# Flexible entanglement-distribution network with an AlGaAs chip for secure communications

F. Appas, F. Baboux, M. I. Amanti, A. Lemaître, F. Boitier, E. Diamanti, S. Ducci

FFlexible entanglement-based secure communication with an AlGaAs chipfor quantum networks

F´elicien Appas, Florent Baboux, Maria I. Amanti, AristideLemaˆıtre, Fabien Boitier, Eleni Diamanti, and Sara Ducci Laboratoire Mat´eriaux et Ph´enom`enes Quantiques,Universit´e de Paris, CNRS-UMR 7162, Paris 75013, France Universit´e Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France Nokia Bell Labs, Nozay, France Sorbonne Universit´e, CNRS, LIP6, 4 place Jussieu, F-75005 Paris, France (Dated: February 10, 2021)Quantum communication networks enable applications ranging from highly secure communica-tion to clock synchronization and distributed quantum computing. Miniaturized, ﬂexible, and cost-eﬃcient resources will be key elements for ensuring the scalability of such networks as they progresstowards large-scale deployed infrastructures. Here, we bring these elements together by combining anon-chip, telecom-wavelength, broadband entangled photon source with industry-grade ﬂexible-gridwavelength division multiplexing techniques, to demonstrate reconﬁgurable entanglement distribu-tion between up to 8 users in a resource-optimized quantum network topology. As a benchmarkapplication we use quantum key distribution, and show low error and high secret key generationrates across several frequency channels, over both symmetric and asymmetric metropolitan-distanceoptical ﬁbered links and including ﬁnite-size eﬀects. By adapting the bandwidth allocation to spe-ciﬁc network constraints, we also illustrate the ﬂexible networking capability of our conﬁguration.Together with the potential of our semiconductor source for distributing secret keys over a 60 nmbandwidth with commercial multiplexing technology, these results oﬀer a promising route to thedeployment of scalable quantum network architectures.

I. INTRODUCTION

Quantum technologies have the potential to enhancein an unprecedented way the security of communica-tions across network infrastructures. Services and ap-plications that large-scale quantum communication net-works can provide span secure communication with secu-rity guarantees impossible to achieve only with classicalsystems [1], delegated and blind quantum computing [2],or distributed quantum computing and sensing [3], even-tually leading to the full capabilities of a quantum inter-net [4]. The choice of network topology and the optimiza-tion of the corresponding required resources are crucialwhen designing the architecture of such infrastructures.This is particularly true for the entangled photonic re-sources that need to be deployed to enable some of themost advanced applications, but also to alleviate the needfor trusted nodes in cryptographic scenarios [5, 6].In this context, a topology that has naturally at-tracted much attention exploits the presence of polariza-tion and time-energy photonic entanglement in state-of-the-art sources to distribute polarization-entangled pho-tons from a single source to multiple receivers (users)that hold energy-matched channel pairs [7–14]. This con-cept was in particular ingeniously pushed in [11], whichproposed and demonstrated a scheme allowing each userof the network to share quantum correlations with ev-ery other user, hence forming a fully connected entangle-ment network. Several challenges come, however, withsuch an architecture. First, the number of required fre-quency channels increases quadratically with the numberof users, making the bandwidth of the source a critical feature for ensuring scalability to large multi-user net-works. Although the scaling was improved in [13], therelatively narrowband polarization entangled photon pairsource based on periodically poled crystals that was usedwould not permit to further increase the number of users.Furthermore, scalability can also be hindered by the useof cascaded passive elements such as dense wavelengthdivision multiplexers (DWDM) and beamsplitters, whichalso come with high losses [11, 13]. Employing insteadadvanced multiplexing techniques based on wavelengthselective switch (WSS) technology oﬀers a number of at-tractive features for ﬂexible entanglement distribution, aswas recently shown in [14], albeit again with a relativelynarrowband entangled photon source.In this work, we demonstrate a scalable approach tothe fully connected architecture of [11]. To this end,we use a broadband source, emitting photon pairs attelecom wavelength that is based on an AlGaAs semi-conductor chip. AlGaAs exhibits direct bandgap, strongelectro-optical eﬀect, and small birefringence, which en-ables an eﬀective miniaturization and the generation ofpolarization-entangled states without any oﬀ-chip com-pensation [15–17]. We show that thanks to the featuresof our source, we can maintain an entanglement ﬁdelityabove 85% over a 60 nm-wide spectral region. To bench-mark the performance of our system, we then run theﬂagship quantum communication application, namelyQKD, and in particular the BBM92 protocol [18], bothin the asymptotic and ﬁnite-size regime, over symmetricand asymmetric ﬁbre-optic links, separating the users byup to 50 km in the symmetric case. We distribute en-tanglement using Flexgrid multiplexing based on WSS a r X i v : . [ qu a n t - ph ] F e b technology, which is the current industry standard inclassical communications, and oﬀers straightforward re-conﬁgurability in terms of central frequency and channelbandwidth adjustment, wavelength-insensitive insertionloss and polarization diversity. This allows us to showthat each of the two-user links in a fully connected en-tangled network of 4, 5 or 8 users, can support high-performance QKD at metropolitan scale distances. Wefurther showcase the ﬂexibility of our scheme in an unbal-anced network scenario, featuring several local users andone distant user. We demonstrate that the bandwidthreallocation enabled by the WSS can be used to equili-brate the resulting rates, hence materializing an elasticnetwork conﬁguration and highlighting its potential forthe deployment of ﬂexible, scalable quantum network ar-chitectures. II. ALGAAS PHOTONIC CHIP ANDEXPERIMENTAL SETUP

