# A Toolbox for Supporting Research on AI in Water Distribution Networks

André Artelt1,2 Marios S. Kyriakou2 Stelios G. Vrachimis2,3 Demetrios G.
Eliades2
Barbara Hammer1&Marios M. Polycarpou2,3
1Faculty of Technology, Bielefeld University, Germany
2KIOS Research and Innovation Center of Excellence, University of Cyprus,
Cyprus
3Department of Electrical and Computer Engineering, University of Cyprus,
Cyprus
{artelt.andre, kiriakou.marios, vrachimis.stelios, eldemet,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Drinking water is a vital resource for humanity, and thus, Water Distribution
Networks (WDNs) are considered critical infrastructures in modern societies.
The operation of WDNs is subject to diverse challenges such as water leakages
and contamination, cyber/physical attacks, high energy consumption during pump
operation, etc. With model-based methods reaching their limits due to various
uncertainty sources, AI methods offer promising solutions to those challenges.
In this work, we introduce a Python toolbox for complex scenario modeling &
generation such that AI researchers can easily access challenging problems
from the drinking water domain. Besides providing a high-level interface for
the easy generation of hydraulic and water quality scenario data, it also
provides easy access to popular event detection benchmarks and an environment
for developing control algorithms.

††Workshop on Artificial Intelligence for Critical Infrastructure (AI4CI 2024)
@ IJCAI’24 , Jeju Island, South Korea, https://sites.google.com/view/aiforci-
ijcai24/, August 4, 2024. Eds: F. Silva, W. Su, R. Glatt, Y. Wang.

##  1 Introduction

The operation of Water Distribution Networks (WDNs) aims to ensure a reliable
supply of drinking water, which is vital for modern societies. WDNs are
typically modeled as graphs where the nodes (i.e. junctions) correspond to
consumer demands (e.g. households consuming water) and the edges correspond to
pipes. Hydraulic models are used to model water flow and pressure in the
network, considering water consumption and operation of pumps and valves.
Moreover, water quality is modeled using partial differential equations
describing the chemical reaction and transport of different substances through
the network. The schematics of a WDN are illustrated in Figure 1. In practice,
WDNs are operated and monitored by humans, which are supported by software,
such as basic control algorithms and event detectors, relying on a few sensors
in the WDN. However, given the rapid growth of urban areas, WDNs are becoming
more complex, and modeling uncertainties prevent their efficient operation.
Besides classic event detection and isolation (e.g. leakage detection),
control tasks such as pump scheduling, as well as
modeling/prediction/forecasting of water quality states in WDNs are becoming
more challenging. Climate change and rapid population growth significantly
affect the quality of source water and the ability of WDN operators to predict
and detect water quality issues. With increasing and time-varying uncertainty,
new AI and data-driven methods provide promising tools for tackling such
challenges.

Figure 1: Illustration of the L-Town Vrachimis et al. (2022) Water
Distribution Network together with the flow at pipe ”p227”.

Currently, non-water experts such as AI researchers face several challenges
when devising practical solutions for water system applications, such as the
unavailability of tools for easy scenario/data generation and easy access to
benchmarks, that hinder the progress of applying AI to this domain. Easy-to-
use toolboxes and access to benchmark data sets are extremely important for
boosting and accelerating research, as well as for supporting reproducible
research, as it was, for instance, the case in deep learning and machine
learning where toolboxes such as TensorFlow and scikit-learn had a significant
impact on boosting research.

The contributions of this paper are two-fold: First, we introduce AI
researchers to emerging problems in the drinking water domain where AI can
have a significant impact. Second, we provide a toolbox for generating and
accessing data to build AI-based solutions for those challenges.

##  2 Related Work

The modeling and simulation of hydraulic and water quality dynamics in water
distribution networks have progressively advanced with the introduction of
simulation software. Notably, EPANET Rossman (2000) and its extension EPANET-
MSX Shang et al. (2008) are foundational tools in this area. These are
complemented by tools that make use of high-level programming languages, such
as the EPANET-MATLAB Toolkit (EMT) Eliades et al. (2016), the Object-Oriented
Pipe Network Analyzer (OOPNET) Steffelbauer and Fuchs-Hanusch (2015), and the
EPANET-Python Toolkit (EPyT) Kyriakou et al. (2023). These tools are
instrumental in facilitating research into WDN resilience and response to
various operational challenges. These tools, however, lack support for the
creation of realistic (benchmark) scenarios by missing implementations of
essential aspects such as realistic fault models (of leakages and sensor
faults), various sensor configurations, custom control modules, and other
events such as changes in water quality caused by external factors. A first
step towards such software for scenario creation is the Water Network Tool for
Resilience (WNTR) Klise et al. (2017), which facilitates the simulation of
hydraulic dynamics, and in addition, it allows the simulation of various
events such as pipe breaks, disasters such as earthquakes, power outages, and
fires. However, it currently does not support quality dynamics and also misses
other crucial aspects such as sensor configurations, and industrial control
modules. The creation and adoption of standardized benchmarks, like LeakDB
Vrachimis et al. (2018) and BattLeDIM Vrachimis et al. (2022) represent
significant strides toward enabling comparative studies and enhancing
reproducibility in the field. LeakDB, created using the EMT, offers a rich set
of realistic leakage scenarios, aiding in the development and evaluation of
anomaly detection algorithms. Similarly, BattLeDIM provides a dataset for
benchmarking algorithms’ ability to detect and isolate (i.e., localize)
leakages, contributing valuable insights into the effectiveness of different
methodological approaches. However, several challenges remain unaddressed. The
field lacks easy accessibility to realistic datasets that encapsulate the
complexity of real-world water distribution systems, limiting the development
and benchmarking of algorithms and methodologies. There’s a critical need for
comprehensive benchmarks and software for scenario generation that can serve
as common grounds for developing, testing, and evaluating algorithms.
Reflecting on the state of the field, it is evident that, while existing tools
have provided significant capabilities, the integration of realistic scenario
generation with real-time data acquisition and advanced analytics into these
platforms is crucial for the next leap in water network management and
research.

