# A reconfigurable entanglement distribution network suitable for connecting
multiple ground nodes with a satellite

Stéphane Vinet  1,* Ramy Tannous  1,2 and Thomas Jennewein1,3  1Institute for
Quantum Computing and Department of Physics & Astronomy, University of
Waterloo, 200 University Ave W, Waterloo, Ontario, N2L 3G1, Canada
2Currently with the National Research Council of Canada, 100 Sussex Drive,
Ottawa, Ontario, K1A 0R6, Canada
3Department of Physics, Simon Fraser University, 8888 University Dr W,
Burnaby, BC V5A 1S6, Canada<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Bridging the long distances between different metropolitan-scale quantum
networks is currently best achieved using satellites, which can cover
thousands of kilometres. The development of efficient approaches to establish
connectivity between ground nodes and a satellite is therefore an essential
next step. We propose a network with dual-functionality: during a brief
satellite pass, the ground network is configured as a multipoint-to-point
topology where all ground nodes establish entanglement with a satellite
receiver. During times when this satellite is not available, the satellite up-
link is rerouted via a single optical switch to the ground nodes, and the
network is configured as a pair-wise ground network. We numerically simulate a
pulsed hyper-entangled photon source and study the performance of the proposed
network configurations for quantum key distribution. We find favourable
resource overheads in the case that the satellite receiver exploits time-
multiplexing whereas the ground nodes utilize frequency-multiplexing. Our
results show high and scalable key rates can be achieved in both
configurations. The scalability, simple reconfigurability, and easy
integration with fibre networks make this architecture a promising candidate
for quantum communication of many ground nodes and a satellite thus paving the
way towards interconnection of ground nodes at a global scale.

##  1 Introduction

A primary objective of quantum communication is the distribution of
entanglement between distant parties [1]. This serves as the foundation for
creating the quantum internet: a quantum network that enables revolutionary
applications, including quantum key distribution (QKD) [2, 3], clock
synchronization [4, 5], distributed quantum computing [6, 7], and long-
baseline interferometry [8]. In recent years, significant research efforts
have been done in expanding the scalability of quantum networks beyond two
communicating parties. To this extent, multiple network architectures have
been proposed, notably trusted-node networks which consists of a mesh of
point-to-point links, where each node acts as a trusted relay to establish a
complete two-party communication set-up [9, 10, 11, 12, 13]. While such a
scheme is compatible with heterogeneous quantum links and can be used to
increase the communication distance arbitrarily, it comes with a significant
security cost as one must ensure the privacy of every relay site.
Alternatively, active switch quantum networks [14, 15, 16, 17, 18] allow for
dynamically reconfigurable networks yet are vulnerable to device failure.
Furthermore, the distribution rate is limited by the switch’s operating speed.
Hence, an all-passive solution is preferred where all users can be connected
simultaneously.

Thus far, two main approaches have been employed to create fully-connected
quantum networks relying either on high-dimensional entanglement or bi-partite
entanglement and multiplexing [19, 9, 20, 21, 22, 23]. In the multipartite
method, each user pair shares a different subspace of the overall network
Hilbert space, which is used to generate their keys. However, adding or
subtracting users from the network requires changing the dimensionality of the
state produced by the source making this approach unscalable. On the other
hand, multiplexing exploits the inherent hypercorrelations produced by the
entangled photon source to deterministically separated correlated photon pairs
to different users. This architecture is easily scalable as the number of
users in the network can be modified without any change to the photon source.
Nevertheless for an $n$-node network to be fully connected,
$\mathcal{O}(n^{2})$ multiplexed channels are required. Consequently, there
remains an important hardware requirement as each channel requires a decoder
and detector module to be fully distinguishable. Otherwise an increase in
accidental counts will deteriorate the performance of the network as was shown
in [9, 22]. Recently, [24, 25, 26, 27] utilized beamsplitters in addition to
multiplexing to increase the scalability of their network, but at the expense
of the key generation rate. An interesting solution proposed by [9], is to use
a pulsed source to recover the channel distinguishability despite limited
detector resources. This implementation requires an additional gating signal
to be sent to each user, and results in noise reduction by a factor equal to
the gating duty cycle. However, this scheme is unsuitable for mobile nodes as
the delays to every other node would need to be continuously compensated by
each user thus restricting the reach of these quantum networks to metropolitan
areas. Continued extensive research is being conducted on fibre-based quantum
repeaters [28, 29, 30] and satellite-based quantum links [31, 32, 33, 34] to
enable the establishment of a global-scale network. While fibre-based links
are limited in distance by exponential losses, the use of a free-space link to
an orbiting satellite reduces the loss scaling from exponential to quadratic,
consequently extending the reach of individual quantum links [35].

