO-sequential.In order to go beyond that phase noise limit, we introduce the idea of self-coherence in phase reference sharingfor CV-QKD implementations based on a local local oscillator. Self-coherent designs consist in ensuring the phasecoherence between pairs of quantum signal and phase reference pulses by deriving both of them from the sameoptical wavefront at emission. This allows to perform relative phase recovery schemes with better sensitivity than inthe LLO-sequential design. In particular, we propose a design, called LLO-delayline, implementing a self-coherentphase sharing design. It ensures the self-coherence using a balanced delay line interferometer split between emitterand receiver sides. We analyze how self-coherence is obtained and study the performance reachable with thisdesign, demonstrating that they exhibit a much stronger resilience to high phase noise than the LLO-sequentialdesign under realistic experimental parameters. While previous experimental proposals of LLO CV-QKD are limitedto slowly varying reference frames regimes (ie. based on very stable lasers or high repetition rates), our newlyintroduced design allows phase reference sharing resilient to high phase noise regimes, using the idea of self-coherence.A second self-coherent design, referred to as LLO-displacement, relies on an original multiplexing allowing totransmit both the quantum signal and the reference pulse within each optical pulse. The simultaneous transmissionof quantum signal and phase reference can be seen as an original cryptographic primitive, considered in [ ] , thatcan be used with different modulation schemes. In particular, this allows to optimize the resources − in terms ofrequired hardware and repetition rate − in LLO-based CV-QKD experiments. We also emphasize that an importantadvantage of our LLO-displacement design is its experimental simplicity as we show that the multiplexing canbe perform numerically on Alice’s variables. As such, no specific hardware devices are required. We study thetheoretical performance of such design and exhibit its limitations.In Sec. II, we review the existing implementations of phase reference sharing CV-QKD. In Sec. III, we introducethe CV-QKD model. In particular, the phase reference sharing issue in CV-QKD is formally introduced and discussedand we also introduce our comprehensive noise model. In Sec. IV, we highlight practical limitations of existinglocal local oscillator based CV-QKD and introduce the idea of self-coherence for reference sharing in CV-QKD. InSec. V and Sec. VI, we respectively introduce the LLO-delayline and LLO-displacement designs and study theirperformance. Conclusion and perspectives are presented in Sec. VII. II. IMPLEMENTING PHASE REFERENCE SHARING IN CV-QKD: PREVIOUS WORK
The procedure used for phase reference sharing in quantum coherent communication is often not tackled explic-itly in experiments. As mentioned in the introduction, this follows from the idea that this question can in principlebe solved independently of the quantum communication protocol itself, with classical techniques. This sometimesmotivates to only perform phase reference sharing by placing emitter and receiver in the same location and usinglocally one single laser source both for quantum signal preparation and as local oscillator. Such proof of princi-ple implementations have been used in early CV-QKD demonstrations [ ] and, more recently, in experimentaldemonstrations of measurement device independent (MDI) CVQKD [ ] .In more realistic experimental demonstrations, emitter and receiver must be placed in distant locations andsome specific design must be used in order to obtain a local oscillator, at the receiver side, phase locked with theemitter laser. The simplest experimental approach is actually to use the laser at the emitter side to generate thelocal oscillator, and to send it to the receiving side, using adapted multiplexing schemes. This procedure is calledthe transmitted LO design (TLO) and we will review its principle and its limitations. As we will see, TLO suffersfrom a fundamental weakness in the cryptographic context of CV-QKD, due to the security loophole associated withLO manipulation as it propagates on a public channel. This has lead to implement CV-QKD with a true local localoscillator (LLO), and we will review the recent work in this direction.3 A M Dopticalchannel Coherentdetection
ALICE BOB
LO pulses signal pulses
FIG. 1: (color) Transmitted local oscillator (TLO) design. In the TLO design, the phase reference (green pulse) and the quantumsignal (red pulse) are derived from the same optical pulse and sent from Alice to Bob using multiplexing / demultiplexing (M / D)techniques.
A. The transmitted local oscillator (TLO) design
In most implementations of CV-QKD performed so far [
12, 29–31 ] , the phase reference is directly transmittedfrom Alice to Bob through the optical channel as a bright optical pulse multiplexed in time and polarization witheach quantum signal pulse and is used as the LO pulse at reception. Such implementation is detailed in Fig. 1 andis referred to as the Transmitted LO (TLO) design. The main advantage of this scheme is the guarantee, by design,of a stable relative phase between quantum signal and LO at reception by producing both of them from a singlelaser L A placed at Alice’s side. An interferometric setup, based on polarization delay-line interferometers, is usedto multiplex (M) and demultiplex (D) the quantum signal and the LO, hence ensuring a low relative phase noise atreception. The only limitation in terms of tolerable phase noise is that the phase of the laser can be considered asstable over the duration of a single optical pulse, resulting in ∆ ν/ f ∼
10, where ∆ ν is the spectral linewidth of thelaser, f is the repetition rate and we assume a typical pulse duration of 0.1 / f . Despite it is the most implementedGMCS protocol, security weaknesses of such implementations have however been demonstrated in practice bymanipulating the LO intensity [ ] or wavelength [ ] on the quantum channel.Furthermore, based on a coherent detection at reception, such protocols rely on the use of a bright LO atreception (around 10 photons per pulse at reception are required to ensure that the coherent detection can beoperated with low electronic to shot noise ratio in [ ] ). For long distance or high speed (where the pulse durationis short), the requirements in terms of launch power at emission creates practical issues. Because of limited powerof lasers as well as Brillouin effect and non-linear effects in optical fibers [ ] , there is a typical limit of fewtens of milliwatts on the launched power of each involved laser for CV-QKD purposes. In the TLO design, this limitis a major limitation of the LO intensity at reception, especially for long distances. This will in particular limit thepossibility of using the TLO design on shared optical fibers at long distance and high-rate operation, i.e. situationswhere the requirements on LO power at emission would be extremely large. B. The local local oscillator (LLO) sequential design
In order to lift the important limitations (both theoretical and practical) of CV-QKD implementations relying onthe TLO design, a new CV-QKD method relying on a “local local oscillator” (LLO) has recently been independentlyintroduced in [ ] . This method, implementing the Gaussian modulated coherent state protocol, consistsin using a second laser at Bob’s side in order to produce local LO pulses for coherent detections. One crucialadvantage of implementing CV-QKD in a LLO configuration is to close, by design, any potential security loopholelinked to the possibility of manipulating the LO as it propagates on the public optical channel between Alice andBob. Implementing LLO CV-QKD allows on the other hand to ensure by design that the LO is fully trusted, andin particular that the LO amplitude (that requires careful calibration) cannot be manipulated. Another importantadvantage of LLO CV-QKD stems from the fact that in this configuration, repetition rate and distance do not affectthe LO intensity at detection. A LO power sufficient to ensure high electronic to shot noise ratio may thus beobtained, independently of the propagation distance.Implementing CV-QKD in the LLO configuration however comes with new experimental challenges. The mainissue in LLO-based CV-QKD is to be able to perform CV-QKD despite the potentially important drift of the relative4 od Coherentdetection+phasecorrection LO LO pulses opticalchannel signal pulse reference pulse
