# Interpretable Traffic Event Analysis with Bayesian Networks

Tong Yuan Data Science and Analytics Thrust, Information Hub, The Hong Kong
University of Science and Technology (Guangzhou); The Hong Kong University of
Science and Technology. Email: {tyuan053<EMAIL_ADDRESS><EMAIL_ADDRESS>Jian Yang 11footnotemark: 1 Zeyi Wen11footnotemark: 1

###### Abstract

Although existing machine learning-based methods for traffic accident analysis
can provide good quality results to downstream tasks, they lack
interpretability which is crucial for this critical problem. This paper
proposes an interpretable framework based on Bayesian Networks for traffic
accident prediction. To enable the ease of interpretability, we design a
dataset construction pipeline to feed the traffic data into the framework
while retaining the essential traffic data information. With a concrete case
study, our framework can derive a Bayesian Network from a dataset based on the
causal relationships between weather and traffic events across the United
States. Consequently, our framework enables the prediction of traffic
accidents with competitive accuracy while examining how the probability of
these events changes under different conditions, thus illustrating transparent
relationships between traffic and weather events. Additionally, the
visualization of the network simplifies the analysis of relationships between
different variables, revealing the primary causes of traffic accidents and
ultimately providing a valuable reference for reducing traffic accidents.

##  1 Introduction

Traffic accidents lead to a tragic loss of life and have a substantial
economic impact. To reduce traffic accidents, it is crucial to predict
accidents happening; meanwhile it is essential to understand the factors that
contribute to accidents interpretably. Recent traffic accident prediction
methods, such as Neural Networks and Random Forest based methods [1], can
yield good quality predictions but lack interpretability. Moreover, other
robust machine learning algorithms, including SVMs, are not easily
interpretable. Similarly, deep learning models [2], which learn patterns by
simulating interconnected neurons with weighted connections, display great
accuracy but lack transparency, limiting their utility in trustworthy traffic
analysis.

The lack of interpretability in prevalent machine learning models [3] poses
challenges in traffic accident analysis, affecting trust actionable insights
and hindering error identification, ultimately compromising reliability and
effectiveness. Several machine-learning models have been explored in the quest
for transparency and interpretability. Decision trees, for instance, offer a
degree of interpretability through their hierarchical structure of decisions.
Rule-based systems, too, provide explicit rules that can be easily understood
and scrutinized. However, these methods often struggle to balance
interpretability with predictive accuracy, particularly when dealing with
complex, high-dimensional data.

Bayesian Networks, on the other hand, have emerged as a promising solution to
this challenge. As an interpretable machine learning approach [4], they
explicitly represent variables and their dependencies, facilitating a clear
understanding of the relationships between different factors. This
interpretability allows us to comprehend the reasoning behind the model’s
predictions, enhancing trust and enabling the correction of potential mistakes
and biases. In the context of traffic accident analysis and prediction, their
ability to provide both interpretability and predictive accuracy makes them a
powerful tool for understanding the factors contributing to traffic accidents
and informing effective prevention strategies.

Figure 1: Pipeline of the proposed learning framework. Started from single
weather and traffic events, they will be paired according to their spatio-
temporal relationships and then learned by Bayesian Network, finally the
network is used to analysis and predict the happening of some accident events
we care about.

This paper utilizes a Bayesian Network framework to predict traffic accidents
and analyze their causes interpretably from open weather and traffic event
dataset. Initially, a dataset for Bayesian Network learning is constructed,
containing potential factors influencing traffic accidents, derived from
cleaned and denoised spatio-temporal events, including weather conditions and
traffic events. After being cleaned, all the available events are paired
according to their spatio-temporal relationships and organized according to
the structure of the proposed Bayesian Network framework. Further, the dataset
is constructed to be balanced to improve learning performance. The Bayesian
Network framework used here varies from the Dynamic Bayesian Network (DBN). We
begin with a predefined network’s initial state based on the natural causal
relationships between variables to learn the Bayesian Network from data. We
use the PC Algorithm with a Conditional Independent (CI) test on a condition
size of 0 using a $\chi^{2}$ test to filter out the independent edges and
learn the network’s structure. Then, we execute parameter learning to learn
the CPDs between variables using Maximum Likelihood estimation and Bayesian
parameter estimation. Once the network is learned from the dataset constructed
in the early stage, we visualize the learned network, especially the strength
of dependencies between different variables based on the $\chi^{2}$ value of
the edges calculated from the dataset. This visualization allows practitioners
to easily understand the relationships between different factors and how they
contribute to traffic accidents and other events. Finally, we apply our
learned Bayesian Network to analyze the causes of traffic accidents and
predict their occurrence. Given the value of other variables, we make the
network predict the occurrence of traffic accidents and congestion. The model
shows similar performance compared with other methods like DNN, KNN, and SVMs
while displaying good interpretability, analyzing the impacts of a single
variable on the probability of another event given the value of the single
variable as the evidence also indicates the impact of different factors on the
probability of traffic events. Our methodology involves setting the values of
certain variables based on real-world conditions and observing the resulting
changes in the probabilities of traffic accidents. Through this process, we
can gain valuable insights into the factors contributing to traffic accidents
and make accurate predictions about their occurrence.

The main contributions of this paper are as follows:

  * •

Dataset construction for Bayesian Network: We propose a data processing method
converting spatio-temporal entries into a Bayesian Network-learnable format,
addressing the lack of a dataset for learning. This approach pairs causal-
related events and matches them to nodes while enhancing data quality through
further analysis and sampling.

  * •

Dynamic Bayesian Network based data mining framework: We present a traffic
safety analysis framework, discovering causal relationships in spatio-temporal
domains using a Dynamic Bayesian Network. Applied to traffic data for accident
prediction and cause discovery, we introduce methods for mining information
such as node-specific inferences, strong relationships, and event influences.

  * •

Visualization: The learned network is visualized to analyze traffic event
causes, aiding in understanding variable relationships and contributions.
Examining nodes and edges allows identifying potential causes and preventative
measures, facilitating data-driven decisions and road safety improvements.

  * •

Interpretable findings and insights: Our framework’s interpretability
encompasses predefined structure, visible variable relationships, and manual
network improvement. Utilizing this approach, we successfully analyze factors
related to traffic accidents and congestion, achieving high prediction
accuracy, particularly for accidents.

The remaining sections of this paper are organized in the following structure:
Section 2 introduces the related work of this paper, Section 3 indicates some
preliminaries used in the following sections, Section 4 introduces how the
dataset used for learning the Bayesian Network is established, Section 6 shows
the methodology used while Section 7 displays the result of the experiments
and Section 8 is a conclusion.

##  2 Related Work

This section presents a comprehensive review of related work, starting with
examining prediction techniques for traffic events such as accidents,
exploring interpretability in machine learning, and discussing Bayesian
Networks as the fundamental basis of our proposed method in this paper.

###  2.1 Traffic Event Prediction

Initial efforts in traffic event prediction relied on simple statistical
models and manual data analysis. K-Nearest Neighbors (KNN) [5] and Support
Vector Machines (SVM) [6] are two early approaches used for classification and
regression. KNN is a non-parametric method that can classify traffic
situations [1] based on historical data and predict simple traffic states like
traffic flow [7]. However, KNN’s performance can degrade with high-dimensional
data, and a meaningful distance function is demanded to ensure good
classification and prediction performance. In the meanwhile, SVM are
supervised learning methods for similar purposes like classifying traffic
patterns [8] and predicting potential incidents [9]; unluckily, when the
dataset becomes more extensive and more complex, the performance of SVM
downgrades. Pattern discovery [10] is another data-driven method that can
identify recurring patterns or sequences from data. Previous work like [11]
has shown its ability to uncover common lines leading to traffic events in
ample amounts of traffic and weather data. A shortcoming of pattern discovery
is that it may struggle with noisy or incomplete data, especially for raw data
collected from the real world. Over time, advancements in computational power
and data collection techniques have enabled the usage of more sophisticated
prediction methods like machine learning. For instance, Deep Learning [12], a
subset of machine learning that employs artificial neural networks with
multiple layers (hence the term deep), has displayed extinguished performance
in traffic event prediction [13, 14].

###  2.2 Interpretable Machine Learning Approach

It is establishing trust in the context of traffic safety hinges upon the use
of interpretable models that provide comprehensible predictions. Models such
as decision trees, linear regression, rule-based systems like the RIPPER
algorithm, and logistic regression inherently provide interpretability. For
instance, decision trees generate outcomes based on a hierarchy of rules [15],
while linear regression quantifies relationships between predictors and
outcomes [16], making these models readily interpretable. Similarly, rule-
based systems employ a simple “if-then” rule set [17], and logistic regression
models the log-odds of the positive class for binary outcomes [18], both
offering transparent decision-making processes. These models deliver
predictions and elucidate the reasoning behind their outcomes, fostering
trust. However, primarily linear or rule-based, these models can struggle with
complex, non-linear relationships, especially when numerous variables are
involved. Thus, pursuing models that can decipher such intricate relationships
is necessary for a more comprehensive analysis of traffic safety data.

###  2.3 Bayesian Network in Traffic Prediction

Bayesian Network (BN) [19] is a probabilistic graphical model containing a set
of interdependent variables, and their conditional dependencies are
represented by a directed acyclic graph (DAG). As the conditional dependencies
[20] among variables of BN are explicitly represented via DAG, the model’s
decision-making process is transparent and interpretable. In other words, BN
is a White-Box model [21], thus providing insights into the underlying causal
mechanisms and contributing to more reliable and explainable predictions,
which significantly benefits traffic areas where trust is essential. In
traffic prediction areas, BN has been widely applied to predict the accident
[22, 23], traffic flow [24], driver’s behavior [25, 26] and can be used to
assist urban design [27] Packages like [28, 29, 30] have provided a series of
tools to learn with Bayesian Networks. Consequently, Bayesian Networks can
capture and model complex relationships among numerous variables and provide
interpretable and trustworthy predictions, making them ideal for applications
in traffic safety and other complex domains.

##  3 Preliminaries and Problem

In this section, we provide the terminology definition and the problem
definition.

###  3.1 Data Entries and Their Inter-relationships

#### Geospatial Entity:

Geospatial entry, represented by $e$, is the most fundamental event unit in
the given entries $E$. One geospatial entry is represented by a tuple
$[\textit{type},\textit{loc},\textit{time}]$, which represents an event of
type type that happened at location loc within the time interval time
$[\textit{start},\textit{end}]$. For traffic data, the location (loc) is
represented by [ latitude, longitude, Street_Name, Zipcode, City, State ], For
weather data, airport code is used to indicate location since the data are
collected from airport weather stations.

#### Spatio-temporal correlation:

Events that occurred nearby temporally and spatially are spatio-temporal
correlations. Temporal correlations can be determined by estimating the mean
of the start time difference of two entries, $e_{1}$ and $e_{2}$, with
$0\leqslant\left|e_{1}.\textrm{start}-e_{2}.\textrm{start}\right|\leq\textrm{T-thresh}$.
Here T-thresh is a time threshold. To ascertain the spatial correlation
between two event entries, we employ the Haversine distance function [31],
denoted as $\textrm{dist}(e_{1},e_{2})$ . This function calculates the
distance based on GPS coordinates. Spatial correlation is determined by
evaluating whether $\textrm{dist}(e_{1},e_{2})$ is less than D-thresh. When
considering pairs of weather and traffic events, we define collocation as a
match between the airport station reporting the weather entity and the station
nearest to the location of the traffic entity.

