# Deep ensembles to improve uncertainty quantification of statistical
downscaling models under climate change conditions

Jose González-Abad & Jorge Baño-Medina
Instituto de Física de Cantabria (IFCA),
CSIC- Universidad de Cantabria
Santander, Spain
<EMAIL_ADDRESS>
###### Abstract

Recently, deep learning has emerged as a promising tool for statistical
downscaling, the set of methods for generating high-resolution climate fields
from coarse low-resolution variables. Nevertheless, their ability to
generalize to climate change conditions remains questionable, mainly due to
the stationarity assumption. We propose deep ensembles as a simple method to
improve the uncertainty quantification of statistical downscaling models. By
better capturing uncertainty, statistical downscaling models allow for
superior planning against extreme weather events, a source of various negative
social and economic impacts. Since no observational future data exists, we
rely on a pseudo reality experiment to assess the suitability of deep
ensembles for quantifying the uncertainty of climate change projections. Deep
ensembles allow for a better risk assessment, highly demanded by sectoral
applications to tackle climate change.

##  1 Introduction

One of the most popular and extended tools for simulating the spatio-temporal
evolution of climate systems are General Circulation Models (GCMs).
Unfortunately, due to the complexity of the physical processes underlying
climate dynamics, they usually incur in a high computational cost, resulting
in coarse resolution climate fields. This hampers their applicability in the
regional-to-local scale, where climate projections are crucial for some
sectorial applications to elaborate adaptation and mitigation plans to climate
change.

Statistical Downscaling (SD) (Maraun & Widmann, 2018) aims to generate high-
resolution fields from coarse climate model outputs by learning a statistical
relationship between a set of predictors (inputs) and predictands (outputs)
representing the coarse and high resolution fields, respectively. Deep
learning (DL) techniques (Goodfellow et al., 2016) have recently emerged as
promising tools for downscaling coarse climate fields, showing excellent
results in reproducing the observed local climate (Pan et al., 2019; Baño-
Medina et al., 2020; Sun & Lan, 2021).

For SD models to reliably compute high-resolution fields in future scenarios,
it is crucial that they generalize to unseen conditions (stationary
assumption), such as those of climate change. Most DL models adapted to the SD
task minimize losses such as the mean squared error, which leads models to fit
the mean of the training data, no providing information about the confidence
in their predictions. Taking this uncertainty into account is extremely
important when applying these models to future scenarios influenced by climate
change, where the stationarity assumption may be violated. Also, being able to
capture extremes is key for planning against extreme weather events (e.g.,
droughts, heat waves, extreme heat) whose frequency is expected to grow due to
climate change. By improving the modeling of the uncertainty of SD models we
can better capture these events when downscaling climate fields under climate
change conditions.

To quantify uncertainty in SD, in (Baño-Medina et al., 2020) authors model the
parameters of a distribution over each gridpoint of the downscaled field,
relying on information about the climate variable to choose the appropriate
distribution (e.g., Gaussian for temperature). Subsequent work (Quesada-Chacón
et al., 2022) follows the same procedure as a way to improve the
distributional properties of the predictions. In (Vandal et al., 2018) authors
rely on Monte Carlo Dropout (Gal & Ghahramani, 2016), an approximation to
Bayesian inference through the use of dropout at test time, to quantify
uncertainty in the downscaling of precipitation, obtaining acceptable results.
Recently, several works explore generative modeling for the statistical
downscaling task (Price & Rasp, 2022; Harris et al., 2022; Accarino et al.,
2021), obtaining benefits from its stochastic properties. Among these works,
only (Baño-Medina et al., 2022) quantifies the uncertainty in climate change
projections.

A simpler way to achieve uncertainty quantification is through deep ensembles.
It consists of training several models in parallel on the same data and then
combining the predictions through an aggregation operation (e.g. mean), which
generally improves performance compared to a single model (Dietterich, 2000).
In (Lakshminarayanan et al., 2017) authors show how this technique also allows
for a better predictive uncertainty estimates, however, its application to SD
has not yet been studied. In this work we explore a natural extension of
(Baño-Medina et al., 2020) through the use of deep ensembles, a simple
technique that does not require tuning hyperparameters or modifying the model
topology. This leads to a better quantification of uncertainty of climate
change projections in future scenarios.

##  2 Experimental Framework

Due to the lack of observations in future periods, we rely on a pseudo reality
experiment to evaluate the effectiveness of deep ensembles for SD. This
experiment is necessary if we need to reveal whether the relationship learned
in a historical period is able to extrapolate to future scenarios (Maraun &
Widmann, 2018). We use the regional climate model (RCM) CanRCM4 (Scinocca et
al., 2016) as pseudo observations (predictand) and its driving GCM (CanESM2)
(Chylek et al., 2011) as predictor. This RCM is part of the Coordinated
Regional Climate Downscaling Experiment (CORDEX111https://cordex.org/)
initiative, which provides multi-model ensembles of RCMs at the continental
level. In addition, the chosen RCM rely on spectral nudging (von Storch et
al., 2000), a technique used to improve the day-to-day correspondence of
dynamical downscaling models (Maraun & Widmann, 2018).

We rely on the DeepESD convolutional model introduced in (Baño-Medina et al.,
2020; 2022) to downscale temperature, which has demonstrated the ability to
accurately reproduce the local climate. It is composed of three convolutional
layers with 50, 25 and 10 kernels (with ReLU activations). The output of the
last convolutional layer is flattened and passed to a dense linear layer to
compute the $\mu$ and $\sigma$ vectors, containing the parameters of the
corresponding Gaussian distribution for each gridpoint of the predictand. As a
result, the neural network defines a mapping $[\mu,\sigma]=f^{w}(x)$
(parameterized by $w$), where $x$ is the input of the model, in our case the
CanESM2 predictors.

We follow previous work (Baño-Medina et al., 2020) and select as predictors 3
large-scale variables (geopotential height, air temperature and specific
humidity) at 4 different vertical levels (500, 750, 800 and 1000 hPa) from
CanESM2 (2.8125∘ resolution). These variables have been shown to be adequate
drivers of local temperature (Maraun et al., 2019; Gutiérrez et al., 2013;
González-Abad et al., 2022). For the target predictand, we lean on the near-
surface air temperature of CanRCM4 (0.22∘ resolution), which justifies
modeling the parameters of a Gaussian distribution. As region to perform the
experiment, we select a western area of United States (25∘ to 55∘ in latitude
and -135∘ to -100∘ in longitude).

The DeepESD model is fitted in a training set spanned by the years 1980-2002
by minimizing the negative log likelihood (NLL) of a Gaussian distribution at
each predictand gridpoint. To avoid overfitting, we follow an early stopping
strategy on a random 10% split of the training set. Following previous work
(Baño-Medina et al., 2021; 2022), the evaluation of these models is performed
in three different future periods (2006-2040, 2041-2070 and 2071-2100).
Predictors are scaled at gridpoint level, taking as reference the mean and
standard deviation computed over the training set.

For the ensemble, we train 10 different models in parallel, following the
procedure described in the previous paragraph. To aggregate the models we
follow (Lakshminarayanan et al., 2017), so the mean and variance of the final
ensemble are given by

$\mu_{*}(x)=\frac{1}{M}\sum_{m}\mu_{\theta_{m}}(x)\quad\quad\sigma^{2}_{*}(x)=\frac{1}{M}\sum_{m}(\sigma^{2}_{m}(x)+\mu^{2}_{m}(x))-\mu^{2}_{*}(x)$
(1)

where $M$ is the number of models composing the ensemble and
$\mu_{\theta_{m}}(x)$ and $\sigma^{2}_{m}(x)$ are the mean and variance of the
$m$ ensemble member.

##  3 Results

Figure 1 shows the RCM target values (in red), as well as the prediction (in
blue) from DeepESD and the DeepESD Ensemble for the year 2100. For the
prediction, the mean and the 95% confidence interval are plotted. The
displayed time series correspond to the gridpoint indicated with a green point
in the upper-left figure, which represents the climatology of the RCM in the
historical period. For each of the models, a time series of the August and
February months is also depicted. As long as the RCM target value falls within
the interval confidence it means that the model is able to account for the
uncertainty of that given value, otherwise it fails to do so. As we can see in
the climatology plot, the selected point falls into an area with warm
temperatures. Overall, DeepESD model fits RCM temperature accurately, however
for the warmer months (June, July and August) we can observe how the mean is
biased toward lower values. If we turn our attention to August, we can observe
how, besides a biased mean, the confidence interval is not able to capture the
RCM values. This is not the case for winter months (e.g. February), where we
see how the mean is not biased, and the confidence interval accurately
captures the RCM values. The DeepESD Ensemble models does not suffer from this
bias in the warmer months, moreover, we can see how in August the confidence
interval is able to capture most of the values. For February, the performance
of DeepESD Ensemble is similar to that of DeepESD.