ALICE BOB L A α S α R k -th pair FIG. 2: (color) Local local oscillator (LLO) sequential design. In the LLO-sequential design, Alice sequentially sends weak quantumsignal (red pulse) and bright phase reference (blue pulse) pulses. At reception, Bob performs consecutive coherent detections of eachpulse received using is own LO pulses (green pulse). phase between Alice’s emitter laser L A and Bob’s local oscillator laser L B , see Fig. 2. The relative phase at receptionis, in the case of LLO-based CV-QKD, the relative phase between the two free-running lasers L A and L B . As such,Bob’s raw measurement outcomes are a priori decorrelated from Alice’s quadratures and a phase correction processhas to be performed in order to allow secret key generation. The goal of the phase reference sharing in the contextof LLO CV-QKD is then to ensure a low enough phase noise so that the excess noise is significantly below thethreshold imposed by security proofs [ ] .Recent works [ ] have demonstrated the possibility of implementing the GMCS protocol using a local localoscillator, by introducing an experimental design, depicted on Fig. 2, that we will call LLO-sequential. In the LLO-sequential design, Alice sequentially sends, at a repetition rate f /
2, consecutive pairs ( | α S 〉 , | α R 〉 ) of coherent stateswhere | α S 〉 is a GMCS quantum signal pulse and | α R 〉 = | E R 〉 is a phase reference pulse with a fixed phase set to 0and an amplitude E R . Phase reference pulses are relatively bright pulses compared to the signal and have a fixedphase in Alice’s phase reference frame, that is publicly known so that it carries information on Alice’s referenceframe. At reception, Bob performs sequential coherent detections of quantum signal and phase reference, usinga single detector, operated with a “local local oscillator”, placed at Bob. Bob can thus estimate the relative phaseusing the phase reference pulse and a phase correction can be performed on Alice and Bob’s signal data in order togenerate secret key.In [ ] a 250 kHz-clocked proof-of-principle experiment of the LLO-sequential designed is performed, howeverwith only one single laser playing the role of both emitter and LO, and two consecutive uses of a homodynedetector used to emulate a heterodyne measurement. In [ ] , another proof-of-principle experiment with twolasers and Alice and Bob connected by a 25 km optical fiber is performed with a 50 MHz-clocked system. Theauthors demonstrate that phase correction can be implemented with a residual excess noise compatible with CV-QKD security threshold. Joint operation of CV-QKD (requiring weak quantum signals) together with the phasecorrection mechanism (requiring bright phase reference pulses) was studied through a simulation, which left asidethe question of the hardware requirements for both CV-QKD and phase reference sharing. [ ] provides a wholeexperimental demonstration of an implementation of LLO-sequential CV-QKD over 25 km, with a 100 MHz-clockedsystem, and a 1 GHz-bandwidth shot-noise limited homodyne detection. We should however emphasize that thesestrong experimental performances have been hove we obtained expensive hardware, namely two low phase noiseECL lasers, as emitter and LO, and an amplitude modulator with 60 dB of dynamics. Such hardware is typically notavailable in standard telecom environment and the issue of considering LLO CV-QKD implementation with realistichardware should be addressed in order to study the ability to be ubiquitously integrate CV-QKD within modernoptical networks. III. CV-QKD: PROTOCOL AND NOISE MODEL
Different CV-QKD protocols have been proposed so far including protocols based on squeezed states of theelectromagnetic field [ ] or on discrete modulations of coherent states [
40, 41 ] . However, squeezed states are ex-perimentally challenging to produce and security analysis for discrete modulation CV-QKD are less advanced. Dueto its experimental convenience [
12, 30, 31 ] and its good security analysis understanding [
42, 43 ] , the Gaussian-modulated coherent states (GMCS) protocol is the most implemented CV-QKD protocol and has reached the stepof commercialization [ ] . 5t is therefore natural to consider GMCS CV-QKD in order to study CV-QKD with a local local oscillator and toperform early experiment, as it has been the case in [ ] . We then also focus our analysis on GMCS CV-QKDand introduce new elements in the noise model in order to account for the important constraints that drive theperformance of CV-QKD in the regime of a local local oscillator. This in particular allows us to discuss the limitationsof LLO-sequential design when implemented with realistic hardwares. A. GMCS protocol and secret key rate
In the Gaussian-modulated coherent states protocol, Alice encodes classical Gaussian variables ( x A , p A ) on themean values of the two conjugate quadratures of coherent states | α 〉 = | x A , p A 〉 . Coherent states are then sent to Bobthrough an insecure channel controlled by an eavesdropper Eve. At reception, Bob performs a coherent detectionof either one quadrature (homodyne detection) or both quadratures (heterodyne detection) of the received pulseand calculates estimators ( x B , p B ) of Alice’s variables. As Eve’s optimal attacks are Gaussian [
42, 43 ] , we can modelthe logical channels between Alice and Bob’s data as additive white Gaussian noise channels [ ] : x B = (cid:113) G δ det · (cid:0) x A + x + x c (cid:1) p B = (cid:113) G δ det · (cid:0) p A + p + p c (cid:1) (1)where ( x c , p c ) is the total noise of the channel and we note ( χ x , χ p ) its variance. In Eq. 1, G is the total intensitytransmission of the channel, δ det stands for the detection used at reception ( δ det = δ det = x and p are Gaussian variables of variance N modelling the shotnoise quadratures. In general [ ] , it is assumed that the channel noise is symmetric and χ = χ x = χ p where thevariance χ is referred to Alice’s input. The variance χ of the total noise can be expressed as [
43, 44 ] : χ = δ det − GG + ξ (2)where the first term is the loss-induced vacuum noise and ξ is the overall excess noise variance of the channelreferred to Alice’s input. Thereby, using Eq. 1 and Eq. 2, we can see that the Gaussian channel between Aliceand Bob is fully characterized by the two parameters G and ξ . In a real-world experiment, Alice and Bob canestimate G and ξ from the correlations between their respective variables by revealing a fraction of their data andare then able to characterize the propagation channel and generate secret key. We discuss and model the differentcontributions to the excess noise ξ in practical CV-QKD in the next paragraph. Finally, the secret key rate availableto Alice and Bob in the reverse reconciliation scheme can be expressed as [
43, 44 ] : k = β · I AB − Q BE (3)where 0 ≤ β ≤ I AB is the mutual information between Alice and Bob’s classicalvariables and Q BE stands for Eve’s maximal accessible information on Bob’s measurements, capturing assumptionson Eve’s behaviour [ ] . In this work, we restrict the security analysis to individual attacks [
43, 44 ] and Q BE isthen the classical information I BE between Bob’s measurements and Eve’s data. In [
19, 30 ] , it is assumed thatEve does not have access to Bob’s electronics. In this work however, we use the stronger security model of [ ] assuming that Eve is able to control the noise of Bob’s detector. B. Noise model
Implementing CV-QKD with a local local oscillator comes with new challenges. The main challenge is related tothe fact that Alice and Bob must use a procedure to compensate efficiently the phase drift between two differentlasers, used respectively as emitter and local oscillator, in order to be able to perform CV-QKD with a tolerablenoise level. Another more specific aspect of the challenge is related to the objective targeted in this paper: proposeand study practical implementation schemes for LLO CV-QKD with shared and affordable hardware: this leads toconsider practical limitations that had been previously overlooked, and allows to study resource trade-off.6 . Relative phase noise.