In this paper we present a scalable quantum network architecture that includes
a mobile satellite node with minimal hardware requirements. To interface the
fibre-optical network with the satellite link, we consider a pulsed non-
degenerate entangled photon source in conjunction with a frequency-to-time
mapping [36, 37]. Using a medium with a high group delay dispersion (GDD), one
can carefully map the frequency distribution of pulsed light onto temporal
modes such that each frequency channel obtains a distinct time delay within
the resolution of the detector. The satellite node is then able to
unambiguously separate the channels while using a single detector module. The
rest of the paper is organized as follows. In Section 2, we introduce the
reconfigurable quantum network architecture. In Section 3 we analyze the
performance of the proposed network configuration and benchmark its potential
by studying its use case for the Canadian QEYSSat quantum satellite mission
[38]. A summary of our results and various extensions are given in Section 4.

##  2 Method

Figure 1: (a) Communication layer and wavelength allocation for a satellite
pass over two ground stations e.g. over Waterloo and Montreal. (b) Waterloo is
in view of the satellite and takes the multipoint-to-point satellite pass
configuration whereas Montreal is in a fully connected ground topology (c). At
a later time, with the satellite now in view of Montreal, the two cities will
adopt the opposite topology by redistributing the frequency-time correlated
pairs $\\{\lambda_{i},\Lambda_{i}\\}$ between the nodes. During the satellite
pass Alice and Bob enjoy are connected by two frequency-time channels and thus
enjoy a higher key rate.

The proposed network architecture is depicted in Fig. 1. It allows two
reconfigurable quantum metropolitan-area networks (QMAN), for example in
Waterloo and Montreal, Canada, to be connected by ground-to-satellite quantum
links. As the satellite passes over Waterloo in Fig. 1b, the network adopts a
multipoint-to-point topology from the $N$ ground nodes to the satellite.
Consequently allowing for a substantive increase of the key generation during
the satellite pass via multiplexing. Meanwhile, the Montreal network in Fig.
1c is arranged into a pair-wise configuration where every user is connected to
each other. As the satellite travels out of reach of Waterloo and within reach
of Montreal, each network reconfigures itself into the opposite topology to
optimize its functionality throughout the day.

This rearrangement is enabled by the hybrid quantum network design in Fig 2.
For a satellite up-link, the idler is kept on the ground and distributed to
the network users. During the satellite pass, the signal is sent through a
frequency-to-time encoder $\uptau(\lambda)$ to assign a time delay to each
wavelength channel. This mapping can be accomplished by wavelength-dependent
group delay dispersion e.g. in a chirped fibre Bragg grating [36, 39, 40]
(CFBG) or by using a fibre Bragg grating (FBG) array with suitable spacing
between each FBG to obtain the desired time delay [37]. Outside of the
satellite pass, the flip mirror redirects the signal to the ground nodes.

The minimal duration of the channel time separation for the frequency-to-time
mapping is determined by the timing jitter of the satellite single-photon
detector and readout electronics which is typically on the order of $\sim 130$
ps for Si-APDs [41, 42, 43, 44]. Thus for a 80 MHz pump repetition rate, the
satellite network configuration is limited to a maximum of $\sim 95$
wavelength-time channels within the 12.5 ns pulse train. To ensure the channel
distinguishability, the GDD must then be greater than $\sim$ 325 ps/nm. For a
given device length, the GDD and the CFBG’s reflection bandwidth are inversely
proportional: a larger dispersion requires higher reflectivities resulting in
a narrower resonance and bandwidth [45, 46]. However, this limitation can be
circumvented by using longer CFBGs or by using a CFBG array i.e. inscribing or
splicing multiple CFBGs with adjacent reflection windows in succession [36].