##  3 AI in Water Distribution Networks

Artificial Intelligence (AI) has emerged as a key tool for WDNs, providing
potential solutions for water demand management, leak detection, pump
optimization, and infrastructure resilience. AI methodologies like machine
learning and deep learning excel at analyzing complex data from diverse
sources. This ability is especially useful in WDNs where uncertainties such as
fluctuating demand, variable water quality supply conditions, increasing
infrastructure complexity, and a relatively small number of measuring devices,
prohibit the extraction of valuable predictive insights and risk assessment.
AI applications extend to the development of surrogate models Zanfei et al.
(2023); Ashraf et al. (2024), which speed up complex simulations and enable
real-time decision support. Moreover, AI can assist in modeling the complex
dynamics of water quality and predicting potential issues, enabling proactive
actions Li et al. (2024). One of the critical applications of AI is in the
detection, isolation, and localization of events like leakages. A recent
example Rajabi et al. (2023) investigates the application of conditional
convolutional generative adversarial networks for real-time leak detection and
localization using hydraulic model-based image conversion and the structural
similarity index. Moreover, events such as contamination Li et al. (2023), and
cyber-physical attacks Taormina and Galelli (2018) have also been benefited by
AI-driven systems that analyze sensor data in real-time, detecting deviations
that may indicate such events, thereby facilitating rapid and informed
responses to mitigate impacts on public health and network integrity.
Additionally, AI can enhance the control of network components like pumps and
valves, optimizing their operation based on predictive models and real-time
data. Such AI-enhanced control systems may be able to dynamically adjust
operational parameters, improving resource efficiency, and reducing costs Guo
et al. (2023).

##  4 EPyT-Flow: A Python Toolbox

In this paper, we propose a Python toolbox called _EPyT-Flow_ that provides a
high-level interface for the easy generation of WDN scenario data, but also
provides access to low-level functions for maximum flexibility as needed by
domain experts – by this we aim to satisfy different needs and make it a
toolbox for ”everybody”. In addition, its design and object-orientated
implementation makes it easy to customize existing functionalities and
implement new ones. Our proposed toolbox _EPyT-Flow_ builds upon EPyT which
itself provides a Python interface to EPANET and EPANET-MSX – see Figure 2 for
an illustration.

Figure 2: Illustration of the functionality of the proposed toolbox _EPyT-
Flow_.

The toolbox currently includes $16$ WDNs that can be used for scenario
generation. However, it also goes beyond pure scenario generation by providing
access to $7$ popular and widely adopted benchmarks on event detection and
localization (incl. their evaluation metrics) – ready to be utilized for
building and evaluating AI models. It also comes with an implemented residual-
based sensor interpolation event detection method that can be used as a
baseline for any event detection benchmark. Furthermore, it also provides an
environment (inspired by the OpenAI Gym) for developing and implementing
control algorithms – i.e. a gym for training reinforcement learning agents to
solve some given control tasks such as energy efficient pump scheduling. In
order to support modeling a wide variety of scenarios, the toolbox comes with
$4$ different event types and a total number of $13$ pre-defined and
implemented events ready to be utilized in custom scenarios: $3$ different
leakages, $3$ actuator events, $5$ sensor faults, and $2$ communication
events. Furthermore, all those events can be easily customized as needed by
the user. Since the quantities in the real world are always subject to
uncertainty, the toolbox comes with $11$ pre-defined types of uncertainties
ranging from classic Gaussian noise to different types of (very) deep
uncertainties that can be applied to hydraulic parameters such as pipe length,
diameter, and roughness, water quality parameters, such as reaction
coefficients, and sensor noise. The proposed toolbox (incl. the detailed
documentation) is available at https://github.com/WaterFutures/EPyT-Flow.

##  5 Use-Case – Event Detection in WDNs

Here, we demonstrate how the proposed toolbox can be utilized to easily set up
an event detection scenario and apply a baseline event detection method.