In LLO-based CV-QKD, the main challenge is to create a reliable phase referencebetween emitter and receiver because the relative phase drift between the two involved lasers may fully decorrelateAlice’s variables and Bob’s measurements thus preventing any secret key rate generation. We define the signalrelative phase θ S as the phase difference between the LO pulse | α LO 〉 and the signal pulse | α S 〉 at reception: θ S = ϕ LO − ϕ S (4)where ϕ S is the signal phase and ϕ LO is the phase of the local oscillator at reception. Using the notations of Eq. 1and in presence of a relative phase θ S , we can write Bob’s measurement outcomes, when performing an heterodyneas: (cid:18) x B p B (cid:19) = (cid:114) G δ · (cid:20)(cid:18) cos θ S sin θ S − sin θ S cos θ S (cid:19) · (cid:18) x A p A (cid:19) + (cid:18) x + x c p + p c (cid:19)(cid:21) (5)where x c and p c capture all excess noise sources but the phase noise. The relative phase θ S acts as the selectorof the measured quadrature. In the TLO design, the relative phase θ S is, by design, always close to 0. However, inthe case of two free-running lasers, θ S depends on the relative phase θ between the two lasers. Assuming that thetwo lasers L A and L B are centered around the same optical frequency and have spectral linewidths ∆ ν A and ∆ ν B ,we can model [ ] the relative phase θ = ϕ B − ϕ A ( ϕ A and ϕ B are respectively the phase of L A and L B ) as aGaussian stochastic process { θ t } t characterized by the variance of the drift between two times t i and t i + :var (cid:0) θ i + | θ i (cid:1) = π · (∆ ν A + ∆ ν B ) · | t i + − t i | (6)where θ i and θ i + correspond to the relative phase at consecutive times t i and t i + .We can see from Eq. 5 that this implies a decorrelation between Alice’s data and Bob’s measurements which canbe seen as a contribution, noted ξ phase , to the excess noise ξ . The principle of phase reference sharing schemesconsidered in the article consists in using a reference pulse to build an estimate ˆ θ S of the actual relative phase θ S of the signal (relative means relative with respect to local oscillator), and to apply a phase correction − ˆ θ S on thesignal, in order to compensate for the phase drift. In a reverse reconciliation scheme, this correction has to beperformed on Alice’s data as a rotation of her data: (cid:18) ˜ x A ˜ p A (cid:19) = (cid:18) cos ˆ θ S sin ˆ θ S − sin ˆ θ S cos ˆ θ S (cid:19) · (cid:18) x A p A (cid:19) (7)We can show from Eq. 2, 5, 7 that the remaining excess noise ξ phase due to phase noise (after correction) dependson the modulation format and is, in general, not symmetric on the two quadratures . However, in the case of theGMCS protocol (with modulation variance V A ) and assuming that the remaining phase noise θ S − ˆ θ S after correctionis Gaussian, the phase noise ξ phase can then be written as: ξ phase = V A · (cid:128) − e − V est / (cid:138) (8)where we define the variance V est of the remaining phase noise (after reference quadrature measurement, relativephase estimation and correction) as: V est ˆ= var (cid:128) θ S − ˆ θ S (cid:138) (9)Eq. 8 (derived in Annex. VIII A) is a generalization of the phase noise expression given in [
20, 26 ] for the case ofsmall phase noise.An important challenge to perform LLO-based CV-QKD is therefore to calculate a precise estimator ˆ θ S of therelative phase θ S in order to minimize V est and thus ξ phase . The general scheme for phase reference sharing designin LLO CV-QKD can be modeled in the following way: Alice generates two coherent states, a quantum signalpulse | α S 〉 and a reference pulse | α R 〉 and sends them on the optical channel using some multiplexing scheme. Atreception, Bob performs demultiplexing, and uses the received reference pulse to derive an estimate ˆ θ R the relativephase θ R between the reference pulse and the local oscillator at reception. The phase sharing designs (cf. sectionsIV, V, VI) give guarantees that the relative phase of the reference pulse is close to the relative phase of the quantumsignal, i.e. that θ R ≈ θ R . Therefore, the estimated value ˆ θ R can be used to approximate and then correct the relativephase of the signal θ S .A general picture of the phase estimation process, and of the sources of deviations, is depicted in Fig. 3. We canexpress the quantum signal relative phase θ S (with respect to local oscillator) as the sum of the relative phase θ AS at emission and the phase θ Sch accumulated by the coherent state | α S 〉 on the optical channel: θ S = θ AS + θ chS (10)7 stimationphase t R t S θ RA θ SA θ drift θ Rch θ Sch θ θ θ θ R RA Rch error = + + θ error θ θ θ S A ch = +
S S signalrelative phasereferencerelative phase Alice Bobopticalchannel
FIG. 3: (color) Schematic representation of a general relative phase estimation process. The relative phase θ S at reception (red),which is defined with respect to the LO phase (Eq. 4), is estimated at reception with the estimator ˆ θ R inferred from specific referencephase information evaluation (blue). Similarly and using the same notations for the reference pulse | α R 〉 , we can express the relative phase θ R atreception of a reference pulse | α R 〉 as: θ R = θ AR + θ chR (11)At reception, Bob measures both quadratures of | α R 〉 using a heterodyne detection (in the remaining of the paper,we only consider heterodyne detections at reception with δ det = ˆ θ R of θ R can be calculated fromthe heterodyne measurement outcomes x ( R ) B and p ( R ) B as: ˆ θ R = tan − (cid:32) p ( R ) B x ( R ) B (cid:33) (12)Due to the fundamental shot noise and to the experimental noise on the heterodyne detection, ˆ θ R differs from ˆ θ R by an error θ error characterized by its variance: V error ˆ= var (cid:128) ˆ θ R − θ R (cid:138) (13)We can show that, in the case of a reference pulse of the form | α R 〉 = | E R 〉 : V error = χ + E (14)where E R = | α R | is the amplitude of the reference pulse and χ is defined in Eq. 2. Finally, Bob uses the relativephase estimate ˆ θ S to apply a phase correction − ˆ θ R to the quantum signal. The overall process is schematicallyrepresented in Fig. 3. It results, after phase correction, to a remaining phase noise V est = var (cid:128) ˆ θ R − θ S (cid:138) which canbe expressed using Eq. 10 and Eq. 11 as: V est = V error + V drift + V channel (15)where: V drift ˆ= var (cid:128) θ AR − θ AS (cid:138) (16) V channel ˆ= var (cid:128) θ chR − θ chS (cid:138) (17)The term V drift corresponds to the variance of the relative phase drift θ drift = θ AR − θ AS between the two free-running lasers L A and L B between time t S at which | α S 〉 is emitted and time t R at which | α R 〉 is emitted. From Eq. 6,we can express the phase noise due to laser phase drift between times t S and t R as: V drift = π · (∆ ν A + ∆ ν B ) · | t R − t S | (18)8e can observe that the time delay between signal and phase reference emissions implies a decorrelationbetween the corresponding relative phases and, thus, introduce a noise on the phase estimation process. Thisleads to the main limitation of the LLO-sequential approach as explained in next section.The term V channel corresponds to the relative phase drift due to the difference of the phase accumulated by | α S 〉 and | α R 〉 during propagation. In practice, we assume that this term is dominated by the difference between theoptical path lengths of | α S 〉 and | α R 〉 .In the remaining of this article we will study and discuss the performance of existing as well as newly introducedLLO based CV-QKD designs, relying on different relative phase sharing designs. For each of these designs, we willexplicit the expressions of the different contributions to the remaining phase noise V est of Eq. 15. b. Electronic to shot noise ratio. Intrinsic electronic noise of Bob’s detector induces