# Collection and Evaluation of a Long-Term 4D Agri-Robotic Dataset

Riccardo Polvara∗, Sergi Molina Mellado, Ibrahim Hroob, Grzegorz Cielniak and
Marc Hanheide *Corresponding author<EMAIL_ADDRESS>the authors are
within the Lincoln Centre for Autonomous Systems (LCAS), University of
Lincoln, Lincoln, UK, LN6 7TS. This work has been supported by the European
Commission as part of H2020 under grant number 871704 (BACCHUS).

###### Abstract

Long-term autonomy is one of the most demanded capabilities looked into a
robot. The possibility to perform the same task over and over on a long
temporal horizon, offering a high standard of reproducibility and robustness,
is appealing. Long-term autonomy can play a crucial role in the adoption of
robotics systems for precision agriculture, for example in assisting humans in
monitoring and harvesting crops in a large orchard. With this scope in mind,
we report an ongoing effort in the long-term deployment of an autonomous
mobile robot in a vineyard for data collection across multiple months. The
main aim is to collect data from the same area at different points in time so
to be able to analyse the impact of the environmental changes in the mapping
and localisation tasks. In this work, we present a map-based localisation
study taking 4 data sessions. We identify expected failures when the pre-built
map visually differs from the environment’s current appearance and we
anticipate LTS-Net, a solution pointed at extracting stable temporal features
for improving long-term 4D localisation results.

##  I Introduction

Agricultural environments, such as an orchard, offer many challenges for
autonomously and safely deploying robots at work [1]. For example, the authors
in [2] compared state-of-the-art SLAM methods in a simulated vineyard
environment and found that plants changing appearance can lead to a
degradation of the localisation module, with the robot drifting from its
course. The main reason why existing SLAM methods fail in guaranteeing robust
and long-term performances in the agriculture domain is that they are
developed and evaluated mainly in urban or indoor environments, which are
consistent across time [3]. Agricultural environments, on the other hand, are
subject to seasonal changes, with plants varying in size and colours, offering
less stable localisation features than buildings and road infrastructures. For
this reason, achieving long-term autonomy for robots in such environments is
both a challenging and interesting research problem [4]. Motivated by this
challenge, we deploy an autonomous mobile robot in the Ktima Gerovassiliou
vineyard (Greece) at specific time intervals and record sensor data to better
capture the map’s variability and, possibly, exploit it later. To do so, we
propose a long-term robust deployment of the robot in a vineyard navigating on
a topological map. We planned to record sensory data from March 2022 until the
harvesting season in October 2022. Differently from [5] and any other dataset
present in the literature [6], our objective is not only to create a dataset
of robot sensor data in the agricultural environment, but we aim at capturing
and studying the long-term nature of such data.

In the remaining of this manuscript, we provide an overview of the recorded
data and the scenario, how the data is stored and how it can be retrieved, an
initial assessment of the degradation of the localisation algorithm over time,
and we briefly discuss ongoing research toward extracting long-term temporally
stable features with the scope of overcoming the shortcomings of existing
LIDAR-based SLAM approaches in a continuously evolving environment.

##  II Data Collection

###  II-A Robot Setup

For the scope of data collection, we are using the Thorvald robot (SAGA
Robotics). It is equipped with a complete sensor suite consisting of an RTK-
GPS (Trimble BX992), two RGBD cameras (Stereolabs ZED2i), one 16-beams 3D
LIDAR (OUSTER OS1-16), IMU (RSX-UM7) and two 4-beams 2D LIDARs (SICK MRS1000)
located at opposite corners of the robot. Figure 1(left) offers a visual
representation of the robot and its sensor suite.

Figure 1: (Left) The Thorvald robot deployed at the Ktima Gerovassiliou
vineyard. (Right) Special icons and GUI have been designed to facilitate the
interaction with the robot.

In order to simplify the operation of the robot and make it more accessible
also to inexperienced users, special desktop icons and a GUI have been
designed (Figure 1(right)), while a touch monitor is mounted on the robot.
More specifically, two icons “Start Robot” and “Robot Stop” allows to launch
and stop, respectively, the basic functionalities of the platform, such as the
roscore, the 2D lasers for obstacle detection, the RTK-GPS for outdoor
localisation, and the topological_navigation[7] module for navigating the
rows. When starting the robot, the ROBOT HEALTH module is also launched to
inform the user about the platform’s status. A red/green palette is used to
communicate if a specific module is currently working or not. Specials icons
allow to start/stop the sensor suite (IMU, 3D LIDAR, and 2 RGBD cameras)
plugged into the additional computer. In an analogous way, icons have been
designed for starting the data recording session and the navigation task.

Figure 2: Topological map with the nodes and the edges followed by the robot
during the data collection.

The area used for data collection includes five adjacent rows within the
vineyard Ktima Gerovassiliou, Greece (lat: 40.450135, lon: 22.924247). By
pressing the “Start Waypoints” icon, the robot traverses autonomously a human-
designed path along the edges connecting the nodes of a topological map
(Figure 2). The path defined is kept the same for all data collection sessions
and is the following: $B\rightarrow G\rightarrow F\rightarrow A\rightarrow
B\rightarrow G\rightarrow H\rightarrow C\rightarrow D\rightarrow I\rightarrow
J\rightarrow E\rightarrow J\rightarrow I\rightarrow D\rightarrow C\rightarrow
H\rightarrow G\rightarrow B$. The length of this entire path is approximately
$500$ m, and it takes, on average, $25$ min for the robot to complete the
path. The average robot speed during the traversal of a row is $0.6$ m/s.

###  II-B Data Recording and Storage

Our objective is to collect data until the end of the harvesting season to
capture the entire crops’ growth. This process started in March, and it is
currently in progress. The data recording is happening at a different
frequency in order to accommodate a variable speed in the crop’s growth
stages, e.g., only one session has been performed in March, while four
sessions (one week apart one from another) have been completed in June. This
is because during winter the plants grow very slowly, not offering significant
visual changes between consecutive weeks. On the other hand, in the warmest
months of the year, the plants drastically change their aspect a few days
apart. A comparison between how the plants look from March to June (left to
right) is offered in Figure 3.

---
Figure 3: Different crop’s growth stage starting from March (left) to June
(right).

Data is recorded and stored in a MongoDB database by using topic_store
(TS)[8]. Unlike ROS bags, TS adds flexibility by serialising all the ROS
messages into a data hierarchy that is easily searchable with database queries
and allows for remote storage. The user simply needs to define a scenario file
listing all the topics to record and the hierarchy to adopt. In our setup, all
the sensory information (2D/3D LIDARs, RGBD, RTK-GPS and IMU) is going to be
saved, together with other navigation data such as motor signals, odometry,
and current location in the topological map.

Data is stored and indexed locally and then synchronised with a remote server.
This can be queried and inspected to retrieve data at a specific time and
location. The most-user-friendly way to inspect the database is by using
MongoDB Compass and establishing a working connection to the database itself.

Figure 4: Overview of MongoDB Compass for inspecting the data collected during
the recording session.

Once the connection to the server has been established, it is possible to have
access to all the databases and their respective collections. By selecting
“kg_recording” we can have an overview of the documents that have been created
during the data collection session, as seen in Figure 4. Each document has the
same hierarchical structure defined in the TS scenario file.

###  II-C Database connection

To connect to the remote server hosting the data collected across all the
recording sessions, a generic user needs first to define a few environmental
variables such as the database’s and collection’s name but also the user
credentials. A dedicated script allows extracting data from the remote
database in the format of a ROS bag. In order to do so, a user needs, first of
all, to define a particular query so as to filter only a portion of data. For
example, a user would be interested to extract only a specific recording
session out of the complete set of data. Hence, after having identified the
ObjectId of the session they are interested in (e.g, 03032022), they can type
in the command line the following:

⬇

rosrun topic_store convert.py -i
”mongodb://$MONGOUSER:$MONGOPASS@files.lar.lincoln.ac.uk:8098/?authSource=bacchus&tls=true&tlsInsecure=true”
\

-c kg_recording \

-q ’{”_ts_meta.session”:”ObjectId(03032022)”}’ \

-o session_03032022.bag

One can also add a projection (-p) to select which fields should be extracted
(e.g., only the laser data).