Figure 1: Mean and 95% confidence intervals of the DeepESD and DeepESD
Ensemble models for the predicted near-surface air temperature (in blue) of
the year 2100. These are compared with the corresponding RCM target values (in
red). In addition, for each model, an enlarged plot of the months of August
and February is shown. The climatology of the historical period (1970-2005)
for the RCM is also shown (upper-left corner). This map shows the gridpoint to
which the time series belongs (green point).

To better understand the advantages of deep ensembles in both mean and
uncertainty quantification, we display in Figure 2 the spatially averaged Root
Mean Square Error (RMSE) and the ratio of the target RCM values falling within
the 95% confidence interval for the three future periods (2006-2040, 2041-2070
and 2071-2100). We focus only on summer days, since it is on these days that
DeepESD and DeepESD Ensemble disagree the most (see Figure 1). RMSE values are
higher as we move forward in time. The improvement caused by adding models to
the ensemble stops around 3-4 models. This improvement is higher as we move
forward in time, from almost no improvement in the 2006-2040 period to a 0.1∘
improvement for 2071-2100. A similar behaviour is observed for the ratios of
target samples within the 95 % confidence interval. In this case, the
improvement also gets larger as we move forward in time. For this metric,
DeepESD Ensemble benefits from a higher number of models in the ensemble
(around 5-6).

Figure 2: Spatial mean of RMSE and ratio of target samples within the
predicted 95% confidence interval for summer days for the three intervals
considered (2006-2040, 2041-2070 and 2071-2100), with respect to the number of
models composing the ensemble.

Figure 3 shows the same ratio over the summer days but without spatially
averaging, that is the value for each gridpoint of the predictand. These are
displayed for the two models (DeepESD and DeepESD Ensemble) in rows and for
the three different intervals (2006-2040, 2041-2070 and 2071-2100), in
columns. We can see how the difference between the two models is remarkable,
especially as we move forward in time, going from an average 4% improvement to
an 11%. Most of the improvements occur at land-based gridpoints. Some areas of
the North Pacific Ocean still do no benefit from the deep ensembles.

Figure 3: Ratio of RCM target values within the predicted 95% confidence
interval for summer days for the three intervals considered (2006-2040,
2041-2070 and 2071-2100) in columns, for the DeepESD and DeepESD Ensemble
models in rows. The number in each panel represents the spatial mean.

##  4 Conclusion

In this work we have extended previous work on SD by focusing on a better
uncertainty quantification in order to capture the uncertainty of climate
change projections. To achieve this, we rely on deep ensembles, a technique
that showed improved performance in the quantification of uncertainty. For the
first time we study the effect of this simple technique in SD, and find that
it provides improvements in the predicted confidence interval, especially in
future scenarios with unseen conditions.

Both models are able to model the uncertainty for the cold months, however for
the warmer months (summer) DeepESD is biased, underestimating and therefore
failing to capture much of these values. However, the DeepESD Ensemble manages
to fit the mean of the RCM, capturing a large part of the positive RCM values
due to its improved confidence interval. This becomes clearer when studying
the RMSE and the ratio on summer days. We see how the ensemble brings
considerable advantages, especially when we move forward in time, where we
expect to find an upward trend in temperature due to climate change and
therefore warmer extremes.

In climate change scenarios, quantifying uncertainty is extremely important
since it helps us to better capture extreme weather events, especially under
climate change conditions, where their intensity and frequency are expected to
worsen. Deep ensembles appear to be a natural and straightforward extension of
current methods for modeling uncertainty, thus helping SD to better tackle
climate change. In this work we validate this technique in a pseudo reality
experiment, in future work we plan to apply it directly to SD and study its
benefits in a real-case scenario. Additionally, we plan to extend this study
to other climate fields, such as precipitation.

### Acknowledgments

J. González-Abad would like to acknowledge the support of the funding from the
Spanish Agencia Estatal de Investigación through the Unidad de Excelencia
María de Maeztu with reference MDM-2017-0765. J. Baño-Medina acknowledges
support from Universidad de Cantabria and Consejería de Universidades,
Igualdad, Cultura y Deporte del Gobierno de Cantabria via the “instrumentación
y ciencia de datos para sondear la naturaleza del universo” project.

## References

  * Accarino et al. (2021) Gabriele Accarino, Marco Chiarelli, Francesco Immorlano, Valeria Aloisi, Andrea Gatto, and Giovanni Aloisio.  Msg-gan-sd: A multi-scale gradients gan for statistical downscaling of 2-meter temperature over the euro-cordex domain.  _AI_ , 2(4):600–620, 2021.
  * Baño-Medina et al. (2020) Jorge Baño-Medina, Rodrigo Manzanas, and José Manuel Gutiérrez.  Configuration and intercomparison of deep learning neural models for statistical downscaling.  _Geoscientific Model Development_ , 13(4):2109–2124, 2020.
  * Baño-Medina et al. (2021) Jorge Baño-Medina, Rodrigo Manzanas, and José Manuel Gutiérrez.  On the suitability of deep convolutional neural networks for continental-wide downscaling of climate change projections.  _Climate Dynamics_ , 57:2941–2951, 2021.
  * Baño-Medina et al. (2022) Jorge Baño-Medina, Rodrigo Manzanas, Ezequiel Cimadevilla, Jesús Fernández, Jose González-Abad, Antonio S Cofiño, and José Manuel Gutiérrez.  Downscaling multi-model climate projection ensembles with deep learning (deepesd): contribution to cordex eur-44.  _Geoscientific Model Development_ , 15(17):6747–6758, 2022.
  * Chylek et al. (2011) P Chylek, J Li, MK Dubey, M Wang, and GJAC Lesins.  Observed and model simulated 20th century arctic temperature variability: Canadian earth system model canesm2.  _Atmospheric Chemistry and Physics Discussions_ , 11(8):22893–22907, 2011.
  * Dietterich (2000) Thomas G Dietterich.  Ensemble methods in machine learning.  In _Multiple Classifier Systems: First International Workshop, MCS 2000 Cagliari, Italy, June 21–23, 2000 Proceedings 1_ , pp. 1–15. Springer, 2000.
  * Gal & Ghahramani (2016) Yarin Gal and Zoubin Ghahramani.  Dropout as a bayesian approximation: Representing model uncertainty in deep learning.  In _international conference on machine learning_ , pp. 1050–1059. PMLR, 2016.
  * González-Abad et al. (2022) Jose González-Abad, Jorge Baño-Medina, and José Manuel Gutiérrez.  Testing interpretability techniques for deep statistical climate downscaling.  In _ICLR 2022 Workshop on AI for Earth and Space Science_ , 2022.
  * Goodfellow et al. (2016) Ian Goodfellow, Yoshua Bengio, and Aaron Courville.  _Deep learning_ , volume 1.  MIT Press, 2016.
  * Gutiérrez et al. (2013) José M Gutiérrez, Daniel San-Martín, Swen Brands, R Manzanas, and S Herrera.  Reassessing statistical downscaling techniques for their robust application under climate change conditions.  _Journal of Climate_ , 26(1):171–188, 2013.
  * Harris et al. (2022) Lucy Harris, Andrew TT McRae, Matthew Chantry, Peter D Dueben, and Tim N Palmer.  A generative deep learning approach to stochastic downscaling of precipitation forecasts.  _Journal of Advances in Modeling Earth Systems_ , 14(10):e2022MS003120, 2022.
  * Lakshminarayanan et al. (2017) Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell.  Simple and scalable predictive uncertainty estimation using deep ensembles.  _Advances in neural information processing systems_ , 30, 2017.
  * Maraun & Widmann (2018) Douglas Maraun and Martin Widmann.  _Statistical Downscaling and Bias Correction for Climate Research_.  Cambridge University Press, January 2018.  ISBN 978-1-108-34064-9.
  * Maraun et al. (2019) Douglas Maraun, Martin Widmann, and José M Gutiérrez.  Statistical downscaling skill under present climate conditions: A synthesis of the value perfect predictor experiment.  _International Journal of Climatology_ , 39(9):3692–3703, 2019.
  * Pan et al. (2019) Baoxiang Pan, Kuolin Hsu, Amir AghaKouchak, and Soroosh Sorooshian.  Improving precipitation estimation using convolutional neural network.  _Water Resources Research_ , 55(3):2301–2321, 2019.
  * Price & Rasp (2022) Ilan Price and Stephan Rasp.  Increasing the accuracy and resolution of precipitation forecasts using deep generative models.  In _International conference on artificial intelligence and statistics_ , pp. 10555–10571. PMLR, 2022.
  * Quesada-Chacón et al. (2022) Dánnell Quesada-Chacón, Klemens Barfus, and Christian Bernhofer.  Repeatable high-resolution statistical downscaling through deep learning.  _Geoscientific Model Development_ , 15(19):7353–7370, 2022.
  * Scinocca et al. (2016) JF Scinocca, VV Kharin, Y Jiao, MW Qian, M Lazare, L Solheim, GM Flato, S Biner, M Desgagne, and B Dugas.  Coordinated global and regional climate modeling.  _Journal of Climate_ , 29(1):17–35, 2016.
  * Sun & Lan (2021) Lei Sun and Yufeng Lan.  Statistical downscaling of daily temperature and precipitation over china using deep learning neural models: Localization and comparison with other methods.  _International Journal of Climatology_ , 41(2):1128–1147, 2021.
  * Vandal et al. (2018) Thomas Vandal, Evan Kodra, Jennifer Dy, Sangram Ganguly, Ramakrishna Nemani, and Auroop R Ganguly.  Quantifying uncertainty in discrete-continuous and skewed data with bayesian deep learning.  In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pp. 2377–2386, 2018.
  * von Storch et al. (2000) Hans von Storch, Heike Langenberg, and Frauke Feser.  A spectral nudging technique for dynamical downscaling purposes.  _Monthly weather review_ , 128(10):3664–3673, 2000.