While the number of channels depends on the entangled photon source’s
brightness and bandwidth, for simplicity we assume four nodes in each QMAN in
Fig. 1. To create a fully connected N-user communication network,
$\binom{N}{2}$ links are needed. Consequently, in the satellite configuration
shown in Fig. 1b the six channels are distributed to the users such that Alice
and Bob each receive two frequency-time channels and therefore attain an
increased secure key rate. For the pair-wise fully-connected topology in Fig.
1c, each node receives 3 wavelength channels to allow communication with every
other node. The network connectivity and the wavelength allocation for the two
different configurations: satellite (flip mirror up) and ground (flip-mirror
down) are summarized in Table 1.

Figure 2: Conceptual physical layout of the hybrid quantum network. The entangled photon source (EPS) consists of a non-linear crystal (NC) pumped by a pulsed laser to create entangled photon pairs via spontaneous parametric down-conversion. During the satellite pass (flip mirror (FM) up) the signal is sent through a frequency-to-time encoder $\uptau(\lambda)$ in an up-link to the satellite (Sat) via the optical ground station (OGS). Outside of the satellite pass (flip mirror down) the signal is rerouted with the idler through telecom fibre to the various ground nodes thus enabling a fully-connected topology. Table 1: Proposed network connectivity and wavelength allocation. Wavelength correlated pairs are denoted as $\\{\lambda_{i},\Lambda_{i}\\}$ for signal and idler respectively. The specific wavelengths considered in Figs. 5-6 are provided in parentheses with their associated ITU channel number. Network | Satellite configuration | Ground configuration
---|---|---
Connection | signal | idler | signal | idler
Alice-Satellite | $\lambda_{1},\lambda_{5}$ | $\Lambda_{1},\Lambda_{5}$ | - | -
Bob-Satellite |  $\lambda_{2}$, $\lambda_{6}$ |  $\Lambda_{2}$, $\Lambda_{6}$ | - | -
Charlie-Satellite | $\lambda_{3}$ | $\Lambda_{3}$ | - | -
Dana-Satellite | $\lambda_{4}$ | $\Lambda_{4}$ | - | -
Alice-Bob | - | - |  $\lambda_{1}$ ($786.75$ nm) |  $\Lambda_{1}$ (Ch40 $=1545.32$ nm)
Bob-Charlie | - | - |  $\lambda_{2}$ ($787.03$ nm) |  $\Lambda_{2}$ (Ch41 $=1544.53$ nm)
Charlie-Dana | - | - |  $\lambda_{3}$ ($787.32$ nm) |  $\Lambda_{3}$ (Ch42 $=1543.73$ nm)
Alice-Dana | - | - |  $\lambda_{4}$ ($787.63$ nm) |  $\Lambda_{4}$ (Ch43 $=1542.14$ nm)
Alice-Charlie | - | - |  $\lambda_{5}$ ($787.93$ nm) |  $\Lambda_{5}$ (Ch44 $=1542.14$ nm)
Bob-Dana | - | - |  $\lambda_{6}$ ($788.24$ nm) |  $\Lambda_{6}$ (Ch45 $=1541.35$ nm)

##  3 Performance estimation

We simulated the key rate for various quantum key distribution scenarios using
the approaches introduced by Ma et al. [47] and Holloway et al. [48]. Our
entangled state is constructed from the following spontaneous parametric down-
conversion (SPDC) Hamiltonian:

$H=i\chi(a_{H}^{\dagger}b_{V}^{\dagger}-a_{V}^{\dagger}b_{H}^{\dagger})+\rm{H.c.},$
(1)

where the subscripts $H$ and $V$ denote the horizontal and vertical
polarizations, H.c. the Hermitian conjugate and $\chi$ is the squeezing
parameter, a coupling constant proportional to the crystal nonlinearity and
the amplitude of the pump beam [49]. Expectation values for the detector
operators applied to the SPDC state are used in a squashing model to determine
the two-fold coincidence rates and quantum bit error rate (QBER) [50, 51]. We
analyze the performance of both bucket and photon-number resolving (PNR)
detectors whose positive operator-valued measure (POVM) are respectively given
by [52]:

$\hat{E}^{(\text{click})}=\sum_{m=0}^{\infty}[1-D(0)(1-\eta)^{m}]\outerproduct{m}{m}.$
(2)

and

$\hat{E}^{(n)}=\sum_{m=0}^{\infty}\sum_{k=0}^{n}D(k)\binom{n-k+m}{m}\eta^{n-k}(1-\eta)^{m}\outerproduct{n+m-k}{n+m-k},$
(3)

where $\eta$ is the detector efficiency and $D(k)$ the dark count
distribution. For the satellite pass configuration, we assume the satellite
detector to be a Si-APD [53] with a detector dead time of 1 $\mu$s whereas we
assume the ground detector to be an SNSPD with a dead time of 10 ns [54, 55].
Due to the dead time, the photon count rate is no longer linearly proportional
to the impinging photon flux but is given by:

$N_{count}=\frac{N_{ideal}}{1+N_{ideal}T_{D}/T_{INT}},$ (4)

where $T_{D}$ corresponds to the dead time, $T_{INT}$ to the integration time,
$N_{ideal}$ to the number of photons impinging on the detector and $N_{count}$
the number of resolved photons [56]. The secure key-rate (SKR) is then
obtained via

$SKR=q\\{Q_{\lambda}[1-f(\delta_{b})H_{2}(\delta_{b})-H_{2}(\delta_{p})]\\}$
(5)

where q is the basis reconciliation factor, the subscript $\lambda$ denotes
for one-half of the expected photon number $\mu$, $Q_{\lambda}$ is the overall
gain, $\delta_{b}$ $(\delta_{p})$ is the bit (phase) error rate, $f(x)$ is the
bidirection error correction efficiency and $H_{2}(x)$ is the binary entropy
function,

$H_{2}(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x).$ (6)

A summary of the experimental parameters assumed throughout the simulation can
be found in Table 2.

Table 2:  Experimental assumptions for QKD modeling Sat-APD dark count rate | 1000 cps
---|---
Ground detector dark count rate | 100 cps
Sat-APD timing jitter | 130 ps
Sat-APD dead time | 1 $\mu$s
Ground detector dead time | 10 ns
coincidence time window | 1 ns
Error correction factor | 1.17
Satellite link loss | 40 dB

At low ground link loss ($<10$ dB) in Fig. 3(a), detector saturation at the
satellite node partially limits the scaling performance. However, in the
higher loss regime one obtains a linear key rate increase whilst maintaining
the same QBER. This scaling stems from the deterministic separation of the
multiplexed channels resulting in independent communication channels with the
same signal-to-noise ratio [9]. To achieve this deterministic scenario, the
frequency-to-time $\uptau(\lambda)$ plays a crucial role. This is particularly
apparent, in Fig. 3(b) where we only consider wavelength multiplexing. In
other words, the idler photons are demultiplexed on the ground but the signal
photons are recorded at the satellite detector without any channel
discrimination. Surprisingly, demultiplexing solely the idler at the ground
stations provides little advantage and in fact results in a maximal tolerable
loss decrease compared to a single channel. This is due to an increase in
accidental coincidences resulting in a loss of fidelity caused by detecting
several channels on the single satellite detector [22]. Therefore, the signal-
to-noise ratio decreases proportionally to the number of users added. Note
that for every channel link setting, the squeezing parameter ($\chi$) was
numerically optimized to maximize the SKR. To compensate for the increase in
accidental counts at higher loss, the photon pair production was decreased
proportionally to the number of channels considered. At low loss, this one-
sided multiplexing approach yields a slight key rate increase due to detector
load management bypassing the dead time of the detector. Nonetheless, for
realistic loss scenarios detector saturation has negligible significance. We
investigate in Fig. 4 the use of PNR detectors at the ground user nodes.
Interestingly, we find that PNR detectors offer no significant key rate
advantage compared to bucket detectors. This can be readily seen in the high
loss regime from considering the series expansion of $\hat{E}^{(1)}$ for
$D(1)\ll 1,\eta\to 0$ at which point one finds that
$\hat{E}^{(1)}\to\hat{E}^{(\text{click})}$. A detailed analysis of this
phenomenon is in preparation [57].