We identified the lack of a standard benchmark for mobile robots deployed in
agriculture domains as one of the biggest challenges faced by the robotic
community. Hence, it is our plan to publicly release the dataset under an
open-source license as soon as we have enough significant data. Even though
the primary application for this dataset is to evaluate localisation
algorithms under various changing conditions, we believe that thanks to its
temporal aspect, the dataset can also be used for phenotyping and crop mapping
tasks (e.g. see [9]).

##  III Data evaluation

Figure 5: Pipeline followed for the localisation evaluation.

As just said, the final objective of the data collection is to create a
dataset of robot sensor data that can be useful for developing and bench-
marking novel navigation and localisation algorithms in the agricultural
domain. An interesting research question to address is about the long-term
deployment of a robot in an environment that is subject to seasonal changes,
offering severe challenges for what concerns the robot’s localisation in such
an environment. Within this scope, we are providing in this manuscript a
comparison performed using data from four different data recording sessions,
the same from which we extracted the frames shown in Figure 3. To be specific,
the recording sessions occurred on the 3rd March 2022, 6th April 2022, 6th May
2022, and 8th June 2022. The goal of this study is to quantify how and if the
localisation performance drops over time when trying to localise the robot by
using a map built in a different moment, e.g., the robot aims at localising
itself in April while using a map built in March. To do so, we follow a 3-step
process as shown in Figure 5, consisting of mapping, localisation, and
evaluation. Before the 3D LIDAR point cloud is fed in the FAST-LIO [10]
mapping and NDT-Localizer [11] algorithms, a people remover filter step is
applied to the raw points. For safety reasons, during the recording process, a
person is always following the robot some meters behind in case an emergency
stop must be triggered. So, in order to remove the person, a 3D area behind
the robot is cropped out of all the scans using the pcl library.

###  III-A 3D Mapping

To this scope, we initially create a 3D map using FAST-LIO (Fast LiDAR-
Inertial Odometry). This uses as input the point cloud data from the Ouster
OS1 3D LIDAR mounted on top of our robot and its IMU data. FAST-LIO is a
computationally efficient and robust LIDAR-inertial odometry package. It fuses
LIDAR feature points with IMU data using a tightly coupled iterated extended
Kalman filter to allow robust navigation in fast-motion, noisy or cluttered
environments where degeneration occurs. The resulting output is a 3D point
cloud in pcd format.

---
Figure 6: Row mapping detail starting from March(top-left) to June(bottom-
right). Moving from late winter across spring and into summer, poles disappear
from the map because the plants are now covered by leaves, which also occlude
LIDAR beams preventing the mapping of the surrounding environment.

A visual representation of the 4 maps obtained using FAST-LIO for the four
sessions is offered in Figure 6. Moving from late winter across spring and
into summer, poles disappear from the map because the plants are now covered
by leaves, which also occlude lidar beams preventing mapping more of the
surrounding environment.

###  III-B Localisation

The resulting maps are then used for localising the robot using a method based
on the Normal Distribution Transform (NDT), which tries to match each point
cloud obtained by the 3D LIDAR with the pre-recorded map. However, this
technique does not try to match the current lidar scan to points in the map;
instead, it tries to match points from the current scan to a 3D grid of
probability functions created from the map. The probability distributions are
obtained by dividing the point cloud into a uniform 3D grid where each 3D cell
(voxel) contains the mean and standard deviation of the points belonging to
it, representing a normal distribution. This way, even if the robot measures a
point a few millimetres away from where the map thinks a point should be,
instead of being completely unable to match those two points, the NDT matching
function connects the measured point to the probability function on the map.
There are 3 factors worth mentioning which affect the performance of the NDT
localizer: (i) the down-sampling method of the input point cloud (voxel grid
filter), (ii) the resolution of the point cloud map, and (iii) the initial
pose given to the robot at the beginning of the process. Regarding (i), all
the points present in each voxel of the input voxel grid will be approximated
(i.e., down-sampled) with their centroid. This approach is a bit slower than
approximating them with the centre of the voxel, but it represents the
underlying surface more accurately. An experimental analysis found that a leaf
(voxel side’s length) of $0.1$ m offers the best trade-off between
computational efficiency and properly capturing salient features in the map,
such as leaves and crops. The resolution parameter (ii) defines the voxel
resolution of the internal NDT grid structure used for representing the
environmental map. This structure is easily searchable, and each voxel
contains the statistical data (e.g., mean and covariance) associated with the
points it contains. The statistical data is used to model the cloud as a set
of multivariate Gaussian distributions and allows us to calculate and optimise
the probability of the existence of points at any position within the voxel.
This parameter needs to be large enough for each voxel to contain at least six
points but small enough to uniquely describe the environment. For this reason,
we found a value of $2.0$ to work for our setting. Finally, for what concerns
the initial pose of the robot (iii), in all the experiments we set it to be
the same from which the mapping task has started, so as to facilitate the
initial localisation of the robot itself. In fact, all the ROS bags have been
extracted from the database by filtering for the same starting position (in
GPS coordinates) so as to spatially align them in an easy way.

###  III-C Evaluation

In order to evaluate the NDT-Localizer performance, we compare the output
produced as estimated odometry against the pose given by the RTK-GPS. We
define a grid-matrix experiment in which, for every month’s map, we compute
the localisation error while playing back the ROS bag of every data recording
session for all the months. As a testbed, we assessed the robot traversing the
first two rows of the research area at Ktima Gerovassiliou. More specifically,
the robot starts at the beginning of the second row, which is traversed
entirely, and then returns it from the first row before traversing the second
row again, completing a loop. Taking as reference Figure 2, the path is
$B\rightarrow G\rightarrow F\rightarrow A\rightarrow B\rightarrow G$. In
total, the traversed distance is $~{}150$ meters.

####  III-C1 Metrics

The metrics used for evaluating the localisation errors are the Absolute Pose
Error (APE), the Relative Pose Error (RPE), and the Root Mean Squared Error
(RMSE). Regarding APE, corresponding poses are directly compared between
estimate and reference, given a pose relation. Then, statistics for the whole
trajectory are calculated. This is useful to test the global consistency of a
trajectory. APE is based on the absolute relative pose between two poses
$P_{est,i},P_{ref,i}\in SE(3)$ at timestamp $i$ [12]:

$E_{i}=P_{est,i}\ominus P_{ref,i}=P_{ref,i}^{-1}P_{est,i}\in SE(3)$ (1)

Where $\ominus$ is the inverse composition operator, which takes two poses and
gives the relative pose. Different pose relations can be used to calculate the
APE, such as only translation, only rotation, or the full transformation:

$APE_{i}=\parallel E_{i}-I_{4\times 4}\parallel_{F}$ (2)

RPE compares instead the relative poses along the estimated and the reference
trajectory. This is based on the delta pose difference. This metric gives
insights into the local accuracy, i.e., the drift.

$\begin{split}E_{i,j}&=\delta_{est_{i,j}}\ominus\delta_{ref_{i,j}}\\\
&=(P_{ref,i}^{-1}P_{ref,j})(P_{est,i}^{-1},P_{est,i})\in SE(3)\end{split}$ (3)

It is possible to use different pose relations to calculate the RPE from
timestamp i to j. For example, if we want to consider the full transformation
as before:

$RPE_{i,j}=\parallel E_{i,j}-I_{4\times 4}\parallel_{F}$ (4)

Then, different statistics can be calculated on the APEs and RPEs of all
timestamps, e.g., the RMSE:

$RMSE=\sqrt{\frac{1}{N}\sum_{\forall_{ij}}APE_{i,j}^{2}}$ (5)

The Evo [13] library has been used for computing the metrics and comparing the
resulting trajectories.