#### Causal Relationship

: For two spatio-temporal correlation geospatial entities $e_{1}$ and $e_{2}$,
their causal relationship can be defined as follows: If both entries are
traffic or weather events, the one with an earlier start is the cause, and the
latter is the result; otherwise, for two entries with different types, only
weather events that start earlier can cause the latter traffic event.

###  3.2 Dataset Processing

Balancing skewed data distributions via sampling techniques is critical for
training robust models. To achieve this, a balanced subset of the dataset will
be sampled from the origin dataset by the undersampling technique, which
selects a subset of the majority of samples while informed undersampling uses
heuristics to remove redundant or noisy samples. Tomek links flag majority
class samples $d_{i}$ that are closest to minority samples $x_{j}$:

(3.1) $dis(d_{i},d_{j})<dis(d_{i},d_{k})\hskip 5.0pt\forall\hskip
5.0ptd_{k}\in D_{maj}$

Where $D_{maj}$ is the majority class and $dis()$ is a distance metric.

###  3.3 Learning from Data via Bayesian Network

Learning from data via Bayesian Networks involves structure learning and
parameter learning. The PC (Peter-Clark) Algorithm [32] is often used for
structure learning, identifying the best network structure to fit observed
data with a set of CT-Tests. Parameter learning estimates the Conditional
Probability Distributions (CPDs) [33] for each variable, typically using
Maximum Likelihood Estimation (MLE) [34].

####  3.3.1 Structure Learning

The PC Algorithm uses statistical tests to identify conditional independencies
in data. A set of CI-Test estimates each edge, and iteratively removed edges
are conditionally independent. The CI Test determines if

(3.2) $X\perp\\!\\!\\!\perp Y|Z\iff P(X,Y|Z)=P(X|Z)P(Y|Z)$

where $X,Y$ are tested variables and $Z$ are conditioning variables.

Edge dependence is examined by the $\chi^{2}$-Test [35] defined as

(3.3) $\chi^{2}=\sum\frac{{(O_{ij}-E_{ij})^{2}}}{{E_{ij}}}$

where $\chi^{2}$ is the test statistic, $O_{ij}$ is the observed frequency,
and $E_{ij}$ the expected frequency. The $p$-value indicates significance as
$p=1-\text{CDF}(\chi^{2},\text{df})$ where CDF is the chi-squared distribution
CDF with $\chi^{2}$ and degrees of freedom df.

####  3.3.2 Parameter Learning

Bayesian networks estimate $P(\mathcal{V}|Pa(\mathcal{V}))$ representing each
variable $\mathcal{V}$ given parents $Pa(\mathcal{V})$. MLE calculates
$P(\mathcal{V}=v_{i}|Pa(\mathcal{V})=pa_{j})$ as

(3.4)
$\frac{N(\mathcal{V}=v_{i},Pa(\mathcal{V})=pa_{j})}{N(Pa(\mathcal{V})=pa_{j})}$

the proportion where $\mathcal{V}=v_{i}$ among those where
$Pa(\mathcal{V})=pa_{j}$.

Bayesian estimation addresses overfitting using Bayes’ theorem

(3.5) $P(\theta|D)=\frac{P(D|\theta)P(\theta)}{P(D)}$

with $P(D|\theta)$ the likelihood, $P(\theta)$ the prior distribution, and
$P(D)$ the evidence, accounting for parameter uncertainty.

###  3.4 Causal Discovery and Inference via Bayesian Network

Causal inference in Bayesian Networks involves determining the probability
distribution of a set of target variables given the observed values of another
set of variables. This is achieved using the concept of the joint probability
distribution.

####  3.4.1 Causal Inference

Given a Bayesian Network represented by graph $G$, given a set of values of n
variables $v={v_{1},...,v_{n}}$ where each node $v_{i}$ has parents
$Pa(v_{i})$, the joint probability distribution is compactly expressed as:

(3.6) $P(v_{1},...,v_{n})=\prod_{i-1}^{n}P({v_{i}|Pa(v_{i})})$

####  3.4.2 Causal Analysis

Now, if we have a set of target variables $T$, observed variables $O$, and
unobserved variables $U$, according to conditional probability, the posterior
distribution of the target variables given the observed variables is computed
as:

(3.7) $P(T|O)=\frac{P(T,O)}{P(O)}=\frac{\sum_{U}P(T,O,U)}{P(O)}$

This process allows us to make probabilistic predictions about the target
variables based on the observed data, thereby enabling causal inference.

###  3.5 Problem Definition

####  3.5.1 Prediction Task

In the prediction task, we provide the network with the observed values of a
set of variables, denoted as $O$. The network’s task is to predict the value
of the target variable $T$, which typically consists of a single variable,
here the target of prediction is the happening of the accident.

####  3.5.2 Analysis Task

In the analysis task, we set the observed variable set $O$ to different
values, then, we need to estimate the CPDs of target variable $T$ under the
different values of $O$ and compare the probability of the happening.

##  4 Dataset

In this section, we describe the dataset used in this paper. The dataset comes
from work introduced in [11], and can be accessed via the link
111https://smoosavi.org/datasets/lstw; in this paper, we construct a new
dataset based on this one that our proposed Bayesian Network framework can
learn.

###  4.1 Traffic Data

The traffic data provided in the dataset were collected in real-time using a
rest API provided by MapQuest 222https://www.mapquest.com from August 2016 to
the end of Dec 2020 for the Contiguous United States and includes about 31.4
million traffic events.

The taxonomy for traffic entities includes the following types: Accident: This
common type involves one or more vehicles and may result in fatalities.
Broken-Vehicle: This type represents a situation where one or more vehicles
are disabled on a road. Congestion: This type signifies a situation where the
speed of traffic is slower than expected. Construction: This type indicates an
ongoing construction or maintenance project on a road. Event: This type
encompasses situations such as sports events, concerts, or demonstrations that
could potentially impact traffic flow. Lane-blocked: This type pertains to
cases where one or more lanes are blocked due to traffic or weather
conditions.

###  4.2 Weather data

The weather data inside the dataset is collected from the weather station all
around the country, and have been processed to a format that contains the
type, severity, and location of the weather events.

In the dataset, weather events are classified as cold, fog, hail, rain, snow,
storm, precipitation, and their severity is classified into different levels
with K-Means clustering algorithm [36], the following types of weather events
have been defined: Temperature: The temperature in the dataset are divided
into five level with cluster center values (degrees are in Celsius)
$-23.7^{\circ}$, $-8.6^{\circ}$, $6.7^{\circ}$, $21.3^{\circ}$, and
$35.8^{\circ}$, which are referred to as severe-cold, cold, cool, warm, and
hot, and here only severe-cold is taken into the dataset. Hail: Solid
precipitation, including ice pellets and hail. Rain: Rain of any type, ranging
from light to heavy, with cluster centers 2.5, 7.1, and 11.6 millimeters.
Snow: Snow of any type, ranging from light to heavy with cluster centers 0.6,
1.7, and 2.5 millimeters. Wind: Wind speeds are classified into three groups
with centers $13.2km/h$, $36.2km/h$, and $60km/h$, respecting calm, moderate,
and storm windy conditions, here we only consider the extremely windy
condition, where the wind speed is at least $60km/h$. Precipitation: Any kind
of solid or liquid deposit different from snow or rain.

Figure 2: Dataset entry $d$ construction process for $e_{weather}$ and
$e_{traffic}$.

###  4.3 Dataset Structure

The dataset $D$ used for learning is constructed from data entities from $E$
according to the co-relations mentioned in Section 3.1, hence it has the same
structure as the predefined BN framework structure, which can be expressed as
$[T_{weather},T_{traffic},T_{weather}^{\prime},T_{traffic}^{\prime}]$ mention
in Section 5.

To construct the dataset $D$ from data $E$, we iterate each data entry $e$
after finding the causal relationships between events, each entry $e$ has two
groups of entries $E_{cause}$ and $E_{result}$ representing events that cause
$e$ and events caused by $e$.

For traffic entries $e_{traffic}$, we set itself at related variable in
$T_{traffic}^{\prime}$ subgroup; then, the weather entries in $E_{cause}$ of
$e_{traffic}$ are put at $T_{weather}^{\prime}$, the traffic entries in
$E_{cause}$ of $e_{traffic}$ are put at $T_{traffic}$, and weather entries in
$E_{cause}$ of $e_{traffic}$s in $T_{traffic}$ are put in $T_{weather}$. Since
traffic events won’t result in weather events, so there is no weather event
contained in the results of traffic events.

For weather entries $e_{weather}$, we set the related variable in
$T_{weather}$ subgroup, the traffic entries in $E_{result}$ are put in
$T_{traffic}$ and the traffic entries in $E_{result}$ of $e_{traffic}$ in
$T_{traffic}$ are put at $T_{traffic}^{\prime}$

Figure 2 described the relationship of entries in parents or children nodes
and variables in the dataset in a more clear approach.

##  5 Bayesian Network Framework

We propose a novel framework based on Bayesian Networks in probabilistic
graphical models. There are two kinds of causal relationships between
geospatial entities $e$s in $E$: weather events to traffic events and traffic
events to other traffic events. Since such pairs of events have temporal
correlations, a variant of Dynamic Bayesian Networks (DBN) can be applied to
model the causal relationships between weather and traffic events in $E$. A
typical DBN utilizes temporal and probabilistic relationships to model the
complex system’s behavior with changing variables over time. The Bayesian
Network framework used in this paper is presented in Fig 3, nodes are
bifurcated into two distinct groups, namely the $T$ and $T^{\prime}$, based on
temporal sequence, while each node $\mathcal{V}_{i}$ in $T$ has its paired
node $\mathcal{V}_{i}^{\prime}$ in $T^{\prime}$. Events in $T$ happen before
those in $T^{\prime}$.
$T_{weather}=\\{\mathcal{V}_{snow},\mathcal{V}_{rain},\mathcal{V}_{wind},...\\}$
and
$T_{traffic}=\\{\mathcal{V}_{acci},\mathcal{V}_{flow},\mathcal{V}_{cong},...\\}$
are two subgroups within group $T$, while
$T_{weather}^{\prime}=\\{\mathcal{V}_{snow}^{\prime},\mathcal{V}_{rain}^{\prime},\mathcal{V}_{wind}^{\prime},...\\}$
and
$T_{traffic}^{\prime}=\\{\mathcal{V}_{acci}^{\prime},\mathcal{V}_{flow}^{\prime},\mathcal{V}_{cong}^{\prime},...\\}$
are corresponding subgroups in $T^{\prime}$. The blue arrows indicate the
causal relationships between subgroups, while the black arrows indicate the
relationship between nodes. According to the situation in the real world, we
predefined four casual relationships between subgroups, from the weather
entries subgroup to the traffic entries subgroup in the same group and from
the former of the weather of traffic entries to the corresponding one.