We load the popular L-Town network Vrachimis et al. (2022) ($785$ nodes and
$909$ pipes – see Figure 1) with the default sensor placement from the
BattLeDIM competition ($29$ pressure and $2$ flow sensors) and demand patterns
for modeling a realistic water consumption behavior:

⬇

cfg = load_ltown(use_realistic_demands=True,

include_default_sensor_placement=True)

We create a new scenario ($14$ days long with $5$min time steps) based on this
configuration:

⬇

sim = ScenarioSimulator(scenario_config=cfg)

params = {

”simulation_duration”: to_seconds(days=14),

”hydraulic_time_step”: to_seconds(minutes=5)}

sim.set_general_parameters(**params)

The first week of data in this scenario does not contain any events, in the
second week, however, we add a small abrupt leakage, and a slightly larger
incipient leakage:

⬇

sim.add_leakage(AbruptLeakage(link_id=”p673”,

diameter=0.001,

start_time=to_seconds(days=7),

end_time=to_seconds(days=8)))

sim.add_leakage(IncipientLeakage(link_id=”p31”,

diameter=0.02,

start_time=to_seconds(days=11),

end_time=to_seconds(days=13),

peak_time=to_seconds(days=12)))

Moreover, a flow sensor fault is included:

⬇

sim.add_sensor_fault(SensorFaultDrift(coef=1.1,

sensor_id=”p227”,

sensor_type=SENSOR_TYPE_LINK_FLOW,

start_time=to_seconds(days=9),

end_time=to_seconds(days=10)))

Finally, we run the simulation to obtain the data (i.e. sensor readings) as
they would be received in an operator’s Supervisory Control and Data
Acquisition (SCADA) system, and are processed by the event detector:

⬇

scada_data = sim.run_simulation()

As a classic baseline, our proposed toolbox already implements a residual-
based interpolation detection method – this method tries to predict the
readings of a given sensor based on all other sensors:
$f:\vec{x}_{t}\setminus\\{i\\}\mapsto(\vec{x}_{t})_{i}$, where $\vec{x}_{t}$
refers to these sensor ratings at time $t$, and $\vec{x}_{t}\setminus\\{i\\}$
denotes these sensor readings without the $i$-th sensor. An alarm is raised
(i.e. event detected) whenever the prediction and the observation of at least
one sensor differ significantly:

$\exists i:\;|f(\vec{x}_{t}\setminus\\{i\\})-(\vec{x}_{t})_{i}|>\theta_{i}$
(1)

where $\theta_{i}>0$ denotes a sensor-specific threshold at which the
difference is considered as significant. For this, the detection method has to
be calibrated (i.e. fitted) to a time window of (ideally event-free) sensor
readings to determine a suitable threshold $\theta$ that does not raise an
alarm when the network is in normal operation (i.e. no events present). We
split the simulated data into half – the first half corresponds to the first
week where no events are active and the second half contains some leakages and
sensor faults as described above.

⬇

X = scada_data.get_data() # 4000 time steps

X_train, X_test = X[:2000, :], X[2000:, :]

The first half of the data is event-free and is used to calibrate (i.e. fit)
the event detector:

⬇

detector = SensorInterpolationDetector()

detector.fit(X_train)

, which then is applied to the second half in order to detect suspicious time
points where an event might have been active:

⬇

suspicious_time_points = detector.apply(X_test)

Figure 3: Evaluation of the event detection on the second week of the
simulated scenario.

The results are visualized in Figure 3. The event detection method does a good
job of detecting the abrupt leakage (first event), as well as the sensor fault
(second event). However, it fails in case of the incipient leakage – note that
incipient leakages are expected to be more challenging due to the gradual
change in the sensor readings instead of an abrupt change. Also, the method
yields a few false positives as well.

An AI researcher could develop and apply more sophisticated event detection
methods and compare their performance against this baseline.

##  6 Conclusion & Summary

In this work, we introduced a Python toolbox for realistic scenario data
generation and access to benchmarks of WDNs, that AI researchers can utilize
to develop methods to support human WDN operators in various real-world
challenges.

Our long-term vision for this toolbox is to split it into three parts to
further facilitate the unfolding of the huge potential of AI in WDNs: 1) A
core part for data generation (i.e., scenario simulation); 2) A _BenchmarkHub_
as a platform for accessing and sharing WDN benchmarks; 3) A _ModelHub_ as a
platform for accessing and sharing AI & classic models and algorithms for
different tasks in WDNs.

## Acknowledgments

This research was supported by the Ministry of Culture and Science NRW
(Germany) as part of the Lamarr Fellow Network. We also gratefully acknowledge
funding from the European Research Council (ERC) under the ERC Synergy Grant
Water-Futures (Grant agreement No. 951424). This publication reflects the
views of the authors only.

## References

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13.3(1.4,15) This manuscript has been co-authored by UT-Battelle, LLC, under
contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US
government retains and the publisher, by accepting the article for
publication, acknowledges that the US government retains a nonexclusive, paid-
up, irrevocable, worldwide license to publish or reproduce the published form
of this manuscript, or allow others to do so, for US government purposes. DOE
will provide public access to these results of federally sponsored research in
accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-
public-access-plan).