(a) With time-frequency mapping

(b) Without time-frequency mapping

Figure 3: Scaling behaviour of the ground-to-satellite QKD scenarios for (a)
with the time-frequency mapping. The time and frequency multiplexing leads to
a linear key rate improvement. (b) Without the $\uptau(\lambda)$ frequency-to-
time encoder, there is no discrimination made between the wavelength channels
jointly recorded at the satellite resulting in an increase in accidental
counts and reduction in key rate. The squeezing parameter $\chi$ in Eq. 1 was
optimized for every channel setting to maximize the SKR. Realistic detector
parameters were considered and can be found in Table 2.  Figure 4: Secure key
rate performance comparison between PNR and Bucket detectors for the ground
user nodes. The simulation parameters considered can be found in Table 2.

To further benchmark the potential of our network proposal, we analyze its
performance for the QEYSSat mission use case. For this purpose, we assume the
entangled photon source consists of a 15 mm long PPMgOLN waveguide pumped by a
mode-locked pulsed laser centered at 521.4 nm with a bandwidth of 2 nm and a
repetition rate of 80 MHz which produces polarization entangled photon pairs
at 787.5 nm (signal) and 1543.2 nm (idler) via spontaneous parametric
downconversion with respectively 15 nm and 39 nm of bandwidth. These
assumptions on the entangled photon source are summarized in Table 3.

Table 3: Entangled photon source assumptions Repetition rate | 80 MHz
---|---
Pump central wavelength | 521.4 nm
Pump bandwidth | 2 nm
Signal central wavelength | 787.5 nm
Signal bandwidth | 15 nm
Idler central wavelength | 1543.2 nm
Idler bandwidth | 39 nm

The required wavelength channels for a hypothetical 4-node ground network and
QEYSSat are illustrated in Fig. 5. The color coding is done in accordance with
the channel allocation depicted in Fig. 1. The idler wavelength channels were
chosen for compatibility with the standard International Telecommunication
Union (ITU) 100 GHz grid. The corresponding correlated signal channel around
$787.5$ nm was selected for low-loss transmission through the atmosphere and
compability with mature high-efficiency Si-APDs [35, 58]. Outside of the
satellite pass, the 787.5 nm signal is rerouted to the ground users through
standard telecom infrastructure. Whilst the telecom fibre is slightly
multimode for the 787.5 nm light, Meyers-Scott et al. have shown that there is
minimal crosstalk between the spatial modes over several kilometers resulting
in a well-preserved polarization state and timing signature [59]. In spite of
the greater fibre losses for the 787.5 nm light, the $32$ km range shown in
Fig. 6 is large enough to span a metropolitan area network. As the ground user
nodes are not resource-limited, we can circumvent the detector timing
limitation via frequency demultiplexing to extend the linear performance
increase to the low-loss regime.

(a)

(b)

Figure 5: The spectrum of the signal (a) and the idler (b) were calculated
from the Sellmeier equations for PPMgOLN assuming a 2 nm pump bandwidth
centered at 521.4 nm and a 15 mm crystal [[60]]. Wavelength correlated pairs
$\\{\lambda_{i},\Lambda_{i}\\}$ are denoted by the same color in both
spectrums and were chosen in accordance with Fig. 1. The idler channels match
the ITU DWDM 100 GHz grid.