# GHz-Pulsed Source of Entangled Photons for Reconfigurable Quantum Networks

Meritxell Cabrejo Ponce<EMAIL_ADDRESS>Christopher
Spiess  Friedrich Schiller University, Max-Wien-Platz 1, 07743 Jena, Germany
Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-
Einstein-str. 7, 07745 Jena, Germany  André Luiz Marques Muniz and Philippe
Ancsin  Fraunhofer Institute for Applied Optics and Precision Engineering,
Albert-Einstein-str. 7, 07745 Jena, Germany  Fabian Steinlechner
<EMAIL_ADDRESS>Fraunhofer Institute for Applied Optics
and Precision Engineering, Albert-Einstein-str. 7, 07745 Jena, Germany  Abbe
Center of Photonics, Friedrich Schiller University Jena, Albert-Einstein-Str.
6, 07745 Jena, Germany

###### Abstract

Entanglement is a universal resource in quantum networks, yet entangled photon
sources are typically custom-made for a specific use case. Versatility, both
in terms of state modulation and tunability of the temporal properties of the
photons, is the key to flexible network architectures and cryptographic
primitives that go beyond quantum key distribution. Here, we report on a
flexible source design that produces high-quality entanglement in continuous-
wave and GHz-rate-pulsed operation modes. Utilizing off-the-shelf optical
components, our approach uses a fiber-based Sagnac loop to generate
polarization-entangled photons at telecom wavelength with high efficiency and
fidelities above 0.99. Phase modulation up to GHz before entangled state
generation is also possible for fast entangled state switching. We show phase
modulation at 100 MHz with an average fidelity of 0.95. Furthermore, the
source 60 nm spectral bandwidth is entirely compatible with fully
reconfigurable wavelength-multiplexed quantum networks.

##  I Introduction

Entangled photons are an essential resource in emerging quantum information
processing and secure quantum communications [1, 2]. In the context of quantum
cryptography, entanglement-based quantum key distribution (QKD) has recently
received considerable attention due to its inherent simplicity. It leverages
the intrinsic randomness of quantum entanglement and can be implemented with
passive optical components. Moreover, it allows for scalable multiuser
operation through the use of auxiliary frequency correlations for multiplexing
[3, 4]. However, richer cryptographic primitives and beyond require more
flexibility from the source. Desiderata for protocols like quantum
teleportation and entanglement swapping are tunability from continuous-wave to
pulsed operation. On the other hand, fast state control is also required for
reconfigurable quantum networks [5] and particularly for protocols such as
three-partite quantum secret sharing (QSS), that need phase modulated
entangled states [6].

Typical sources are those of polarization entangled qubits, often consisting
of free-space interferometers based on bulk optics [7, 8, 9]. They provide
stable operation and high photon-pair rates by employing spontaneous
parametric down-conversion (SPDC) processes in $\chi^{(2)}$-nonlinear crystals
or waveguides [10, 11, 12, 13]. However, the dimensions of the bulky
components make it difficult for the source to be reconfigurable and
especially fast. They are optimized to be solely used for continuous wave or
in pulsed regimes and are not easily interchangeable. Furthermore, pump and
down-converted photons frequently need to propagate along the same paths,
involving the use of expensive dual-wavelength components to achieve high
visibility [9, 14, 15, 16].

A lot of effort has been directed toward the realisation of polarization
entangled photon sources (EPS) with both pump and correlated photons in
telecom wavelengths. In this manner, inexpensive high-speed optical modulators
and communication through the already existing global fiber network could be
exploited. The natural choices for those systems are $\chi^{(3)}$ materials
via spontaneous four wave mixing, since they enable both pump and photon-pairs
in the same wavelength bands [17, 18]. Nevertheless, their bulk nonlinearity
is $10^{4}-10^{5}$ orders of magnitude smaller than cascaded $\chi^{(2)}$
processes (i.e. second harmonic generation and SPDC in the same $\chi^{(2)}$
material) [19, 20, 21].

Figure 1: Schematic of our polarization entangled photon-pair source: a
telecom pump (continous-wave or pulsed) is sent to the fiber-based Sagnac
interferometer (blue box). It consists of a fiber loop with bidirectional up-
and down-conversion stages. After second harmonic generation (SHG), phase
modulation $\vartheta(t)=\theta_{\text{early}}(t)-\theta_{\text{late}}(t)$ can
be implemented (purple box) with arbitrary electrical signals and same period
$\Delta t$ as the optical pulses. In that case, a delay $\delta t$ is employed
to temporally separate and independently address the SHG pulses. The emerging
state is rerouted with a circulator (CIRC) to our detection system. Single-
mode (SM) and polarization-maintaining (PM) fibers are shown in yellow and
blue color, respectively. EOM: electro-optic modulator, PBS: polarizing beam
splitter, PC: polarization controller, ppLN: periodically poled Lithium
Niobate.

In this letter, inspired by the work in [22, 23, 24, 25, 26] for faint-pulse
sources (i.e., a classical laser attenuated to the single photon level), we
propose a novel EPS at telecom wavelengths based on a fiber-based Sagnac
interferometer with two $\chi^{(2)}$ nonlinear waveguides. In this
configuration, a telecom pump enters the fiber loop and follows a cascaded up-
and down-conversion process in both directions. The immediate advantage is
that the source does not need dual-wavelength components like the free-space
versions. Additionally, the design allows the incorporation of fiber-
integrated optical modulators to i) easily switch the pump from continuous-
wave (CW) to GHz-pulsed regimes, and ii) to modulate the phase of the
generated state up to GHz rate. To demonstrate the versatility of this source,
we characterize it in CW and GHz-pulsed regimes in terms of photon pair
generation and visibility as a function of repetition rate and duty cycle for
coarse and dense wavelength division multiplexing (WDM) arrangements. We
report Bell-state fidelities above 0.99 for different pump configurations and
high brightness, with millions of photons at emission per nm of bandwidth. The
spectrum spans over 60 nm, offering the possibility to increase the number of
users via wavelength demultiplexing or to actively allocate the necessary
bandwidth in a practical scenario [27].

Finally, to showcase the ultimate flexibility of our design, we continuously
switch the generated entangled state from a basis of four different states. In
this proof-of-concept experiment, we achieve average state fidelities of 0.95
at 100 MHz rate. These results are a step forward towards a fully
reconfigurable entangled photon source and may pave the way for active quantum
networks.