# Procrustean entanglement concentration in quantum–classical networking

Hsuan-Hao Lu<EMAIL_ADDRESS>Muneer Alshowkan  Quantum Information Science
Section, Computational Sciences and Engineering Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Jude Alnas  School of Electrical
and Computer Engineering, Duke University, Durham, North Carolina 27708, USA
Joseph M. Lukens  Quantum Information Science Section, Computational Sciences
and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  Research Technology Office and Quantum Collaborative, Arizona
State University, Tempe, Arizona 85287, USA  Nicholas A. Peters  Quantum
Information Science Section, Computational Sciences and Engineering Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

The success of a future quantum internet will rest in part on the ability of
quantum and classical signals to coexist in the same optical fiber
infrastructure, a challenging endeavor given the orders of magnitude
differences in flux of single-photon-level quantum fields and bright classical
traffic. We theoretically describe and experimentally implement Procrustean
entanglement concentration for polarization-entangled states contaminated with
classical light, showing significant mitigation of crosstalk noise in dense
wavelength-division multiplexing. Our approach leverages a pair of
polarization-dependent loss emulators to attenuate highly polarized crosstalk
that results from imperfect isolation of conventional signals copropagating on
shared fiber links. We demonstrate our technique both on the tabletop and over
a deployed quantum local area network, finding a substantial improvement of
two-qubit entangled state fidelity from approximately 75% to over 92%. This
local filtering technique could be used as a preliminary step to reduce
asymmetric errors, potentially improving the overall efficiency when combined
with more complex error mitigation techniques in future quantum repeater
networks.

Introduction.—Quantum-classical coexistence within fiber-optic resources will
streamline the implementation of quantum networks, enabling efficient
utilization of infrastructure and reducing deployment costs. One promising
pathway relies on distinguishing classical and quantum channels by wavelength.
Although originally pursued through coarse wavelength-division multiplexing
(CWDM) Townsend (1997); Patel  _et al._ (2012); Huang  _et al._ (2015); Mao
_et al._ (2018); Burenkov  _et al._ (2023); Thomas  _et al._ (2023),
quantum–classical coexistence efforts are increasingly targeting dense
wavelength-division multiplexing (DWDM) with channels of 200 GHz or less, for
even greater spectral efficiency Peters  _et al._ (2009); Eraerds  _et al._
(2010); Choi  _et al._ (2010); Qi  _et al._ (2010); Patel  _et al._ (2014);
Chapman  _et al._ (2023a, b); Honz  _et al._ (2023). Given the stark contrast
in brightness between classical and quantum signals, such ultratight spacings
are highly susceptible to imperfect filter isolation and unwanted noise
infiltrating the quantum output Peters  _et al._ (2009); Eraerds  _et al._
(2010); Chapman  _et al._ (2023a); Honz  _et al._ (2023). At whatever level of
crosstalk is present, the strong temporal correlations between entangled
photons can be leveraged to some extent for filtering uncorrelated noise by
reducing the coincidence detection window. Nevertheless, the timing jitter
between two nodes ultimately determines how narrow this window can be before
genuine coincidences are lost, a limitation becoming especially pronounced in
geographically separated nodes Alshowkan  _et al._ (2022a); Burenkov  _et al._
(2023).

In this Letter, we introduce a crosstalk mitigation method tailored to
polarization-entangled photons tainted by highly polarized classical signals,
a practical source of error from typically polarized conventional
communications signals. For appropriate orientations, the noisy density matrix
maps approximately to the class of maximally entangled mixed states (MEMS)
Munro  _et al._ (2001); Wei  _et al._ (2003); Peters  _et al._ (2004a), whose
entanglement can be concentrated through local “Procrustean” filtering Bennett
_et al._ (1996); Kwiat  _et al._ (2001); Peters  _et al._ (2004a); Wang  _et
al._ (2006). Additionally, our approach resembles the polarization dependent
loss (PDL) compensation possible for Bell states Jones  _et al._ (2018); Kirby
_et al._ (2019); Riccardi  _et al._ (2021), whereby reductions in concurrence
due to PDL on one photon can be completely compensated (in postselection) by
applying PDL to the other. This situation facilitates the filtering of noisy
quantum states while retaining maximal entanglement in the Bell state portion
and suppressing nonentanlged terms. After simulating a general model, we
experimentally demonstrate the approach with programmable polarization-
dependent loss emulators (PDLEs; OZ Optics) applied to entangled photons
contaminated with crosstalk from copropragating lasers. In all cases
examined—both on the optical table and over a deployed network—fidelity is
found to increase in good agreement with theory. Overall, our method provides
an additional error-mitigation layer for quantum–classical coexistence
networks that can be applied, e.g., for dynamic removal of asymmetric noise.

Motivation.—Figure 1 illustrates the proposed scheme. Consider an ideal Bell
state $\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{HH}+\ket{VV})$ as the input,
where $\ket{H}$ ($\ket{V}$) denotes a horizontally (vertically) polarized
state. Following generation, signal and idler photons are physically separated
into two optical arms destined for Alice ($A$) and Bob ($B$), each combined
with highly polarized classical traffic through a DWDM multiplexer (MUX). Upon
reaching the receivers, photons are wavelength-demultiplexed using another
DWDM demultiplexer (DEMUX). Due to imperfect spectral filtering and channel
isolation, the quantum channel now includes a small portion of classical
crosstalk noise. Alice and Bob send their respective signals through a PDLE
module designed to controllably replicate PDL in optical links Gisin (1995);
Mecozzi and Shtaif (2002). The PDLE module spatially separates the input into
two orthogonal polarizations, applying user-defined attenuation to one of
them, referred to as the PDL axis, while passing the other unaltered. The two
polarizations are then recombined into a single fiber-optic spatial mode at
the output.