Table 4 shows the estimated key produced per month between ground users
(ground configuration), and between a satellite and ground users (satellite
configuration). Assuming that over a one-month period, one would typically
have around 10 satellite passes with a typical 100 seconds quantum link
duration, we estimate the total key generated monthly in both configurations
by a user located 16 kilometres equidistant from all other users in the
network and the OGS. In the ground configuration, the $95$ channels can
connect up to 14 users with two preferred connections. The average key
generated by a given user pair is then $\sim 1.9\times 10^{11}$ bits. For the
satellite configuration, the link loss will vary throughout the pass. We
compute a conservative estimate by assuming an average link loss of $40$ dB
for all 10 passes which corresponds to a generated key of $\sim 7.5\times
10^{7}$ bits. The proposed design presents a considerable scaling potential as
the entire SPDC spectrum can be utilized. Assuming detector time resolutions
as low as 35 ps [61], a $80$ MHz repetition rate and an ultra-dense WDM
(UDWDM) with 12.5 GHz spacing, one could employ up to $\sim 350$ channels. The
number of available frequency-time channels could be further increased by
reducing the pulse repetition rate but at the expense of the single channel
key rate.

Figure 6: Scaling behaviour of the ground configuration ( Fig. 1c) for the QEYSSat compatible scenario considered in Table 3 with the detector parameters given in Table 2. Table 4: Generated key rate over a one-month period for both network configurations User-to-user ground distance | 16 km
---|---
Number of ground users in the network | 14
User to OGS distance | 16 km
Average number of passes | 10
Average satellite pass duration | 100 s
Average satellite link loss | 40 dB
Total key generated per ground user pair | $1.9\times 10^{11}\hskip 1.00006pt\rm bits$
Total key generated between a ground node and satellite | $7.5\times 10^{7}\hskip 1.00006pt\rm bits$

##  4 Conclusion

We proposed and analyzed an architecture suitable for serving both
metropolitan networks or ground-satellite links, reconfigurable with minimal
modifications. Our approach is uniquely designed to optimize the efficiency of
satellite up-link scenarios such as QEYSSat, where the satellite receiver is
only available for a few minutes every 24 hours. During the satellite pass,
the secure key rate is improved through time and frequency multiplexing with
no additional hardware requirements on the satellite. Additionally, the
network’s utility outside of the satellite pass is optimized by rerouting the
signal to form a ground-based metropolitan network. The satellite can be
exploited as a trusted node or entanglement swapping station to interconnect
two metropolitan areas and extend the reach of quantum networks to a global
scale. The scalability and simplicity render our network architecture a good
candidate for commercial quantum communication networks.

Funding This work is supported by the High Throughput and Secure Networks
(HTSN) challenge by the National Research Council Canada, project HTSN-630.
The authors further acknowlege support from the Natural Sciences and
Engineering Research Council of Canada (NSERC), the Canadian Foundation for
Innovation (CFI), the Ontario Research Fund (ORF). SV would like to thank the
NSERC PGS-D for personal funding. Acknowledgments The authors would like to
thank Duncan England for discussions on the frequency-to-time encoder.
Disclosures The authors declare no conflicts of interest related to this
article.. Data availabilityThe simulation code used in this paper is available
in the supplementary materials, Ref. [62].

##  Appendix A Quantum Key Distribution Simulation

The simulation code was written using the QuTIP toolbox in Python [63, 48]. We
assume the protocol is symmetric with respect to the $X$ and $Z$ bases thus we
find that $\delta_{b}=\delta_{p}$ and calculate the detection probabilities
for the $Z$ basis according to the squashing model below with the following
parameters:
$\chi$ : squeezing parameter
loss_a: Efficiency of the ground quantum channel
loss_b: Efficiency of the satellite quantum channel
n_factor_a: Dark count rate in ground detectors
n_factor_b: Dark count rate in satellite detectors
dim: Fock space dimension
N: number of multiplexed channels
$\tau$: Detector dead time
f_ec: error correction factor.