The schematic of our experimental setup is presentedin Fig. 1 (c) and consists of three stages: entanglementgeneration with an AlGaAs chip, frequency demultiplex-ing/multiplexing (demux/mux) of the generated signaland a distribution stage. Note that two diﬀerent fre-quency demultiplexing/multiplexing approaches can beimplemented, depending on the spectral region of inter-est: in the telecom C-Band (1530 nm to 1565 nm) weuse a WSS and in the L-Band (1565 nm to 1625 nm) acoarse wavelength division multiplexing unit (CWDM)with 13 nm wide channels followed by a tunable ﬁlter.The WSS enables the implementation of ﬂexible reconﬁg-urable fully-connected multi-user entanglement networksas shown in Fig. 1 (b).The AlGaAs chip, depicted in Fig. 1 (a), consists ofa Bragg reﬂection ridge waveguide emitting photon pairsthrough type II spontaneous parametric down conversion(SPDC) in the C+L-Band [10]. The sample is pumpedwith a tunable CW diode laser leading to a pair produc-tion rate of the order of 10 pairs/s at the chip output,corresponding to a brightness of 3 . × pairs/s/mW(see Appendix A for details).Thanks to the dispersion and nonlinear properties ofthe AlGaAs platform, the quantum state produced bythe chip presents several features making it appealing forthe implementation of quantum networks. Our chip gen-erates time-energy and polarization entangled photonsemitted in a | Ψ + (cid:105) polarization Bell state [10, 15]. Thepairs being generated in a broad spectral band, we canmultiplex the generated photons in a large number ofwavelength channels using standard telecom componentsand share polarization entanglement between users re-ceiving energy-matched channels [10]. Besides, the groupvelocity mismatch between orthogonally polarized pho-tons in the AlGaAs chip is so small that no oﬀ-chipwalk-oﬀ compensation is required to obtain polarizationentanglement [15, 16]. This is a key feature enabling the direct use of the emitted pairs at the output of the chip,opening the way to its easy integration into simple androbust architectures.The biphoton bandwidth has been measured by inject-ing the generated photon pairs into a ﬁbered 50/50 beamsplitter (BS) with a ﬁbered C+L-Bands tunable ﬁlter in-serted in one of its output ports and detecting the co-incidences between the two output ports via supercon-ducting nanowire single photon detectors (SNSPD) anda time-to-digital converter (TDC). The result is shownin the inset of Fig. 2: the full-width at half-maximumof the output signal is 60 nm, corresponding to approxi-mately 72 ITU 100 GHz channels, 36 on each side of thequantum state degeneracy frequency.In order to assess the eﬀective biphoton bandwidthfor entanglement distribution we have selected symmet-ric output channels with respect to the biphoton degen-eracy frequency and we have estimated the ﬁdelity ofthe transmitted photon pairs to the | Ψ + (cid:105) Bell state asa function of the channel number. In the spectral re-gion around degeneracy (C-Band) the measurement hasbeen performed by deﬁning channels using the WSS,while in the spectral region far from degeneracy (L-Band)we have used the CWDM followed by a ﬁbered tun-able ﬁlter. As shown in Fig. 1 (c), after demultiplex-ing, the photons are sent through a polarization anal-ysis stage consisting of a set of λ/ , λ/ , λ/ X = {| H (cid:105) , | V (cid:105)} and Z = {| D (cid:105) = √ ( | H (cid:105) + | V (cid:105) ) , | A (cid:105) = √ ( | H (cid:105) − | V (cid:105) ) } . A lower bound to the ﬁdelity F to the | Ψ + (cid:105) Bell state is estimated by coincidence measurementsin those two bases by recording the number of coincidencecounts between the two detectors in 8 diﬀerent conﬁg-urations : C HH , C HV , C V H , C

V V , C DD , C DA , C AD , C AA (see Appendix C for details). From these quantities, wecan access the diagonal elements of the density matrixin both MUBs. For ( α, β ) ∈ B × B , with B ∈ { X , Z } ,the matrix element ρ αβ corresponding to the ﬁrst (sec-ond) photon in α ( β ) polarization state is calculated as ρ αβ = C αβ / ( (cid:80) B×B C α (cid:48) β (cid:48) ). Note that the normalizationfactor has to be calculated independently for the twobases. The lower bound to F is then obtained throughthe relation [19, 20]: F ≥

12 ( ρ HV + ρ V H − ρ HH − ρ V V (1)+ ρ DD + ρ AA − ρ DA − ρ AD ) . The experimental results, along with the ﬁdelity of thenumerically simulated quantum state emitted by the Al-GaAs chip, are shown in Fig. 2. The observed lowerbound of the ﬁdelity is above 95 % over a 26 nm spec-tral range corresponding to the ﬁrst 13 pairs of 100 GHzITU channels around degeneracy, and stays above 85 %for up to 38 channels pairs around degeneracy spanning a λ /4 λ /2 λ /4 SNSPDSNSPD 25 km SMF2825 km SMF28 λ /4 λ /2 λ /4 Demux / Mux Demux / MuxMO MO HPF TFCWDMWSSLaser 780 nm input port 1port 2port 1port 2inputinput port 1port 2

AlGaAs chip

A BCDESource AliceBobCharlieDaveEmilyDemux MuxPhysical architecture Quantum correlationsA BCD A B CDEFGHSM ﬁ berPM ﬁ ber10 μ m FPBSFPBS FIG. 1. (a) Artist view of an AlGaAs chip under operation. An incoming pump photon is depicted on the top left corner,at the input of the waveguide. The generated entangled two-photon state is represented on the bottom right corner. (b)Schematic view of the entanglement network architecture. The physical structure of the network consists of an entangledphoton source, a demux/mux stage and a distribution stage. The resulting quantum correlation topology takes the form of afully-connected graph of variable size (here N = 4 , , WSS

CWDM +Tunable Filter

Detuning to biphoton degeneracy (nm) B e ll s t a t e f i de li t y ( ne t ) Simulation (no cavity effect)Simulation (with cavity effect)Experiment

100 GHz channel number offset C o i n c . C oun t s ( a . u . ) FIG. 2. Main graph: Lower bound to the ﬁdelity to a (cid:12)(cid:12) Ψ + (cid:11) Bell state as a function of the detuning to biphoton degen-eracy wavelength (1556 .

55 nm) in nm (lower x-axis) and inunits of 100 GHz channels (upper x-axis). Error bars takeinto account the Poissonian statistics of both signal countsand subtracted background counts. The diﬀerent color ar-eas correspond to the two diﬀerent wavelength demultiplex-ing schemes (see main text for details). Inset: Coincidencecounts as a function of ﬁlter wavelength. total 60 nm wavelength range. Since the channels of theCWDM are not centered around the biphoton degener-acy wavelength, measurements in the L-Band are limitedto the conjugate channels that fall within the transmis-sion window of the CWDM ports. For this reason only3 data points have been acquired in the orange region ofFig. 2.Close to degeneracy, the experimental data are ingood agreement with theoretical predictions describingthe SPDC process in our sample in a simple pass con-ﬁguration (solid lines). However, moving away from de-generacy, cavity eﬀects arising from facets reﬂectivity ofthe AlGaAs waveguide have to be taken into account.The inclusion of these eﬀects in our theoretical modelleads to the calculated ﬁdelity represented with a dashedline, reproducing the experimental data with a very goodquantitative agreement. Details on the calculation of theﬁdelity can be found in Appendix B.The ability to maintain high quality polarization en-tanglement over a broad spectral region establishes thehigh potential of AlGaAs-based devices as ready-to-useminiature sources of broadband polarization entangledbiphoton states. A sy m p t. K e y R a t e ( b i t s / s )

100 GHz channel number offset

Detuning to biphoton degeneracy (nm) Q BE R ( % ) Positive key rate threshold

FIG. 3. Asymptotic key rate (upper panel) and QBER (lowerpanel) as a function of the detuning to biphoton degeneracyin nm (lower x-axis) and in units of 100 GHz channels (upperx-axis). The 11 % error threshold for a positive key rate isgiven for an error correcting code operating at the Shannonlimit.