Figure 1: Concept of Procrustean filtering for polarization-entangled photons.
Crosstalk suppression can be achieved by aligning the PDLE axes and classical
traffic polarizations accordingly.

To optimize the noise suppression possible with PDLEs, intuition suggests
orienting the PDL axes orthogonally ($H/V$) with respect to the initial
reference frame to ensure equal attenuation of the terms of interest in
$\ket{\Psi^{+}}$, while aligning the classical traffic to these axes for
greatest suppression. In this scenario, a reasonable approximation of the
initial noisy quantum state (before any PDL) is
$\rho\approx\gamma\ket{\Phi^{+}}\bra{\Phi^{+}}+(1-\gamma)\ket{HV}\bra{HV}$,
where $\gamma\in[0,1]$ defines the relative weight of accidentals stemming
from the noise photons in comparison to the genuine coincidences between the
entangled pairs. For $\gamma\in[2/3,1]$ this state is precisely that of the
MEMS I subclass Munro  _et al._ (2001); Wei  _et al._ (2003); Peters  _et al._
(2004a), which is a sufficient (though not necessary) criterion for our
concentration method.

With the introduction of two PDLEs imparting variable transmissivities
$T_{H},T_{V}\in[0,1]$ along their respective PDL axes, the output density
matrix becomes
$\rho\propto\gamma\left(\sqrt{T_{H}}\ket{HH}+\sqrt{T_{V}}\ket{VV}\right)\left(\sqrt{T_{H}}\bra{HH}+\sqrt{T_{V}}\bra{VV}\right)+(1-\gamma)T_{H}T_{V}\ket{HV}\bra{HV}$.
Notably, the unwanted term $\ket{HV}\bra{HV}$ decreases more rapidly than the
rest of the matrix entries as $T_{H}$ and $T_{V}$ decrease. In particular, the
fidelity with respect to the maximally entangled ideal
($\operatorname{\mathcal{F}}=\bra{\Phi^{+}}\rho\ket{\Phi^{+}}$) approaches
unity as $T_{H}=T_{V}\rightarrow 0$, implying an arbitrary increase (at the
expense of flux). Now, this model is only approximate in that it neglects
other noise terms, such as coincidences _between_ crosstalk and input photons,
but it summarizes the basic concept. In the following, we introduce the more
complete model accounting for all accidental contributions in the system of
interest.

Model.—Consider polarization-entangled photons generated in the state
$\ket{\Phi^{+}}$ at rate $\mu$. Subsequently, the idler (signal) photon is
separated into optical arm $A$ ($B$) with pathway efficiency $\eta_{A}$
($\eta_{B}$) encompassing all loss effects from generation through detection.
The rate of photon detection from classical crosstalk is $\nu_{A}$ ($\nu_{B}$)
in arm $A$ ($B$) when the polarization analyzer is aligned to $H$ ($V$) and
$T_{H}=1$ ($T_{V}=1$). Additionally, we consider dark counts at detector $A$
($B$) with rate $d_{A}$ ($d_{B}$). The coincidence window is set to $\tau$,
with the assumption that this window is significantly longer than the two-
photon temporal correlations.

Figure 2: Numerical simulation of Procrustean filtering with fixed parameters
$\\{\mu,\eta_{A},\eta_{B},d_{A},d_{B}\\}=\\{7\times
10^{6},0.02,0.02,100,100\\}$. (a) Variable noise rates
$\nu_{A}=\nu_{B}=\nu\in[2\times 10^{5},3.6\times 10^{6}]$ with a coincidence
window $\tau=10^{-9}$. (b) Variable coincidence window $\tau\in[2\times
10^{-10},3.4\times 10^{-9}]$ with a noise rate of $\nu_{A}=\nu_{B}=10^{6}$.
Solid curves represent initial noisy states (right) and optimized filtered
states (left). Dashed curves illustrate state evolution as $T_{H}=T_{V}$ is
scanned from $1$ to $10^{-3}$.

Under these conditions, the complete received quantum state can be written as

$\rho=\frac{1}{R}\Bigg{\\{}\frac{\mu\eta_{A}\eta_{B}}{2}\left(\sqrt{T_{H}}\ket{HH}+\sqrt{T_{V}}\ket{VV}\right)\left(\sqrt{T_{H}}\bra{HH}+\sqrt{T_{V}}\bra{VV}\right)\\\
+\tau\left[\frac{\mu\eta_{A}}{2}\left(T_{H}\ket{H}\bra{H}+\ket{V}\bra{V}\right)+\nu_{A}T_{H}\ket{H}\bra{H}+d_{A}I_{2}\right]\\\
\otimes\left[\frac{\mu\eta_{B}}{2}\left(\ket{H}\bra{H}+T_{V}\ket{V}\bra{V}\right)+\nu_{B}T_{V}\ket{V}\bra{V}+d_{B}I_{2}\right]\Bigg{\\}},$
(1)