####  III-C2 Results

TABLE I: Loop Experiment: Absolute Pose Error (APE – expressed as mean and standard deviation) and Root Mean Square Error (RMSE) while cross-testing the NDT-Localizer on seasonal data. The best result is in bold. | BAG | March | April | May | June
---|---|---|---|---|---
APE | RMSE | APE | RMSE | APE | RMSE | APE | RMSE
MAP | March | 0.31(0.17) | 0.36 | 0.29(0.18) | 0.35 | 0.35(0.20) | 0.40 | 0.44(0.22) | 0.49
April | 0.51(0.29) | 0.59 | 0.37(0.19) | 0.42 | 0.39(0.20) | 0.44 | 0.50(0.52) | 0.72
| May | 0.40(0.31) | 0.51 | 0.41(0.16) | 0.43 | 0.24(0.21) | 0.32 | 0.23(0.43) | 0.48
| June | – | – | – | – | 0.22(0.08) | 0.25 | 0.16(0.07) | 0.17
TABLE II: Loop Experiment: Relative Pose Error (RPE – expressed as mean and standard deviation) and Root Mean Square Error (RMSE) while cross-testing the NDT-Localizer on seasonal data. The best result is in bold. | BAG | March | April | May | June
---|---|---|---|---|---
RPE | RMSE | RPE | RMSE | RPE | RMSE | RPE | RMSE
MAP | March | 0.03(0.01) | 0.12 | 0.03(0.03) | 0.04 | 0.03(0.06) | 0.07 | 0.05(0.06) | 0.08
April | 0.04(0.14) | 0.14 | 0.02(0.06) | 0.07 | 0.03(0.02) | 0.04 | 0.06(0.16) | 0.17
| May | 0.05(0.19) | 0.19 | 0.03(0.02) | 0.03 | 0.05(0.02) | 0.03 | 0.05(0.33) | 0.34
| June | – | – | – | – | 0.03(0.07) | 0.08 | 0.03(0.04) | 0.05

The results are reported Table I and Table II for the APE and the RPE,
respectively. It is possible to see how, in general, the localisation error
grows in those situations in which the map and the ROS bag belong to different
months. For example, the first row of Table I shows that, while localising on
the map built in March, the localisation error for May ($0.35$) and June
($0.44$) is bigger than the one for March and April (which are comparable –
$~{}0.30$). More specifically, it can be noted that the smallest error is
always located on the main diagonal of the resulting matrix, meaning that
localising a ROS bag on the map built out of the same bag always leads to
better performance. In other words, localising on a map built in the same
month in which the robot is deployed (or around the same date) achieves the
lowest localisation error.

The most interesting result obtained, but not entirely unexpected, is that
testing the ROS bags belonging to the March and April sessions on the summer
map of June does fail to the point that metrics could not be computed. An
intuitive explanation of such failure is that the sensor readings in March and
April are quite sparse because the vines are still in a “dormant” stage and no
foliage is present on them. Therefore, the only features present in the point
cloud belong to the buildings and the structure which is located far from the
robot. However, these features are not present in June’s map due to
occlusions. In fact, during the summer months, the crops are in a growth stage
so advanced that almost prevents any laser beam to leave the row in which the
robot is located, impacting the mapping quality of the surrounding area.

###  III-D Adding Long-Term Stability Mapping

Capturing only the stable features in a given environment is the essence of
enabling long-term operation. With this objective in mind, we propose LTS-
Net111This work is currently under review., an end-to-end self-supervised
learning model. The approach is data-driven and learns implicitly the long-
term stable features by means of a neural network based on PointNet++ [14].
The training data is generated in an unsupervised way by having multiple
observations of the same environment taken at different times, and producing a
pointwise labelling representing each point’s long-term stability. The network
is a regression model trained by using the stability score as a training
signal. This approach was initially evaluated in the NCLT dataset [15], which
provides data over multiple months. The results showed that the network is
able to implicitly learn the long-term stable features of that dataset.
However, the nature of the NCTL dataset, collected inside a university campus
and characterised by a large number of temporally stable features such as
buildings and street lamps, offers limited possibilities to learn the long-
term stability compared to the dataset we are proposing.

TABLE III: Loop Experiment: Absolute Pose Error (APE – expressed as mean and standard deviation) and Root Mean Square Error (RMSE) while cross testing the NDT-Localizer on seasonal data and map cleaned by LTS-Net. The best result is in bold. | BAG | March | April | May | June
---|---|---|---|---|---
APE | RMSE | APE | RMSE | APE | RMSE | APE | RMSE
MAP | June | – | – | – | – | 0.22(0.08) | 0.25 | 0.16(0.07) | 0.17
June-filtered | – | – | 0.16(0.06) | 0.17 | 0.20(0.05) | 0.21 | 0.07(0.04) | 0.08
TABLE IV: Loop Experiment: Relative Pose Error (RPE – expressed as mean and standard deviation) and Root Mean Square Error (RMSE) while cross testing the NDT-Localizer on seasonal data and map cleaned by LTS-Net. The best result is in bold. | BAG | March | April | May | June
---|---|---|---|---|---
APE | RMSE | APE | RMSE | APE | RMSE | APE | RMSE
MAP | June | – | – | – | – | 0.03(0.07) | 0.08 | 0.03(0.04) | 0.05
June-filtered | – | – | 0.02(0.03) | 0.04 | 0.02(0.02) ) | 0.03 | 0.04(0.05) | 0.06
Figure 7: Evaluation pipeline with the introduction of the LTS-Net
---
Figure 8: Visual comparison of stable(blue) and unstable (green to red)
features in the map built out of the data recording session of June
before(left) and after(right) the filtering performed by LTS-Net.

Therefore, we repeat the same experiment done in the previous section using
the map from June after being filtered of the unstable points by using the
target signal which LTS-Net is trained with (see the update pipeline in Figure
7). A visual comparison of the original map against the ground-truth filtered
one is presented in Figure 8 (the ground points have been removed for
clarity). The results of the experiments are reported in Table III and Table
IV, in which we also report the metrics obtained using the original map to
facilitate the comparison.

This final experiment shows that, by removing the temporally unstable points,
is now possible to localise the ROS bag recorded in April on June’s map, with
a discretely low error too ($0.16$). This validates the idea that, out of a
point cloud map, LTS-Net can extract only those points which belong to
temporal stable features such as poles and trunks, which are also the only
features present in April’s bag. Interestingly, also the localisation’s
performance while using May and June’s bags has seen a consistent improvement:
more specifically, the APE for June drastically dropped from $0.16$ to $0.07$.

Besides the increased localisation performance, another important advantage
brought by LTS-NET is the possibility to reduce the number of maps the robot
needs to build before being deployed autonomously. This is a crucial aspect
when targeting outdoor robotic applications, in particular navigating an
orchard which can extend for multiple square kilometres, making mapping a
tedious and expensive task. In our experiments, we found that out of the four
maps used, having only two maps (e.g., March and June) would be sufficient for
achieving a precise localisation across four months. This is already a
positive result because guarantees there is no need for mapping the vineyard
every time the robot needs to be used for data collection or inspection. In
addition, we showed that LST-NET can come in handy in extracting only those
features which are stable across time (such as poles and trunks), removing
from the map those points belonging to grass and leaves. In this way, it is
possible not only to improve the localisation accuracy in general but to
further extend the time required between two mapping sessions, with a saving
in time and cost. Unfortunately, we noted that it was not possible to achieve
a successful localisation while using the ROS bag recorded in March while
using a map built in June. Hence, future directions of this work will be
towards improving the features extraction capabilities of LTS-NET by also
integrating the additional data that is planned to be recorded until the end
of the next harvesting season.

##  IV Conclusion

The present manuscript aims at discussing the problem of the long-term
deployment of an autonomous robot in a continuously changing environment, such
as a vineyard. To this scope, we established a robust and reproducible data
recording pipeline, starting the data collection in March 2022 and with plans
to continue until the end of the harvesting season in October 2022. Some of
the data recording sessions have been used as a benchmark for testing mapping
and localisation performance across time. The results show that re-using maps
built previously in time produces a drop in performance in map-based
localisation algorithms, mainly due to the natural seasonal changes of the
plants and crops. However, we showed that by extracting stable features out of
the maps, we can improve the localisation error and reduce the frequency at
which news maps must be created for guaranteeing a correct and efficient
deployment of the robotic platform.