Figure 3: An example of proposed Bayesian Network framework. $T$ and
$T^{\prime}$ represent two groups, while $T_{weather}$ and $T_{traffic}$
represent subgroups in $T$, $T_{weather}^{\prime}$ and $T_{traffic}^{\prime}$
represent subgroups in $T^{\prime}$. The blue arrows indicate the causal
relationships between subgroups, and the black edges indicate causal
relationships between nodes.

##  6 Method

This section introduces how we use our proposed framework to learn from
traffic data and perform prediction and analysis.

###  6.1 Baselines

#### Logistic Regression (LR) [37]

This approach has been identified as working well in accident prediction by
previous work [38]. Thus, we employ it as a feasible baseline in our binary
prediction task.

#### Deep Neural Network (DNN)

We employ a four-layer feed-forward neural network with three hidden layers
(512, 256, 64), ReLU activation, softmax output, batch normalization [39]
after the second and third layers, and Adam optimizer [40] with a 0.01
learning rate to evaluate the performance of our framework.

#### Support Vector Machine (SVM)

In machine learning classification tasks, Support Vector Machines (SVM) are a
go-to choice for which are highly skilled at managing related data and
intricate decision boundaries.

#### K Nearest Neighbors (KNN)

KNN, or k-nearest neighbors, is another popular algorithm in ML used for
classification and regression tasks.

###  6.2 Dataset Construction & Processing

To simplify the analysis, prediction, and visualization process, weather
events with multiple levels of severity in the dataset will be merged into
binary events. This will allow for easier understanding of the data. To
analyze the impact of this operation, we will test the probability of an
accident under different severity of rain and snow. By converting all the
events to binary ones (only with state YES and NO), the dataset is constructed
according to the format mentioned in Section 4.3. The constructed dataset will
be divided into several subsets according to the cities data entries belong
to; then, to reduce the bias contained in data and thus improve the prediction
performance, each subgroup is sampled to make the number of accident
$d_{Accident}$ and non-accident entries $d_{Non-accident}$ equal. In the
experiment, traffic accident prediction will be done on selected subsets and
impact analysis on the whole dataset.

###  6.3 Casual Relationship Visualization

During the structure learning process, the $\chi^{2}$ value as well as
$p$-value of each edge have been obtained, and the visualization process is
determined based on them. The edges are divided into multiple groups according
to their $\chi^{2}$ value since their degree of freedom is the same. This
allows us to filter the edges according to their $\chi^{2}$ values or the
nodes they connect. Three kinds of relationships are analyzed: 1) the
strongest relations between the nodes, in which only the strongest edges are
preserved; 2) the cause of accidents, in which only the edges directly or
indirectly pointing to accidents should remain; 3) the cause of congestion, in
which only the edges directly or indirectly pointing to congestion remain.
Then the general trend of the relationships and some distinguished causal
patterns are extracted from the network to show how the events affect each
other.

###  6.4 Traffic Event Prediction

In this phase, we employ the learned Bayesian Network to forecast traffic
accidents; subsequently, we evaluate the accuracy of these predictions and
compare the result with other machine learning approaches introduced in Sec
6.1. To accomplish this, we sample a series of test datasets from selected
subdatasets from cities including Atlanta (AT), Austin (AU), Charlotte (CH),
and Dallas (DA), each of the test sets containing 1000 positive and 1000
negative entries for the prediction target, accident. We provide the network
with the observed values of a set of variables, denoted as $O$. The network’s
task is to predict the value of the target variable $T$, which is accident
here. The target variable represents either an accident or congestion.
Finally, we evaluate the accuracy of the network’s predictions by three
metrics: Accident (Acc) Prediction Accuracy, No-Accident Prediction Accuracy,
and Weighted Average (W-Ave), or F1-Score, which is expressed as
$\begin{aligned} \@add@centering\centering\text{ Precision }&=\frac{\text{
true positive }}{\text{ true positive }+\text{ false positive }}\\\ \text{
Recall }&=\frac{\text{ true positive }}{\text{ true positive }+\text{ false
negative }}\\\ \text{ F1-Score }&=\frac{2\times\text{ Precision }\times\text{
Recall }}{\text{ Precision }+\text{ Recall }}\end{aligned}$

###  6.5 Event Influence Analysis

To analyze the influence of single factors on the probability of accidents, we
set the single factor as the observed value of variable set $O$; then, we need
to estimate the CPDs of target variable $T$ under the different values of $O$
and compare the probability of the happening. If the probability of $T$
increase when the value of $O$ switch from NO to YES, we can say $O$ could
increase the probability of $T$. This paper studies the influence of other
traffic and weather events with an edge directly connected to accidents.

##  7 Experiments and Results

This section contains the experiments and their results. Firstly we
investigate the influence of the severity of weather events rain and snow
traffic events accident and congestion. Then we visualize the learned network
to analyze the relationship between variables. Further, the performance of the
prediction of the model is tested. Finally, the influence of single events on
another is investigated. This section is implemented by pgmpy [30] Python
library.

###  7.1 Influence of Different Severity of Rain and Snow

(a) Influence of rain or snow on accident

(b) Influence of rain or snow on congestion

Figure 4: The probability of accident (a) and congestion (b) under different
severity of rain or snow

When we first define the elements in the dataset, we divide the rain and snow
events into several levels according to their severity, and we investigate
whether the influence of the weather events varies with different severity. We
investigated the probability of accidents and congestion under different
severity of rain or snow. The results are shown in Fig 4; we can see that in
Fig 4(a), the probability of accident remains almost the same under different
severity of rain and drops when there is no rain; the same result also
appeared under different severity of snow. From Fig 4(b), we can find out that
the probability of congestion remains almost the same under different severity
of rain and snow and decreases a little when there is no rain and snow. From
this experiment, we conclude that the severity of rain and snow won’t affect
the probability of accident and congestion and we can consider rain and snow
as binary variables like others.

###  7.2 Network Structure and Relationship

(a) Strong Co-relations

(b) Edges point to accident

(c) Edges point to congestion

Figure 5: Relationships between different variables: in (a), the strongest
relationships remained; in (b), nodes with edges directly or indirectly point
to accident and related edges remained; in (c), nodes with edges directly or
indirectly point to congestion and related edges are retained. Foe the edges,
different linewidth indicated different level of p-value clustered by
k-nearest neiburs. For the nodes, Acci = Accident, Cong = Congestion, Flow =
Flow-Incident, Cons = Construction, Lane = Lane-Blocked, Bro = Borken-Vehicle,
Pre = Precipitation

In the structure learning session, we execute a series of CI-Tests ($\chi^{2}$
test) with the size of conditional set $Z$ equal to 0 to every edge we
predefined, meanwhile, the $p$-values of the edges are computed and, the edge
with $p$-value larger than 0.05 are removed in this session. Since all the
variables here are binary, so the degree of freedom of each CI-Test is the
same as 1, we can directly compare the $\chi^{2}$ value of each CI-Test to
compare the strength of the relationships. We divided the edges into several
groups according to their $\chi^{2}$ values according to k-nearest neighbors
clustering, and indicating then by different linewidth of edges.

Therefore, according to the $\chi^{2}$ values, we visualize the variables and
the edges, and in each figure, edges, and nodes are filtered in different
methods to catch different kinds of relationships between variables. In Fig
5(a), the edges with $\chi^{2}$ larger than 10000 remain; from the edges in
the figure, we could find out that the weather event rain has a strong impact
on multiple events including accident, lane-blocked, borken-vehicle and have a
strong indirect impact to congestion via affecting accident, we can also find
out that congestion events have high potential to cause other congestion
events. In Fig 5(b), only events that may finally directly or indirectly
affect the accident event are retained. From the figure, we can find out that
weather events, including rain, fog, and traffic events including congestion,
are the main cause of traffic accidents; meanwhile, events like precipitation,
snow, severe cold, flow incident, congestion, and other accidents, which is
potentially caused by Rain or Fog could also lead to the happening of traffic
accidents. And in Fig 5(c) we investigate the cause of congestion via
filtering related edges and nodes; the figure shows that the main causes of
congestion are other traffic events like previous congestion, accidents,
broken-vehicle, and construction, while such traffic may be caused by weather
events like rain, fog, and snow.

###  7.3 Result of Traffic Event Prediction

Table 1: Inference performance in different metrics for accident of our proposed framework and baselines. For the cities, AT = Atlanta, AU = Austin, CH = Charlotte, and DA = Dallas Model | Metric | AT | AU | CH | DA
---|---|---|---|---|---
LR | Acc | 0.54 | 0.58 | 0.56 | 0.3
Non-Acc | 0.91 | 0.93 | 0.91 | 0.94
W-Ave | 0.83 | 0.87 | 0.83 | 0.87
DNN | Acc | 0.62 | 0.62 | 0.61 | 0.36
Non-Acc | 0.89 | 0.92 | 0.87 | 0.94
W-Ave | 0.83 | 0.87 | 0.82 | 0.87
SVM | Acc | 0.75 | 0.80 | 0.69 | 0.75
Non-Acc | 0.96 | 0.95 | 0.97 | 0.97
W-Ave | 0.47 | 0.62 | 0.27 | 0.47
KNN | Acc | 0.50 | 0.78 | 0.61 | 0.72
Non-Acc | 0.43 | 0.90 | 0.59 | 0.89
W-Ave | 0.46 | 0.73 | 0.53 | 0.48
BN (ours) | Acc | 0.65 | 0.73 | 0.60 | 0.65
Non-Acc | 0.76 | 0.90 | 0.31 | 0.78
W-Ave | 0.61 | 0.67 | 0.69 | 0.59

Based on the prediction results in the table 1, our proposed BN framework
performs comparably well against other machine learning models in predicting
accident cases and achieves competitive weighted average F1 scores.
Specifically, BN achieves higher accuracy than LR, DNN, and KNN in predicting
accident cases (Acc) on the majority of datasets (AT, AU, and DA) and closes
the SVM which is the highest one, demonstrating its advantage in capturing
complex relationships between variables and deal with tabular dataset compare
with DNN. For non-accident prediction accuracy (Non-Acc), BN outperforms
others on the AT dataset but is weak in others, suggesting a better data
balancing method should be found to reduce the bias of BN. While SVM achieves
the best individual metrics on some datasets, its accuracy and weighted
averages varies a lot between different cities, showing a lack of robustness.
In contrast, BN delivers more stable predictions with leading accuracy across
accident and non-accident cases within selected cities, with the highest
average F1 scores on the CH dataset and relatively high ones on other
datasets.

Figure 6: Inference performance in different metrics for accident of our
proposed framework and baselines.

Figure 6 indicates a more general result of inference performance as the
average of four cities; as we can see, our proposed BN framework shows the
most balanced performance within all the methods, specially the average W-Ave
(F1-Score) of SVM is greatly pulled down by low score in CH, which shows its
lake of rubustness within different cities.

In summary, by incorporating prior knowledge about conditional dependencies,
the Bayesian network can achieve overall balanced, robust and competitive
performance compared to other general models. This confirms the benefit of BN
in safety-critical applications like accident prediction under its ability to
model complex relationships in the data.

###  7.4 Analysis of the Influence of Single Variable to Another

(a) weather to accident.