where $I_{2}$ is the $2\times 2$ identity and $R$ denotes the coincidence rate
over all outcomes in a basis, such that $\Tr\rho=1$. The quantum state
contains a correlated term from photons in the same pair (scaled by
$\mu\eta_{A}\eta_{B}$) and an uncorrelated term (scaled by $\tau$), which can
be derived following standard arguments for the rate of accidentals as the
product of the rates on the individual detectors multiplied by $\tau$ Eckart
and Shonka (1938); Pearson and Jackson (2010)—valid in the limit of small
detection probabilities. The noise term thereby consists of a product of the
marginal states at each detector, in brackets $[\cdot]$ in Eq. (1), each of
which contains three contributions: (i) partially polarized light from the
filtered entangled photons (scaled by $\mu\eta_{A(B)}$), (ii) attenuated
linearly polarized light from classical crosstalk ($\nu_{A(B)}$), and (iii)
white noise from detector dark counts ($d_{A(B)}$).

Due to the physical effects reflected in (i) and (iii), Eq. (1) no longer
matches a true MEMS I form, yet the classical crosstalk term (proportional to
$\nu_{A}\nu_{B}$) can still be selectively suppressed relative to the
correlated term. If $T_{H}=T_{V}=T$, for example, the flux from the former
drops quadratically ($T^{2}$), while the latter only linearly ($T$). As $T$
decreases, at some point the multipair (i) and dark count effects (iii) will
take over; e.g., the extreme case of $T_{H}=T_{V}=0$ would completely suppress
the crosstalk but lead to a separable state. Consequently, for any state of
the form in Eq. (1), there exists an optimal pair of PDLE filters
$(T_{H},T_{V})$ to maximize $\operatorname{\mathcal{F}}$.

In Fig. 2(a), we model these tradeoffs in a scenario with $\mu=6\times 10^{6}$
s-1, $\eta_{A}=\eta_{B}=0.02$, $d_{A}=d_{B}=100$ s-1, and $\tau=1$ ns. We
introduce variable noise in both arms, with
$\nu_{A}=\nu_{B}=\nu\in[2,36]\times 10^{5}$ s-1. The five points (Roman
numerals i–v) in Fig. 2(a) correspond to distinct crosstalk levels $\nu$ with
no filtering applied ($T_{H}=T_{V}=1$) and represent initial noisy quantum
states. For each of these cases, we numerically find the $(T_{H},T_{V})$ pair
that maximizes $\operatorname{\mathcal{F}}$, which are are shown as primed
values (i′–v′) and reach
$\operatorname{\mathcal{F}}\in\\{0.98,0.95,0.90,0.85,0.80\\}$. Our results
consistently indicate the optimal filtering occurs when $T_{H}=T_{V}$,
aligning with the case for ideal MEMS I states. We also illustrate the state
evolution concerning PDLE filtering by mapping the fidelity/rate trajectories
for $T_{H}=T_{V}=T\in[0.001,1]$, as represented by the dashed curves. The
maximum attainable fidelity, along with the optimal filtering levels, depends
on the noisiness of the initial states. Beyond this threshold,
$\operatorname{\mathcal{F}}$ decreases as the multipair effect (the unfiltered
$\ket{VH}\bra{VH}$ term) increasingly dominates.

Likewise, the influence of crosstalk varies as the coincidence window $\tau$
changes. In Fig. 2(b), we consider a scenario where photon flux and background
levels are fixed at
$(\mu,\eta_{A},\eta_{B},\nu_{A},\nu_{B},d_{A},d_{B})=(7\times
10^{6}~{}\mathrm{s}^{-1},0.02,0.02,10^{6}\,\mathrm{s}^{-1},10^{6}\,\mathrm{s}^{-1},100\,\mathrm{s}^{-1},100\,\mathrm{s}^{-1})$,
while $\tau$ varies in $[0.2,3.4]$ ns. As expected, the optimal fidelity is
contingent upon the noisiness of the original state. Nevertheless, the optimal
filtering configuration ($T_{H},T_{V}$) remains consistent across all cases.

Tabletop experiments.—Figure 3 illustrates the experimental setup. Broadband,
polarization-entangled photons areseparated by a C/L-band demultiplexer, with
signal and idler photons covering the optical C-band (1530–1565 nm) and L-band
(1565–1625 nm), respectively [detailed design outlined in Ref. Alshowkan  _et
al._ (2022b); Lu  _et al._ (2023a)]. In each fiber arm, we program a
wavelength-selective switch (WSS; Finisar) operated in reverse to accept a 25
GHz-wide energy-matched channel pair: 193.9 THz in the C-band and 189.1 THz in
the L-band. For all scenarios tested, we measure coincidences (5 s per point
in the tabletop experiments, 10 s in the deployed tests) in an overcomplete
set of 36 polarization projections using a pair of motor-controlled
polarization analyzers Alshowkan  _et al._ (2022b); Lu  _et al._ (2023a).
Bayesian quantum state tomography (QST) Blume-Kohout (2010); Lukens  _et al._
(2020), following the refinements outlined in Ref. Lu  _et al._ (2022), is
then applied to recover density matrix samples. Without any crosstalk and a
coincidence window $\tau=1.4$ s, we find the mean density matrix in Fig. 3,
with fidelity $\operatorname{\mathcal{F}}=0.988(1)$.