III. SECRET KEY SHARING BETWEEN USERPAIRSA. BBM92 protocol with polarization-entangledphotons

Next, we exploit the broadband polarization entangledbiphoton state generated with the AlGaAs chip to imple-ment the BBM92 QKD protocol [18]. In this scenario,two users share a maximally entangled Bell state, in ourcase: | Ψ + (cid:105) = √ ( | H (cid:105) s | V (cid:105) i + | V (cid:105) s | H (cid:105) i ) with signal ( s )and idler ( i ) denoting the high and low energy photonrespectively. We deterministically separate the photonsof each pair into two ﬁber paths according to their fre-quency ( s or i ) using the WSS. As explained previously,each photon is then sent to a measurement station, con-sisting of a polarization analysis stage and a SNSPD asshown in Fig. 1 (c). In the full BBM92 protocol, the ba-sis choice is random, each user having 4 detectors, onefor each measurement outcome ( H, V, D, A ). The ran-dom basis choice can be implemented by using, for in-stance, passive 50/50 BS [21] or polarization-to-time-binconversion [12]. However, in our proof-of-principle mea-surement, we choose for simplicity to manually conﬁgurethe waveplates in each path ( s and i ) and record the coin-cidence counts in the same 8 conﬁgurations as presentedin section II. This allows us to assess the performance ofour QKD scheme. B. Performance after the WSS stage

To evaluate the intrinsic QKD performance of ourscheme, we measure the ﬁgures of merit of the BBM92protocol right after the demultiplexing stage: a) theasymptotic secret key rate R key , which corresponds tothe number of secret bits per second established by thetwo users following basis reconciliation (sifting), errorcorrection and privacy ampliﬁcation, and b) the quan-tum bit error rate e (QBER), which is the fraction oferroneous bits in the raw key. We repeat the measure-ment for 13 diﬀerent choices of signal and idler 100 GHzchannel pairs. Note that this spectral range is limited bythe upper cutoﬀ wavelength (1565 nm) of the WSS cor-responding to the boundary of the C-Band, and not bythe spectral bandwidth of the generated biphoton state.We ﬁrst express R key and e in terms of the recordedcoincidence counts [12]. The raw coincidence counts ineach basis ( X and Z ) read: C (raw) X = C HH + C HV + C V H + C V V (2) C (raw) Z = C DD + C DA + C AD + C AA (3)The QBER can be readily obtained by taking the ratioof accidental counts against the total number of recordedraw coincidence counts: e = C HH + C V V + C DA + C AD C (raw) X + C (raw) Z . (4)Next, we compute the raw key rate as the arithmeticmean between the raw count rate in the X and in the Z basis: R raw = 12 C (raw) X + C (raw) Z τ , (5)with τ the integration time. Finally, the asymptotic keyrate is obtained as [22, 23]: R key ≥ R raw

12 (1 − f ( e ) H ( e ) − H ( e )) . (6)where H ( e ) = − e log ( e ) − (1 − e ) log (1 − e ) is thebinary entropy function. This corresponds to the max-imum number of key bits per unit time that can be se-curely extracted by the two parties in the limit of inﬁ-nite key length. The factor 1 / f ( e ) H ( e ) is the portion of the key that has to be usedfor error correction, with f ( e ) ≥ f ( e )[22] assuming the standard bidirectionnal code presentedin ref. [24]. Finally, the last term in H ( e ) is the lowerbound of the fraction of the raw key that is lost afterprivacy ampliﬁcation [25].The calculated R key and QBER are plotted in Fig. 3.We see that both quantities are stable over the 13 ITU100 GHz channels spanning a 10 . − f ( e ) H ( e ) − H ( e ) =0, assuming an error-correcting code operating at theShannon limit ( f ( e ) = 1). This stability can be at-tributed to the ﬂatness of the source spectrum in thisspectral region (inset of Fig. 2) and to the wavelength-independent insertion losses of the WSS. The entangle-ment quality of the emitted quantum state, combinedwith the high conversion eﬃciency of AlGaAs, yields highasymptotic key rates of 28 to 39 bits/s, over a very broadspectral range. The ability to support high key rateswith low error over many frequency channels makes ourAlGaAs entangled-photon source an ideal technology fora chip-based QKD network, as will be illustrated in Sec-tion IV. C. Performance across long-distance ﬁber links

We assess the long-distance performance of our ar-chitecture by estimating R key and e after adding 25 kmSMF28 ﬁber spools between the demux/mux stage andthe user’s measurement station. This measurement isperformed in two conﬁgurations: symmetric (both usersseparated from the source by 25 km of ﬁber) and asym-metric (one user at 0 km and the other at 25 km) in or-der to compare the performances and provide the op-timized solution for secure communication. All mea-surement runs are performed by selecting entangled pho-tons from 100 GHz ITU channels 23 (1558 .

98 nm) and29 (1554 .

13 nm) which are considered representative ofall the available channels. The results are shown asblack symbols on Fig. 4 (a) and (b) for the symmetricand asymmetric conﬁgurations respectively.To estimate the maximum distance of a repeaterlesslink we also perform this measurement by replacing theﬁber spools by variable channel attenuation programmedon the WSS. The attenuation can then be converted intoequivalent length of SMF28 ﬁbers assuming a standardvalue of 0 .