##  II Entangled photon source

Our fiber-based Sagnac loop is pictured in Fig. 1. After pump preparation (see
Methods), the pump polarization is set with polarization controllers (PC)
right before a polarization beam splitter (PBS). By launching 45∘-linearly
polarized optical pulses,
$|\Psi_{p}\rangle=\frac{1}{\sqrt{2}}(|H\rangle_{p}+|V\rangle_{p})$, the fiber
PBS accordingly separates H-and V-components and sends them over polarization-
maintaining (PM) fibers, whereby each output is aligned to the slow axis of
the fiber. Thus, both H and V polarizations travel along the same axis such
that the group velocity mismatch is compensated. The outputs of the PBS are
connected to two fiber-pigtailed periodically poled lithium niobate (ppLN)
waveguides (Covesion MSHG1550 with 40mm long type-0 crystal). These are
commercially available second harmonic generation (SHG) modules, where a 1560
nm PM fiber is directly coupled to a ppLN waveguide, while a 780 nm PM fiber
is coupled at the other end. They have slightly different SHG conversion
efficiencies due to fabrication imperfections, $10\%$ and $11\%$ respectively
for 300 mW of pump power. Optimum SHG conversion is achieved by tuning the
temperature of the ppLN waveguides #1 and #2 (53°C and 48°C respectively),
which simultaneously produces broadband pair generation around a degeneracy
wavelength of 1560 nm. The SPDC spectrum spans over the C-band and beyond
(from 1530 to 1590nm) with the full width at half maximum (FWHM) of 58 nm and
68 nm for waveguide #1 and #2 respectively, as shown in Fig. 2. Finally, both
ppLN modules are interconnected with a 780 nm PM fiber to close the loop. As a
result, the second harmonic generated light of one waveguide is sent to the
opposite one to generate SPDC and vice versa. This situation occurs in both
clockwise and counterclockwise directions, giving rise to a polarization-
entangled state. Additionally, any perturbance on the fiber loop affects
simultaneously both propagation directions, maximizing phase stability of the
entangled state. For active state switching, as explained at the end of this
letter, we can introduce here a delay line to temporally separate the SHG
pulses, and an electro-optic modulator (EOM) to modulate their relative phase
$\vartheta(t)$ (purple box in Fig. 1).

Figure 2: SPDC spectra of ppLN waveguides (wg) #1 and #2. The coloured areas
indicate the channel bandwidth of the signal and idler for our coarse (8 nm)
and dense demultiplexing (1 nm) settings. Yet, our source is also compatible
with the international telecom union (ITU) standard coarse wavelength division
multiplexing channels (CWDM, 20 nm grid).

In this configuration, SPDC photon pairs from both ppLN modules are combined
at the output of the PBS and emerge in the entangled state
$|\phi\rangle=\alpha|H_{s}H_{i}\rangle+e^{i\vartheta(t)}\beta|V_{s}V_{i}\rangle$,
where $H_{s(i)}$ is the signal (idler) photon in the $H$ polarization, and
similarly for $V$. In the remaining, we suppress the subscripts of the state.
Here, $\alpha$ and $\beta$ are the probabilities of generating SPDC photon-
pairs clockwise and counterclockwise, respectively, while $\vartheta(t)$
denotes the detuning phase between both directions. For a 45° linearly
polarized pump and without SHG modulation, $\alpha=\beta=1/\sqrt{2}$ and
$\vartheta=0$, resulting in the maximally entangled Bell state
$|\phi^{+}\rangle$. The outgoing light from the PBS is rerouted via the
circulator (CIRC) followed by a 100-dB notch filter, with 1.6 nm bandwidth at
1560 nm, that removes any remaining fraction of the pump or additional noise.
Finally, the signal and idler correlated photons are separated by a
demultiplexer (DEMUX) to be sent to independent users. Here, we directly send
them to our detection system (see Methods).

##  III Performance analysis

We first characterize our source excluding the active state control elements.
In our narrowband WDM scheme, the signal and idler channels selected by the
programmable filter are separated by 3.2 nm with a bandwidth of
$\Delta\lambda=$ 1 nm, as shown in Fig. 2. We report in Fig. 3a the photon-
pair coincidence counts per nm of spectral bandwidth, a parameter known as the
spectral brightness, as a function of the average pump power ($P_{\text{in}}$)
right before the PBS, for CW and pulsed operation at repetition rates of 1 GHz
and duty cycles of 9%, 25% and 49%. These measurements are performed over
either H- or V-polarized signal and idler SPDC photons, which represent the
performance of a single direction of the loop. The coincidence counts grow
with a power of 2 and are proportional to $P_{\text{in}}^{2}$ (fitted in Fig.
3a), according to the SHG power conversion. Furthermore, shorter duty cycles
yield stronger SHG at the same average but higher peak pump power, and
therefore, higher SPDC brightness. The photon-pair heralding, defined as the
ratio of coincidences over single-photon counts, improves with pump power and
shorter duty cycle because of a relative reduction of additional noise sources
(see Supplementary Material). It amounts to $H_{s}=1.6-2.9\%$ and
$H_{i}=2.9-5.2\%$ for signal and idler, respectively, differences that can be
explained by the accumulated and asymmetric losses in the system as well as
the coupling efficiency of the ppLN waveguide modules. With a total loss of
$\sim$ 12.85 dB (see Methods), the actually emitted heralding is almost
20-fold higher.

To verify the quality of the entanglement, polarization projecting
measurements are performed in two mutually unbiased bases, H/V and D/A, for
different pump parameters. In the H/V basis and for the state
$|\phi^{+}\rangle$, the visibility per nm of spectral bandwidth is computed
as:

$V_{HV}=\frac{R_{HH}-R_{VH}-R_{HV}+R_{VV}}{R_{HH}+R_{VH}+R_{HV}+R_{VV}}$ (1)

where $R_{ij}$ denotes the coincidence rate measured with i, j settings, and
similarly for D/A. We find that for a 1 GHz repetition rate and $9\%$ duty
cycle, we achieve visibilities of $99.5\%$ in H/V and $99.0\%$ in D/A, with no
subtraction of accidentals. Using the fidelity witness
$F\geq(V_{HV}+V_{DA})/2$ [28], we lower-bound the Bell-state fidelity to
$F=99.25\%$.

Figure 3: Analysis of the source performance under CW and pulsed regimes at 1 GHz and various duty cycles: (a) Coincidence counts per nm of spectral bandwidth as a function of average pump power at 1560 nm, where the lines correspond to a power of 2 fit. The photon pair rate increases quadratically with pump power due to the SHG process. Moreover, shorter duty cycles also enhance the SHG conversion per pulse, resulting in greater coincidence events. (b) CAR results for the same system settings, where the lines correspond to our fitted model including noise. Notice that lower duty cycles shift the peak towards lower powers, but also increase CAR values thanks to improved SHG and lower noise counts.  Table 1: Detected and estimated photon pair rates ($R^{\text{exp}}$ and $R^{\text{est}}$) and heralding values for signal and idler photons ($H_{s}^{\text{exp}}$, $H_{i}^{\text{exp}}$ and $H_{s}^{\text{est}}$, $H_{i}^{\text{est}}$) at the output of the entangled photon source (i.e. after the PBS) for one polarization projection $|\Psi_{\text{proj}}\rangle=|HH\rangle$. Notice that the heralding is limited by the coupling of the fibers to the ppLN waveguides. $P_{\text{in}}$: input pump power right before the PBS, $\Delta\lambda$: signal and idler channel bandwidth.  Pump rep. rate |  Duty cycle (%) |  $P_{\text{in}}$ (dBm) |  $\Delta\lambda$ (nm) |  $R^{\text{exp}}$ (cps) |  $H_{s}^{\text{exp}}$ (%) |  $H_{i}^{\text{exp}}$ (%) |  $R^{\text{est}}$ (Mcps) |  $H_{s}^{\text{est}}$ (%) |  $H_{i}^{\text{est}}$ (%)
---|---|---|---|---|---|---|---|---|---
CW |  100 | 7.9 | 1 | 535 | 1.6 | 2.9 | 0.20 | 39.8 | 44.2
1 GHz |  9 | 7.9 | 1 | 3672 | 2.9 | 5.2 | 1.36 | 71.6 | 79.0
49 | 7.9 | 1 | 963 | 2.1 | 3.8 | 0.36 | 51.4 | 57.6
4 GHz |  44 | 7.9 | 1 | 901 | 2.1 | 3.8 | 0.33 | 51.7 | 57.3
8 GHz |  50 | 7.9 | 1 | 565 | 1.9 | 3.3 | 0.21 | 45.5 | 50.0
50 | 9.0 | 8 | 8027 | 1.7 | 3.6 | 2.98 | 41.6 | 54.0

Next, we assess the contribution of additional noise sources to the
experimentally observed heralding efficiency. We evaluate the coincidence-to-
accidental ratio (CAR) for CW and pulses at 1 GHz repetition rates with the
same duty cycles as before and display the results in Fig. 3b. The CAR
parameter is defined by the ratio between true photon-pair coincidences
generated within the same SPDC process and accidental coincidences with
subsequent pairs and noise photons. The accidental detections are proportional
to the single-photon counts of each channel as well as the total coincidence
window. Accordingly, we can model the CAR as a function of input power as:

$CAR=\frac{\mu\eta_{s}\eta_{i}}{((\mu+N_{noise})\eta_{s}+d_{s})((\mu+N_{noise})\eta_{i}+d_{i})}-1$
(2)

where $\mu$ denotes the average photon-pair number per coincidence window,
$\eta_{s(i)}$ stand for the detection efficiencies and $d_{s(i)}$ are the dark
count probabilities for the signal (idler) channels. $N_{noise}$ is an
additional noise term accounting for excess single-photon contributions
attributed to spontaneous Raman scattering (SpRS), further discussed in the
Supplementary Material. Detection efficiency, dark counts ($\sim$300 Hz), and
noise photon numbers are expected to be similar for both channels. The
relation of SPDC and SpRS noise with pump power $P_{\text{in}}$ follows as
$\mu=\gamma_{\text{SPDC}}\gamma_{\text{SHG}}P_{\text{in}}^{2}$ and
$N_{\text{noise}}=\gamma_{\text{noise}}P_{\text{in}}$ [29], with $\gamma$
including the respective generation efficiencies. Given the large number of
parameters, we fit the measurements in our model and confirm the linear growth
of $N_{\text{noise}}$. It is notable that pulsed pump at constant average
power presents better CAR than CW regime due to improved SHG generation and
thus higher photon pairs per pulse, while the accidentals from
$N_{\text{noise}}$ are slightly lower (see Supplementary). Furthermore,
stronger conversion efficiencies shift the CAR peak towards lower pump power.

We also quantify the quality of entanglement for a coarse WDM scheme. The
broadest filter we can set for our source with our programmable filter is 8 nm
bandwidth, although broader channels should also be possible with different
DEMUX. We use this setting for both the signal and idler, separated by a notch
of 1.6 nm (as shown in the light gray area in Fig. 2). The pump power is set
to 9.0 dBm just before the PBS at an 8-GHz pulsed regime and 50% duty cycle,
the fastest pump rate we can set with our current electrical devices. We
observe visibilities of $V_{HV}=99.4\%$ and $V_{DA}=97.7\%$, with no
subtraction of accidentals. The lower bounded fidelity is thus $F\geq 98.6\%$.

The detected pair rate at 12.85 dB loss is $R_{HH}=$8027 and $R_{VV}=$7812,
with a heralding of 1.7% for the signal and 3.4% for the idler photon with
respect to the averaged $|HH\rangle$ and $|VV\rangle$ projection settings. The
actual emission rates can be estimated by compensating for the loss such that
$R^{\text{est}}=R^{\text{exp}}/\eta_{s}\eta_{i}$, with $R^{\text{exp}}$ the
experimental measurement, $\eta_{s}$=6.6%, $\eta_{i}$=4.1% (see Methods), and
amounts to 5.9 Mcps at this power and channel bandwidth. By replacing several
parts of our detection system with commercially available low-loss components,
the losses could be reduced at least by 10 dB. In that case, the usable photon
pairs would be in the order of 1.6 Mcps with a heralding of 20-30%, although
such high photon rates would already saturate all current single photon
detectors. Typical InGaAs avalanche photodiodes have deadtimes of the order of
10-20 $\mu$s, while SNSPDs deadtimes are around 1 $\mu$s. Therefore, one would
need to work with lower power values. The emitted brightness with respect to
the 1560 nm pump is 93.3 kcps/mW². However, to compare our system with other
$\chi^{(2)}$ sources, the estimated power of the second harmonic at 780 nm can
be used for normalization, which is 3 $\mu$W per SHG module at this pump
power. In this case, the emitted total pair rate corresponds to 1 Gcps/mW. To
conclude, we summarize the results of this section in Table 1.

This characterization shows that brightness and heralding in our source
improve with shorter duty cycle and that our design can be used for any
repetition rate (as long as fiber dispersion does not affect). In our
experiment, the highest repetition rate (8 GHz) and shortest pulse width
($\sim$90ps) are driven at the limit of our electrical pulse generator. Thus,
deteriorated electrical signals eventually deform the optical pulse, leading
to lower spectral brightness. Yet, our design is not only restricted to pulsed
operation, since good figures of merit are also obtained for CW pump. In that
case, cheap laser diodes could be employed instead of our involved pulse
preparation scheme (see Methods). Additionally, since our source spans over 60
nm of bandwidth, multiple demultiplexed channels can be used with large photon
numbers and potentially high interference visibility. We note that a better
spectral overlap between both ppLN modules would be needed to achieve high
visibility.

Figure 4: Reconstructed density matrices of the four selected states with
overall fidelities of 95%. On the first row, the real part of the density
matrix is displayed, while the second raw shows the imaginary part. Notice how
the diagonal terms are always the same, while the off-diagonal terms indicate
the phase of the state.

##  IV Active state control

Finally, we investigate the fast switching capabilities of the polarization-
entangled state phase as a basic prerequisite for active quantum networks. For
this purpose, a phase modulator and a PM fiber delay line are inserted between
the ppLN waveguides, as shown in Fig. 1. In this manner, the SHG pulses that
travel along the clockwise direction traverse the EOM first, which introduces
a phase $\theta_{\text{early}}$, followed by the delay line. The pulses
traveling in the counter-clockwise direction encounter an additional delay
before they acquire the phase $\theta_{\text{late}}$ from the EOM. Since the
length of the interferometer is the same in both directions, the SPDC
probability amplitudes of both paths are perfectly recombined at the output
even after SHG modulation.

The final state depends on the relative phase between the SHG pulses.
Therefore, it is enough to modify the phase of only one of them (lower right
inset in Fig.1). When no voltage is supplied to the EOM, and if a diagonal
pump polarization is fed into the loop as described previously, the state
remains the same:

$|\phi^{+}\rangle=\frac{1}{\sqrt{2}}\left(|HH\rangle+|VV\rangle\right)$ (3)

However, if $V_{\pi}$ is applied to one of the pulses, i.e.
$\vartheta=\theta_{\text{early}}-\theta_{\text{late}}=\pi$, then the entangled
state becomes:

$|\phi^{-}\rangle=\frac{1}{\sqrt{2}}\left(|HH\rangle-|VV\rangle\right)$ (4)

which is another maximally entangled Bell state. Nonetheless, we can also
generate other states that are necessary for cryptographic primitives such as
quantum secret sharing [5]. We choose the “$i+/i-$” states:

$|\varphi^{+}\rangle=\frac{1}{\sqrt{2}}\left(|HH\rangle+i|VV\rangle\right)$
(5)

$|\varphi^{-}\rangle=\frac{1}{\sqrt{2}}\left(|HH\rangle-i|VV\rangle\right)$
(6)

that just require $\pm V_{\pi/2}$ voltage for the EOM. If the phase is
introduced to the first pulse, the sign of the voltage transfers to the phase
in the superposition. If introduced to the second pulse, then the sign of the
phase is inverted. One can also restrict only to positive voltages by applying
$+V_{\pi/2}$ either to the first or the second SHG pulse to alternate between
these two states.

We perform the phase modulation experiment by using the second channel of our
electrical signal generator, which allows us to fine-tune the time and
amplitude of the RF electrical pulses that feed the EOM. These pulses are
generated at a 100 MHz rate and have a duty cycle of $5\%$. The pulse
amplitude is chosen from the voltage set
$\\{0,+V_{\pi},-V_{\pi/2},+V_{\pi/2}\\}$, targeting the second and delayed SHG
pulse (see Methods). The optical pulses are generated at the same repetition
rate and duty cycle of $1\%$, with an average power of 5 dBm before the PBS.
Note that the losses of the EOM account for 3.5 dB, but only affect the SHG
stage and do not deteriorate the final entangled state. The filter settings
are again 8 nm bandwidth for the signal and idler channels, as in Fig. 2, and
their polarization state is analyzed in our detection system (see Methods).
The measured coincidence rate reaches 4.5 kcps per polarization setting, again
after $\sim$12.85 dB loss in our setup. We measure the full-state tomography
to reconstruct the density matrix and apply maximum likelihood estimation to
obtain a physical one [30, 31]. The final results are presented in Fig. 4.