(b) traffic to accident

Figure 7: (a) Accuracy, precision, recall and F1 score of inference result for
variable accident-latter with our proposed Bayesian Network framework, as well
as KNN and SVM. (b) The probability of an accident under the evidence that
weather event is happening or not happening. (c) The probability of an
accident when other traffic events happen former. In the figure Acci =
Accident, Cong = Congestion, Flow = Flow-Incident, Cons = Construction, Lane =
Lane-Blocked, Bro = Borken-Vehicle, Pre = Precipitation

After inferencing, we further dive into the learned model itself. For the two
most concerned variables, accident, and congestion, we study how much their
probability vary when given the exact value of one of the remaining variables.

Fig 7(a) show the impact of different weather events on the accident events,
we can see that the existence of all the weather events, including rain, snow,
cold, storm, fog, precipitation, and hail could significantly affect the
probability of accident, especially when raining, the probability of accident
almost reach 60%, and when there is no rain, the probability of accident drow
to the lowest of seven to under 30% which means rain is one of the most potent
factors that affect the happening of accidents. The impact of other traffic
events on the probability of accident is depicted in Fig 7(b). The influence
of congestion and flow-incidents on accident is minor. In contrast, a
previously occurred accident decreases the probability of a subsequent
accident, potentially due to congestion caused by the accident. Other events
such as Lane-Blocked, Broken-Vehicle, and Construction also increase the
probability of accident by approximately 10% for the first two and 5% for the
last one.

##  8 Conclusion

This paper presented an interpretable Bayesian network framework for traffic
accident analysis and prediction. Through constructing a spatially and
temporally aware dataset capturing causal relationships between weather,
traffic events and accidents, the proposed framework learns network structure
via the PC-algorithm with $\chi^{2}$ conditional independence tests. Parameter
learning then estimates probabilities from the dataset. Visualizing the
learned Bayesian network elucidates strong relationships between variables and
causes of accidents/congestion, demonstrating the model’s interpretability
advantage. Performance evaluation against baseline methods showed that our
framework achieved comparable prediction accuracy while maintaining a
relatively lower false negative rate, which is an important property for
safety-critical applications. Notably, the learned network successfully
analyzed impacts of different events on accident probabilities. As an
inherently interpretable ”white-box” model achieving state-of-the-art
performance, our approach uniquely enables trustworthy insights for reducing
traffic dangers. Future work will incorporate more sophisticated data pre-
processing and mining of the learned network within our framework to further
enhance prediction and understanding of causality in traffic systems. Overall,
the results illustrate Bayesian networks’ potential for interpretable and
effective modelling of complex transportation safety problems.

## References

  * [1] S. L. Karri, L. C. De Silva, D. T. C. Lai, and S. Y. Yong, “Classification and prediction of driving behaviour at a traffic intersection using svm and knn,” _SN computer science_ , vol. 2, pp. 1–11, 2021.
  * [2] I. Goodfellow, Y. Bengio, and A. Courville, _Deep learning_. MIT press, 2016.
  * [3] Z. C. Lipton, “The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery.” _Queue_ , vol. 16, no. 3, pp. 31–57, 2018.
  * [4] B. Mihaljević, C. Bielza, and P. Larrañaga, “Bayesian networks for interpretable machine learning and optimization,” _Neurocomputing_ , vol. 456, pp. 648–665, 2021.
  * [5] T. Cover and P. Hart, “Nearest neighbor pattern classification,” _IEEE Trans. Inf. Theory_ , vol. 13, no. 1, pp. 21–27, Jan. 1967.
  * [6] C. Cortes and V. Vapnik, “Support-vector networks,” _Mach. Learn._ , vol. 20, no. 3, pp. 273–297, Sep. 1995.
  * [7] L. Zhang, Q. Liu, W. Yang, N. Wei, and D. Dong, “An Improved K-nearest Neighbor Model for Short-term Traffic Flow Prediction,” _Procedia - Social and Behavioral Sciences_ , vol. 96, pp. 653–662, Nov. 2013.
  * [8] A. Este, F. Gringoli, and L. Salgarelli, “Support vector machines for tcp traffic classification,” _Computer Networks_ , vol. 53, no. 14, pp. 2476–2490, 2009.
  * [9] J. Tang, L. Zheng, C. Han, W. Yin, Y. Zhang, Y. Zou, and H. Huang, “Statistical and machine-learning methods for clearance time prediction of road incidents: A methodology review,” _Analytic Methods in Accident Research_ , vol. 27, p. 100123, 2020.
  * [10] J. Han, H. Cheng, D. Xin, and X. Yan, “Frequent pattern mining: current status and future directions,” _Data mining and knowledge discovery_ , vol. 15, no. 1, pp. 55–86, 2007.
  * [11] S. Moosavi, M. H. Samavatian, A. Nandi, S. Parthasarathy, and R. Ramnath, “Short and Long-term Pattern Discovery Over Large-Scale Geo-Spatiotemporal Data,” in _KDD ’19: Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. New York, NY, USA: Association for Computing Machinery, Jul. 2019, pp. 2905–2913.
  * [12] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” _Nature_ , vol. 521, pp. 436–444, May 2015.
  * [13] C. Dong, C. Shao, J. Li, and Z. Xiong, “An improved deep learning model for traffic crash prediction,” _Journal of Advanced Transportation_ , vol. 2018, pp. 1–13, 2018.
  * [14] Z. Yuan, X. Zhou, and T. Yang, “Hetero-convlstm: A deep learning approach to traffic accident prediction on heterogeneous spatio-temporal data,” in _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 2018, pp. 984–992.
  * [15] M. S. Tehrany, B. Pradhan, and M. N. Jebur, “Spatial prediction of flood susceptible areas using rule based decision tree (dt) and a novel ensemble bivariate and multivariate statistical models in gis,” _Journal of hydrology_ , vol. 504, pp. 69–79, 2013.
  * [16] K. Kumari, S. Yadav _et al._ , “Linear regression analysis study,” _Journal of the practice of Cardiovascular Sciences_ , vol. 4, no. 1, p. 33, 2018.
  * [17] A. Abraham, “Rule-based expert systems,” _Handbook of measuring system design_ , 2005.
  * [18] V. Carey, S. L. Zeger, and P. Diggle, “Modelling multivariate binary data with alternating logistic regressions,” _Biometrika_ , vol. 80, no. 3, pp. 517–526, 1993.
  * [19] J. Pearl, “Bayesian netwcrks: A model cf self-activated memory for evidential reasoning,” in _Proceedings of the 7th conference of the Cognitive Science Society, University of California, Irvine, CA, USA_ , 1985, pp. 15–17.
  * [20] A. P. Dawid, “Conditional independence in statistical theory,” _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , vol. 41, no. 1, pp. 1–15, 1979.
  * [21] P. J. Driscoll, G. S. Parnell, and D. L. Henderson, _Decision making in systems engineering and management_. John Wiley & Sons, 2022.
  * [22] M. Hossain and Y. Muromachi, “A bayesian network based framework for real-time crash prediction on the basic freeway segments of urban expressways,” _Accident Analysis & Prevention_, vol. 45, pp. 373–381, 2012.
  * [23] J. Sun and J. Sun, “A dynamic bayesian network model for real-time crash prediction using traffic speed conditions data,” _Transportation Research Part C: Emerging Technologies_ , vol. 54, pp. 176–186, 2015.
  * [24] S. Sun, C. Zhang, G. Yu, N. Lu, and F. Xiao, “Bayesian network methods for traffic flow forecasting with incomplete data,” in _Machine Learning: ECML 2004: 15th European Conference on Machine Learning, Pisa, Italy, September 20-24, 2004. Proceedings 15_. Springer, 2004, pp. 419–428.
  * [25] X. Zhu, Y. Yuan, X. Hu, Y.-C. Chiu, and Y.-L. Ma, “A bayesian network model for contextual versus non-contextual driving behavior assessment,” _Transportation research part C: emerging technologies_ , vol. 81, pp. 172–187, 2017.
  * [26] G. Xie, H. Gao, B. Huang, L. Qian, and J. Wang, “A driving behavior awareness model based on a dynamic bayesian network and distributed genetic algorithm,” _International Journal of Computational Intelligence Systems_ , vol. 11, no. 1, pp. 469–482, 2018.
  * [27] J. Tang, H. Heinimann, K. Han, H. Luo, and B. Zhong, “Evaluating resilience in urban transportation systems for sustainability: A systems-based bayesian network model,” _Transportation Research Part C: Emerging Technologies_ , vol. 121, p. 102840, 2020.
  * [28] J. Jiang, Z. Wen, A. Mansoor, and A. Mian, “Fast parallel exact inference on bayesian networks,” in _Proceedings of the 28th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming_ , 2023, pp. 425–426.
  * [29] E. Taskesen, “Learning Bayesian Networks with the bnlearn Python Package.” Jan. 2020. [Online]. Available: https://erdogant.github.io/bnlearn
  * [30] A. Ankan and A. Panda, “pgmpy: Probabilistic graphical models using python,” in _Proceedings of the 14th Python in Science Conference (SCIPY 2015)_. Citeseer, 2015.
  * [31] J. d. Mendoza y Rios _et al._ , “Memoria sobre algunos métodos nuevos para calcular la longitud por las distancias lunares, y aplicación de su teórica a la solución de otros problemas de navegación/por don joseph de mendoza y rios,” _Madrid : Imprenta Real_ , 1795.
  * [32] P. Spirtes and C. Glymour, “An algorithm for fast recovery of sparse causal graphs,” _Social science computer review_ , vol. 9, no. 1, pp. 62–72, 1991\.
  * [33] S. M. Ross, “Introduction to probability models fifth edition,” 1993.
  * [34] R. J. Rossi, _Mathematical statistics: an introduction to likelihood based inference_. John Wiley & Sons, 2018.
  * [35] K. Pearson, “X. on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling,” _The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science_ , vol. 50, no. 302, pp. 157–175, 1900.
  * [36] A. Likas, N. Vlassis, and J. J. Verbeek, “The global k-means clustering algorithm,” _Pattern recognition_ , vol. 36, no. 2, pp. 451–461, 2003.
  * [37] “MapQuest,” Jun. 2023, [Online; accessed 25. Jun. 2023]. [Online]. Available: https://www.mapquest.com
  * [38] L.-Y. Chang, “Analysis of freeway accident frequencies: Negative binomial regression versus artificial neural network,” _Saf. Sci._ , vol. 43, no. 8, pp. 541–557, Oct. 2005.
  * [39] S. Ioffe and C. Szegedy, “Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift,” in _International Conference on Machine Learning_. PMLR, Jun. 2015, pp. 448–456. [Online]. Available: http://proceedings.mlr.press/v37/ioffe15.html
  * [40] D. P. Kingma and J. Ba, “Adam: A Method for Stochastic Optimization,” _arXiv_ , Dec. 2014.