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# Nonlocality enhanced precision in quantum polarimetry via entangled photons

Ali Pedram<EMAIL_ADDRESS>Department of Physics, Koç University,
Istanbul, Sarıyer 34450, Türkiye  Vira R. Besaga<EMAIL_ADDRESS>Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-
Universität Jena, 07745 Jena, Germany  Frank Setzpfandt  f.setzpfandt@uni-
jena.de Institute of Applied Physics, Abbe Center of Photonics, Friedrich-
Schiller-Universität Jena, 07745 Jena, Germany  Fraunhofer Institute for
Applied Optics and Precision Engineering IOF, 07745 Jena, Germany  Özgür E.
Müstecaplıoğlu<EMAIL_ADDRESS>Department of Physics, Koç University,
Istanbul, Sarıyer 34450, Türkiye  TÜBİTAK Research Institute for Fundamental
Sciences, 41470 Gebze, Türkiye  Faculty of Engineering and Natural Sciences,
Sabancı University, Tuzla 34956, Türkiye

###### Abstract

We present a nonlocal quantum approach to polarimetry, leveraging the
phenomenon of entanglement in photon pairs to enhance the precision in sample
property determination. By employing two distinct channels, one containing the
sample of interest and the other serving as a reference, we explore the
conditions under which the inherent correlation between entangled photons can
increase measurement sensitivity. Specifically, we calculate the quantum
Fisher information (QFI) and compare the accuracy and sensitivity for the
cases of single sample channel versus two channel quantum state tomography
measurements. The theoretical results are verified by experimental analysis.
Our theoretical and experimental framework demonstrates that the nonlocal
strategy enables enhanced precision and accuracy in extracting information
about sample characteristics more than the local measurements. Depending on
the chosen estimators and noise channels present, theoretical and experimental
results show that noise-induced bias decreases the precision for the estimated
parameter. Such a quantum-enhanced nonlocal polarimetry holds promise for
advancing diverse fields including material science, biomedical imaging, and
remote sensing, via high-precision measurements through quantum entanglement.

quantum metrology; quantum optics; polarimetry; quantum state tomography

##  I Introduction

The concept of light polarization is fundamental, depicting the directional
behavior of electric field oscillations within the electromagnetic context. In
cases where light encompasses diverse polarization states, it remains
unpolarized. Polarizers, materials adept at transforming unpolarized light
into a single polarization, play a vital role in this context. Meanwhile,
analyzing how light’s polarization evolves upon interaction with materials
offers insights into their inherent properties, forming the core of
polarimetry [1, 2].

In the realm of contemporary metrology, the pursuit of refined techniques for
enhancing precision measurements is unceasing. One intriguing avenue is
quantum enhanced metrology in which the goal is to use quantum resources such
as entanglement and squeezing, for transcending classical precision bounds [3,
4, 5, 6, 7, 8, 9, 10]. Entangled photon pairs, generated through processes
like spontaneous parametric down-conversion [11, 12], have enabled remarkable
applications in quantum communication and cryptography [13, 14, 15, 16, 17,
18, 19, 20]. Due to this fact, quantum entanglement, a phenomenon where
particles achieve intrinsic correlation regardless of spatial separation, has
captured particular attention in optical quantum metrology and it is revealed
that prudent use of entanglement can be effective in mitigating classical
noise, potentially elevating precision beyond classical limits [21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32]. This leads to the natural question,
whether the integration of quantum entangled light could potentially amplify
the precision of polarimetric measurements.

To reveal the sensitivity of quantum states of light for estimating physical
parameters of optical samples one needs to treat light-matter interaction in a
fully quantum manner. To this end attempts have been made to describe the
polarization states within the framework of quantum optics [33, 34, 35, 36,
37, 38]. It is well established that the classical definition for degree of
polarization fails to capture polarization characteristics of quantum states
of light and quantum states can contain so called ”hidden polarization” or
higher order polarization to which higher moments of the Stokes vectors
contribute. Several works have been published to demonstrate the discrepancy
between quantum and classical concepts of polarization and also new measures
for quantifying the quantum degree of polarization have been proposed [39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

One can describe polarization transformation of light interacting with a
medium using Jones calculus or more generally using Mueller calculus [51].
Interaction of the optical signal with a medium can induce various effects on
the signal including attenuation due to absorption or reflection,
diattenuation of the signal due to dichroism, rotation of its Stokes vector
due to birefringence or depolarization due to random scattering. In the
classical theory of polarimetry, in case that the Mueller matrix is not
singular, the Lu-Chipman decomposition (polar decomposition) can be utilized
to effectively establish an input-output relation between input and output
Stokes vectors decomposing the original Mueller matrix into three elementary
matrices with well defined optical properties and parameterized by their
respective physical parameters (rotation angles, depolarization factors, etc.)
[52]. It is important to note that there has been attempts to generalize the
framework of classical Mueller polarimetry and ellipsometry into the quantum
domain [53, 54]. Finding the ultimate measurement sensitivity of these
parameters using optimal quantum probes and measurements is of utmost
importance. It is shown that NOON states and anti-coherent states can saturate
the Heisenberg limit for rotation sensing [55, 56, 57, 58, 59, 60]. The
problem of rotation sensing is also studied considering different polar
decompositions with optical depolarization and diattenuation noise channels
[61]. Furthermore, Fock and two-mode squeezed vacuum (TMSV) states are optimal
for sensing loss channels and outperform coherent states for estimating the
loss parameter in the bosonic channels, however they can’t beat the shot noise
limit [62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. For a depolarization
channel it is shown that Heisenberg limit scaling is not possible, however
using correlated probes and ancilla-assisted schemes increases the estimation
accuracy [74, 75, 76, 77]. In general, it was argued that no parameter which
is encoded on the density matrix by a non-unitary operator or a convex
combination of unitary operators can be estimated beyond the shot-noise limit
[78]. However, later research revealed that controlled and error-corrected
metrology and environmental monitoring can be utilized to restore the
Heisenberg limit for certain problems [79, 80, 81, 82]. Recently it was
demonstrated that using an entangled photonic source, one can non-
destructively probe single-layer cell cultures and differentiate between
different optical elements in the setup with a fewer number of projective
measurements, [83, 84]. However, quantum advantage in terms of metrology is
not studied in detail these works. Additionally, ghost polarimetry using
polarization entangled photons and utilizing correlations between the Stokes
operators in the reference and sample beams have gained attraction in recent
years because of increased sensitivity of parameter estimation, specifically
for the cases in which having a few number of photons is advantageous in order
not to perturb the material that is being probed [85, 86, 87, 88, 89].

Figure 1: ((Top Panel) Nonlocal Quantum Polarimetry Using Entangled Photons.
In this setup, a pair of photons initially prepared in a polarization-
entangled state $\rho_{\text{in}}^{(rs)}$ is directed through two distinct
spatial channels. One of these paths incorporates a sample denoted as $(s)$,
while the other path serves as the reference denoted as $(r)$. The sample
channel can induce changes in the polarization state of the photon, a
phenomenon described by a single-parameter operation $\hat{U}_{s}(\alpha)$. On
the contrary, the $(r)$ channel applies an identity operation, $\hat{I}_{r}$,
to the photon it contains. By employing quantum state tomography (QST) and
conducting measurements in various polarization bases, the resulting output
state $\rho_{\text{out}}^{(rs)}$ is reconstructed. (Bottom Panel) Local
Quantum Polarimetry Using Single Photons. This scenario involves a single
photon that is prepared in the reduced state of the two-photon entangled
state, specifically
$\rho_{\text{in}}^{(s)}=\text{Tr}_{r}(\rho_{\text{in}}^{(rs)})$. Subsequently,
this photon traverses the sample channel. Employing QST allows for the
reconstruction of the output state.

Within the framework of our quantum polarimetry approach, we endeavor to
harness the unique nonlocal attributes inherent to entangled photons.
Illustrated in Fig. 1, our approach employs two separate pathways: a channel
containing the sample (s) and a reference channel (r), with each accommodating
one photon and the photons in the two channels being entangled. By
capitalizing on the nonlocal correlations between these photons, we aim to
discern subtle polarization changes with enhanced precision. This nonlocal
methodology has the potential to mitigate local noise and disturbances that
traditionally challenge single-channel polarimetry methods. Through
comprehensive theoretical analysis, we examine the viability of the nonlocal
approach, particularly by evaluating the QFI, in comparison to the single-
channel approach. Hereby, our work contributes to the understanding of
fundamental nonlocal quantum phenomena in metrology and proposes a practical
avenue for utilizing nonlocal quantum states in polarimetry. This avenue could
potentially impact high-sensitivity measurements across scientific
disciplines.

##  II Preliminaries: Local and Nonlocal Polarimetry

We consider an entangled pair of photons, signal and idler, generated by
spontaneous parametric down conversion (SPDC). They are prepared in a Bell-
type entangled state in the canonical polarization basis. We assume the
initial state can be ideally prepared as a pure state such that

$\left|\psi_{\text{in}}\right\rangle=\frac{1}{\sqrt{2}}(\left|HV\right\rangle+\left|VH\right\rangle),$
(1)

where $H$ and $V$ refers to horizontal and vertical polarizations,
respectively. For the sake of brevity, we adopt a convenient notation
$\left|AB\right\rangle=\left|A\right\rangle_{r}\otimes\left|B\right\rangle_{s}$
with $A,B\in\\{H,V\\}$ such that the first letter corresponds to the photon’s
state in the reference channel, while the second letter pertains to the sample
channel’s photon state.