a noise of variance v elec inshot noise unit (SNU) on Bob’s quadrature measurements. As the shot noise value is linear with the LO intensity,it is however possible to reduce the effective electronic to shot noise ratio ξ elec by increasing the LO intensity. Wemodel ξ elec , referred to Alice’s input, as: ξ elec = δ det G · E · v elec E (19)where E is the photon number in the LO at which the electronic noise is v elec at Bob side and E is the actualphoton number per LO pulse at reception. Furthermore, we consider that Eve is able to manipulate the electronicnoise which corresponds to a strong security scenario [ ] . c. Amplitude modulator finite dynamics. Amplitude modulators efficiency are limited by their dynamics re-stricting the range of the achievable transmission coefficient. Recent works [ ] have proposed to conjointlycommunicate weak quantum signals and relatively bright reference pulses using a single experimental setup and,in particular, a single amplitude modulator (AM). This directly adresses the issue of the AM dynamics at emission,limitating the maximal amplitude that Alice can output and introducing a leakage on the amplitude modulated.The ratio between the maximal and minimal amplitudes E max and E min that Alice can output is characterized bythe dynamics dyn dB of the AM defined as: dyn dB = · log (cid:130) E max E min (cid:140) (20)From this equation, one can model Alice’s modulator imperfection as an amplitude leakage on each optical pulseresulting in an excess noise which can be approximated as: ξ AM = E · − dyn dB / (21)where E max is the maximal amplitude to be modulated. The finite dynamics of Alice’s AM thus adds a noiseproportional to the amplitude E max . This imperfection is then a limitation to the maximal amplitude of the phasereference pulses in LLO-based CV-QKD designs. d. Linearity range of the reception detector. In practice, Bob’s detector response is linear with the input numberof photons within a finite range. Beyond a threshold, the output of the detector is no longer linear and the securitycan be broken [ ] . Thereby, this threshold can be seen as a limitation on the amplitude of the reference pulseused to transmit the phase reference. However, as discussed in Annex. VIII B, typical values of this thresholdare sufficiently large to allow precise relative phase estimation and, thus, are not a limitation to the referenceamplitude. e. Technical noise. In order to simulate the experimental imperfections that one can not calibrate within atypical CV-QKD implementations, we introduce a technical excess noise which is typically a fraction of the shotnoise ( ξ tech = N ).The necessity of exchanging both weak and intense optical signals in CV-QKD based on a local local oscillatorusing only one experimental setup is limited by the finite AM dynamics. These effects are not considered in standardTLO designs because only weak quantum signals are modulated and detected but we will show that they are keyparameters in order to compare realistic implementations of LLO-based CV-QKD in terms of secret key rate. To ourknowledge, the amplitude modulator and linearity range issues have not been taken into account so far in CV-QKDanalysis. Based on this refined model, we first analyze practical limitations of the LLO-sequential design and, then,we compare the LLO-sequential implementations with our newly proposed designs.9
20 40 6010 −4 −3 −2 −1 Distance [km] S e c r e t k e y r a t e [ b i t. pu l s e − ] V drift = 0.001 dyn dB = 80dBdyn dB = 60dBdyn dB = 45dBdyn dB = 30dB −4 −3 −2 −1 Distance [km] S e c r e t k e y r a t e [ b i t. pu l s e − ] V drift = 0.2 dyn dB = 80dBdyn dB = 60dBdyn dB = 45dBdyn dB = 30dB FIG. 4: (color) Secret key rates of the LLO-sequential design for two different values of V drift in presence of AM finite dynamics.Simulations are performed in the individual attacks and pessimistic (Eve can control Bob’s detection) model [ ] . The values of V A and E R are chosen to optimize the secret key rate with β = , η = , v elec = and ξ tech = . IV. TOWARDS IMPROVED PHASE REFERENCE SHARING DESIGNS
In this section, we highlight limitations of the LLO-sequential design in terms of tolerable phase noise due to theunderlying phase reference sharing scheme. We then propose the novel idea of self-coherence to go beyond thatphase noise limit.
A. Limitations of the LLO-sequential design
Since signal and reference pulses follow the same optical path, the estimation process in the LLO-sequentialdesign can be schematically shown using Fig. 3 where θ chS = θ chR . Thus, we have V channel ≈ V est = V drift + V error .One important motivation of the present work is related to the fact that there exists a minimal amount of phasenoise V est (Eq. 15) that can be reached with the LLO-sequential design. The main limitation is due to the factthat signal and reference pulses are emitted with a time delay 1 / f , leading to a phase noise that cannot becompensated, of variance V drift = π · ∆ ν A + ∆ ν B f (22)The phase variance associated to reference pulse phase estimation error, V error can be minimized by choosingthe amplitude E R as large as possible. However the value E R that can be chosen in practice is limited by the finitedynamics of her amplitude modulator, and the existence of an associated optical leakage, whose excess noise ξ AM is proportional to the amplitude E R as discussed in Eq. 21. This in practice leads to a compromise regarding thevalue of E R , in order to minimize the total excess noise.The excess noise due to imperfect phase reference sharing reads as (Eq. 8): ξ phase = V A · ( − e − V est / ) (23)In the regimes of low V est , it simplifies to ξ phase = V A · V est . In order to ensure a tolerable value ξ phase ≤ [
25, 29 ] ),this imposes that V drift (cid:174) / V A .We can finally express a lower bound on the total excess noise, sum of the excess noise ξ phase and the excessnoise ξ AM in the LLO-sequential design as: ξ phase + ξ AM ≥ V A · (cid:130) V drift + χ + E (cid:140) + E · − dyn dB / (24)10 esign Trusted LO Tolerable phase noise Hardware requirements Transmitted LO (Fig. 1) No ∆ ν/ f ∼
10 Stable interferometric set-up [
12, 29–31 ] LLO-sequential (Fig. 2) Yes V drift ∼ − (60dB AM) High AM dynamics [ ] V drift ∼ − (30dB AM)TABLE I: Summary of the advantages and drawbacks of all the different CV-QKD designs considered in this work.
We can quantitatively understand from Eq. 24 that increasing the amplitude E R can reduce the ξ phase contributionat the cost of increasing the ξ AM contribution. The LLO-sequential design thus requires to the experimental regimeswhere V drift (cid:28) f below 100 MHz. This imposes in return strong requirements on the spectral linewidth ofthe lasers that can be used in order to perform LLO-sequential CV-QKD: the linewidth of the lasers must be at mostof 200 kHz, to ensure an excess noise ξ phase lower than 0.1. As a consequence, only very low phase noise lasers,such as external-cavity lasers (ECL), whose typical spectral linewidth is of a few kHz, are suitable to implementthe LLO-sequential design. This is actually illustrated in [ ] , where the performance analysis is made in the lowphase noise regime where f (cid:29) ∆ ν A + ∆ ν B (ie. in a regime where V drift ≈
0) and in [ ] with the experimentalchoice ultra low noise of ECL lasers of 1.9 kHz linewidth.Another issue is actually that finite modulation dynamics has not been taken into account in [ ] , allowing theauthors to choose arbitrary large amplitudes E R . For instance, they show that, by choosing E = V A , a distanceof 40 km is achievable while a more realistic value E = · V A ( ξ AM ≈ − for dyn dB =