# Quantum Teleportation Coexisting with Conventional Classical Communications
in Optical Fiber

Jordan M. Thomas  Center for Photonic Communication and Computing,
Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
<EMAIL_ADDRESS>Fei I. Yeh  International Center for
Advanced Internet Research, Northwestern University, 750 N. Lake Shore Drive,
Chicago, IL 60611, USA  Jim Hao Chen  International Center for Advanced
Internet Research, Northwestern University, 750 N. Lake Shore Drive, Chicago,
IL 60611, USA  Joe J. Mambretti  International Center for Advanced Internet
Research, Northwestern University, 750 N. Lake Shore Drive, Chicago, IL 60611,
USA  Scott J. Kohlert  Ciena Corporation, 7035 Ridge Road, Hanover, MD 21076,
USA  Gregory S. Kanter  Center for Photonic Communication and Computing,
Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA  NuCrypt,
LLC, 1460 Renaissance Drive, Suite #205, Park Ridge, IL 60068, USA  Prem Kumar
Center for Photonic Communication and Computing, Northwestern University, 2145
Sheridan Road, Evanston, IL 60208, USA  Department of Physics and Astronomy,
Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3112, USA

###### Abstract

The ability for quantum and classical networks to operate in the same optical
fibers would aid the deployment of quantum network technology. However,
quantum performance can be susceptible to noise photons generated by
spontaneous Raman scattering of high-power coexisting classical light. Quantum
teleportation is a fundamental operation in quantum networking, but has yet to
be demonstrated in fibers populated with high data rate conventional optical
signals. In this paper, we demonstrate a three-node quantum state
teleportation system coexisting with 400-Gbps C-band classical communications
in 30.2 km of fiber. To protect quantum fidelity, Raman noise rates are
suppressed using optimized O-band quantum channels and filtering in multiple
degrees of freedom. Fidelity is shown to be well maintained with elevated
classical powers as high as 18.7 dBm, which could support multiple classical
channels with many terabits/s aggregate data rates. These results show the
feasibility of advanced quantum and classical network applications operating
within a unified fiber infrastructure.

††journal: opticajournal

##  1 Introduction

Figure 1: (a) Conceptual diagram of the experiment. Alice prepares a qubit
$|\psi\rangle_{A}$ that she wishes to transfer onto one photon of an entangled
Bell state photon pair at Bob via quantum teleportation. Alice encodes
$|\psi\rangle_{A}$ onto a single photon, which is multiplexed into a fiber of
length $L_{AC}$ to co-propagate with a high-rate classical communications
signal to a node at Charlie. The classical signal by-passes Charlie’s node to
continue propagating over a fiber of length $L_{BC}$. Two photons are prepared
in an entangled Bell state at Bob’s node. One photon counter-propagates with
the classical signal over this fiber to Charlie to undergo a Bell state
measurement (BSM) with Alice’s qubit. After a BSM detection, Alice’s state is
destroyed whilst the other entangled photon at Bob’s node is projected onto
the state $\sigma|\psi\rangle_{A}$, where $\sigma$ is a unitary transformation
unique to the BSM result. (b) Experimental implementation. Quantum and
classical signals coexist in 30.2  km of fiber ($L_{AC}=15.2\text{ km}$,
$L_{BC}=15.2\text{ km}$). Alice and Bob both generate photon pairs with
wavelengths of 1290 nm and 1310 nm via non-degenerate spontaneous parametric
down-conversion. Alice’s source heralds a single photon, whereas Bob prepares
a two-photon entangled state. Qubits are encoded in polarization and the
1290-nm photons from each source are transmitted to Charlie for the BSM. The
classical signal has a rate of 400-Gbps and is allocated to the 1547.32 nm
C-band wavelength channel. This signal travels an additional 24-km of deployed
fiber before and after multiplexing, giving a total classical link distance of
78.2 km (FPC = fiber polarization controller, DWDM = dense wavelength division
multiplexer, FWDM = O-band/C-band WDM, PPLN = periodically poled lithium
niobate waveguide, $\lambda/2$ = half-wave plate, $\lambda/4$ = quarter-wave
plate, LP = linear polarizer, LCR = liquid crystal retarder, PBS = polarizing
beam splitter, FBS = 50:50 fiber coupler, VDL = variable optical delay line,
FBG = fiber Bragg grating, CIR = circulator, $\text{D}_{j}$ = superconducting
nanowire single photon detector, C = common port, P = pass port, R = reflect
port, EDFA = erbium doped fiber amplifier).

The optical fiber infrastructure and telecommunications technology that
underlie the Internet have been widely adopted by researchers seeking to
develop quantum networks capable of applications such as quantum-enhanced
cryptography, sensing, and networked quantum computing [1, 2]. However, since
the majority of the existing fiber infrastructure is populated with classical
network signals, whether or not quantum networking can be realized on a large
scale will depend on the ability for both quantum and classical networks to
operate in the same fibers due to limited existing fiber connections and the
high cost of new installation.
Quantum and classical signals can readily share a single fiber via standard
wavelength division multiplexing. However, inelastic scattering of high-power
classical light can generate noise photons that obscure the detection of
typically sub-photon-level quantum signals. Spontaneous Raman scattering
(SpRS) is most prominent due to its broadband spectrum that spans the telecom
bands. Without careful design, classical signals require significant
attenuation to perform quantum operations, resulting in a trade-off between
achievable classical data rates or quantum fidelity.
Quantum-classical integration has been studied for various systems, such as
weak coherent states (WCSs) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], entangled
photon pairs [13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Bell state measurements
(BSMs) on WCSs [23, 24], continuous variables [25], and squeezed light [26].
However, many systems that are integral to future quantum network
architectures, such as those utilizing quantum teleportation, remain
unexplored. Using the non-local properties of entanglement, quantum
teleportation enables the transfer of a quantum state between two distant
physical systems without direct transmission of the systems themselves [27].
It plays a foundational role in advanced applications such as quantum relays
[28], quantum repeaters [29], networking quantum computers [30], and other
applications in quantum science and technology [31].
Recent years have seen impressive progress towards implementing teleportation-
based systems in quantum-dedicated fiber networks [32, 33, 34, 35, 36, 37,
31], but teleportation over fibers carrying conventional classical
communications signals has yet to be demonstrated. The ability to perform
teleportation alongside modern classical networking without sacrificing
fidelity would be an important step in ensuring advanced quantum networks can
make full use of the existing fiber infrastructure.
In this paper, we demonstrate a three-node quantum state teleportation system
operating over 30.2 km of optical fiber that simultaneously carries high-power
C-band classical communications at a rate of 400 Gbps. Teleportation is
performed via a joint BSM on a single photon and one member of an entangled
Bell state photon pair [38]. Utilizing wavelength multiplexing of quantum and
classical signals, a heralded single photon carrying a polarization encoded
qubit and one photon of an entangled Bell state photon pair are distributed
over roughly 15 km of fiber to undergo a BSM near the midpoint of the fiber,
whilst classical light traverses the full 30.2 km link plus an additional 48
km of deployed fiber.
We implement multiple techniques to minimize the degradation of quantum
fidelity by SpRS noise. At the quantum source nodes, we generate and
distribute photons with O-band wavelengths <1300 nm. Due to the multi-mode
SpRS spectrum, these are the telecom wavelengths with nearly the minimum SpRS
photon generation probability from the C-band classical source [17]. At
quantum receiver nodes, we employ narrow-band spectral filtering and
coincidence detection temporal filtering to reject uncorrelated in-band noise
photons. This combines various approaches used in high-power quantum-classical
coexistence studies of WCSs [9, 10, 11, 12], entangled photons [17], and BSMs
on two WCSs [24], but has not been investigated in the context of quantum
teleportation.
We first demonstrate that our system’s exceptionally low noise rates allow for
no significant impact on entanglement distribution to the BSM nor Hong-Ou-
Mandel (HOM) interference [39] between independent photons, which are the
underlying operations of the photonic quantum teleportation protocol [38]. We
then perform quantum state teleportation, where we measure an average state
transfer fidelity of $89.8\pm 3.1$% with elevated classical C-band powers as
high as 74 mW (18.7 dBm) launched into the fiber link. We discuss methods to
improve both raw teleportation fidelity and noise-resilience as well as the
implications of our results for future quantum networking.
This experiment demonstrates approaches for integrating many key classical and
quantum network components, including high-rate C-band classical signals and
optical amplifiers, single photon and entangled photon pair distribution, and
multi-qubit operations such as BSMs all operating simultaneously in the same
fibers.

##  2 Experimental Design

A conceptual diagram of the experiment is shown in Fig. 1(a). Alice generates
a single photon to encode the quantum state
$|\psi\rangle_{A}=\alpha|0\rangle+\beta|1\rangle$ that she wishes to teleport
to Bob. Alice’s photon is multiplexed into an optical fiber of length $L_{AC}$
to co-propagate with a classical communications signal to a BSM node at
Charlie. The classical signal is de-multiplexed just prior to the BSM and then
re-multiplexed to bypass Charlie’s node. The classical signal then travels
along another fiber of length $L_{BC}$, counter-propagating with respect to
one photon from an entangled Bell state photon pair generated at Bob’s node.
During the BSM at Charlie, both photons are irreversibly destroyed by the
detection whilst Bob’s other photon of the entangled pair is projected onto
the state $|\psi\rangle_{B}=\sigma|\psi\rangle_{A}$, where $\sigma$ is a
unitary operation that is unique to the BSM result and can be classically
communicated to Bob to reconstruct Alice’s state [27].
Fig. 1(b) shows the physical realization of the experiment. In our case,
Alice’s state is encoded onto a heralded single photon’s polarization,
$L_{AC}$ is 15.2 km, and $L_{BC}$ is 15.0 km. Thus, the quantum state transfer
distance, and the distance in which quantum and classical signals coexist, is
30.2 km.
The classical source consists of a 400-Gbps C-band transceiver operating at
1547.32 nm (Ciena WaveLogic 5 Nano 400ZR). To reduce the amount of SpRS noise
photons that are generated into the quantum channels, we strategically place
our quantum signals in the O-band so that they have an anti-Stokes frequency
detuning from the C-band as well as suppressed Raman gain at the far offset
detuning between the C- and O-bands [4]. Instead of the commonly used 1310-nm
wavelength allocation [9, 10, 11, 12, 24], our transmitted quantum signals are
at a wavelength of 1290 nm. This channel has roughly an order of magnitude
lower SpRS noise due to the non-uniform SpRS spectrum across the O-band [17],
resulting in a higher tolerance to C-band power.
Alice and Bob generate their respective quantum signals via type-0 cascaded
second harmonic generation-spontaneous parametric down conversion (c-SHG-SPDC)
photon pair generation [40]. Alice and Bob both have independent periodically-
poled lithium niobate (PPLN) waveguides that are each phase matched for SHG at
650 nm, which pumps the SPDC process to generate a two-mode squeezed vacuum
state with a non-degenerate joint spectrum centered around 1300 nm. This state
approximates a single photon pair given the probability of generating a pair
per mode is $\ll 1$. The cascaded second order nonlinear process is analogous
to four-wave mixing, which we pump using a 1300-nm distributed feedback laser
that is intensity modulated using a lithium niobate modulator to generate
pulses with an approximate 65-ps temporal full-width at half maximum (FWHM)
and a repetition rate of 500 MHz. This pump is then split and directed to
Alice and Bob’s nodes.
Alice’s qubit is initially encoded onto a heralded single photon. Her PPLN
waveguide is pumped directly to generate photon pairs, where the detection of
one photon indicates the presence of a single photon in the other mode. After
the waveguide, we use a 100-GHz 1300-nm dense wavelength division multiplexer
(DWDM) as a notch filter for the pump, followed by a 1310-nm DWDM to separate
each photon into different fibers via the DWDM’s pass and reflect ports. The
1310-nm photon is filtered by a 10.5-GHz FWHM bandwidth fiber Bragg grating
(FBG) filter and subsequently detected by a single photon detector. Due to
energy conservation, this heralds a single photon with a center wavelength of
1290 nm. Alice’s qubit is then encoded onto the horizontal (H) and vertical
(V) components of the heralded photon’s polarization
$|\psi\rangle_{A}=\alpha|H\rangle+\beta|V\rangle$ using polarization
waveplates. This photon is then multiplexed using a standard O-band/C-band WDM
to co-propagate with the 400-Gbps C-band classical signal over 15.2 km of
spooled optical fiber (SMF-28(R) ULL) to Charlie.
Bob’s entangled photon pair source is designed similarly, except his PPLN
waveguide is placed inside a polarization Sagnac loop to generate the two-
photon quantum state
$\frac{1}{\sqrt{2}}(|{H_{s}H_{i}}\rangle+e^{i\delta}|{V_{s}V_{i}}\rangle)$.
Bob’s 1290-nm photon is transmitted to counter-propagate with the 400-Gbps
signal over 15.0 km of spooled fiber (SMF-28(R) ULL) to Charlie. However,
Bob’s 1310-nm photon is kept locally to act as the target photon for
transferring Alice’s state to Bob’s node. This photon is sent to a free-space
polarization module which first compensates rotations in the fibers and
prepares the $|\Psi^{-}\rangle_{B}$ Bell state using alignment signals and
coincidence detection [41], and then uses a quarter-waveplate, half-waveplate,
and polarizing beam splitter (PBS) combination to allow for arbitrary
projective measurements of the polarization of Bob’s target photon.
Subsequently, this photon is filtered by a 9.8-GHz FBG filter and then
detected by a single-photon detector.