A more explicit connection to the Bell state can be established by delving
further into the polarization states. These states can be effectively
expressed in terms of Fock number states, wherein the quantities $n_{H}$ and
$n_{V}$ denote the counts of horizontally ($H$) and vertically ($V$) polarized
photons. Specifically,
$\left|H\right\rangle=\left|n_{H}=1,n_{V}=0\right\rangle\equiv\left|1_{H}\right\rangle$
and
$\left|V\right\rangle=\left|n_{H}=0,n_{V}=1\right\rangle\equiv\left|1_{V}\right\rangle$.
The introduction of a vacuum state encompassing both polarizations, denoted as
$\left|\text{vac}\right\rangle:=\left|0,0\right\rangle\equiv\left|0\right\rangle$,
facilitates the introduction of creation and annihilation operators
corresponding to $H$ and $V$ polarized photons. This can be mathematically
articulated as follows:

$\left|n_{k}\right\rangle=\frac{(\hat{a}_{k}^{\dagger})^{n_{k}}}{\sqrt{n_{k}!}}\left|0\right\rangle,$
(2)

where $k=H,V$. The translation of polarization mode number states into spin
states $\left|S,S_{z}\right\rangle$ is achieved through the utilization of the
total spin, defined as $S=(n_{H}+n_{V})/2$, and the
$S_{z}=(n_{H}-n_{V})/2$-component of spin (or magnetic quantum number). This
translation assumes that the $z$-axis is chosen as the quantization axis.
Consequently, the horizontal polarization state,
$\left|H\right\rangle=\left|n_{H}=1,n_{V}=0\right\rangle=\left|S=1/2,S_{z}=1/2\right\rangle\equiv\left|\uparrow\right\rangle$,
can be identified as the spin-up state of a spin-$1/2$ particle. Similarly, we
can equate $\left|V\right\rangle$ with the spin-down state,
$\left|\downarrow\right\rangle$. In light of this, Equation (1) can be
interpreted as a Bell state, associated with the entanglement of two
spin-$1/2$ particles; although the photons are not spin-$1/2$ particles. For
transverse waves, away from radiation sources in the far field, we can thus
use an effective spin-$1/2$ representation for the polarization states of a
single photon.

In the context of a single photon polarization and spin-$1/2$ analogy, the
utilization of polarization mode creation and annihilation operators as
Schwinger boson representation of spin operators allows for the identification
of generators corresponding to the SU(2) spin algebra. These generators serve
as observables that capture the quantum polarization attributes of photons.
Named after their resemblance to classical Stokes parameters, these
observables are referred to as Stokes operators. In the context of canonical
polarizations, they are expressed as follows:

$\displaystyle\hat{S}_{0}$
$\displaystyle=\frac{1}{2}(\hat{a}_{H}^{\dagger}a_{H}+a_{V}^{\dagger}a_{V}),$
(3) $\displaystyle\hat{S}_{1}$
$\displaystyle=\frac{1}{2}(\hat{a}_{H}^{\dagger}a_{V}+a_{H}a_{V}^{\dagger}),$
(4) $\displaystyle\hat{S}_{2}$
$\displaystyle=-\frac{i}{2}(a_{H}^{\dagger}a_{V}-\hat{a}_{V}^{\dagger}a_{H}),$
(5) $\displaystyle\hat{S}_{3}$
$\displaystyle=\frac{1}{2}(\hat{a}_{H}^{\dagger}a_{H}-a_{V}^{\dagger}a_{V}),$
(6)

where $\hat{S}=(\hat{S}_{1}\hat{S}_{2}\hat{S}_{3})^{T}$ constitutes the
quantum Stokes vector. Notably, this vector commutes with $\hat{S}_{0}$, while
its individual components adhere to the spin algebra:
$[\hat{S}_{i},\hat{S}_{j}]=i\epsilon_{ijk}\hat{S}_{k}$. Within the domain of a
single photon state, the Stokes operators serve as comprehensive descriptors
of the photon’s density matrix, thus enabling quantum state tomography (QST)
of the single photon state through measurement of these operators.
Consequently, any alterations in quantum polarization attributes resulting
from photon-sample interactions manifest as changes in the expectation values
of the Stokes operators, denoted as $S_{k}=\text{Tr}[\rho\hat{S}_{k}]$.

The Bell state arising from Eq. (1) embodies a spin triplet state
$\left|S=1,S_{z}=0\right\rangle$, emerging from the amalgamation of two
spin-$1/2$ entities. This insight prompts the proposal of extending the
concept of single photon Stokes parameters to scenarios involving two or more
photons, thereby encapsulating the polarization attributes of composite
multiphoton systems. We remark that multiphoton nonlocal generalizations of
Stokes parameters are different than generalized Stokes operators written for
spin-1 atomic Bose Einstein condensate to describe their magnetization
properties or for photons that can have longitudinal polarization components
in near or intermediate radiation zones, where transverve wave approximation
fails. Such local generalizations of Stokes operators uses generators of SU(3)
algebra, introduced by Gell-Mann to explore strong interaction in particle
physics [90]. In contradistinction to the ”local” Stokes operators linked with
individual photons, this ”nonlocal” extension of Stokes parameters introduces
a heightened level of complexity in terms of visualization and interpretation.
Given the equivalence between Stokes parameter measurements and the direct
determination of density matrices, multiphoton Stokes parameters can be
equated with the process of quantum state tomography (QST) applied to a
multiphoton density matrix [91]. Remarkably, the density matrix itself offers
a means to quantify the degree of polarization. This is achieved by assessing
the Q function across angular coordinates, integrating it across the entire
angular domain, and subsequently gauging the deviation from the Q function of
unpolarized photons [41]. Furthermore, the density matrix of a single photon
can be decomposed using Stokes operators, which intriguingly coincide with the
decomposition obtained via Pauli matrices. This property can be further
generalized to encompass multiphoton scenarios.

$N$-photon generalized nonlocal Stokes operators are proposed by direct
generalization of single photon Stokes operators [38], which are equivalent to
Pauli spin-$1/2$ operators $\hat{S}_{i}=\hat{\sigma}_{i}$ with $i=0,1,2,3$ to

$\hat{S}_{i_{1}...i_{N}}=\hat{\sigma}_{i_{1}}\otimes...\otimes\hat{\sigma}_{i_{N}}.$
(7)

Here, $\hat{\sigma}_{0}=\hat{I}$ is the identity operator, and we have
$\text{Tr}(\hat{\sigma}_{i}\sigma_{j})=2\delta_{ij}$. Simlar to the Pauli
operator decomposition of single photon density matrix, the $N$-photon density
matrix can be written as

$\hat{\rho}=\frac{1}{2^{N}}\sum_{i_{1}=0}^{3}...\sum_{i_{N}=0}^{3}S_{i_{1}...i_{N}}\hat{S}_{i_{1}...i_{N}},$
(8)

where $S_{i_{1}...i_{N}}=\text{Tr}(\rho\hat{S}_{i_{1}...i_{N}})$ are the
multiphoton nonlocal Stokes parameters.

In our case, we have $N=2$ photons in an initially entangled state. One photon
will interact with the sample in the $s$-channel while the other channel is
empty and taken as the reference ($r$-channel). Accordingly, we determine the
output state by evaluating the map

$\left|\psi_{\text{out}}\right\rangle=\left(\hat{I}_{r}\otimes\hat{U}_{s}(\alpha)\right)\left|\psi_{\text{in}}\right\rangle.$
(9)

In this formulation, we account for the scenario wherein the sample under
consideration could either take the form of a linear polarizer (LP) or a
quarter wave plate (QWP), each characterized by a distinct angular parameter
denoted as $\alpha$. In the case of the LP, $\alpha$ signifies the
transmission angle relative to the vertical axis within the laboratory frame.
Conversely, for the QWP, $\alpha$ denotes the angle defining the orientation
of the fast axis of the birefringent crystal concerning the vertical axis
within the laboratory frame.