Subsequently, we introduce classical crosstalk using a pair of tunable,
linearly polarized continuous-wave (CW) lasers, operating at frequencies of
189.125 THz [CW (L) in Fig. 3] and 193.925 THz [CW (C) in Fig. 3],
respectively, i.e., one frequency slot away from their respective quantum
channel on the 25 GHz ITU grid (ITU-T Rec. G.694.1). To emulate the narrowband
filtering necessary for future coexistence networks, both lasers are heavily
attenuated. At each receiver, we record $\nu\approx 4.5\times 10^{5}$ s-1,
chosen to introduce an aggressive amount of accidentals while avoiding
saturation of the superconducting nanowire single-photon detectors
(SNSPDs)—the regime of interest for our Procrustean method.

Figure 3: Experimental setup and Bayesian mean density matrix of the initial
entangled resource. See text for details. Figure 4: Bayesian state fidelities
and collected coincidences versus PDLE local filtering. Measured density
matrices at (i,ii) 0 dB ($T_{H}=T_{V}=1$) and (i′,ii′) 10 dB
($T_{H}=T_{V}=0.1$) of PDL . Coincidence window $\tau$: (a) 1.4 ns, (b) 2.2
ns.

To demonstrate the method for maximum improvement relies on two levels of
coordination in polarization: first, the PDL axis at each receiver must be
aligned to the polarization of the classical crosstalk; second, the
polarization correlations for the crosstalk contributions must be orthogonal
to the correlations in the Bell state of interest. Once the former is
achieved, the latter can in principle be realized through local rotation of
only one photon from the maximally entangled pair (prior to multiplexing)
Wilde (2017); Valencia  _et al._ (2020). Nonetheless, for ease of
implementation here, we manually tune the polarization of all four inputs
relative to fixed PDL axes [via polarization controllers (PCs) 1–4 in Fig. 3];
this allows us to produce a state in the form of Eq. (1), up to a relative
phase between $\ket{HH}$ and $\ket{VV}$ in the Bell state term. Any residual
phase is automatically compensated by the local rotations we apply to the
inferred density matrices in post-processing, chosen to maximize fidelity with
respect to $\ket{\Phi^{+}}$ Alshowkan  _et al._ (2022b). The resulting noisy
state measured with $\tau=$1.4 ns is depicted in Fig. 4(a) (labeled i) and
resembles a MEMS I state Munro  _et al._ (2001); Wei  _et al._ (2003); Peters
_et al._ (2004a).

For the above and all experimental results below, we estimate the output
density matrix $\rho$ and flux $R$ with no reference to Eq. (1); the states
are inferred from a completely uniform Bures prior without assuming any
particular noise model Lu  _et al._ (2022). However, to extract critical
parameters of interest for validating our model, for this particular case only
($T_{H}=T_{V}=1$ and $\tau=1.4$ ns) we perform an additional inference step in
which we temporarily assume Eq. (1) to infer the parameters
$(\mu,\eta_{A},\eta_{B},\nu_{A},\nu_{B})$, which are difficult to measure
independently. For this purpose, we develop a customized Bayesian model
patterned after Ref. Lu  _et al._ (2019) that considers both single-detector
events and coincidences, takes as input the independently measured dark count
rates at $d_{A}=d_{B}=100$ s-1, and places normal priors on the four unknowns
of interest $(\mu,\eta_{A},\eta_{B},\nu_{A},\nu_{B})$, with means given by our
initial estimates from the count data, and standard deviations equal to 10% of
these values. Under this “open box” model Lu  _et al._ (2019, 2023b), we
obtain $(\mu,\eta_{A},\eta_{B},\nu_{A},\nu_{B})=(5.889(3)\times
10^{6}\,\mathrm{s}^{-1},2.113(1)\times 10^{-2},1.375(2)\times
10^{-2},4.152(1)\times 10^{5}\,\mathrm{s}^{-1},4.653(2)\times
10^{5}\,\mathrm{s}^{-1})$, which, in conjunction with Eq. (1), serve as the
basis for subsequent theory curves.

Figure 4(a) depicts the state evolution as the PDLE filtering level is
increased from 0 dB ($T=1$) to 12 dB ($T=0.063$) in 2 dB steps. The fidelity
advances steadily from
$\operatorname{\mathcal{F}}_{0~{}\mathrm{dB}}=83.4(3)\%$ to
$\operatorname{\mathcal{F}}_{10~{}\mathrm{dB}}=94.2(7)\%$ [Fig. 4(a)], at the
expense of reduced flux [Fig. 4(b)]. Incidentally, the linear entropy
decreases from $0.384(5)$ to $0.10(2)$, while the tangle increases from
$0.57(1)$ to $0.81(3)$, confirming true entanglement “concentration” as
defined as increases in both entanglement and purity Thew and Munro (2001);
Peters  _et al._ (2004a). We halt the investigation at $12$ dB as we have
observed that, for our particular system parameters, the fidelity reaches a
plateau within the range of 10–12 dB, as predicted by the accompanying theory.