###  A.1 Squashing Model

def measure_2folds_4modes_squashing_optimized(dim,$\chi$,proj,proj2,$\tau$):
vacc = basis(dim,0)
det_exp = zeros((2,2,2,2))
H_sq =
1j*$\chi$*(tensor(create(dim),create(dim))+tensor(destroy(dim),destroy(dim)))
U_sq = H_sq.expm()
spdc =U_sq*tensor(vacc,vacc)
psi_i = tensor(spdc,spdc)
out_i = reshape(transpose(reshape(psi_i.full(),
(dim,dim,dim,-1)),(0,3,2,1)),(dim*dim*dim*dim,-1))
psi_i = Qobj(out_i,dims = [[dim, dim, dim, dim], [1, 1, 1, 1]])
for i in range(2):
for j in range(2):
for k in range(2):
for l in range(2):
det_exp[i][j][k][l] =
abs(expect(tensor(proj[i],proj[j],proj2[k],proj2[l]),psi_i))
HH $=\frac{(\rm
det\\_exp[0][1][0][1]+0.5*(det\\_exp[0][0][0][1]+det\\_exp[0][1][0][0])+0.25*det\\_exp[0][0][0][0])}{\rm(1+\tau*(det\\_exp[0][1][0][1]+0.5*(det\\_exp[0][0][0][1]+det\\_exp[0][1][0][0])+0.25*det\\_exp[0][0][0][0]))}$
VV
$=\frac{\rm(det\\_exp[1][0][1][0]+0.5*(det\\_exp[0][0][1][0]+det\\_exp[1][0][0][0])+0.25*det\\_exp[0][0][0][0])}{\rm(1+\tau*(det\\_exp[1][0][1][0]+0.5*(det\\_exp[0][0][1][0]+det\\_exp[1][0][0][0])+0.25*det\\_exp[0][0][0][0]))}$
HV$=\frac{\rm(det\\_exp[0][1][1][0]+0.5*(det\\_exp[0][0][1][0]+det\\_exp[0][1][0][0])+0.25*det\\_exp[0][0][0][0])}{\rm(1+\tau*(det\\_exp[0][1][1][0]+0.5*(det\\_exp[0][0][1][0]+det\\_exp[0][1][0][0])+0.25*det\\_exp[0][0][0][0]))}$
VH$=\frac{\rm(det\\_exp[1][0][0][1]+0.5*(det\\_exp[0][0][0][1]+det\\_exp[1][0][0][0])+0.25*det\\_exp[0][0][0][0])}{\rm(1+\tau*(det\\_exp[1][0][0][1]+0.5*(det\\_exp[0][0][0][1]+det\\_exp[1][0][0][0])+0.25*det\\_exp[0][0][0][0]))}$
return [HH,HV,VH,VV]

###  A.2 Secure key rate calculation

def simqkdentanglement($\chi$,loss_a,loss_b,n_factor_a,n_factor_b,dim,N):
vacc = basis(N,0)
qber_list=[]
twofolds_list=[]
for i in range(N):
a_det = BucketDetector_realistic_detector(N,loss_a,n_factor_a)
b_det = BucketDetector_realistic_detector(N,loss_b,n_factor_b)
probs2f_i=
measure_2folds_4modes_squashing_optimized(N,$\chi$,a_det,b_det,$\tau$)
twofolds_i=probs2f_i[0]+probs2f_i[1]+probs2f_i[2]+probs2f_i[3]
twofolds_list.append(twofolds_i)
qber_i=(probs2f_i[0]+probs2f_i[3])
qber_list.append(qber_i)
twofolds=sum(twofolds_list)
qber=sum(qber_list)/sum(twofolds_list)
if qber$>$0:
H2=-qber*log2(qber) - (1-qber)*log2(1-qber)
else:
H2 = 0
f_e = f_ec + qber
skr=-1.0*real(twofolds*0.5*(1-(1+f_e)*H2))
return [skr,qber,twofolds]

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