The effect of a QWP in the canonical polarization basis can be described by

$\hat{U}_{\text{QWP}}(\alpha)=\begin{bmatrix}\cos^{2}\alpha+i\sin^{2}\alpha&(i-1)\sin\alpha\cos\alpha\\\
(i-1)\sin\alpha\cos\alpha&i\cos^{2}\alpha+\sin^{2}\alpha\end{bmatrix},$ (10)

while for an LP we have

$\hat{U}_{\text{LP}}(\alpha)=\begin{bmatrix}\sin^{2}\alpha&\sin\alpha\cos\alpha\\\
\sin\alpha\cos\alpha&\cos^{2}\alpha\end{bmatrix}.$ (11)

In the forthcoming section, our analysis is dedicated to the evaluation of the
output state, which arises from a predetermined initial entangled state and
distinct models characterizing the sample. The density matrix components of
this output state are ascertained via QST. These elements, when combined in
specific ways, give the nonlocal $2$-photon Stokes parameters. Our objective
is to examine the potential manifestation of a quantum advantage that may
arise due to quantum nonlocality. More specifically, we endeavor to discern
whether the adoption of this nonlocal quantum approach offers any discernible
advantages over conventional local measurements, which are confined to single
channels. After finding the output states, we evaluate the QFI to answer this
question.

##  III Evaluation of the output state

The determination of the output state follows a straightforward matrix product
procedure. For the specific case of a linear polarizer (LP) sample, the
outcome can be succinctly expressed as:

$\left|\psi_{\text{out}}\right\rangle=A(\left|HH\right\rangle+\left|VV\right\rangle)+C\left|HV\right\rangle+B\left|VH\right\rangle,$
(12)

Here, the coefficients are assigned as $A=\sin\alpha\cos\alpha$,
$B=\sin^{2}\alpha$, and $C=1-B=\cos^{2}\alpha$.

Discrepancies between the theoretical predictions and experimental outcomes
for the output state can be attributed to a couple of distinct factors.
Firstly, the experimental initial state possesses slight deviations from the
ideal entangled state, necessitating its description through a density matrix
$\rho_{\text{in}}$, accounting for these imperfections. Consequently, these
deviations propagate to the output state, warranting a depiction as the
density matrix $\rho_{\text{out}}$. Secondly, real-world sample conditions
diverge from the ideal LP or QWP, introducing diattenuation and depolarization
influences on the photon’s polarization state. These alterations can be
effectively modeled through supplementary operations alongside
$\hat{U}_{s}(\alpha)$, formalized using Mueller matrices in accordance with
the Lu-Chipman decomposition [52], extended to the quantum domain [53].

In the present analysis, the modest impact of these effects, coupled with the
close proximity of theoretical and experimental outcomes, has led us to omit a
detailed exploration of imperfections in the results. Nevertheless, it is
noteworthy that such imperfections could potentially enrich the polarimetric
analysis of an unknown sample. The constellation of parameters encompassing
retardation angle, diattenuation, and depolarization characteristics can serve
as distinctive signatures for materials, offering a means of characterization
and classification vis-à-vis datasets from other samples.

Further comparison between the theoretical output state and experimental
findings has consistently yielded nearly perfect agreement in the case of the
QWP as well. An essential divergence between the LP and QWP scenarios lies in
the emergence of imaginary components within the density matrix elements in
the latter case.

The input state for the local, single $s$-channel scenario is given by the
reduced state

$\rho_{\text{in}}^{(s)}=\frac{1}{2}(\left|H\right\rangle_{s}\left\langle
H\right|_{s}+\left|V\right\rangle_{s}\left\langle V\right|_{s}),$ (13)

which is maximally mixed, unpolarized, state with
$S_{0}=1/2,S_{1}=S_{2}=S_{3}=0$. After the LP, the output state becomes

$\rho_{\text{out}}^{(s)}=B\left|H\right\rangle_{s}\left\langle
H\right|_{s}+C\left|V\right\rangle_{s}\left\langle
V\right|_{s}+A(\left|H\right\rangle_{s}\left\langle
V\right|_{s}+\left|V\right\rangle_{s}\left\langle H\right|_{s}).$ (14)

The Stokes parameters of the output state are given by
$S_{0}=1/2,S_{1}=A,S_{2}=0,S_{3}=B-1/2$, which shows that LP can polarize the
initially unbpolarized photon.

Following a similar procedure we find the output of the two photon entangled
state after being acted upon by QWP to be

$\left|\psi_{\text{out}}\right\rangle=\frac{(i-1)A}{\sqrt{2}}(\left|HH\right\rangle+\left|VV\right\rangle)+\frac{iC+B}{\sqrt{2}}\left|HV\right\rangle+\frac{C+iB}{\sqrt{2}}\left|VH\right\rangle.$
(15)

For the case of local polarimetry, the state at the output of QWP becomes

$\rho_{\text{out}}^{(s)}=\frac{1}{2}(\left|H\right\rangle_{s}\left\langle
H\right|_{s}+\left|V\right\rangle_{s}\left\langle V\right|_{s}).$ (16)

It is evident that the utilization of both single-channel local and two-
channel non-local quantum polarimetry, encompassing the examination of local
and nonlocal Stokes parameters, or equivalently, the elements of local or
nonlocal states, furnishes a viable means to ascertain the unknown parameters
associated with the sample, represented in this context by the angle $\alpha$.
However, which strategy proves more precise in this pursuit? In order to
address this question rigorously, we proceed to evaluate the QFI in the
subsequent section.

##  IV  Comparison of the precision

QFI quantifies the maximum achievable information about a parameter using a
parameterized density matrix. The Quantum Cramer-Rao bound (QCRB) establishes
a lower bound on the variance of any unbiased estimator $\hat{\alpha}$ of the
parameter $\alpha$ such that for a single repetition of the measurement of
$\theta$, the variance of the estimator is lower bounded by the reciprocal of
the QFI [92]. The mathematical expression of the QCRB for an unbiased
estimator $\hat{\alpha}$ is

$\text{Var}(\alpha)\geq\frac{1}{mF_{Q}},$ (17)

in which $m$ is the number of observations and $F_{Q}$ is the QFI. For a pure
state QFI can be expressed as [92]

$F_{Q}=4[\langle\partial_{\alpha}\psi|\partial_{\alpha}\psi\rangle-|\langle\psi|\partial_{\alpha}\psi\rangle|^{2}].$
(18)

For a mixed state the QFI becomes [92]

$\mathcal{F}=\sum_{i}\frac{(\partial_{\alpha}\lambda_{i})^{2}}{\lambda_{i}}+2\sum_{i\neq
j}\frac{(\lambda_{i}-\lambda_{j})^{2}}{\lambda_{i}+\lambda_{j}}|\langle
e_{i}|\partial_{\alpha}e_{j}\rangle|^{2}.$ (19)

We aim to find the estimation precision of the parameter $\alpha$ using QFI
for both local and non-local scenarios. For the case of the LP, the output
state for the single channel has the matrix form

$\rho_{\text{out}}=\begin{bmatrix}B&A\\\ A&1-B\end{bmatrix},$ (20)

whose eigenvalues are $\lambda_{0}=0$ and $\lambda_{1}=1$. The corresponding
eigenvectors can be written as $e_{0}=(\cos\alpha-\sin\alpha)^{T}$ and
$e_{1}=(\sin\alpha\cos\alpha)^{T}$. For the case of an entangled initial state
and two-channel nonlocal polarimetry, the output state is a pure state as
given in Eq. 12. Substituting them into the quantum Fisher information formula
gives

$\displaystyle\mathcal{F}_{\text{LP}}^{\text{L}}=4,$ (21)
$\displaystyle\mathcal{F}_{\text{LP}}^{\text{NL}}=8.$

Both expressions are independent of $\alpha$. Hence we find
$\mathcal{F}_{\text{LP}}^{\text{NL}}=2\mathcal{F}_{\text{LP}}^{\text{L}}$.
Similar calculations for local metrology for the case of QWP gives

$\displaystyle\mathcal{F}_{\text{QWP}}^{\text{L}}=0,$ (22)
$\displaystyle\mathcal{F}_{\text{QWP}}^{\text{NL}}=8.$

It is clear that the output state for the local case is parameter independent
as given in Eq. 16 and we can’t use it to give any estimate of $\alpha$.
Therefore the non-local approach is far superior than the local one for the
case of QWP. It is instructive to check the QFI for equally weighted
superposition states of vertical and horizontal polarization modes for the two
photon polarimetry scheme. Using such a state for two photon LP and QWP
settings we get

$\begin{gathered}\mathcal{F}_{\text{LP}}^{\text{SP}}=\frac{16}{(\text{cos}(\alpha)+\text{sin}(\alpha))^{4}}-(\frac{4(\text{cos}(\alpha)-\text{sin}(\alpha))}{\text{cos}(\alpha)-\text{sin}(\alpha))^{3}})^{2},\\\
\mathcal{F}_{\text{QWP}}^{\text{SP}}=16-16\text{cos}^{2}(\alpha).\end{gathered}$
(23)

Both expressions in Eq. 23 depend on $\alpha$ and for $0\leq\alpha\leq\pi/2$
they are less than 8. For $\mathcal{F}_{\text{QWP}}^{\text{SP}}$ the minimum
of QFI is zero at $\alpha=\pi/4$ and the maximum is 16 at $\pi/2$. For
$\mathcal{F}_{\text{LP}}^{\text{SP}}$ the minimum of QFI is 1 at
$\alpha=\pi/4$ and the maximum is 4.74074 at $\pi/12$ and $5\pi/12$.
Therefore, polarimetry for QWP and LP with entangled photons not only yields a
larger result for QFI, but entanglement is also necessary to gain information
independent of the specific choice of $\alpha$.