22 dB km − for the SMF28 optical losses. Re-sults for a symmetric and asymmetric link are shown inFig. 4 (a) and (b) respectively as color symbols. We com-pare the experimental value of the asymptotic key rateto theoretical predictions (continuous lines) for a BBM92scheme with passive basis selection following the modelof refs. [23, 26].In order to highlight the relevance of these results forreal-world implementations, we perform a ﬁnite key se-curity analysis of our scheme. As mentioned in the pre-vious section, the secret key rate in Eq. (6) is only validin the limit of inﬁnite key length. However, in practice,raw key bits are exchanged in blocks of ﬁnite size. Toguarantee a certain security threshold for a given blocksize, we resort to the framework developed in refs. [5, 27].We set correctness and security bounds to (cid:15) corr = 10 and (cid:15) sec = 10 respectively for a 10 min block size. Theresulting key rates are shown in Fig. 4 (a) and (b) as tri- angles for experimental values and dashed lines for theo-retical predictions.We see that the measurements performed with vari-able attenuation closely follow our theoretical predictions(solid and dashed lines) while measurements with ﬁberspools (black symbols) deviate slightly from our model.Since the QBER does not increase in ﬁber-based exper-iments (black ﬁlled circles), this small discrepancy canbe attributed solely to the insertion losses induced bythe presence of extra ﬁber connectors between the WSSand the ﬁber spools and not to the potential thermal andmechanical instability of our ﬁber link.Based on this data, we can compare experimentallythe long-distance performance of the BBM92 protocol insymmetric and asymmetric conﬁgurations. We observethat, in both scenarios, the key rate including ﬁnite sizeeﬀects stays positive for distances of up to 75 km. How-ever, we also observe that, at very long distances, theasymptotic key rate for a symmetric link stays positiveup to 250 km, while in an asymmetric link, it drops ataround 215 km. Indeed, in the latter conﬁguration, astrong attenuation in the link between the source andthe distant user induces a strong increase of the QBER asthe signal approaches the noise background. On the con-trary, in the symmetric case, the losses are distributed be-tween the two users and this critical situation is reachedat higher levels of attenuation. In the ﬁnite key securityregime in which we are operating, the ﬁnite key rates be-come negative before the diﬀerence between symmetricand asymmetric links comes into play. This result haspractical implications in the context of deployed QKDschemes; indeed this proves that a scenario involving anentangled pair source connected to one local user andone distant user will not be detrimental to the BBM92protocol eﬃciency.Furthermore, the performance of our device beingpractically insensitive to wavelength in the C-band, asshown in Fig. 3, the measured performances for the twoparticular channels (23 and 29) can be extrapolatedto the 12 other available 100 GHz channel pairs. Thismeans that each channel pair can support a long-distanceﬁber link, making our scheme compatible with large-scaleﬁbered QKD networks. These results establish a solidground for a future deployment of our scheme in ﬁberedmetropolitan networks. IV. MULTI-USER ENTANGLEMENTNETWORKS WITH FLEXIBLE BANDWIDTHALLOCATIONA. Reconﬁgurable multi-user entanglementnetworks

Building on the previously presented results, we nowdemonstrate the operation of a scalable multi-user net-work architecture by taking advantage of the broad en-tanglement bandwidth of our source and of the ﬂexible

Equivalent SMF28 fiber link distance (km) -8 -6 -4 -2 S e c r e t k e y r a t e ( b i t s / s ) Q BE R ( % ) Asymptotic rateFinite key rateQBER

Total attenuation (dB)

Equivalent SMF28 fiber link distance (km) -8 -6 -4 -2 S e c r e t k e y r a t e ( b i t s / s ) Q BE R ( % ) Asymptotic rateFinite key rateQBER

Total attenuation (dB)

50 km ﬁ ber 25 km ﬁ ber Symmetriclink Asymmetriclink

FIG. 4. Asymptotic and ﬁnite size key rates and QBER for a (a) symmetric and an (b) asymmetric two-user link supportedby 100 GHz ITU channels 23 (1558 .

98 nm) and 29 (1554 .

13 nm) as a function of SMF28 ﬁber distance (lower x-axis) andattenuation (upper x-axis). Symbols represent experimental data and continuous lines theoretical predictions for a BBM92scheme. Finite key rates are estimated assuming 10 min block size. frequency management enabled by the WSS.We consider a simple physical network consisting ofone central server node which hosts the entangled pho-ton source, and N user nodes. Each user has a polar-ization analyzing device and single photon detectors tocarry out the BBM92 protocol. We focus on a particularclass of such networks, where each of the N users sharesan entangled state with the remaining N − N = 4 , , N is N ( N − /

2. Hence to get a completeentanglement network, one needs to establish N ( N − / N ( N − / N ( N −

1) frequency channels. Thenwe recombine those channels into single optical ﬁbers,one for each user. This is done via the WSS which im-plements both operations. Each user then receives pho-tons from N − N − N . In this context, we see thatthe broadband entanglement of the biphoton state emit-ted by the AlGaAs chip is a crucial asset. In addition, theuse of a WSS at the demultiplexing/multiplexing stageenables a reconﬁgurable multi-user network where thecentral frequency and the bandwidth of each channel can be adjusted. In our setup we can increase the number ofusers in the network by simply reducing the bandwidthallocated to each channel. Using all the available wave-length overlap between the biphoton state and the WSSoperating range, our reconﬁgurable complete-graph net-work can accommodate from 4 users, with 200 GHz chan-nels, to 8 users, where each link is supported by a 50 GHzconjugate channel pair.We sequentially characterize the N ( N − / N = 4 , N = 5 and N = 8. Due to the lack of additional single photon detec-tors, we did not run real-time QKD sessions for N usersin parallel. Instead, we measure the time-correlation his-tograms for all N ( N − / N = 4 , , e and R key using the protocol de-

50 100 150

Temporal bin (81 ps) C o i n c i den c e c oun t s i n s ABACBCBDCDDA T o t a l c oun t s

400 500 600 700 800

Temporal bin (81 ps) C o i n c i den c e c oun t s i n s ABACADAECBCDDBEBECED T o t a l c oun t s A sy m p t. k e y r a t e ( b i t s / s ) Channel width (GHz) Q BE R ( % )

200 300 400 500 600 700 800 900 1000

Temporal bin (81 ps) C o i n c i d e n c e c o un t s i n s ABACADAEAFAGAHBFBGBHCBCDCGDB EBECEDEGEHFCFDFEFGFHGDGHHCHD T o t a l c o un t s A BCD A BCDEA B CDEFGH

FIG. 5. Reconﬁgurable multi-user entanglement network. (a-c): Time-correlation histograms for all two-user links in a fullyconnected network of size (a) N = 4 with 200 GHz channels, (b) N = 5 with 100 GHz channels and (c) N = 8 with 50 GHzchannels. (d): Measured asymptotic key rate and QBER for ITU channels 23 (1558 .