40 dB) restricts the protocolto less than 10 km. This indicated that AM dynamics is an important parameter for analyzing CV-QKD within theLLO framework and the LLO-sequential design requires expensive optical equipments, which is not practical interms of large-scale deployment for next-generation CV-QKD.In Fig. 4, we plot the secret key rates of the LLO-sequential design with finite AM dynamics. We can see thatthe AM dynamics is an important parameter as it allows to recover the relative phase with good efficiency whileensuring a low excess noise ξ AM . Below an AM dynamics of 30 dB, no secret key rate can be produced beyond adistance of around 20 km, even for a moderate relative phase drift V drift = − . On the other hand, because of thefundamental limit V drift the LLO-sequential design has to be run at a minimal repetition rate to produce secret key,even in large AM dynamics regimes.In Table. I, we summarize the main characteristics of the two existing implementations of the GMCS protocol pro-posed so far. Although strong security loopholes have been demonstrated on the TLO implementation, the GMCSprotocol has mainly been implemented by directly sending the LO from Alice to Bob. Recent works have howeverintroduced the idea of LLO-based CV-QKD by proposing the experimental LLO-sequential design, hence fixing secu-rity weaknesses by generating the LO pulses at Bob side. We have however shown that the LLO-sequential designhas strong limitations in terms of implementability in realistic regimes. In the next sections, we investigate howthese limitations in term of hardware requirements can be lifted by proposing the idea of self-coherence for phasereference sharing designs. B. Self-coherent phase reference sharing schemes
Performing CV-QKD protocols in the LLO regime can be seen as the issue of conjointly − in the sense of using thesame hardware − sharing a phase reference between two remote lasers and performing CV-QKD between the twoparties Alice and Bob holding lasers L A and L B . A first method to perform such task is the LLO-sequential design [ ] . Indeed, the specific modulation of the sequential optical pulses allows one to perform CV-QKD on signalpulses while sharing the phase reference on specific pulses. We have however shown in Sec. IV A a fundamentallimitation in terms of tolerable relative phase drift in the LLO-sequential design. As an unspeakable information,the phase reference has to be encoded over physical carriers, photons in this case. However, by design, the timedelay between the emission of quantum signal photons and phase reference photons introduce a decoherencebetween signal and reference which can prevent any secret key generation in high phase noise regimes.We now introduce the novel idea of self-coherence for quantum coherent communication protocols. In order toprevent the phase decorrelation between signal and reference due to sequential emissions, we propose to deriveboth the signal and reference from the same optical wavefront at emission thus ensuring the physical coherencebetween signal and phase reference pulses, ensuring that the relative phase drift from Eq. 15 is V drift =
0. Wecall self-coherent such a design. The relative phase between signal and phase reference is then not affected by the11elative phase drift of the lasers and a stable relation between the quantum signal and the LO phases at receptioncan be provided. As the relative phase estimation does no longer depend on the relative phase drift, self-coherentdesigns allow Alice and Bob to perform more efficient phase reference sharings. This new method however comeswith the challenge of coherently sending − ie. by conserving the stable phase relation − the quantum signal andthe phase reference from Alice to Bob. This challenge can be seen as a multiplexing issue. In the remaining ofthis work, we propose two designs to realize GMCS CV-QKD relying on such self-coherent phase reference sharingdesigns.Our first proposal to implement self-coherent CV-QKD is to split a single optical pulse into two pulses used torespectively carry signal and reference information. As output of the same optical pulse, the relative phase betweenthe two pulses at reception only depends on the phases accumulated on their optical paths between emission andreception. This phase reference sharing design relies on the balancing of remote delay line interferometers and werefer to it as the LLO-delayline design. We describe and study its performance in Sec. V.A second idea, that we first introduced in [ ] , to directly ensure self-coherence between signal and phasereference is to encode both of them within the same optical pulse at emission while recovering both information atreception. In Sec. VI, we propose such a design, the LLO-displacement, in which the phase reference informationis encoded over a displacement of the quantum signal modulation. Altough we show that LLO-displacement isrestricted to low phase noise, an advantage of this design is that the experimental setup is drastically simplifiedcompare to LLO-delayline, which is a major advantage in the optics of the integration of LLO-based CV-QKD. V. SELF COHERENT DESIGN BASED ON DELAY-LINE INTERFEROMETER
The idea of the LLO-delayline design is to derive consecutive pulse pairs with fixed relative phase, using abalanced delay line interferometer, hence ensuring a self-coherence property. This design does not suffer from thedrift limitation of LLO-sequential and can allow Bob to recover the relative phase with better precision.
The protocol.
The protocol can be decomposed in successive cycles at the repetition rate f /
2. We note 2 τ = / f the time interval between two consecutive cycles. Each cycle consists in producing and measuring a self-coherentpair of pulses: one quantum signal pulse and one phase reference pulse. We here describe the protocol for onecycle while Fig. 5 details the overall design.At the beginning of a cycle, Alice produces a coherent state | α source 〉 which has an optical phase ϕ A source . Fromthat single optical pulse, she derives two coherent optical pulses in the following way: she splits the state | α source 〉 into two optical pulses, using an unbalanced delayline interferometer:• The weak pulse | α S 〉 is modulated as the GMCS quantum signal and propagates through an optical path oflength l A .• The strong pulse | α R 〉 is delayed by a time τ = / f on a optical path of length l A + δ l A and is referred to asthe reference pulse.Alice then recombines the two pulses | α S 〉 and | α R 〉 resulting in consecutive optical pulses. A major point is thatthe relative phase between phase reference and quantum signal does no longer depend on the phase drift betweenAlice and Bob’s lasers. An other advantage of this scheme is that the amplitude modulator only modulates thequantum signal which removes the constraints on the AM dynamics. The two optical pulses are then successivelysent to Bob through the optical channel, resulting in a repetition rate f .At reception, Bob produces coherent LO pulse pairs using a similar delay line technique used at Alice side. Heproduces an optical pulse | β source 〉 with phase ϕ B source and derives two pulses on a 50 /
50 beamsplitter:• The pulse | β S 〉 that goes through an optical path of length l B .• The pulse | β R 〉 that is delayed and follows an optical path of length l B + δ l B .Bob uses the | β S 〉 and | β R 〉 pulses as LO pulses to successively measure the received | α S 〉 and | α R 〉 pulses. Thisexperimental setup can thus be seen as a remote delay-line interferometer split between Alice and Bob sides.The reference pulse measurement outcomes allows Bob to calculate an estimation ˆ θ R of the relative phase θ R atreference measurement and, thus, infer an estimation of the relative phase θ S at signal measurement. Alice canthen correct her data to decrease the induced excess noise according to Eq. 7. Excess noise evaluation.