##  V Quantum State Tomography

QST is the process of reconstructing a quantum state from the measurement
outcomes of experiments on an ensemble of identical quantum states [91, 93,
94, 95]. Considering that the measurements are represented by the positive
operator-valued measurements (POVM)
$\hat{\Pi}_{i}=\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|$, we
know that measuring $\hat{\Pi}_{i}$ will produce an outcome with label $i$
whose probability is calculated using Born rule

$p_{i}=\text{Tr}[\hat{\rho}\hat{\Pi}_{i}].$ (24)

Experimentally, one measures the POVM $\mathcal{N}$ times using $\mathcal{N}$
copies of the unknown state $\hat{\rho}$. The average number of coincidence
counts will be given by

$n_{i}=\mathcal{N}\text{Tr}[\hat{\Pi}_{i}\hat{\rho}].$ (25)

Therefore in QST one approximates $p_{i}$ by $f_{i}=n_{i}/\mathcal{N}$. For
$\mathcal{N}\rightarrow\infty$ the probability for these measurement outcomes
is given by $p_{i}=n_{i}/\mathcal{N}$. Assuming that the Hilbert space is
finite dimensional and taking ${\hat{E}_{i}}$ as a set of orthonormal
Hermitian basis of traceless operators, we can express the density matrix as:

$\hat{\rho}=\frac{\hat{I}}{d}+\sum_{i=1}^{d^{2}-1}r_{i}\hat{E}_{i}.$ (26)

Here, $r_{i}$ are real numbers and $d$ is the dimensionality of the Hilbert
space. Substituting Eq. 27 in Eq. 24 and approximating the probabilities with
frequencies we get.

$f_{i}\approx\frac{\text{Tr}[\hat{\Pi}_{i}]}{d}+\sum_{i=1}^{d^{2}-1}r_{i}\text{Tr}[\hat{\Pi}_{i}\hat{E}_{i}].$
(27)

Assuming that $\tilde{f}_{i}=f_{i}-\text{Tr}[\hat{\Pi}_{i}]{d}$ and
$\tilde{\Pi}_{i,j}=\text{Tr}[\hat{\Pi}_{i}\hat{E}_{i}]$ we can write this
equation in a matrix form as $\bm{\tilde{f}}\approx\bm{\tilde{\Pi}r}$. The
goal is to find the vector of expansion coefficients $\bm{r}$ using
$\bm{\tilde{f}}$ which is only possible if the matrix $\bm{\tilde{\Pi}}$ is
invertible. This condition is satisfied for informationally complete sets of
POVMs that span the space of density matrices acting on the Hilbert space.

In what follows in this section, we will present a basic review of optical
polarization QST for a single and two qubit systems. In polarization QST the
experimental data is a collection of coincidence counts of the photodetectors
implementing a collection of projective measurements. In polarization
measurements, the usual states used for projective measurements are horizontal
($\left|H\right\rangle$) and vertical ($\left|V\right\rangle$) linear
polarized states, right ($\left|R\right\rangle$) and left
($\left|L\right\rangle$) circular states and diagonal ($\left|D\right\rangle$)
and anti-diagonal ($\left|A\right\rangle$) polarization states. Hence, the
goal is to estimate the state $\hat{\rho}$ based on the information of the
coincidence counts $n_{i}$ for each projective measurement
$\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|$. For a single qubit
Eq. 8 can be written as

$\hat{\rho}=\frac{1}{2}\sum_{i=0}^{3}\frac{S_{i}}{S_{0}}\sigma_{i}.$ (28)

Writing the Stokes operators in terms of the POVMs one can find their
expectation values as

$\begin{gathered}\mathcal{N}=n_{H}+n_{V}=n_{D}+n_{A}=n_{R}+n_{L},\\\
S_{1}=\frac{1}{\mathcal{N}}(n_{D}-n_{A}),\\\
S_{2}=\frac{1}{\mathcal{N}}(n_{R}-n_{L}),\\\
S_{3}=\frac{1}{\mathcal{N}}(n_{H}-n_{V}).\end{gathered}$ (29)

Upon substituting these equations into Eq. 28 and assuming $S_{0}=1$ (no
photon loss) one can estimate the one qubit density matrix using the
coincidence numbers of the six POVMs. For two qubits Eq. 8 can be written as

$\hat{\rho}=\sum_{i=1}^{16}r_{i}\hat{\Gamma_{i}},$ (30)

in which the basis $\hat{\Gamma_{i}}$ are the two-qubit Stokes operators. In
Eq. 30 $r_{i}=\text{Tr}[\hat{\Gamma_{i}}\hat{\rho}]$ are the expansion
coefficients of the density matrix for their respective two-qubit Stokes
operators given as a $16\times 1$ vector. The coincidence counts and the
expansion coefficients are connected via

$n_{i}=\mathcal{N}\sum_{i=1}^{16}(B^{-1})_{i,j}n_{j}.$ (31)

The $16\times 16$ matrix $B$ is defined as
$B_{i,j}=\left\langle\psi_{i}\right|\Gamma_{j}\left|\psi_{i}\right\rangle$. It
is shown that the density matrix can be expanded as

$\hat{\rho}=\frac{1}{\mathcal{N}}\sum_{i=1}^{16}\hat{M}_{i}n_{i},$ (32)

in which the matrix $\hat{M}_{i}$ are $4\times 4$ matrices defined via

$\hat{M}_{i}=\sum_{i=1}^{16}(B^{-1})_{i,j}\Gamma_{j}.$ (33)

To partially mitigate the detrimental effects of experimental systematic
errors on the reconstructed density matrices, one can utilize maximum
likelihood estimation. In maximum likelihood estimation, the objective is to
explore the entire space of permissible, physically valid density matrices in
order to identify the one that maximizes the likelihood of yielding the
observed measurement outcomes. During this optimization process, to ensure
that the resulting states are physical, the density matrices are written as
their normalized Cholesky decomposition

$\hat{\rho}(\vec{t})=\frac{\hat{T}^{\dagger}\hat{T}}{\text{Tr}[\hat{T}^{\dagger}\hat{T}]}.$
(34)

By construction, Eq. 34 yields a positive, Hermitian and normalized density
matrix. The operator $\hat{T}$ is a triangular matrix and is parametrized as

$\hat{T}(\vec{t})=\begin{bmatrix}t_{1}&0&\cdots&0\\\
t_{2^{n}+1}+it_{2^{n}+1}&t_{2}&\cdots&0\\\ \cdots&\cdots&\cdots&0\\\
t_{4^{n}-1}+it_{4^{n}}&t_{4^{n}-3}+it_{4^{n}-2}&t_{4^{n}-5}+it_{4^{n}-4}&t_{2^{n}}.\end{bmatrix}$
(35)

The distribution of coincidence numbers is of Poisson type but for large
coincidence counts, due to the central limit theorem, one can approximate this
using a normal distribution. For a normally distributed coincidence counts,
the likelihood that the parametrized density matrix Eq. 34 produces the
measured counts is

$P(\vec{n},\vec{t})=\frac{1}{N_{norm}}\prod_{i}\text{exp}\left[\frac{(\mathcal{N}\text{Tr}[\hat{\Pi}_{i}\hat{\rho}(\vec{t})-n_{i}])^{2}}{2\mathcal{N}\text{Tr}[\hat{\Pi}_{i}\hat{\rho}(\vec{t})]}\right].$
(36)

The task is to maximize this likelihood function by finding the optimal values
for the parameters in $\vec{t}$. To simplify the procedure, one can minimize
the negative of the log-likelihood function instead of maximizing the
likelihood. Doing so, the function that MLE minimizes becomes

$\mathcal{L}(\vec{n},\vec{t})=\sum_{i}\text{exp}\left[-\frac{(\mathcal{N}\text{Tr}[\hat{\Pi}_{i}\hat{\rho}(\vec{t})-n_{i}])^{2}}{2\mathcal{N}\text{Tr}[\hat{\Pi}_{i}\hat{\rho}(\vec{t})]}\right].$
(37)

Hence, through the process of MLE quantum state tomography, due to Eq. 34 one
is guaranteed to get a valid density matrix which minimizes the penalty
function Eq. 37 and therefore is most likely to produce the experimentally
measured coincidence counts.