98 nm) and 29 (1554 .

13 nm) as a functionof channel width. scribed in section III B for ITU channels 23 and 29 as afunction of channel bandwidth. The result is shown inFig. 5 (d). We see that the QBER is essentially insen-sitive to the channel bandwidth and the correspondingasymptotic key rates show a linear dependence with thebandwidth, as expected. We conclude that these valuesfor R key and QBER, alongside with those of Fig. 3, pro-vide an estimate for the achievable QKD performancesin the various N -user graphs presented in Fig. 5 (a-c).Our scheme presents several advantages in terms ofscalability over passive WDM QKD schemes. Indeed, theﬂexibility oﬀered by the WSS makes it possible to fullyreconﬁgure the network without modifying the hardware.On the contrary, using passive WDM demux/mux the ad-dition of more users to the network requires either chang-ing the complex combination of cascaded WDM ﬁlters ornon-deterministically splitting some channels by combin-ing WDM ﬁlters and multiport ﬁber splitters [13, 28],which, either way, comes at the cost of extra opticallosses. In contrast, the ﬁxed insertion losses of the WSSmakes it possible to extend the network without degrad-ing the signal, a major asset in a fully deployed networkscenario. B. Towards a smart quantum network with ﬂexiblebandwidth allocation

Finally, we exploit the ﬂexibility of our scheme todemonstrate a quantum network where the signal of eachlink is optimized following a structural constraint. Thisexperiment represents a ﬁrst step towards a fully au-tonomous smart network, where communication ratesare automatically optimized based on bidirectional in-formation exchange between the users and the networkprovider. We consider a fully-connected entanglementnetwork where one of the user nodes is located far apartfrom the entangled photon source, while all the othernodes are located close to the source as schematicallydepicted in Fig. 6 (a). We implement this situation for N = 5 by separating node B’s detector from the sourceby a 25 km-long SMF28 ﬁber spool. If all two-user linkshave the same ﬁxed bandwidth, the key rate of the fourlinks connecting the distant user to the rest of the net-work, i.e. , AB CB DB EB, will be lower than the others.To avoid this problem, we reallocate the state bandwidthby changing the channel widths, assigning more band-width to long links, and less bandwidth to short links,as schematically depicted in Fig. 6 (b). We use a simplealgorithm and distribute the available bandwidth with a12 . AB AC CB CD CE DA DB DE EA EB010002000300040005000 C o i n c i den c e c oun t s i n s Before reallocationAfter reallocation

25 km SMF28 source

A BCDE

Signal/idler wavelength B e f o r e r e a ll o c a t i o n A f t e r r e a ll o c a t i o n Degeneracy(CH26, 1556.55 nm)

FIG. 6. Illustration of ﬂexible bandwidth allocation. (a)Sketch of the experiment. In a fully connected network with N = 5 users, user B is separated from the source by a 25 kmlong SMF28 ﬁber spool. (b) Schematic of the bandwidth dis-tribution between all two-user links before and after applyingthe bandwidth reallocation algorithm. After reallocation, thefour links (red, dark green, light blue and dark blue) that con-nect user B to the rest of the network are allowed a broaderbandwidth. (c) Coincidence counts for each two-user link be-fore (squares) and after (triangles) bandwidth reallocation. the measured signal across the whole network, we mea-sure one by one the 10 two-user coincidence counts ratesfor each link. The results are shown in Fig. 6 (c). Redsquares represent the raw coincidence counts of the 10two-user links for ﬁxed 100 GHz channels and blue trian-gles are the raw coincidence counts recorded after band-width reallocation. We observe that, starting from avery unbalanced initial conﬁguration, we can bring allusers to a similar level of signal. The same techniquecould be applied to other ends, such as boosting the sig-nal across speciﬁc links according to user needs. Thisproof-of-principle experiment shows that QKD is fullycompatible with state-of-the-art telecom network man-agement, opening the way for ﬂexible metropolitan-scaleentanglement distribution. V. DISCUSSION AND OUTLOOK

We have demonstrated a scalable approach to a fully-connected entanglement distribution network using anAlGaAs chip emitting broadband polarization-entangled photon pairs in the telecom band. The lower bound onentanglement ﬁdelity of the quantum state generated byour chip stays above 95 % over a 26 nm wide spectralrange around biphoton degeneracy and above 85 % overa 60 nm range. We deterministically separate the pho-tons of each pair into energy-matched frequency channelsand distribute them to the network users using a wave-length selective switch. We benchmark the performanceof our quantum network by running the entanglement-based BBM92 QKD protocol. We perform key distri-bution between two users with a QBER below 2% acrossﬁbered optical links of up to 50 km including ﬁnite-key ef-fects and we extrapolate a positive key rate for distancesof up to 75 km in both symmetric and asymmetric con-ﬁgurations. We have extended our study to the multi-user case, taking advantage of the ﬂexibility oﬀered byour setup. By reconﬁguring the frequency grid, we showthat our network can accommodate up to 8 users over 50GHz ITU channels. We further demonstrate the insensi-tivity of the QBER with respect to channel width, indi-cating that every two-user link in the network can sup-port high-performance QKD at metropolitan-scale dis-tances. Finally, we demonstrate that the bandwidth re-allocation enabled by the WSS can be used to equilibrateunbalanced network scenarii, which is compatible withan elastic network conﬁguration. Future work includesthe implementation of our architecture in a metropolitanquantum communication network to test its performanceand robustness in a real-world situation.Further progress is possible in diﬀerent ways. On onehand, an extreme miniaturization of the AlGaAs sourcecan be achieved thanks to its compliance with electri-cal injection as already shown in [29], giving a clear ad-vantage to our approach in terms of portability, energyconsumption and integration with future quantum tech-nologies. On the other hand, the cavity eﬀects presentlylimiting the ﬁdelity to a polarization Bell state far fromdegeneracy can be avoided by applying an anti-reﬂectioncoating to the waveguide facets. An optimization of thedesign can also be implemented to reduce the modalbirefringence in order to further broaden the entangledphoton pair bandwidth [30] and to place the degeneracywavelength at the center of the telecom C-band to fullyexploit the WSS bandwidth. Using WSS technology cov-ering both the C and L bands would further allow toexploit the full bandwidth of our source. Besides, thebiphoton bandwidth and brightness could be optimizedby choosing the optimal sample length taking into ac-count SPDC eﬃciency and propagation losses.In addition to the improvement of the photon-pairsource, future investigations of diﬀerent entanglement-distribution topologies can beneﬁt from the ﬂexibility ofour setup. In particular, using the full potential of our60 nm entanglement bandwidth, non-fully connected net-works which can host up to 28 users over 50 GHz ITUchannels could readily be implemented. In this topol-ogy, the network is broken up into independent fully-connected 4-user subnets which are all interconnectedvia extra links. Owing to the broad bandwidth of oursource, this scheme can be demonstrated without theneed to resort to passive BS multiplexing, as suggestedin refs. [13, 28] thus avoiding the reduction of the signalcaused by extra optical losses.Finally, in another approach we could take advantageof the cavity eﬀects present in our sample to generateand coherently control biphoton frequency combs [31],opening the way to the utilization of frequency for quan-tum information processing [32, 33] and to the combinedexploitation of both polarization and frequency entangle-ment for future quantum networks.