In order to study the performance of the LLO-delayline design and calculate theachievable secret key rate, one can note that Alice modulates the quantum signal according to the standard GMCSmodulation. Thus, the usual secret key rates formulas of [ ] can be used. We then have to express the excess12 /2f balanced delay line interferometerL A AM Alice signal pulsereference pulse PM Bob coherent detection digital acquisitionand phase detection opticalchannel
FIG. 5: (color) Full experimental scheme of the LLO-delayline design. Alice sends consecutives phase coherent signal / reference pulsespairs to Bob based on a balanced delay line interferometer. On his side, Bob uses his own laser as the LO for coherent detections usingthe same delay line technique to produce phase coherent LO pulses. Phase estimation and phase correction are digitally performedafter measurement acquisition. noise of the propagating channel in this design, in particular the amplitude modulator noise and the remainingrelative phase noise V est .In the LLO-delayline design, the finite dynamics of Alice’s amplitude modulator only induces a small contributionto the excess noise ξ in affordable hardware regimes. As it only modulates the quantum signal, the maximal ampli-tude E max of Eq. 21 does not depend on the reference pulse amplitude. The excess noise ξ AM is then independant ofthe reference amplitude E R and the intensity E of Eq. 21 only has to be a few times larger than V A [ ] , resultingin a moderate contribution of ξ AM to the excess noise ξ ( ξ AM ∼ − for V A = E = V A and dyn dB = ξ phase by expressing the remaining phase noise V est on Bob’s estimation of therelative phase. By design, the simultaneous emission on the source pulse of | α S 〉 and | α R 〉 , ie. t S = t R , implies θ drift = V drift =
0, which corresponds to the self-coherence property. The variance V est can then bewritten as the sum of the phase estimation efficiency V error (given in Eq. 14) and the variance V channel (Eq. 17) ofthe difference between the accumulated phases on the channel: V est = V error + V channel (25)In this design, signal and phase reference pulses propagate through different optical path. Then, the former termdepends on the stability of the delayline interferometer. As introduced in Sec. III B, this corresponds to the variance: V channel = var (cid:128) θ chR − θ chS (cid:138) (26)where θ chS and θ chR respectively correspond to the phases accumulated by | α S 〉 and | α R 〉 through their propagation.Therefore, one wants to express the quantity θ channel = θ chR − θ chS . Using the definition of the relative phase of Eq. 4,we can first write the relative phase as the difference between the phases respectively accumulated by the twointerfering LO and signal pulses: θ chS = ϕ acc β ,S − ϕ acc α ,S θ chR = ϕ acc β ,R − ϕ acc α ,R (27)13here, for instance, ϕ acc β ,S stands for the phase accumulated by the LO pulse | β S 〉 during its propagation. Wemodel the accumulated phase as a linear function ϕ acc ( l ) of the optical path length l , then we can derive thefollowing expressions: ϕ acc α ,S = ϕ acc ( l A ) ϕ acc α ,R = ϕ acc ( l A ) + ϕ acc ( δ l A ) ϕ acc β ,S = ϕ acc ( l B ) ϕ acc β ,R = ϕ acc ( l B ) + ϕ acc ( δ l B ) (28)Using the previous equations, we can finally express θ channel = θ chR − θ chS as: θ channel = ϕ acc ( δ l B ) − ϕ acc ( δ l A ) (29)As we can see, the relative phase drift θ channel only depends on the difference of the accumulated phases betweenthe delayline optical paths δ l A and δ l B . Due to experimental imperfections as thermal fluctuations, we model δ l A and δ l B as a stochastic processes over time around the same mean value 〈 δ l A 〉 = 〈 δ l B 〉 = c τ ( c being the speedof the light). The phase θ channel then only depends on the fluctuations of the processes δ l A and δ l B correspondingto the interferometer balancing efficiency. An important point is that previous experimental demonstrations of CV-QKD [ ] with transmitted LO, that rely on such delay line interferometers, have proven that phase fluctuations(with frequency typically of order of Hz) associated to interferometer path length fluctuations can be kept low infrequency and amplitude when sampled at CV-QKD repetition rate and do not prevent to perform CV-QKD withrepetition rates in the MHz (or above) and we consider that V channel = var ( θ channel ) ≈
0. We can then consider thatthe variance V est from Eq. 25 is dominated by the phase measurement efficiency: V est = V error (30)The LLO-delayline design ensures self-coherence at interference using delayline interferometers and the depen-dence on the relative phase drift of the lasers is removed. Bob gets self-coherent outcome measurements and isable to estimate the phase drift in the same way as in the LLO-sequential design but with higher efficiency. Finally,this results in the following excess noise: ξ phase = V A · ( − e − V error / ) (31) Performance analysis.
Based on the previous excess noise analysis, we can now study the performance of theLLO-delayline design in terms of secret key rate and compare its performance with the LLO-sequential design. Asthe quantum signal modulation is the same as in the LLO-sequential, we can equivalently compare the achievablesecret key rate or the excess noise contributions.The LLO-delayline design allows to remove the relative phase drift V drift from the excess noise expression. How-ever, the relative phase drift between the two lasers should be stable within the duration of a single optical pulseand imposes that V drift (cid:174)
10 (Sec. II A). The remaining phase noise is only limited by the efficiency V error of the phasereference estimation. In particular, the LLO-delayline allows to perform CV-QKD stronger phase noise regime thanthe LLO-sequential design. Furthermore, as the amplitude modulator excess noise ξ AM does not depend on thephase reference amplitude, the AM dynamics do no longer restrict the relative phase measurement efficiency V error .The reference amplitude E R can then be chosen as large as possible in the limit of the saturation limit of Bob’sdetector and of the launched power limit without increasing the excess noise ξ AM . In practice, these limits allowan very efficient phase measurement.In Fig. 6, we plot the expected key rates for both the LLO-sequential and LLO-delayline designs for differentrelative phase drift and AM dynamics. We can see that the LLO-delayline design is more resilient to both a decreaseof the AM dynamics and to an increase of the relative phase drift. Tthe self-coherence between quantum signal andphase reference allows to reach stronger phase noise regime than LLO-sequential with similar optical hardwares.We can see on the left figure of Fig. 6 that with a 50 dB AM, the LLO-delayline design allows secret key generationat 50 km with V drift = ξ AM . Thus, the relative phase is estimated with better precision reducing the inducedexcess noise. For instance, the LLO-delayline allows to perform CV-QKD at a distance of 50 km with affordable30 dB amplitude modulators even in the regime of standard DFB lasers (linewidths of order of MHz) which isnot possible with the LLO-sequential design. We have thus shown that the LLO-delayline allows to perform LLO-based CV-QKD in the regime of affordable optical hardware regimes, which is an improvement towards LLO-basedCV-QKD based on standard optical equipments. 14
20 40 6010 −4 −3 −2 −1 K e y r a t e [ b i t. pu l s e − ] Distance [km]V drift = 0.1 0 20 40 6010 −4 −3 −2 −1 Distance [km]V drift = 0.001
LLO−delayline : 50 dBLLO−delayline : 40 dBLLO−delayline : 30 dBLLO−sequential : 50 dBLLO−sequential : 40 dBLLO−sequential : 30 dB
FIG. 6: (color) Secret key rate comparaison between the LLO-sequential and the LLO-delayline designs for different AM dynamicsand relative phase drift V drift . VI. SELF COHERENT DESIGNS BASED ON A MODULATION DISPLACEMENT
In this section, we propose a second design (firstly proposed in [ ] ), the LLO-displacement design, implement-ing CV-QKD with a self-coherent phase reference sharing. This design is based on a method for jointly encodingboth quantum signal and phase reference information over each optical pulse produced by Alice’s laser L A atemission. The protocol.
The main idea is to displace, in the phase space, the modulation sent by Alice with a fixeddisplacement of amplitude ∆ and phase φ ∆ . Given her standard GMCS variables ( x A , p A ) , Alice produces andsends to Bob the displaced coherent state: | α 〉 −→ ∆ = | x A + ∆ cos φ ∆ , p A + ∆ sin φ ∆ 〉 (32)The amplitude ∆ and the phase φ ∆ of the displacement are publicly known so that it carries information on Alice’sphase reference. At reception, Bob measures both ˆ x and ˆ p quadratures of each received optical pulse using aheterodyne detection and gets measurement outcomes ( x B , p B ) : x B = (cid:198) G · (cid:2) ( x A + ∆ cos θ ∆ ) · cos θ + ( p A + ∆ sin θ ∆ ) · sin θ + x + x c (cid:3) p B = (cid:198) G · (cid:2) − ( x A + ∆ cos θ ∆ ) · sin θ + ( p A + ∆ sin θ ∆ ) · cos θ + p + p c (cid:3) (33)Using the displacement of Alice’s modulation, Bob is able to measure an estimator ˆ θ S of the relative phase θ S byusing his measurement outcomes ( x B , p B ) as detailed in Sec. III B. Furthermore, using the i indexes for successivepulses, Bob can calculate a more precise estimator ˆ θ ( i ) filter by averaging each estimator ˆ θ ( i ) S with the previous filteredestimator ˆ θ ( i − ) filter using optimized weighted coefficients. Finally, the estimator ˆ θ ( i ) filter allows Alice and Bob to correcttheir data using Eq. 7. Security of the protocol.