Figure 2: (a) Noise photon single photon count rate in the Bell state
measurement detectors $D_{1}$ and $D_{2}$ as a function of the 400-Gbps C-band
signal’s launch power $P_{\text{cl}}$ injected into the 30.2-km fiber link.
(b) Visibility of non-local two-photon interference fringes for Bob’s
entanglement distribution over 15.0 km to the Bell state measurement node as a
function of $P_{\text{cl}}$ (top) and the measured two-fold coincidence count
(CC) interference fringes in the vertical (V) and anti-diagonal (A) bases for
$P_{\text{cl}}$ = 74 mW (bottom). (c) Hong-Ou-Mandel interference visibility
between heralded photons from Alice and Bob as a function of $P_{\text{cl}}$
(top) and an interference dip measurement when $P_{\text{cl}}$ = 74 mW
(bottom).

At Charlie’s node, a BSM is performed on Alice and Bob’s incident polarization
qubits using conditional HOM interference at a 50:50 beam splitter [38]. Since
HOM interference only occurs between identical photons, the fidelity of the
BSM is governed by their spatial and spectro-temporal indistinguishability. To
achieve this, a single-mode fiber 50:50 coupler is used for spatial
indistinguishability and a variable optical delay in Alice’s pump path is used
to ensure that both photons arrive simultaneously at the coupler. Furthermore,
the 1290-nm photons are filtered by 10.8-GHz FBGs. Together with the 1310-nm
FBGs, this extends the photons’ coherence times relative to the pump’s and
increases the probability that each photon is in a single identical spectral
mode [42].
These narrow-band spectral filters also play a critical role in improving
tolerance to classical power due to the SpRS noise photons being spectrally
independent of the quantum signal. Another potential source of noise is
residual C-band photons due to insufficient isolation when filtering the
O-band quantum signals. This is prevented by placing two cascaded
O-band/C-band WDMs before the FBGs and a 1290-nm DWDM, which altogether
provides >190 dB isolation from C-band light.
To perform the BSM, we place two PBSs in each output arm of 50:50 coupler
before the single photon detectors $D_{1}$ and $D_{2}$, which are set to
project onto the orthogonal polarization states $|H\rangle_{D_{1}}$ and
$|V\rangle_{D_{2}}$. A two-fold coincidence detection at opposite ports of the
50:50 splitter indicates that Alice and Bob’s photons were in the
$|\Psi^{-}\rangle_{AB}$ Bell state due to HOM interference that occurs if they
were in any of the three other Bell states [38]. Due to the non-local
correlations of Bob’s entangled state $|\Psi^{-}\rangle_{B}$, a successful BSM
detection results in the target photon at Bob’s node being equivalent to
Alice’s input state $|\psi\rangle_{A}$ [38]. We note that teleportation via
detecting $|\Psi^{-}\rangle_{AB}$ can be performed without the PBSs. However,
basis-selective Bell state analyzers are most common as they can double the
success probability of a BSM by also measuring the $|\Psi^{+}\rangle$ Bell
state, which could be done here if detectors were added to the other output
ports of the PBSs [43]. Thus, this design allows us to evaluate how coexisting
classical signals impact BSM nodes most likely to be implemented in future
networks. We also note the PBSs at the BSM naturally filter SpRS noise in the
polarization degree of freedom. Due to polarization mode dispersion, SpRS
photons generated over long fibers are largely unpolarized [44, 17] which
means polarizing elements can reduce noise rates by roughly a factor of two in
each detector.
Teleportation is evaluated by measuring the polarization of Bob’s target
photon at detector $D_{3}$ conditioned on a three-fold coincidence detection
between the heralding and BSM detectors, requiring four-fold coincidence
detection. Photons are detected by superconducting nanowire single photon
detectors (SNSPD), each having >90$\%$ detection efficiency and a dark count
rate of $\sim$100 counts/s. Photon arrival times are determined using a time-
to-digital converter and coincidences are registered when the arrival times in
each channel fall within a 500-ps wide time window. The coincidence logic can
also be viewed as the classical channel in the teleportation protocol which
informs Bob that a successful BSM occurred.
Since the modulation of the classical signal is uncorrelated with the pulsed
quantum system, SpRS photons are randomly distributed in time relative to the
quantum signals. As a result, narrow detection windows around the expected
quantum signal arrival times act as a temporal filter on the noise.
Furthermore, the inherent temporal correlations in photon pair detection can
provide another layer of timing information that can increase the signal-to-
noise ratio compared to sources such as WCSs [45].
The quantum teleportation-classical coexistence study is conducted in a
laboratory located at Northwestern University’s Evanston campus. However, the
classical transceiver is located in a building at the Northwestern University
Chicago campus, which requires transmission over a 24-km deployed fiber pair
to and from the Evanston location to operate. Thus, the classical signal
travels an additional 48 km of fiber giving a total link distance of 78.2 km.
The transmission loss over the 30.2-km fiber is 4.9 dB at the classical
signal’s 1547.32-nm wavelength and the 1290-nm quantum signals have a total
loss of 10.1 dB (5.1 dB and 5.0 dB for Alice and Bob’s photons, respectively).
At Charlie, the bypassing of the BSM by the classical signal using WDMs adds
an additional 1.2 dB insertion loss.
The total loss when transmitting the classical signal over the full 78.2-km
fiber is 22.8 dB. Due to a receiver sensitivity of -18 dBm, the transceiver’s
output power of -9 dBm requires amplification for the system to operate error-
free. To achieve this, we place two cascaded erbium doped fiber amplifiers
(EDFA) at the Evanston location just before multiplexing into the 30.2-km link
to co-propagate with Alice’s photon. The minimum power launched into the
30.2-km fiber to achieve error-free communications could be as low as $-3$ dBm
(0.5 mW) for the receiver located 30.2 km + 24 km away from the EDFA, and
-10.7 dBm (85 $\mu$W) if the receiver were instead located directly at the end
of the 30.2-km link. However, in this study we explore elevated launch powers
to emulate more hostile environments that could be encountered in the field
such as those found in high-capacity long-distance DWDM classical network
systems.

##  3 Experimental Results

###  3.1 Noise rates, entanglement distribution, and Hong-Ou-Mandel
interference

We now evaluate the performance of our quantum system. First, we characterize
how much noise is introduced into the detectors $D_{1}$ and $D_{2}$ as a
function of the classical power $P_{\text{cl}}$ that is multiplexed into the
30.2-km fiber. Fig. 2(a) shows the single photon count rates as we increase
$P_{\text{cl}}$, where the SNSPD dark count rates have been subtracted. The
extra noise scales linearly with power as 79.0 counts/s/mW and 97.9
counts/s/mW in $D_{1}$ and $D_{2}$, respectively, which differ due to slightly
different detection system efficiencies. These results demonstrate that SpRS
noise is significantly suppressed, as even mW power levels introduce
extraneous noise on the order of the SNSPD dark count level. Since the
400-Gbps system can operate at $P_{\text{cl}}>0.5$ mW, this means that the
SNSPDs hardly notice any rise in background counts when the system is
operating at the minimum necessary classical power.
As teleportation requires quantum entanglement as a resource and the ability
to have indistinguishable photons for the BSM, we independently characterize
the quality of Bob’s entanglement distribution to Charlie and HOM interference
as we vary $P_{\text{cl}}$. We set Alice and Bob’s mean photon pairs per pulse
and polarization qubit $\mu$ generated inside each PPLN waveguide to
approximately $\mu_{A}$ = $0.018$ and $\mu_{B}$ = $0.013$, respectively, which
are chosen to balance rates with reduced performance due to multi-pair
emission during SPDC [46].
Fig. 2(b) shows how Bob’s entanglement source’s non-local two-photon
interference fringe visibility in the vertical and anti-diagonal bases is
impacted after distributing one photon over 15.0 km to Charlie as we increase
the classical power launched into the full 30.2-km link. The visibility is
defined as $V_{\rm ent}=(R_{\rm max}-R_{\rm min})/(R_{\rm max}+R_{\rm min})$,
where $R_{\rm max(\rm min)}$ are the maximum (minimum) two-fold coincidence
counts of the interference fringe. Fig. 2(b) shows no noticeable degradation
of $V_{\rm ent}$ up to $P_{\text{cl}}=$ 74 mW, in which we measure
visibilities of $V_{V}=97.5\pm 0.1\%$ and $V_{A}=95.3\pm 0.2\%$. All error
bars in this paper are determined via the Monte-Carlo method assuming Poisson
photon counting statistics. These results indicate that entanglement
distribution is well preserved at high classical powers, and that visibility
is mainly limited from unity due to multi-pair emission during SPDC. These
values are also well above the $1/\sqrt{2}$ bound for demonstrating the non-
local nature of quantum entanglement [47]. Thus, this state is more than
sufficient to use as a resource for quantum teleportation.
Next, we characterize HOM interference between heralded photons from Alice and
Bob’s sources as we increase $P_{\text{cl}}$. To ensure polarization
indistinguishably, we align the PBSs in the BSM setup to instead project onto
$|H\rangle_{D_{1}}|H\rangle_{D_{2}}$ and prepare Alice and Bob’s heralded
photons in the state $|H\rangle$. We then record four-fold coincidences as we
vary the relative time-of-arrival of each photon using Alice’s variable
optical delay. The quality of the interference is determined by the visibility
of the fringe $V_{\rm HOM}=(R_{\rm max}-R_{\rm min})/R_{\rm max}$, where
$R_{\rm max(min)}$ are the maximum (minimum) four-fold coincidence rates of
the HOM interference fringe. The results are shown in Fig. 2(c). We measure a
visibility of $V_{\rm HOM}=(R_{\rm max}-R_{\rm min})/R_{\rm max}=82.9\pm
4.5\%$ without the classical source, which is comparable to the $V_{\rm
HOM}=83.1\pm 1.4\%$ visibility we obtain without distribution over the 30.2-km
fiber. We measure $V_{\rm HOM}=80.3\pm 3.8\%$ with $P_{\text{cl}}$ set to 74
mW, showing interference is not significantly degraded by high-power
coexisting C-band light. Each visibility is well above the classical bound of
50% [48], indicating our photons are sufficiently indistinguishable for
performing quantum teleportation.