ACKNOWLEDGEMENTS

This work has been supported by ANR (AgenceNationale de la Recherche) through the QUANTIFYProject (Project No. ANR-19-ASTR-0018-01) and byParis ˆIle-de-France R´egion in the framework of DIMSIRTEQ through the LION Project. The authors ac-knowledge Perola Milman for fruitful discussions andYves Jaouen for the C+L band tunable ﬁlter loan.

APPENDIX A: SAMPLE AND EXPERIMENTALSETUP

The sample consists of a 6-periodAl . Ga . As/Al . Ga . As Bragg reﬂector (lowermirror), a 366 nm Al . Ga . As core and a 2-periodAl . Ga . As/Al . Ga . Bragg reﬂector (uppermirror). Waveguides are fabricated using wet chemicaletching to deﬁne 5 µ m wide and 790 nm deep ridges ; thewaveguide length is 4 mm. The sample is pumped with atunable CW diode laser (TOPTICA TM Photonics DLpro 780) which is coupled into the waveguide through amicroscope objective (NA = 0.95, 63 × ); light emergingfrom the opposite end is collected with a second identicalmicroscope objective and sent to a ﬁbre coupler, afterﬁltering out the pump wavelength with a high pass ﬁlter.A thermocouple and a Peltier cooler, connected to a PIDcontroller, monitor and keep the waveguide temperatureconstant at 19 . ° C, ﬁxing the wavelength degeneracy ofthe photon pairs to 1556 .

55 nm, which corresponds tothe center of the ITU 100 GHz channel number 26. Thedemux/mux stage is realized either with a WavelengthSelective Switch (model Finisar 4000s) or with a CoarseWavelength Division Multiplexing unit (model FS 78163)followed by a tunable C+L band ﬁlter (model AlnairLabs CVF-220-CL) depending on the spectral regionof interest. After the analysis and distribution stagephotons are detected with superconducting nanowiresingle photon detectors (SNSPD, Quantum Opus) andtemporal correlation measurement are performed with atime-to-digital converter (TDC, quTools).

APPENDIX B: THEORY

In this section, we give the expression of the quantumstate of the source and derive the theoretical ﬁdelity F as a function of channel frequency. A. Quantum state emitted by the AlGaAs source

We start from the generic form of the two-photon stategenerated by collinear Type-II parametric downconver-sion: | ψ (cid:105) = (cid:90) (cid:90) + ∞−∞ d ω d ω C ( ω , ω ) | ω , H (cid:105) | ω , V (cid:105) (7)where | ω, α (cid:105) = a † α ( ω ) | vac (cid:105) denotes the state of the elec-tromagnetic ﬁeld with one photon in a mode of po-larization α = H, V and angular frequency ω . Thecomplex function C ( ω , ω ) is called the joint spec-tral amplitude (JSA) and it is normalized to unity (cid:82)(cid:82) d ω d ω |C ( ω , ω ) | = 1. By rewriting the quan-tum state in the rotated basis: ω + = ω + ω and ω − = ω − ω , the JSA takes the form [34]: C ( ω , ω ) = f + ( ω + ) f − ( ω − ). The ﬁrst term of the product corre-sponds to energy conservation during the downconver-sion process and the second term is related to momen-tum conservation (phase matching). The latter can benumerically calculated using the dispersion properties ofthe guided modes involved in the downconversion pro-cess. Cavity eﬀects arising from the nonzero reﬂectiv-ity of the AlGaAs waveguide facets can be included inthe numerically-calculated JSA using a two-mode Airydistribution [31]. Since the waveguide is pumped by anarrow-linewidth CW laser, one can approximate f + toa Dirac delta function centered on the pump laser angularfrequency f + ( ω + ) = δ ( ω + − ω p ) and the resulting gener-ated quantum state is strongly frequency-anticorrelated.In this case, the state can be rewritten: | ψ (cid:105) = (cid:90) + ∞−∞ dΩΦ(Ω) | ω d + Ω , H (cid:105) | ω d − Ω , V (cid:105) , (8)where we deﬁned for convenience ω d = ω p / ω − / f − ( ω − ).We then rewrite the state as a continuous superposi-tion of bipartite polarization-entangled states. To do so,we split the summation in Eq. (8) into two parts usingthe identity: (cid:82) ∞∞ = (cid:82) ∞ − (cid:82) −∞ then make the changeof variable Ω → − Ω into the second term and ﬁnallyrecombine the two integrals to obtain: | ψ (cid:105) = (cid:90) ∞ dΩ [Φ(Ω) | ω d + Ω , H (cid:105) | ω d − Ω , V (cid:105) + Φ( − Ω) | ω d − Ω , H (cid:105) | ω d + Ω , V (cid:105) ] . (9)0 B. Bell state ﬁdelity