In order to study the security of the design LLO-displacement and calculate the secretkey rate, one can observe that security proofs for the GMCS protocols [
42, 43 ] do not rely on the mean value ofAlice’s quadrature because it is fully described using the covariance matrices formalism. However, as we will show,the excess noise induced by the phase noise on a displaced modulation is asymmetric. The secret key rates thenhas to be calculated using a specific method which is detailed in Annex. VIII C.We now quantify the remaining phase noise V est of Eq. 15. As both quantum signal and phase reference areencoded and transmitted within the same optical pulse, we can directly write V drift = V channel =
0. Finally, theonly term contributing to the remaining phase noise V est is the variance V error . Due to the the particular modulationscheme however, the variance of the estimates ˆ θ ( i ) S is not expressed as Eq. 17. In this case, Alice’s modulation canbe seen as a noise in the phase estimation process, resulting in: V error = V A + χ + ∆ (34)15
20 40 6010 −4 −2 K e y r a t e [ b i t. pu l s e − ] Distance [km]V drift = 0.0001 0 20 40 6010 −4 −2 Distance [km]V drift = 0.005
LLO−seq (30dB)LLO−seq (40dB)LLO−seq (50dB)LLO−displ (30dB)LLO−displ (40dB)LLO−displ (50dB)
FIG. 7: (color) Secret key rate comparaison between the LLO-sequential and the LLO-displacement designs for two relative phasedrift V drift and for different AM dynamics.
Furthermore, the filtering technique − hence based on all previous relative phase estimates − allows to use corre-lations of the phase drift over time to recover the relative phase with better precision than V error . In the asymptoticregime (when i is large), we can show that the successive variances V ( i ) filter tend to an asymptotic limit and, finally,one can write: V est = (cid:112) V error · (cid:112) V drift (35)where V drift is the relative phase drift between lasers L A and L B between two consecutive pulses, ie. it is expressedas Eq. ?? .We can now express the excess noise ξ phase due to phase noise in the LLO-displacement regime (using An-nex. VIII A). It can be simplified when V est (cid:28) ξ ( x ) phase = ( V A + ∆ · sin φ ∆ ) · V est ξ ( p ) phase = ( V A + ∆ · cos φ ∆ ) · V est (36)where the expression of V est is defined using Eq. 35. A crucial point in the noise analysis of the LLO-displacementdesign is that the displacement of Alice’s modulation creates an asymmetry on the excess noise on each quadrature.For instance, if φ ∆ = ˆ x quadrature), the displacement induces an increasing ofthe excess noise on the ˆ p quadrature.Furthermore, in this design, the maximal amplitude E max of the AM excess noise (Eq. 21) can be approximate as (cid:112) V A + ∆ , resulting in: ξ AM = ( V A + ∆ ) · − dyn dB (37)However, one can note that the excess noise ξ phase is a more restricting limitation to the displacement amplitudethan the excess noise ξ AM for realistic parameters and, in practice, the AM dynamics do not restrict the amplitudedisplacement. Performance analysis. In [ ] , we only considered the ξ ( x ) phase contribution in the case of φ ∆ =
0, resulting ina too optimistic key rate. Although the displacement decreases the variance V error and, thus, the remaining phasenoise V est , it also increases its impact on the excess noise according to Eq. 36. Unfortunately, this result is a stronglimitation to the achievable ∆ and, thus, to the tolerable phase noise in the LLO-displacement design.We here consider the case where φ ∆ = ξ ( x ) phase + ξ ( p ) phase does not depend on φ ∆ ). From Eq. 36, one can note that the phase excess noise induced on the ˆ p quadrature is proportional to the displacement mean photon number ∆ . This sets a strong constraints on theachievable value of ∆ and, thereby, on the achievable value of V error . There is a trade-off in terms of the ∆ valuebetween the remaining phase noise V est − the displacement ∆ decreases V error − and the excess noise ξ ( p ) phase . Anoptimal value for the displacement can be found and is calculated in our simulations. However, the optimal value16 esign Trusted LO Tolerable phase noise Hardware requirements Transmitted LO [
12, 29–31 ] No ∆ ν/ f ∼
10 Stable interferometric set-up(Section II A)LLO-sequential [ ] Yes V drift ∼ − High AM dynamics(Section II B)LLO-delayline(Section V) Yes V drift ∼
10 Stable interferometric set-upLLO-displacement(Section VI) Yes V drift ∼ − (cid:59) TABLE II:
Summary of all the CV-QKD designs discussed in this work. We compare them in terms of tolerable phase noise and ontheir experimental limitations. of the ∆ only allows secret key generation for low values of V drift . This means that, solely based on the singleestimates ˆ θ S , the LLO-displacement design does not allow a low enough excess noise ξ phase and, as such, requiresstrong correlations, ie. low values of V drift , between consecutive relative phases to recover phase information basedon filtering techniques.In Fig. 7, we plot the expected key rates for both the LLO-sequential and the LLO-displacement designs. Thedisplacement value is optimized to maximize the overall secret key rate. In low phase noise regimes, we cansee that the LLO-displacement is better in terms of secret key rate generation. This is due to the fact that theLLO-displacement relies on the whole repetition rate to generate secret key and the filtering technique combinedto the displacement allows a more efficient relative phase recovery. For higher phase noise regimes however, thedisplacement value required to estimate the relative phase is limited by the excess noise ξ ( p ) phase and, finally, therelative phase estimation can not be performed with a good efficiency.We have shown that the coherence coming from the simultaneous encoding of both the quantum signal and phasereference in this design comes with new challenges. In particular, the displacement of the modulation increases theexcess noise induced by the relative phase noise and creates an asymmetry on the excess noise on each quadratures.An interesting issue is then to optimize the LLO-displacement design in order to increase its performance, especiallyin terms of phase noise resilience. This study is however kept for future works. We however emphasize that LLO-displacement design relies on an extremely convenient experimental scheme as the phase reference encoding isperformed simultaneously with the quantum signal modulation. To our knowledge, the simultaneous quantumsignal and phase reference transmission introduced in the LLO-displacement design is a new primitive which hasnot been studied so far in quantum communication regimes and can be applied to different signal modulationsas BPSK or higher order modulations. Since both quantum signal and phase reference information are sent overthe same optical pulse, the key rate obtained with the LLO-displacement design is moreover not lowered by timemultiplexing, as it is the case with LLO-sequential. Thereby, unlike all other proposals for locally generated localoscillator based CV-QKD designs, it allows to use the whole repetition rate for secret key generation. VII. CONCLUSION AND PERSPECTIVES
In order to lift security loophole issues, the local oscillator should not be directly sent through the optical channelin CV-QKD experiments and LLO-based CV-QKD protocols have been introduced. We have however shown that theonly design proposed to date, the LLO-sequential design [ ] , requires to use ultra low noise lasers and highdynamics modulators. This strong requirements in terms of hardware performance are a limitation to the abilityto deploy LLO-based CV-QKD over large-scale optical networks. In this work, we have addressed the issue ofperforming CV-QKD with a local local oscillator, using affordable hardware.The main challenge of LLO CV-QKD is that the phase drift between emitter laser and local oscillator laser, placedat Bob, induces a phase noise on the quantum communication, that has to be efficiently corrected. In Table II,we summarize the performance and requirements of existing as well as newly proposed designs for CV-QKD. TheLLO-sequential design is intrinsically limited to low phase noise regimes. This puts important constraints on thetype of lasers that can be used both as emitter and LO. An other limitation of the LLO-sequential is the efficiency17f the relative phase estimation process, which is limited in practice by Alice’s amplitude modulator dynamics.Our results imply that next generation CV-QKD, implemented with a local LO, is possible even with low costDFB lasers and standard amplitude modulators. Such features are made possible by the newly introduced self-coherent phase reference sharing design, the LLO-delayline design, and are essential to progress towards photonicintegration and wide deployment of CV-QKD. The LLO-delayline design relies on the self-coherence between thequantum signal and phase reference and on the interferometric stability of two delay lines at short time scale.The finite dynamics of the amplitude modulator does no longer restrict the reference pulse amplitude and, byconsequence, the relative phase estimation process. These characteristics allow the LLO-delayline design to bemore resilient to phase noise than the previously proposed LLO-sequential design. In Fig. 6, we can see that theLLO-delayline design is able to reach a distance of 50 km in a regime of high phase noise, V drift = [
29, 30 ] paving the way to the demonstration of LLO CV-QKD with cheap hardware,using the LLO-delayline design.As another contribution, we have investigated a scheme, LLO-displacement, allowing to simultaneously transmitthe quantum signal and the phase reference information on the same optical pulse. We have however observedthat the implementation of the LLO-displacement design with Gaussian modulated coherente states (GMCS CV-QKD protocol) leads to an overall excess noise that increases with displacement, which restricts its use to lowphase noise regimes. Simultaneous transmission of quantum and phase reference information had however notbeen studied so far and our results can be of interest in view of performing a joint optimization of classical andquantum coherent communication systems, operating with the same hardware. The optimization of such protocolsis then an interesting open question and is kept for future works. Aknowledgements.