With little optimization effort, we achieve fidelities of
$95.0\%,95.0\%,95,8\%$ and $94.3\%$ for the states $|\phi^{+}\rangle$,
$|\phi^{-}\rangle$, $|\varphi^{+}\rangle$ and $|\varphi^{-}\rangle$
respectively. These measurements are performed in one run by integrating over
100 ms per polarization setting and averaging over 10 samples, while the same
periodic RF electrical pulse is employed continuously to modulate the state
phase. To change the state, we only change the pulse sequence at the
electrical pulse generator. Nevertheless, our signal generator and common RF
sources can also be fed with a very long sequence of arbitrary pulses among
the set of four. In that case, a state switching at a 100 MHz repetition rate
would occur. Even higher switching rates up to GHz are possible, but are
eventually limited to the propagation time of the pulses in the phase
modulator.

##  V Conclusion

We have developed and experimentally evaluated a fiber-based Sagnac
interferometer that utilizes standard telecom components to generate high-
quality polarization-entangled states for a variety of pulse rates and duty
cycles. We show that our source can be easily employed in CW up to GHz-pulsed
regimes, with the latter being more suitable due to the stronger SHG and SpRS
noise reduction and hence higher brightness and CAR. We obtain fidelities of
0.99 and generate millions of photon pairs per nm of spectral bandwidth, while
the source spectrum extends 60 nm. Although our detection rates are
considerably affected by the loss in the optical components used to
characterize the source, most notably the DEMUX, these numbers would be
substantially improved using commercially available low-loss components. We
also prove great entanglement quality after active state switching at 100 MHz
rates with fidelities up to 0.95. While not experimentally demonstrated here,
we note that the source scheme is also ideally suited to the generation of
hyperentanglement in the time-energy and polarization degrees of freedom, as
well as other hybrid quantum states. To prove this, only the detection scheme
should be adapted.

Combined with the simplicity of the design and its compact footprint, these
results make the source an ideal device for future field experiments on
quantum entanglement tests and quantum communication networks. Particularly,
the use of standard telecom components is well suited for integration in
compact CubeSat platforms for satellite-based quantum networks. The versatile
spectrally broadband source could be used for multi-user operation with
variable temporal and spectral bandwidth allocation or high-capacity backbone
channels with frequency division multiplexing, making it an ideal candidate
for integration into flexible quantum network architectures of the future.

## Methods

Pump preparation A continuous-wave telecom laser operating at a center
wavelength of 1560 nm is used as an initial pump source. If optical pulses are
required for a certain application, they are carved from the laser by an
intensity modulator (IM), controlled by an arbitrary waveform generator (AWG
Tektronik 70002B), and amplified by an erbium-doped fiber amplifier (EDFA).
The IM bias control and the total optical power are monitored by tapping a
fraction of the optical power via fiber beam splitters (FBS). A tunable
optical bandpass filter (BPF) placed thereafter spectrally purifies the pulses
by removing the out-of-band amplified spontaneous emission (ASE) originating
from the amplifier. It is worth noting that the BPF (0.2 nm bandwidth) placed
after the EDFA is narrower than the notch filter after the Sagnac loop to
minimize any pump noise coupled to the detection system.

Detection system A polarization-independent programmable filter (Finisar
WaveShaper 16000A) is used to separate the signal and idler photons for the
convenience of wavelength tunability. To analyze the polarization entanglement
of the photon pairs, we use a free-space polarization analysis module (PAM),
consisting of a half-wave plate (HWP) and a free-space PBS. This setting
allows us to have a constant polarization at the output of the PAM, which is
more suitable for efficient detection with our superconducting nanowire
single-photon detectors (SNSPD). The SNSPDs (SingleQuantum) are optimized for
C-band operation and have specified detection efficiencies of $\eta\approx
74\%$. Lastly, a time-to-digital converter (TDC; Qutools QuTAG with 1 ps
resolution and 4.2 ps RMS time jitter) analyzes the photon-pair coincidences.

Loss characterization From the photon pair sources to the detection system, an
average total loss of 12.85 dB are estimated, which breaks down as follows: 1
dB for the CIRC, 1.3dB for the notch filter, 6 dB for the programmable optical
filter, 3.25 dB insertion loss for the PAM and 1.3 dB detection loss. However,
small asymmetries on each path lead to an overall detection efficiency for the
signal channel $\eta_{s}=$-11.8 dB, whereas for the idler channel it is
$\eta_{i}=$-13.9 dB. These significant losses can be reduced by at least 10 dB
with low-loss components, particularly the DEMUX and PAM parts, but are
tolerable to demonstrate the source functionality. We note that DEMUX with
less than 1 dB loss are already commercially available.

Active state control The phase modulator has $V_{\pi}$ = 4 V at 50 kHz - 2 GHz
(iXblue, specs). The peak voltages used for our phase modulator are
$\\{0,+V_{\pi},-V_{\pi/2},+V_{\pi/2}\\}$ = $\\{0,250,-112.5,112.5\\}$ mV. The
pulses are then amplified with an RF amplifier with $\sim$28 dB of fixed gain.
To analyse the polarization of the entangled state, we include now in our
previous PAM a quarter wave plate before the HWP to access the phase
information.

## Funding

This work was financially supported by the Federal Ministry of Education and
Research of Germany (BMBF) through the QuNET initiative.

###### Acknowledgements.

The presented results were partially acquired using facilities and devices
funded by the Free State of Thuringia within the application center Quantum
Engineering. M.C.P and C.S. are part of the Max Planck School of Photonics
supported by BMBF, Max Planck Society, and Fraunhofer Society.