To calculate the ﬁdelity to a | Ψ + (cid:105) Bell state as a func-tion of channel frequency, we derive the reduced densitymatrix in polarization space after spectral ﬁltering. Atthe demultiplexing stage, signal and idler photons lyingin energy matched frequency channels are sent onto sep-arate ﬁber paths A and B . Namely, all signal photonswithin the frequency window [ ω d +(Ω − ∆ / , ω d +(Ω +∆ / A and all idler photons within thefrequency window [ ω d − (Ω + ∆ / , ω d − (Ω − ∆ / B , where ∆ is the channel width and Ω the de-tuning of the channel central frequency with respect todegeneracy. The resulting post-selected quantum statetakes the form: | ψ (cid:48) (cid:105) = (cid:90) ∞ dΩ f (Ω) [Φ(Ω) | ω d + Ω , H (cid:105) A | ω d − Ω , V (cid:105) B + Φ( − Ω) | ω d − Ω , H (cid:105) B | ω d + Ω , V (cid:105) A ] (10)with f (Ω) the ﬁlter lineshape, which in our case is as-sumed to be rectangular: f (Ω) = (cid:26) , for Ω ∈ [Ω − ∆ / , Ω + ∆ / , elsewhere . (11)The corresponding density operator is ˜ ρ = | ψ (cid:48) (cid:105) (cid:104) ψ (cid:48) | . Byfollowing the approach of ref. [35], we compute the re-duced polarization density matrix ρ by tracing out thefrequency part of the density operator: ρ = 1 N (cid:90) (cid:90) d ω (cid:48) d ω (cid:48)(cid:48) A (cid:104) ω (cid:48) | B (cid:104) ω (cid:48)(cid:48) | ˜ ρ | ω (cid:48)(cid:48) (cid:105) B | ω (cid:48) (cid:105) A . (12)with N a normalization constant. After some straight-forward algebra, one obtains: ρ = α | HV (cid:105) ABAB (cid:104) HV | + D | HV (cid:105) ABAB (cid:104)

V H | + D ∗ | V H (cid:105)

ABAB (cid:104) HV | + β | V H (cid:105)

ABAB (cid:104)

V H | , (13)where the 4 non-zero matrix elements are: α = 1 N (cid:90) ∞ dΩ f (Ω) | Φ(Ω) | (14) β = 1 N (cid:90) ∞ dΩ f (Ω) | Φ( − Ω) | (15) D = 1 N (cid:90) ∞ dΩ f (Ω)Φ(Ω)Φ ∗ ( − Ω) (16) and the normalization constant is set to N = (cid:82) ∞ dΩ f (Ω) (cid:2) | Φ(Ω) | + | Φ( − Ω) | (cid:3) such that Tr ρ = 1. Fi-nally, the ﬁdelity to a | Ψ + (cid:105) Bell state can be evaluatedfrom the deﬁnition: F = (cid:0) Tr (cid:112) √ ρ | Ψ + (cid:105) (cid:104) Ψ + | √ ρ (cid:1) . APPENDIX C: COINCIDENCE HISTOGRAMSAND BELL CORRELATION CURVES

In this section, we display an example of data for the8 projective measurements that are used to estimate theBell state ﬁdelity and the corresponding Bell correlationcurves.The time-correlation histograms that have beenrecorded for 100 GHz ITU channels 23 and 29 are dis-played in Fig.7. The number of coincidence counts ineach conﬁguration is given by the sum of the of the countsin the 6 central time-bins. The width of the time binsis 81 ps corresponding to the temporal resolution of ourTDC.We deﬁne the coincidence to accidental ratio (CAR)as the mean number of counts in the 6 central bins overthe mean number of counts in 6 bins outside the coinci-dence window. In the conﬁgurations where the numberof counts is maximum (

AA, DD, HV, V H ) we achieve aCAR of the order of 4 × .In addition, for each pair of 100 GHz channel, we mea-sured the Bell correlation curves in both X and Z bases.To do so, we recorded coincidence counts when project-ing the polarization of the signal photon onto an axis ofangle θ with respect to the horizontal axis of the labo-ratory frame, for values of θ spanning [0 ° , ° ], and thepolarization of the idler photon on a ﬁxed axis, either H (0 ° ) or D (45 ° ). The visibility of the obtained two-photon interference fringes is an indicator of the qualityof entanglement. As an illustration, we show the correla-tion curves obtained between ITU channels 23 and 29 inFig. 8. Solid lines are least-square ﬁts to the expression C = a sin ( θ − θ ) + b where θ and θ are the anglesbetween the horizontal axis of the laboratory frame andthe projection axes of signal and idler polarization re-spectively. We measured raw visibilities of 98 . X basis and of 97 . Z basis. [1] E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, Practicalchallenges in quantum key distribution, npj Quantum In-formation , 16025 (2016).[2] A. Gheorghiu, T. Kapourniotis, and E. Kasheﬁ, Veriﬁ-cation of quantum computation: An overview of exist-ing approaches, Theory of Computing Systems , 715 (2019).[3] S.-R. Zhao, Y.-Z. Zhang, W.-Z. Liu, J.-Y. Guan,W. Zhang, C.-L. Li, B. Bai, M.-H. Li, Y. Liu, L. You,J. Zhang, J. Fan, F. Xu, Q. Zhang, and J.-W. Pan, Fielddemonstration of distributed quantum sensing withoutpost-selection, 2011.02807v1. FIG. 7. Measured coincidence histograms for the 8 projective measurements performed to obtain the lower bound on F for100 GHz ITU channels 23 (1558 .

98 nm) and 29 (1554 .