Figure 3: Four-fold coincidence counts as we scan Bob’s polarization analyzer
setting along the Bloch sphere for Alice transmitting $|{D}\rangle$ (blue) and
$|A\rangle$ (red) when the launch power of the 400-Gbps classical signal into
the 30.2-km fiber is set to 74 mW. Figure 4: Real and imaginary components of
the density matrices obtained by single-qubit quantum state tomography on
Bob’s target photon, conditioned on a successful BSM, whilst the launch power
of the 400-Gbps C-band signal into the 30.2-km fiber is set to 74 mW. (a-d)
show the results for Alice transmitting $|\psi\rangle_{A}=|H\rangle$,
$|V\rangle$, $|D\rangle$, and $|A\rangle$, respectively.

###  3.2 Quantum teleportation coexisting with high-rate classical
communications

Looking at the performance of entanglement distribution and HOM interference,
we find that the key operations underlying quantum teleportation are well
maintained at high C-band power levels. To demonstrate that this translates to
the ability to perform noise-resilient teleportation, we evaluate the
teleportation of various qubits while simultaneously transmitting 74 mW of
C-band classical power.
Fig. 3 shows four-fold coincidence counts as we scan Bob’s polarization
analyzer setting along the Bloch sphere for Alice transmitting
$|{D}\rangle=\frac{1}{\sqrt{2}}(|{H}\rangle+|{V}\rangle)$ and
$|{A}\rangle=\frac{1}{\sqrt{2}}(|{H}\rangle-|{V}\rangle)$. The coincidences
exhibit sinusoidal fringes that coincide with Alice’s prepared qubit. We
obtain fringe visibilities of $V_{D}=81.3\pm 5.4\%$ and $V_{A}=74.7\pm 4.7\%$,
both values being above the $V>1/3$ classical limit [49].
Next, we perform maximum likelihood quantum state tomography [50] on Bob’s
target photon to reconstruct the density matrix $\rho_{B}$ of his quantum
state conditioned on a BSM detection. Fig. 4 shows $\rho_{B}$ for Alice
transmitting the states $|H\rangle$, $|V\rangle$, $|D\rangle$, and
$|A\rangle$. The fidelity of $\rho_{B}$ to Alice’s ideal qubit
$|\psi\rangle_{A}$ is determined by calculating
$F_{\psi}=\langle\psi_{A}|\rho_{B}|\psi_{A}\rangle$. State transfer in the H/V
basis does not require HOM interference due to the projection onto
$|H\rangle_{D_{1}}|V\rangle_{D_{2}}$ in the Bell state analyzer [51], where
raw fidelity is mainly limited from unity due to multi-photon pair
degradation. In this basis, we measure $F_{H}=97.5\pm 1.2\%$ and
$F_{V}=95.8\pm 2.5\%$ for $|{\psi}\rangle_{A}=|{H}\rangle$ and $|{V}\rangle$,
respectively. However, transferring coherent superpositions of $|H\rangle$ and
$|V\rangle$ requires interference and thus exhibit more decoherence due to an
imperfect spectro-temporal indistinguishability between Alice and Bob’s
photons. For Alice transmitting $|{\psi}\rangle_{A}=|{D}\rangle$ and
$|{A}\rangle$, we obtain fidelities of $F_{D}=87.5\pm 3.9\%$ and
$F_{A}=85.5\pm 3.7\%$, respectively. Knowing that SpRS noise rates are
independent of Alice’s prepared qubit, assuming symmetry of the equatorial
states we calculate the average fidelity for an arbitrary qubit via $F_{\rm
avg}=(F_{H}+F_{V}+2(F_{D}+F_{A}))/6$ in which we obtain $F_{\rm avg}=89.9\pm
3.1\%$. Furthermore, we measure $F_{\rm avg}=90.4\pm 2.7\%$ without sending
the photons over the 30.2-km link, demonstrating that there is minimal
degradation of fidelity from distribution over the fiber link nor the
inclusion of the classical light. All of the measured fidelities are shown in
Fig. 5. Since all are well above the classical bound of $F=2/3$ [49], we have
clearly demonstrated that non-classical teleportation can be performed
alongside high-rate conventional classical communications.

Figure 5: Fidelity of the reconstructed density matrix of Bob’s state to
Alice’s initially prepared state for four different qubits as well as the
average fidelity when the power of the 400-Gbps classical signal is set to 74
mW. The horizontal line shows the $F>2/3$ limit for demonstrating non-
classical teleportation.

##  4 Discussion and Outlook

Since teleportation was conducted alongside much higher powers than necessary
for the 400-Gbps classical system, this indicates that substantially higher
data rates could have propagated through the fiber. The classical power used
was 148 times greater than necessary for a single 400-Gbps channel, and 870
times greater than if the receiver were located directly at the end of the
30.2-km fiber. Thus, a multi-channel WDM classical system could easily have
carried many terabits/s aggregate data rates without having a significant
impact on quantum fidelity. The demonstrated powers are also on the order of
the highest achieved in state-of-the-art single [10, 11, 12] and entangled
[17] photon coexistence studies. This indicates the potential for
teleportation-based quantum networking systems, such as quantum repeaters,
operating alongside classical backbone networks [10].
Since our system uses two different photon pair sources, multiple nodes, and
four-fold coincidence detection, our results indicate the potential for more
complex systems operating alongside classical networks. For example, simple
modifications to Alice’s source could allow entangled photon pair generation.
A BSM at Charlie would then result in entanglement swapping [52], establishing
entanglement between initially independent photons at Alice and Bob’s nodes.
Due to our system’s high tolerance to classical power, it is reasonable to
conclude that a similarly designed entanglement swapping system could operate
alongside high-rate coexisting classical communications. Further, our multi-
node configuration with the BSM performed near the midpoint of a longer fiber
link is a key element for multi-node networking with a central quantum
processor [34] and to implement quantum repeaters [29]. We note that these
applications will also require quantum memories to store incident photons, the
degree to which SpRS noise will impact various platforms and applications will
require future investigation.
The state teleportation fidelity we obtained is comparable to recent
experiments conducted over quantum-dedicated fiber networks [34, 33, 37], but
could be further improved via standard methods. We showed that opposed to SpRS
noise from coexisting classical light, our fidelity was mainly limited by
multi-photon pair emission during SPDC and imperfect photon indistinguishably
during the BSM. Raw fidelity could thus be improved by reducing the the photon
pair generation probability [46] and increasing indistinguishability by
decreasing the ratio between the source’s pump spectral bandwidth to the
photon filter bandwidths [42]. When considering SpRS noise, narrower filters
would simultaneously increase indistinguishability and reject more SpRS noise.
Altogether, employing these approaches could allow our teleportation fidelity
to approach more closely to unity and have an even higher tolerance to
classical power.
Finally, classical signals are often used for quantum networking purposes.
Some applications include communicating measurement results between nodes
(eg., the BSM), synchronizing distant pulsed sources and detectors to a common
clock, and channel monitoring. Similarly, quantum-classical coexistence can
enhance functionality without sacrificing potentially limited fiber resources
[16, 53, 22, 21]. Since these signals can typically operate well below the
powers demonstrated here (sub-mW levels), our results show that including
these signals in our system would have negligible impact if allocated to the
C- or L-bands.
In conclusion, we have demonstrated a three-node quantum state teleportation
system operating in a 30.2-km fiber link that simultaneously carries 400-Gbps
conventional classical communications. By employing various methods to
suppress the impact of SpRS noise photons, teleportation fidelity was well
maintained with elevated classical powers capable of transmitting many
terabits/s aggregate data rates. Since we have clearly shown that
appropriately designed teleportation systems can operate alongside modern
high-rate data traffic, this work demonstrates a significant step towards
ensuring future quantum networking can be realized anywhere in the existing
fiber infrastructure.

Funding This work is funded by Subcontract No. 664603 from Fermi Research
Alliance, LLC (FRA) to Northwestern University issued under Prime Contract No.
DE-AC02-07CH11359 between FRA and the U.S. Department of Energy (DOE).
Although the work is supported by the DOE’s Advanced Scientific Computing
Research Transparent Optical Quantum Networks for Distributed Science program,
no government endorsement is implied.

Acknowledgments The authors would like to acknowledge the support of StarLight
International/National Communications Exchange Facility and Ciena Corporation.
We further thank the members of the Advanced Quantum Networks for Scientific
Discovery (AQNET-SD) project, which is a collaboration between Northwestern
University, Caltech, Fermilab National Laboratory, Argonne National
Laboratory, the Jet Propulsion Laboratory, and University of Illinois Urbana-
Champaign towards advancing quantum network technology and applications.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement Data may be obtained from the authors upon
reasonable request.