## References

  * Steane [1998] A. Steane, Quantum computing, Reports on Progress in Physics 61, 117 (1998).
  * Slussarenko and Pryde [2019] S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Applied Physics Reviews 6, 041303 (2019).
  * Wengerowsky  _et al._ [2018] S. Wengerowsky, S. K. Joshi, F. Steinlechner, H. Hübel, and R. Ursin, An entanglement-based wavelength-multiplexed quantum communication network, Nature 564, 225 (2018).
  * Joshi  _et al._ [2020] S. K. Joshi, D. Aktas, S. Wengerowsky, M. Lončarić, S. P. Neumann, B. Liu, T. Scheidl, G. C. Lorenzo, Ž. Samec, L. Kling, A. Qiu, M. Razavi, M. Stipčević, J. G. Rarity, and R. Ursin, A trusted node–free eight-user metropolitan quantum communication network, Science Advances 6, eaba0959 (2020).
  * Williams  _et al._ [2019] B. P. Williams, J. M. Lukens, N. A. Peters, B. Qi, and W. P. Grice, Quantum secret sharing with polarization-entangled photon pairs, Physical Review A 99, 062311 (2019).
  * Grice  _et al._ [2019] W. P. Grice, M. Beck, D. Earl, D. Mulkey, and J. Schaake, Reconfigurable entangled photon source (Conference Presentation), in  _Quantum Technologies and Quantum Information Science V_, edited by M. T. Gruneisen, M. Dusek, J. G. Rarity, and P. M. Alsing (SPIE, Strasbourg, France, 2019) p. 15.
  * Anwar  _et al._ [2021a] A. Anwar, C. Perumangatt, F. Steinlechner, T. Jennewein, and A. Ling, Entangled photon-pair sources based on three-wave mixing in bulk crystals, Review of Scientific Instruments 92, 041101 (2021a).
  * Brassard  _et al._ [2000] G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, Limitations on Practical Quantum Cryptography, Physical Review Letters 85, 1330 (2000).
  * Lee  _et al._ [2021] Y. S. Lee, M. Xie, R. Tannous, and T. Jennewein, Sagnac-type entangled photon source using only conventional polarization optics, Quantum Science and Technology 6, 025004 (2021).
  * Levine  _et al._ [2011] Z. H. Levine, J. Fan, J. Chen, and A. L. Migdall, Polarization-entangled photon pairs from a periodically poled crystalline waveguide, Optics Express 19, 6724 (2011).
  * Fiorentino  _et al._ [2005] M. Fiorentino, C. E. Kuklewicz, and F. N. C. Wong, Source of polarization entanglement in a single periodically poled KTiOPO4 crystal with overlapping emission cones, Optics Express 13, 127 (2005).
  * Martin  _et al._ [2010] A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength, New Journal of Physics 12, 103005 (2010).
  * Caspani  _et al._ [2017] L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, Integrated sources of photon quantum states based on nonlinear optics, Light: Science & Applications 6, e17100 (2017).
  * Kim  _et al._ [2019] H. Kim, O. Kwon, and H. S. Moon, Pulsed Sagnac source of polarization-entangled photon pairs in telecommunication band, Scientific Reports 9, 5031 (2019).
  * Li  _et al._ [2015] Y. Li, Z.-Y. Zhou, D.-S. Ding, and B.-S. Shi, CW-pumped telecom band polarization entangled photon pair generation in a Sagnac interferometer, Optics Express 23, 28792 (2015).
  * Vergyris  _et al._ [2017] P. Vergyris, F. Kaiser, E. Gouzien, G. Sauder, T. Lunghi, and S. Tanzilli, Fully guided-wave photon pair source for quantum applications, Quantum Science and Technology 2, 024007 (2017).
  * Suo  _et al._ [2015] J. Suo, S. Dong, W. Zhang, Y. Huang, and J. Peng, Generation of hyper-entanglement on polarization and energy-time based on a silicon micro-ring cavity, Optics Express 23, 3985 (2015).
  * Zhang  _et al._ [2019] M. Zhang, L.-T. Feng, Z.-Y. Zhou, Y. Chen, H. Wu, M. Li, S.-M. Gao, G.-P. Guo, G.-C. Guo, D.-X. Dai, and X.-F. Ren, Generation of multiphoton quantum states on silicon, Light: Science & Applications 8, 41 (2019).
  * Chou  _et al._ [1999] M. Chou, I. Brener, M. Fejer, E. Chaban, and S. Christman, 1.5-$\mu$m-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides, IEEE Photonics Technology Letters 11, 653 (1999).
  * Jiang and Tomita [2007] Y.-K. Jiang and A. Tomita, The generation of polarization-entangled photon pairs using periodically poled lithium niobate waveguides in a fibre loop, Journal of Physics B: Atomic, Molecular and Optical Physics 40, 437 (2007).
  * Arahira  _et al._ [2011] S. Arahira, N. Namekata, T. Kishimoto, H. Yaegashi, and S. Inoue, Generation of polarization entangled photon pairs at telecommunication wavelength using cascaded $\chi$^(2) processes in a periodically poled LiNbO_3 ridge waveguide, Optics Express 19, 16032 (2011).
  * Agnesi  _et al._ [2019] C. Agnesi, M. Avesani, A. Stanco, P. Villoresi, and G. Vallone, All-fiber self-compensating polarization encoder for quantum key distribution, Optics Letters 44, 2398 (2019).
  * Li  _et al._ [2019a] Y.-P. Li, W. Chen, F.-X. Wang, Z.-Q. Yin, L. Zhang, H. Liu, S. Wang, D.-Y. He, Z. Zhou, G.-C. Guo, and Z.-F. Han, Experimental realization of a reference-frame-independent decoy BB84 quantum key distribution based on Sagnac interferometer, Optics Letters 44, 4523 (2019a).
  * Li  _et al._ [2019b] Y. Li, Y.-H. Li, H.-B. Xie, Z.-P. Li, X. Jiang, W.-Q. Cai, J.-G. Ren, J. Yin, S.-K. Liao, and C.-Z. Peng, High-speed robust polarization modulation for quantum key distribution, Optics Letters 44, 5262 (2019b).
  * Alibart  _et al._ [2016] O. Alibart, V. D’Auria, M. D. Micheli, F. Doutre, F. Kaiser, L. Labonté, T. Lunghi, É. Picholle, and S. Tanzilli, Quantum photonics at telecom wavelengths based on lithium niobate waveguides, Journal of Optics 18, 104001 (2016).
  * Tanzilli  _et al._ [2012] S. Tanzilli, A. Martin, F. Kaiser, M. De Micheli, O. Alibart, and D. Ostrowsky, On the genesis and evolution of Integrated Quantum Optics: Integrated quantum optics, Laser & Photonics Reviews 6, 115 (2012).
  * Appas  _et al._ [2021] F. Appas, F. Baboux, M. I. Amanti, A. Lemaítre, F. Boitier, E. Diamanti, and S. Ducci, Flexible entanglement-distribution network with an AlGaAs chip for secure communications, npj Quantum Information 7, 118 (2021).
  * Anwar  _et al._ [2021b] A. Anwar, C. Perumangatt, F. Steinlechner, T. Jennewein, and A. Ling, Entangled photon-pair sources based on three-wave mixing in bulk crystals, Review of Scientific Instruments 92, 041101 (2021b).
  * Eraerds  _et al._ [2010] P. Eraerds, N. Walenta, M. Legré, N. Gisin, and H. Zbinden, Quantum key distribution and 1 Gbps data encryption over a single fibre, New Journal of Physics 12, 063027 (2010).
  * James  _et al._ [2001] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Physical Review A 64, 052312 (2001).
  * Kwiat Quantum Information Group [2021] Kwiat Quantum Information Group, Quantum-Tomography (2021), https://github.com/KwiatQIM/Quantum-Tomography.
  * Lin  _et al._ [2007] Q. Lin, F. Yaman, and G. P. Agrawal, Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization, Physical Review A 75, 023803 (2007).
  * Agrawal [2013] G. P. Agrawal,  _Nonlinear fiber optics_ , fifth edition ed. (Elsevier/Academic Press, Amsterdam, 2013).
  * Hu  _et al._ [2020] Y.-Y. Hu, X. Lin, S. Wang, J.-Q. Geng, Z.-Q. Yin, W. Chen, D.-Y. He, W. Huang, B.-J. Xu, G.-C. Guo, and Z.-F. Han, Quantum random number generation based on spontaneous Raman scattering in standard single-mode fiber, Optics Letters 45, 6038 (2020).

*

##  Appendix A Single-photon noise analysis

The Sagnac configuration requires the 1560 nm pump to be introduced in the
interferometer via the same components used to extract the SPDC. This implies
that, even over short distances, the pump and the SPDC may co-propagate along
the same fibers. In this setting, spontaneous four-wave mixing (SFWM) and
spontaneous Raman scattering (SpRS) could potentially occur. SFWM results as a
parametric interaction of two pump photons with a third-order nonlinear media,
such as optical fibers, generating two other new photons at neighboring
frequencies [13]. However, this process can be discarded in our experiment
since no time-energy correlations were found among the noise photons. This
observation agrees with the discussions in Refs. [32, 29]. To quantify the
amount of generated SFWM under phase-matching conditions, the product $\gamma
P_{0}L$ can be used, where $\gamma$ is the nonlinear fiber parameter
(typically $\gamma\sim 1W^{-1}km^{-1}$ for standard fibers), $P_{0}$ denotes
the pump power, and $L$ stands for the fiber length. For any substantial
contribution of SFWM, the product should be $\gamma P_{0}L>0.1$ [32]. In our
system, $L\sim 10$ m and $P_{0}=10$ mW, which leads to the product value of
$\sim$0.0001, indicating minor SFWM contribution. While SFWM noise is
negligible, we do expect a substantial noise contribution of SpRS. This
scattering process is known to be an inelastic effect, where a pump photon
interacting with the material is annihilated, creating a photon at a lower
frequency (Stokes) and a phonon to conserve energy and momentum. Photons at
higher frequencies (anti-Stokes) can also be generated when existing phonons
transfer energy to pump photons. These scattering events occur in any
direction and due to the non-crystalline nature of silica fibers, they extend
over a large frequency range (up to 40 THz or 300 nm), surpassing the SPDC
signal [33]. Theoretical models referred to in [29] predict a linear growth of
SpRS as a function of initial power and fiber length (i.e. restricted only for
short fiber distances).

Figure 5: Spectrum of spontaneous Raman scattering after pump propagation
through 12m of optical fiber and at room temperature. The 8nm channels of
signal and idler from our entanglement characterization are shown in grey for
comparison. They coincide with the lowest spectral SpRS contribution.

Although SpRS photons are uncorrelated events and do not directly contribute
to coincidence detections, a high number of noise photons increases the
probability of accidental detections and deteriorate figures of merit, such as
CAR and heralding. To investigate the behaviour of the noise alone, we isolate
it from the SPDC photons by replacing the fiber-loop with a 10 m SMF. In this
way, we skip the PBS and the up-and down-conversion processes and characterize
the noise generated as a function of the initial power and pulse duty cycle.
Note that we leave the second fiber end, terminated in APC (angle-polished
connector), disconnected and no reflections should be expected from here. Fig.
5 displays the noise spectrum at room temperature generated over approximately
12 m of accumulated optical fiber from the BPF to the circulator and the 10 m
SMF. For the reflection curve (i.e., the second end of the 10 m SMF is left
disconnected), a dip in the spectrum at 1560 nm is caused by the spectral
filtering of the pump with the notch filter. In contrast, for the transmission
case (i.e., the second end of the 10m SMF is directly connected to notch
filters and to the spectrometer), the pump power is too high and is not
completely removed. Similar SpRS spectrum traces have been found in other
works [34].