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Homodyne detector E LO2 photonsE S2 photons N e =E s E LO electrons optical pulses FIG. 8: (color) Scheme of a typical homodyne detector. Signal and local oscillator pulses interfer on a 50 /
50 beamsplitter (BS). Bothresulting fields are detected on two photodiodes (PD1 and PD2) which convert photons into electrons. The electrical pulses (purplepulses) produced by the photodiodes are then substracted (-) and the resulting quadrature electrical pulse intensity is measured usinga integrator circuit. The outcome of the integrator circuit is proportionnal to N e = E S · E LO up to an intensity threshold N sat . VIII. ANNEXA. Excess noise due to phase noise
Alice sends the coherent state | α 〉 = | x A + x , p A + p 〉 , where, in the general case, we suppose that x A ∼ N ( V x ) and p A ∼ N ( V p ) while Bob uses a heterodyne detection at reception. In order to estimate the phase noise inducedexcess noise, we consider in this analysis that the relative phase noise is the only noise source. Bob then gets thefollowing measurement outcomes ( x m , p m ) : (cid:18) x m p m (cid:19) = (cid:114) G · (cid:20)(cid:18) cos θ sin θ − sin θ cos θ (cid:19) · (cid:18) x A + x p A + p (cid:19)(cid:21) (38)We suppose that Bob gets an estimator ˆ θ ∼ N ( θ , V ϕ ) . He sends his estimator to Alice which corrects her dataand, in the reverse reconciliation scheme, Alice then estimates Bob’s measurements as: (cid:18) ˜ x A ˜ p A (cid:19) = (cid:114) G · (cid:18) cos ˆ θ sin ˆ θ − sin ˆ θ cos ˆ θ (cid:19) · (cid:18) x A + x p A + p (cid:19) (39)We can then express the excess noise on each quadrature as: ξ x = var ( x m − ˜ x A ) ξ p = var ( p m − ˜ p A ) (40)These two quantities depends on the remaining relative phase ϕ = θ − ˆ θ . Assuming that the variable ϕ is aGaussian variable such that ϕ ∼ N ( V ϕ ) , the above expressions can be calculated from the characteristic functionof the Gaussian function and, after calculations, we obtain the following expressions: ξ ( x ) phase = V x · ( + e − V ϕ − e − V ϕ / ) + ( V x + x ) · ( + e − V ϕ − e − V ϕ ) + ( V p + p ) · ( − e − V ϕ ) ξ ( p ) phase = V p · ( + e − V ϕ − e − V ϕ / ) + ( V p + p ) · ( + e − V ϕ − e − V ϕ ) + ( V x + x ) · ( − e − V ϕ ) (41) B. Linearity range of the coherent detector
We consider that Bob relies on a single coherent detector which addresses the issue of the linearity range of thedetector when considering both quantum signal and phase reference transmission. A typical homodyne detectoris presented in Fig. 8. The response of the integrator circuit is proportional to the incoming number of electronsover a finite range. Beyond a certain threshold, the response of the integrator circuit is no longer linear and thesecurity can be broken by specific attacks [ ] . We define this threshold as the maximal number of electrons N sat per electrical pulses that can be detected in a linear regime. The number of electrons in each electrical pulse is N e = · G · E S · E LO (see Fig. 8) where E S and E LO are the amplitudes of the signal and the LO so that the saturationhypothesis imposes: G · E S · E LO ≤ N sat (42)20
20 40 60 80 10010 −5 −4 −3 −2 −1 Distance [km] S e c r e t k e y r a t e [ b i t. pu l s e − ] N sat = 10 N sat = 5 ⋅ N sat = 2 ⋅ N sat = 10 FIG. 9: (color) Expected secret key rates for the LLO-sequential design for different value of linearity threshold N sat . A example of saturation threshold N sat = has been experimentally evaluated in [ ] . For quantum signal ofintensity of order of the shot noise, this threshold is not important and has not been considered so far in CV-QKD analysis. In LLO-based CV-QKD however, the relatively large amplitude of phase reference pulses imposes toconsider the saturation threshold as a limit on the reference pulses amplitude.Eq. 42 implies a trade-off, in the LLO-sequential design, between the signal amplitude − in particular the refer-ence amplitude E R − and the local oscillator amplitude − used to decrease the electronic to shot noise ratio. If wewant to maximize these two quantities, one has to saturate Eq. 42 by choosing: E LO = G · N sat E R (43)The electronic to shot noise ratio of Eq. 19 is then written as: ξ elec = G · · E N (44)where v elec = E = . In Fig. 9, we plot the expected key rate of the LLO-sequential design fordifferent values of the threshold N sat . As we can see, only low values of N sat (two order of magnitude lower thanthe experimental value of [ ] ) are limitations to this design. As a typical value of N sat = photons is sufficientlylarge to allow a precise relative phase sharing, the saturation threshold will not be a limitation to the referenceamplitude. C. Secret key rate formulas for CV-QKD
In this work, we focus on the Gaussian-modulated coherent state (GMCS) protocol. In this protocol, Aliceencodes zero-mean gaussian classical variables x A and p A on both the ˆ x and ˆ p quadratures of coherent states [ ] before sending them to Bob through an insecure optical channel controlled by an eavesdropper Eve. In the Gaussianmodel, the channel between Alice and Bob is fully characterized by the intensity transmission G and the excessnoise ξ x and ξ p ( a priori different on each quadratures. We also assume that Bob uses a heterodyne detection atreception. Symmetric channel.
We first detail the secret key rate formulas used for symmetric excess noise ξ = ξ x = ξ p .This formulas are used for the LLO-sequential and the LLO-delayline designs. We consider that Eve controls thewhole excess noise and we also consider individual attacks. The secret key rates is written as [ ] : k = β · I AB − I BE (45)where: I AB = · log (cid:18) V + χ + χ (cid:19) (46)21 BE = · log (cid:18) G · ( V + χ ) · ( V + χ E )( χ E + ) · ( V + ) (cid:19) (47)with: V = V A + χ = − GG + ξ (49) χ E = G · ( − ξ ) ( (cid:112) − G + G ξ + (cid:112) ξ ) + Asymmetric channel.
We have shown in Sec. VI that the excess noise induced by the phase noise in the caseof the displaced modulation is asymmetric in the two quadratures. We then need to derive specific secret key rateexpressions. From [
43, 44 ] , we can write: k = k x + k p (51)where k x and k p represent the respective key rates on the logical channel corresponding to each quadrature. Eachof these two key rates can then be obtained using the secret key formulas from Eq. 45, using the correspondingexpression ξ x and ξ pp