## References

  * [1] H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
  * [2] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” Science 362, eaam9288 (2018).
  * [3] P. D. Townsend, “Simultaneous quantum cryptographic key distribution and conventional data transmission over installed fibre using wavelength-division multiplexing,” Electronics Letters 33, 188–190(2) (1997). Publisher: Institution of Engineering and Technology.
  * [4] T. E. Chapuran, P. Toliver, N. A. Peters, J. Jackel, M. S. Goodman, R. J. Runser, S. R. McNown, N. Dallmann, R. J. Hughes, K. P. McCabe, J. E. Nordholt, C. G. Peterson, K. T. Tyagi, L. Mercer, and H. Dardy, “Optical networking for quantum key distribution and quantum communications,” New Journal of Physics 11, 105001 (2009).
  * [5] 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).
  * [6] B. Qi, W. Zhu, L. Qian, and H.-K. Lo, “Feasibility of quantum key distribution through a dense wavelength division multiplexing network,” New Journal of Physics 12, 103042 (2010).
  * [7] K. A. Patel, J. F. Dynes, I. Choi, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Coexistence of High-Bit-Rate Quantum Key Distribution and Data on Optical Fiber,” Physical Review X 2, 041010 (2012).
  * [8] J. F. Dynes, W. W.-S. Tam, A. Plews, B. Fröhlich, A. W. Sharpe, M. Lucamarini, Z. Yuan, C. Radig, A. Straw, T. Edwards, and A. J. Shields, “Ultra-high bandwidth quantum secured data transmission,” Scientific Reports 6, 35149 (2016).
  * [9] L.-J. Wang, K.-H. Zou, W. Sun, Y. Mao, Y.-X. Zhu, H.-L. Yin, Q. Chen, Y. Zhao, F. Zhang, T.-Y. Chen, and J.-W. Pan, “Long-distance copropagation of quantum key distribution and terabit classical optical data channels,” Phys. Rev. A 95, 012301 (2017).
  * [10] Y. Mao, B.-X. Wang, C. Zhao, G. Wang, R. Wang, H. Wang, F. Zhou, J. Nie, Q. Chen, Y. Zhao, Q. Zhang, J. Zhang, T.-Y. Chen, and J.-W. Pan, “Integrating quantum key distribution with classical communications in backbone fiber network,” Opt. Express 26, 6010–6020 (2018).
  * [11] J.-Q. Geng, G.-J. Fan-Yuan, S. Wang, Q.-F. Zhang, Y.-Y. Hu, W. Chen, Z.-Q. Yin, D.-Y. He, G.-C. Guo, and Z.-F. Han, “Coexistence of quantum key distribution and optical transport network based on standard single-mode fiber at high launch power,” Opt. Lett. 46, 2573–2576 (2021).
  * [12] P. Gavignet, E. Pincemin, F. Herviou, Y. Loussouarn, F. Mondain, A. J. Grant, L. Johnson, R. I. Woodward, J. F. Dynes, B. Summers, A. J. Shields, K. Taira, H. Sato, R. Zink, V. Grempka, V. Castay, and J. Zou, “Co-propagation of qkd & 6 tb/s (60x100g) dwdm channels with 17 dbm total wdm power in single and multi-span configurations,” Journal of Lightwave Technology pp. 1–7 (2023).
  * [13] C. Liang, K. F. Lee, J. Chen, and P. Kumar, “Distribution of fiber-generated polarization entangled photon-pairs over 100 km of standard fiber in oc-192 wdm environment,” in _2006 Optical Fiber Communication Conference and the National Fiber Optic Engineers Conference,_ (2006), pp. 1–3.
  * [14] A. Ciurana, V. Martin, J. Martinez-Mateo, B. Schrenk, M. Peev, and A. Poppe, “Entanglement Distribution in Optical Networks,” IEEE Journal of Selected Topics in Quantum Electronics 21, 37–48 (2015).
  * [15] F. Hipp, M. Hentschel, S. Aleksic, A. Poppe, and H. Huebel, “Demonstration of a coexistence scheme between polarization-entangled QKD and classical data channels,” in _Quantum Optics,_ vol. 9900 J. Stuhler and A. J. Shields, eds., International Society for Optics and Photonics (SPIE, 2016), p. 99000P.
  * [16] K. Kapoor, S. Xie, J. Chung, R. Valivarthi, C. Pena, L. Narvaez, N. Sinclair, J. P. Allmaras, A. D. Beyer, S. I. Davis, G. Fabre, G. Iskander, G. S. Kanter, R. Kettimuthu, B. Korzh, P. Kumar, N. Lauk, A. Mueller, M. Shaw, P. Spentzouris, M. Spiropulu, J. M. Thomas, and E. E. Wollman, “Picosecond Synchronization System for the Distribution of Photon Pairs through a Fiber Link between Fermilab and Argonne National Laboratories,” IEEE Journal of Quantum Electronics pp. 1–1 (2023).
  * [17] J. M. Thomas, G. S. Kanter, and P. Kumar, “Designing noise-robust quantum networks coexisting in the classical fiber infrastructure,” Opt. Express 31, 43035–43047 (2023).
  * [18] Y.-R. Fan, Y. Luo, Z.-C. Zhang, Y.-B. Li, S. Liu, D. Wang, D.-C. Zhang, G.-W. Deng, Y. Wang, H.-Z. Song, Z. Wang, L.-X. You, C.-Z. Yuan, G.-C. Guo, and Q. Zhou, “Energy-time entanglement coexisting with fiber-optical communication in the telecom $c$ band,” Phys. Rev. A 108, L020601 (2023).
  * [19] X. Jing, C. Qian, X. Zheng, H. Nian, C. Wang, J. Tang, X. Gu, Y. Kong, T. Chen, Y. Liu, C. Sheng, D. Jiang, B. Niu, and L. Lu, “Coexistence of multiuser entanglement distribution and classical light in optical fiber network with a semiconductor chip,” Chip p. 100083 (2024).
  * [20] R. Wang, R. Yang, M. J. Clark, R. D. Oliveira, S. Bahrani, M. Peranić, M. Lončarić, M. Stipčević, J. Rarity, S. K. Joshi, S. K. Joshi, R. Nejabati, and D. Simeonidou, “Field trial of a dynamically switched quantum network supporting co-existence of entanglement, prepare-and-measure qkd and classical channels,” in _49th European Conference on Optical Communications (ECOC 2023),_ (Institution of Engineering and Technology, 2023), p. 1682 – 1685.
  * [21] S. J. B. Yoo, S. K. Singh, M. B. On, G. Gul, G. S. Kanter, R. Proietti, and P. Kumar, “Quantum wrapper networking,” IEEE Communications Magazine pp. 1–7 (2024).
  * [22] A. Rahmouni, P. S. Kuo, Y. Li-Baboud, Y. S. I. A. Burenkov, M. V. Jabir, N. Lal, D. Reddy, M. Merzouki, L. Ma, A. Battou, S. V. Polyakov, O. Slattery, and T. Gerrits, “Metropolitan-scale entanglement distribution with co existing quantum and classical signals in a single fiber,” Arxiv (2024). ArXiv:2402.00617.
  * [23] R. Valivarthi, P. Umesh, C. John, K. A. Owen, V. B. Verma, S. W. Nam, D. Oblak, Q. Zhou, and W. Tittel, “Measurement-device-independent quantum key distribution coexisting with classical communication,” Quantum Science and Technology 4, 045002 (2019).
  * [24] R. C. Berrevoets, T. Middelburg, R. F. L. Vermeulen, L. D. Chiesa, F. Broggi, S. Piciaccia, R. Pluis, P. Umesh, J. F. Marques, W. Tittel, and J. A. Slater, “Deployed measurement-device independent quantum key distribution and Bell-state measurements coexisting with standard internet data and networking equipment,” Communications Physics 5, 186 (2022).
  * [25] R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable QKD with intense DWDM classical channels,” New Journal of Physics 17, 043027 (2015).
  * [26] J. C. Chapman, A. Miloshevsky, H.-H. Lu, N. Rao, M. Alshowkan, and N. A. Peters, “Two-mode squeezing over deployed fiber coexisting with conventional communications,” Opt. Express 31, 26254–26275 (2023).
  * [27] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
  * [28] D. Collins, N. Gisin, and H. D. Riedmatten, “Quantum relays for long distance quantum cryptography,” Journal of Modern Optics 52, 735–753 (2005).
  * [29] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
  * [30] A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
  * [31] X.-M. Hu, Y. Guo, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Progress in quantum teleportation,” Nature Reviews Physics 5, 339–353 (2023).
  * [32] O. Landry, J. A. W. van Houwelingen, A. Beveratos, H. Zbinden, and N. Gisin, “Quantum teleportation over the swisscom telecommunication network,” J. Opt. Soc. Am. B 24, 398–403 (2007).
  * [33] R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum teleportation across a metropolitan fibre network,” Nature Photonics 10, 676–680 (2016).
  * [34] Q.-C. Sun, Y.-L. Mao, S.-J. Chen, W. Zhang, Y.-F. Jiang, Y.-B. Zhang, W.-J. Zhang, S. Miki, T. Yamashita, H. Terai, X. Jiang, T.-Y. Chen, L.-X. You, X.-F. Chen, Z. Wang, J.-Y. Fan, Q. Zhang, and J.-W. Pan, “Quantum teleportation with independent sources and prior entanglement distribution over a network,” Nature Photonics 10, 671–675 (2016).
  * [35] Q.-C. Sun, Y.-F. Jiang, Y.-L. Mao, L.-X. You, W. Zhang, W.-J. Zhang, X. Jiang, T.-Y. Chen, H. Li, Y.-D. Huang, X.-F. Chen, Z. Wang, J. Fan, Q. Zhang, and J.-W. Pan, “Entanglement swapping over 100 km optical fiber with independent entangled photon-pair sources,” Optica 4, 1214 (2017).
  * [36] R. Valivarthi, S. I. Davis, C. Peña, S. Xie, N. Lauk, L. Narváez, J. P. Allmaras, A. D. Beyer, Y. Gim, M. Hussein, G. Iskander, H. L. Kim, B. Korzh, A. Mueller, M. Rominsky, M. Shaw, D. Tang, E. E. Wollman, C. Simon, P. Spentzouris, D. Oblak, N. Sinclair, and M. Spiropulu, “Teleportation systems toward a quantum internet,” PRX Quantum 1, 020317 (2020).
  * [37] S. Shen, C. Yuan, Z. Zhang, H. Yu, R. Zhang, C. Yang, H. Li, Z. Wang, Y. Wang, G. Deng, H. Song, L. You, Y. Fan, G. Guo, and Q. Zhou, “Hertz-rate metropolitan quantum teleportation,” Light: Science & Applications 12, 115 (2023).
  * [38] S. L. Braunstein and A. Mann, “Measurement of the bell operator and quantum teleportation,” Phys. Rev. A 51, R1727–R1730 (1995).
  * [39] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
  * [40] 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 linbo3 ridge waveguide,” Opt. Express 19, 16032–16043 (2011).
  * [41] S. X. Wang and G. S. Kanter, “Robust multiwavelength all-fiber source of polarization-entangled photons with built-in analyzer alignment signal,” IEEE Journal of Selected Topics in Quantum Electronics 15, 1733–1740 (2009).
  * [42] J. G. Rarity, “Interference of single photons from separate sources,” Annals of the New York Academy of Sciences 755, 624–631 (1995).
  * [43] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
  * [44] J. C. Chapman, J. M. Lukens, M. Alshowkan, N. Rao, B. T. Kirby, and N. A. Peters, “Coexistent quantum channel characterization using spectrally resolved bayesian quantum process tomography,” Phys. Rev. Appl. 19, 044026 (2023).
  * [45] M. T. Liu and H. C. Lim, “Transmission of o-band wavelength-division-multiplexed heralded photons over a noise-corrupted optical fiber channel,” Opt. Express 21, 30358–30369 (2013).
  * [46] M. Takeoka, R.-B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New Journal of Physics 17, 043030 (2015).
  * [47] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Physical Review Letters 23, 880–884 (1969).
  * [48] H. Paul, “Interference between independent photons,” Rev. Mod. Phys. 58, 209–231 (1986).
  * [49] S. Massar and S. Popescu, “Optimal extraction of information from finite quantum ensembles,” Phys. Rev. Lett. 74, 1259–1263 (1995).
  * [50] J. Altepeter, E. Jeffrey, and P. Kwiat, “Photonic state tomography,” (Academic Press, 2005), pp. 105–159.
  * [51] F. Basso Basset, F. Salusti, L. Schweickert, M. B. Rota, D. Tedeschi, S. F. Covre da Silva, E. Roccia, V. Zwiller, K. D. Jöns, A. Rastelli, and R. Trotta, “Quantum teleportation with imperfect quantum dots,” npj Quantum Information 7, 7 (2021).
  * [52] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, ““event-ready-detectors” bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
  * [53] I. A. Burenkov, A. Semionov, Hala, T. Gerrits, A. Rahmouni, D. Anand, Y.-S. Li-Baboud, O. Slattery, A. Battou, and S. V. Polyakov, “Synchronization and coexistence in quantum networks,” Opt. Express 31, 11431–11446 (2023).