To examine noisier state, we widen the coincidence window to $\tau=$2.2 ns,
resulting in a roughly 60% increase in accidentals [cf. Eq. (1)]. The results,
depicted in Fig. 4(b), show Procrustean fidelity improvements from
$\operatorname{\mathcal{F}}_{0~{}\mathrm{dB}}=75.8(3)\%$ to
$\operatorname{\mathcal{F}}_{10~{}\mathrm{dB}}=92.2(8)\%$. Additionally, the
overall trend and optimal filtering level closely match those observed in the
case of Fig. 4(a) when $\tau=$1.4 ns, which aligns with our findings in Fig.
2(b).

_Deployed network tests.—_ We next apply our method in a deployed quantum
local area network on the Oak Ridge National Laboratory campus Alshowkan  _et
al._ (2021, 2022a). The source and components in the idler arm remain in the
original lab (Alice in Alshowkan  _et al._ (2021, 2022a)), while PDLE2 and the
polarization analyzer in the signal arm are relocated to another building (Bob
in Alshowkan  _et al._ (2021, 2022a)). A deployed fiber link, approximately
250 m long with a round-trip loss of $\sim$5 dB, connects the two buildings.
After polarization projections, signal photons are routed back to Alice for
photon detection. We transmit Bob’s projected photons back to Alice—rather
than detect them at Bob—due to the availability of SNSPDs; Bob’s avalanche
photodiodes considered in previous experiments Alshowkan  _et al._ (2021,
2022a) approach saturation at the level of crosstalk noise of interest here.

As a further modification from the tabletop experiment, we tune the classical
crosstalk lasers from their original positions to the center of their
respective quantum channels at 189.1 THz and 193.1 THz. Now, the Procrustean
filtering approach described here places no requirements on the specific
wavelength of the crosstalk, as long as it is polarized and copropagating with
the demultiplexed quantum output. Thus, whether stemming from imperfect
isolation of an nearby frequency (emulated in Fig. 4) or from background light
that truly spectrally overlaps with the quantum signal (emulated in Fig. 5),
all that matters is its overall polarized contribution to the total rate of
received photons.

Figure 5 summarizes the experimental results obtained from our deployed
network. In this case, Procrustean filtering improves the entangled state
resource from $\operatorname{\mathcal{F}}_{0~{}\mathrm{dB}}=81.7(3)\%$ to
$\operatorname{\mathcal{F}}_{6~{}\mathrm{dB}}=91.5(5)\%$, which so happens to
transition the distributed entangled state from a regime where quantum key
distribution (QKD) is impossible to one where it may be performed. The BB84
QKD protocol using one-way communication possesses an 11% quantum bit error
rate (QBER) threshold Shor and Preskill (2000); Koashi and Preskill (2003); Ma
_et al._ (2007); the QBERs associated with the initial state are 13.8(4)% and
12.2(4)% for the rectilinear and diagonal bases, respectively, but they
decrease to 6.6(6)% and 3.8(6)% for the 6 dB filtered state.

Figure 5: (a) Bayesian-estimated fidelities and flux versus PDLE local
filtering in a deployed network setting. (b) Measured density matrices at (i)
0 dB and (i′) 6 dB PDL.

Discussion.—In this study, we have employed fidelity as the key metric to
assess the effectiveness of Procrustean filtering. However, our model can be
adapted to optimize other metrics that are computable from the density matrix
as well Peters  _et al._ (2004b); Vidal and Werner (2002); Plenio (2005);
Devetak and Winter (2005). Expanding our analysis, we could employ constrained
nonlinear optimization to explore more complex scenarios, such as maximizing
total coincidence rates while constraining fidelity
$\operatorname{\mathcal{F}}$ above a predefined application-specific threshold
$\operatorname{\mathcal{F}}_{\mathrm{th}}$ Alnas  _et al._ (2022); Vardoyan
and Wehner (2022). Such an approach can be particularly valuable in photon-
starved applications where optimizing both throughput and quality of
entanglement are essential.

Procrustean filtering should not be construed as an alternative to CWDM/DWDM
but rather an additional layer that can supplement such strategies to further
mitigate the impact of classical crosstalk in shared optical fiber. By
coordinating the relative orientations of entangled photons and coexisting
classic traffic, the two-photon correlations possessed by the quantum and
classical portions are distinct and can be selectively filtered when
multiplexed in shared fiber. Accordingly, while our procedure does presuppose
polarization coordination between coexisting quantum and classical signals,
the benefits it engenders are quite general and apply to a variety of
quantum–classical operating conditions in which the dominant crosstalk is
polarized. Such cases can lead to asymmetric errors in the quantum state,
making this local filtering technique a potentially valuable preliminary step,
in tandem with other error reduction techniques, for improving entanglement
distribution efficiency in future quantum repeater networks.

###### Acknowledgements.

We thank Y. Zhang for helpful discussions regarding QBER thresholds. This work
was performed in part at Oak Ridge National Laboratory, operated by UT-
Battelle for the U.S. Department of energy under contract no. DE-
AC05-00OR22725. Funding was provided by the U.S. Department of Energy, Office
of Science, Advanced Scientific Computing Research (Field work Proposals
ERKJ381, ERKJ353).

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