13 nm).FIG. 8. Measured correlation curves for 100 GHz ITUchannels 23 and 29. Squares correspond to experimental rawcounts and solid lines to a ﬁt to a sine square.[4] S. Wehner, D. Elkouss, and R. Hanson, Quantum inter-net: A vision for the road ahead, Science , eaam9288(2018).[5] J. Yin, Y.-H. Li, S.-K. Liao, M. Yang, Y. Cao, L. Zhang,J.-G. Ren, W.-Q. Cai, W.-Y. Liu, S.-L. Li, R. Shu, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A.Chen, C.-Y. Lu, X.-B. Wang, F. Xu, J.-Y. Wang, C.-Z.Peng, A. K. Ekert, and J.-W. Pan, Entanglement-basedsecure quantum cryptography over 1,120 kilometres, Na- ture , 501 (2020).[6] E. Diamanti, A step closer to secure global communica-tion, Nature , 494 (2020).[7] H. C. Lim, A. Yoshizawa, H. Tsuchida, and K. Kikuchi,Wavelength-multiplexed distribution of highly entangledphoton-pairs over optical ﬁber, Optics Express , 22099(2008).[8] I. Herbauts, B. Blauensteiner, A. Poppe, T. Jennewein,and H. H¨ubel, Demonstration of active routing of en-tanglement in a multi-user network, Optics Express ,29013 (2013).[9] D. Aktas, B. Fedrici, F. Kaiser, T. Lunghi, L. Labont´e,and S. Tanzilli, Entanglement distribution over 150 kmin wavelength division multiplexed channels for quan-tum cryptography, Laser and Photonics Reviews , 451(2016).[10] C. Autebert, J. Trapateau, A. Orieux, A. Lemaˆıtre,C. Gomez-Carbonell, E. Diamanti, I. Zaquine, andS. Ducci, Multi-user quantum key distribution with en-tangled photons from an AlGaAs chip, Quantum Scienceand Technology , 01LT02 (2016).[11] S. Wengerowsky, S. K. Joshi, F. Steinlechner, H. H¨ubel,and R. Ursin, An entanglement-based wavelength-multiplexed quantum communication network, Nature , 225 (2018).[12] E. Y. Zhu, C. Corbari, A. Gladyshev, P. G. Kazansky,H.-K. Lo, and L. Qian, Toward a reconﬁgurable quantumnetwork enabled by a broadband entangled source, Jour-nal of the Optical Society of America B , B1 (2019).[13] S. K. Joshi, D. Aktas, S. Wengerowsky, M. Lonˇcari´c, S. P.Neumann, B. Liu, T. Scheidl, G. C. Lorenzo, ˇZ. Samec,L. Kling, A. Qiu, M. Razavi, M. Stipˇcevi´c, J. G. Rarity,and R. Ursin, A trusted node–free eight-user metropoli-tan quantum communication network, Science Advances , eaba0959 (2020).[14] N. B. Lingaraju, H.-H. Lu, S. Seshadri, D. E. Leaird,A. M. Weiner, and J. M. Lukens, Adaptive bandwidthmanagement for entanglement distribution in quantumnetworks, 2010.10369v1.[15] A. Orieux, A. Eckstein, A. Lemaˆıtre, P. Filloux, I. Favero,G. Leo, T. Coudreau, A. Keller, P. Milman, and S. Ducci,Direct bell states generation on a III-v semiconductorchip at room temperature, Physical Review Letters (2013).[16] R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S.Helmy, and G. Weihs, Monolithic source of photon pairs,Physical Review Letters (2012).[17] D. Kang, A. Anirban, and A. S. Helmy, Monolithic semi-conductor chips as a source for broadband wavelength-multiplexed polarization entangled photons, Optics Ex-press , 15160 (2016).[18] C. H. Bennett, G. Brassard, and N. D. Mermin, Quan-tum cryptography without bell’s theorem, Physical Re-view Letters , 557 (1992).[19] X.-Y. Chang, D.-L. Deng, X.-X. Yuan, P.-Y. Hou, Y.-Y. Huang, and L.-M. Duan, Experimental realization ofan entanglement access network and secure multi-partycomputation, Scientiﬁc Reports , 10.1038/srep29453(2016).[20] B. B. Blinov, D. L. Moehring, L.-M. Duan, and C. Mon-roe, Observation of entanglement between a singletrapped atom and a single photon, Nature , 153(2004).[21] R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach,H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner,T. Jennewein, J. Perdigues, P. Trojek, B. ¨Omer,M. F¨urst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbi-eri, H. Weinfurter, and A. Zeilinger, Entanglement-basedquantum communication over 144 km, Nature Physics ,481 (2007).[22] N. L¨utkenhaus, Security against individual attacks forrealistic quantum key distribution, Physical Review A (2000).[23] X. Ma, C.-H. F. Fung, and H.-K. Lo, Quantum key distri-bution with entangled photon sources, Physical ReviewA (2007).[24] G. Brassard and L. Salvail, Secret-key reconciliation bypublic discussion, in Advances in Cryptology — EURO-CRYPT ’93 , edited by T. Helleseth (Springer Berlin Hei-delberg, Berlin, Heidelberg, 1994) pp. 410–423. [25] C. H. Bennett, G. Brassard, C. Crepeau, and U. M.Maurer, Generalized privacy ampliﬁcation, IEEE Trans-actions on Information Theory , 1915 (1995).[26] T. Scheidl, R. Ursin, A. Fedrizzi, S. Ramelow, X.-S.Ma, T. Herbst, R. Prevedel, L. Ratschbacher, J. Koﬂer,T. Jennewein, and A. Zeilinger, Feasibility of 300 kmquantum key distribution with entangled states, NewJournal of Physics , 085002 (2009).[27] M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Ren-ner, Tight ﬁnite-key analysis for quantum cryptography,Nature Communications , 10.1038/ncomms1631 (2012).[28] X. Liu, X. Yao, R. Xue, H. Wang, H. Li, Z. Wang, L. You,X. Feng, F. Liu, K. Cui, Y. Huang, and W. Zhang, Anentanglement-based quantum network based on symmet-ric dispersive optics quantum key distribution, APL Pho-tonics , 076104 (2020).[29] F. Boitier, A. Orieux, C. Autebert, A. Lemaˆıtre, E. Ga-lopin, C. Manquest, C. Sirtori, I. Favero, G. Leo, andS. Ducci, Electrically injected photon-pair source at roomtemperature, Physical Review Letters (2014).[30] H. Chen, K. Laiho, B. Pressl, A. Schlager, H. Suchomel,M. Kamp, S. H¨oﬂing, C. Schneider, and G. Weihs, Op-timizing the spectro-temporal properties of photon pairsfrom bragg-reﬂection waveguides, Journal of Optics ,054001 (2019).[31] G. Maltese, M. I. Amanti, F. Appas, G. Sinnl,A. Lemaˆıtre, P. Milman, F. Baboux, and S. Ducci, Gen-eration and symmetry control of quantum frequencycombs, npj Quantum Information , 13 (2020).[32] J. M. Lukens and P. Lougovski, Frequency-encoded pho-tonic qubits for scalable quantum information processing,Optica , 8 (2016).[33] N. Fabre, G. Maltese, F. Appas, S. Felicetti, A. Ket-terer, A. Keller, T. Coudreau, F. Baboux, M. I. Amanti,S. Ducci, and P. Milman, Generation of a time-frequencygrid state with integrated biphoton frequency combs,Phys. Rev. A , 012607 (2020).[34] G. Boucher, T. Douce, D. Bresteau, S. P. Walborn,A. Keller, T. Coudreau, S. Ducci, and P. Milman,Toolbox for continuous-variable entanglement produc-tion and measurement using spontaneous parametricdown-conversion, Physical Review A (2015).[35] A. Schlager, B. Pressl, K. Laiho, H. Suchomel, M. Kamp,S. H¨oﬂing, C. Schneider, and G. Weihs, Temporally ver-satile polarization entanglement from bragg reﬂectionwaveguides, Optics Letters42