Even though the noise spectral brightness is not very high, the extremely
broad bandwidth already hints at its impact on the source. In fact, the total
number of circulating photons in the system due to SpRS alone is so high that
the single-photon detectors are easily saturated.

Figure 6: Analysis of SpRS behaviour for different pump configurations: (a)
Single-photon detection of SpRS only (i.e. the EPS has been disconnected and
replaced with a 10 m fiber with open end). The increase is linear with pump
power.

To uncover the main noise trends while avoiding saturation, we introduce now a
variable optical attenuator before the SNSPD in the same setup as before, the
10 m SMF instead of the fiber loop, and we measure the single-photon noise in
a reflection configuration. For clear visualization, we show the noise-photon
generation as a function of repetition rate (50 % duty cycle), and average
pump power in Fig. 6(a), and verify the linear increase with power. In Fig.
6(b), we summarize and normalize the obtained generation rates for different
pulse rates and duty cycle. Note that a constant average pump power ensures
the same number of pump photons per second, even when the pulse settings are
different. Given that SpRS is a stochastic process, the number of scattering
events should be proportional to the number of pump photons and thus, it
should also be constant. Interestingly, we found that the CW regime presents
the highest noise rate, suggesting that the probability of generating SpRS
photons is higher than for pulsed signals. Instead, pulsed signals with
decreasing duty cycles reveal a slight reduction of the noise, e.g. the 8-GHz
pulsed regime appears to be the best scenario with the lowest noise
contribution.

In sum, we can conclude that to diminish the impact of SpRS noise, which grows
linearly with power, a sufficient strong pump should be used to improve the
SHG and thus SPDC processes, which grow quadratically. As already shown in the
main article, this regime can be found before generating multiphoton states or
stimulated processes that would decrease our visibility measurements.
Nonetheless, and perhaps more importantly, spectral filtering is required. For
example, by spectrally filtering the 60 nm broad SPDC band, the remaining 200
nm of almost uniformly distributed SpRS noise can be easily suppressed. This
procedure can be further optimized if narrower channels are used for
filtering, which is perfectly suitable for dense wavelength division
multiplexing (DWDM) schemes.


the sophisticated details as the aforementioned one, we believe that the
formalism suitable for blockchain-specific validity properties deserves a
standalone paper.

###  D.3. Validity

We start by restating the definition of process-proposal pairs. A _process-
proposal_ pair is a pair $(P,v)$, where (1) $P\in\Pi$ is a process, and (2)
$v\in\mathcal{V}_{I}$ is a proposal. Given a process-proposal pair
$\mathit{pp}=(P,v)$, $\mathsf{proposal}(\mathit{pp})=v$ denotes the proposal
associated with $\mathit{pp}$.

An _input configuration_ is a tuple
$\big{[}\mathit{pp}_{1},\mathit{pp}_{2},...,\mathit{pp}_{x},\rho\big{]}$ of
$x$ process-proposal pairs and a set $\rho\subseteq\mathcal{V}_{I}$, where (1)
$n-t\leq x\leq n$, (2) every process-proposal pair is associated with a
distinct process, and (3) if $x=n$, $\rho=\emptyset$. Intuitively, an input
configuration represents an assignment of proposals to correct processes, as
well as a “part” of the input space known to the adversary. For example, an
input configuration
$\big{[}(P_{1},v),(P_{2},v),(P_{3},v),\\{v,v^{\prime},v^{\prime\prime}\\}\big{]}$
describes an execution in which (1) only processes $P_{1}$, $P_{2}$, and
$P_{3}$ are correct, (2) processes $P_{1}$, $P_{2}$, and $P_{3}$ propose the
same value $v$, and (3) faulty processes know only $v$, $v^{\prime}$, and
$v^{\prime\prime}$.

We denote by $\mathcal{I}$ the set of all input configurations. For every
input configuration $c\in\mathcal{I}$, we denote by $c[i]$ the process-
proposal pair associated with process $P_{i}$; if such a process-proposal pair
does not exist, $c[i]=\bot$. Moreover, we define by $\mathsf{pool}(c)$ the set
of input values associated with $c$ (the “$\rho$” field of $c$). Next,
$\pi(c)=\\{P_{i}\in\Pi\,|\,c[i]\neq\bot\\}$ denotes the set of all processes
included in $c$. Finally,
$\mathsf{correct\\_proposals}(c)=\\{v\in\mathcal{V}_{I}\,|\,\exists
i\in[1,n]:c[i]\neq\bot\land\mathsf{proposal}(c[i])=v\\}$ denotes the set of
all proposals of correct processes (as specified by $c$).

Given (1) an execution $\mathcal{E}\in\mathit{execs}(\mathcal{A})$, where
$\mathcal{A}$ is an algorithm which exposes the
$\mathsf{propose}(\cdot)/\mathsf{decide}(\cdot)$ interface, and (2) an input
configuration $c\in\mathcal{I}$, we say that $\mathcal{E}$ _corresponds_ to
$c$ if and only if (1) $\pi(c)=\mathit{Corr}_{\mathcal{A}}(\mathcal{E})$, (2)
for every process $P_{i}\in\mathit{Corr}_{\mathcal{A}}(\mathcal{E})$,
$P_{i}$’s proposal in $\mathcal{E}$ is $\mathsf{proposal}(c[i])$, and (3)
$\mathcal{P}(\mathcal{E})=\mathsf{pool}(c)$. We denote by
$\mathsf{corresponding}(\mathcal{E})=c$ the input configuration to which
$\mathcal{E}$ corresponds.

A validity property $\mathit{val}$ is a function $\mathit{val}:\mathcal{I}\to
2^{\mathcal{V}_{O}}$ such that, for every input configuration
$c\in\mathcal{I}$, $\mathit{val}(\mathit{c})\neq\emptyset$. Algorithm
$\mathcal{A}$, where $\mathcal{A}$ exposes the
$\mathsf{propose}(\cdot)/\mathsf{decide}(\cdot)$ interface, _satisfies_ a
validity property $\mathit{val}$ if and only if, in every execution
$\mathcal{E}\in\mathit{execs}(\mathcal{A})$, no correct process decides a
value
$v^{\prime}\notin\mathit{val}\big{(}\mathsf{corresponding}(\mathcal{E})\big{)}$.
That is, an algorithm satisfies a validity property if and only if correct
processes decide only admissible values.

##### Assumptions on executions.

Lastly, we introduce two assumptions that conclude our proposal for the
extended formalism.

###### Assumption 1.

For every execution $\mathcal{E}$ of any algorithm $\mathcal{A}$ which solves
the Byzantine consensus problem with some validity property, if a correct
process $P$ decides a value $v^{\prime}\in\mathcal{V}_{O}$ in $\mathcal{E}$,
then
$v^{\prime}\in\mathsf{discover}\big{(}\mathsf{correct\\_proposals}(c)\cup\mathsf{pool}(c)\big{)}$,
where $\mathsf{corresponding}(\mathcal{E})=c$.

Assumption 1 states that correct processes can only decide values which are
“discoverable” using all the proposals of correct processes and the knowledge
of the adversary. For example, if every correct process proposes the same
value $v\in\mathcal{V}_{I}$ and the adversary pool contains only
$v^{\prime}\in\mathcal{V}_{I}$, then a correct process can only decide a value
from $\mathsf{discover}(\\{v,v^{\prime}\\})$.

Next, we introduce an assumption concerned only with the canonical executions
(executions in which faulty processes do not take any computational step).

###### Assumption 2.

For every canonical execution $\mathcal{E}$ of any algorithm $\mathcal{A}$
which solves the Byzantine consensus problem with some validity property, if a
correct process $P$ decides a value $v^{\prime}\in\mathcal{V}_{O}$ in
$\mathcal{E}$, then
$v^{\prime}\in\mathsf{discover}\big{(}\mathsf{correct\\_proposals}(c)\big{)}$,
where $\mathsf{corresponding}(\mathcal{E})=c$.

Intuitively, Assumption 2 states that, if faulty processes are silent, correct
processes can only decide values which can be discovered using their own
proposals. In other words, correct processes cannot use “hidden” proposals
(possessed by the “silent” adversary) to discover a decision.

Finally, we underline that these two assumptions do not completely prevent
“unreasonable” executions. For example, given these two assumptions, a
(correct or faulty) process is still able to send a message with a value which
cannot be discovered using the proposals of correct processes and the
adversary pool. Hence, an assumption that prevents such an execution should be
introduced. Thus, due to the complexity we envision for the extended
formalism, we leave it out of this paper. In the future, we will focus on this
interesting and important problem.