##  VI Experimental Implementation

Figure 2: Optical arrangement used in the experiments. (a) Nonlocal quantum
polarimetry using a pair of polarization-entangled photons (Bell state in the
form of${1/\sqrt{2}}(\left|HV\right\rangle+\left|VH\right\rangle$). (b) Local
quantum polarimetry. The partner photon from the entangled pair acts as a
heralding one. PSA: polarization state analyzer; $\alpha$: angle of sample’s
principal optical axis orientation (fast axis for quarter-wave plate,
transmission axis for linear polarizer); SPCM: single photon counting module.

The experimental setup that we used within the reported studies is outlined in
Fig. 2. A polarization Mach-Zehnder interferometer with two periodically-poled
KTP (KTiOPO4) crystals serves as the source of polarization-entangled photon
pairs [96]. The wavelength-degenerate photons at 810 nm are produced through
spontaneous parametric down-conversion (SPDC) in type-II phase matching
configuration. At the exit of the interferometer the photons are split into
two spatially separated channels. In one of the channels we employ a
combination of quarter- and half-wave plates to ensure the Bell state in the
form of $\frac{1}{\sqrt{2}}(|HV\rangle+|VH\rangle)$. The preparation of the
entangled photon pairs is not detailed in the figure.

The performance of the source is evaluated via QST [91] using one detector per
channel. For projective polarization detection we introduce rotating QWP and
LP in both channels and fiber-coupled single photon counting modules. By using
a time tagging device we acquire the coincidence events which are then used
for reconstructing the density matrix by the maximum likelihood method [93].
The experimentally achieved state reaches a concurrence of 0.94 and a fidelity
with the designed Bell state of 0.96.

During the sample evaluation, a QWP or a LP is introduced into the sample
channel at several orientations. The angles between the optical axis of the
sample (fast axis of the QWP, transmission axis of the LP) and the vertical
axis in the laboratory frame were 0, 45∘, 90∘, and a randomly generated angled
of 37∘. Following the concept from Fig. 1 we performed measurements using the
one- and two-channel configurations. For the latter, the same procedure as for
source performance evaluation was used, but with a sample influencing the
polarization of one of the photons from the pair.

For the one-photon measurement we utilized the same source but polarization
sensing was performed only in the sample channel. Taking into account that the
number of coincidence events is much less than the single photon counts, the
coincidence based measurement in two-channel configuration can not be directly
compared with the single-photon experiment by simply using the same
integration time. To overcome this issue we performed heralded single-channel
measurements. For this, the QWP and LP in the reference arm were removed and
the projective measurements were performed only in the sample channel. Also
here, we counted the coincidence events between two channels, now the
reference photon acting as a heralding one, and used these counts for one-
photon density matrix reconstruction [93]. Hereby we kept the number of
photons contributing to the useful signal on the same level as in the two-
channel measurement mode.

Figure 3: Representative density matrices of the probe state (”source”); two-
photon state after the signal photon interacted with a linear polarizer (LP)
and a quarter wave plate (QWP) oriented with their optical axis at 37∘
(averaged over 10 experimental runs); signal photon state after interaction
with LP and QWP oriented at 37∘ (averaged over 10 experimental runs).

The representative examples of the density matrix of the two-photon state
entering the experimental arrangement, and outcomes from the two- and one-
channel measurements are shown in Fig. 3. For both modes of the experiments we
measured the density matrix of the quantum state altered by the sample, each
repeated 10 times.

##  VII  Theoretical Analysis of the Data

In order to compare the experimental results given in Fig. 3 with the
theoretical predictions, in Fig. 4 we calculated the expected output density
matrices for each polarimetry task, which are shown in Fig. 4. We have used
the open source package QuTiP for our calculations [97].

Figure 4: Theoretical results for the density matrix after each experiment
shown as a histogram. Real (a) and imaginary (b) components of the density
matrix for the non-local scheme with LP. Real (c) and imaginary (d) components
of the density matrix for the non-local scheme with QWP. Real (e) and
imaginary (f) components of the density matrix for the local scheme with LP.
Real (g) and imaginary (h) components of the density matrix for the local
scheme with QWP. Both LP and QWP are oriented at $37^{\circ}$.

Upon comparing Fig. 3 and Fig. 4 we see that although they are qualitatively
very close, the measured and calculated density matrix differ slightly. This
is expected due to several sources of error and noise including state
preparation error, ambient dark noise, imperfections in the optical elements,
fluctuations of the photon source and errors in photon counting and post-
processing. The errors are more pronounced for the local and non-local
polarimetry experiments on LP. To gain a better understanding of the errors
with respect to the theoretical predictions, in Fig. 5 the fidelity of the
experimental density matrices with their theoretical counterparts are given.
Here we clearly see that the aforementioned experimental data are more distant
to their theoretical counterparts. The goal in these experiments is to predict
the value of $\alpha$ which we set to be equal to 37 degrees. We consider two
different estimators to find the rotation angle of the optical devices.

Figure 5: Fidelity between the tomographically reconstructed experimental
density matrix and the theoretical density matrix for each experiment. Black
diamonds, green circles, blue triangles and red stars represent the results
for local QWP, non-local QWP, non-local LP and local LP respectively.

From Eq. 12, Eq. 14 and Eq. 15 we can use the following expressions as
estimators for $\alpha$.

$\displaystyle\hat{\alpha}_{1}^{\text{LP}_{\text{NL}}}=\text{arctan}(\sqrt{\frac{\rho_{11}}{\rho_{22}}}),$
(38)
$\displaystyle\hat{\alpha}_{1}^{\text{QWP}_{\text{NL}}}=\text{arccos}(\sqrt{2(\rho_{33}-\rho_{44})}),$
$\displaystyle\hat{\alpha}_{1}^{\text{LP}_{\text{L}}}=\text{arctan}(\sqrt{\frac{\rho_{11}}{\rho_{22}}}).$

We can also define another estimator based on the value of angle that
maximizes the fidelity between the experimental density matrix and the density
matrix obtained by transforming the input state (entangled state for the non-
local and maximally mixed state for the local) by that angle, using the
corresponding transformation matrix related to the experiment

$\hat{\alpha}_{2}=\\{\alpha^{*}\in[0,\pi/2]:\forall\alpha\in[0,\pi/2],\text{fidelity}(\rho_{\text{exp}},\varepsilon_{\alpha^{*}}(\rho_{\text{in}}))\geq\text{fidelity}(\rho_{\text{exp}},\varepsilon_{\alpha}(\rho_{\text{in}}))\\}.$
(39)

Here, $\varepsilon_{\alpha}$ represents a transformation operator using QWP or
LP acting on the initial state and giving the final state and parameterized by
$\alpha$. So all the estimators given in Eq. 38 have their counterparts in Eq.
39. In Fig. 6 the estimated angles given by these two sets of estimators are
given.

Figure 6: Estimated angle $\alpha$ for each experiment using two different
estimators. (a) Estimated angles with $\hat{\alpha}_{1}$ using the
trigonometric relations between the elements of the density matrices given in
Eq. 38. The red stars and red dotted line are the angles estimated for local
LP experiment and their average. The blue triangles and blue dot dashed line
are the angles estimated for non-local LP experiment and their average. The
green circles and green dashed line are the angles estimated for non-local QWP
experiment and their averages. (b) Estimated angles with $\hat{\alpha}_{2}$
using the fidelity maximization procedure given in Eq. 39. The description of
the data points and their averages are similar to part (a). The solid black
line is the actual value of the angle at 37 degrees. The variances for the
estimated angles in (a) are
$\text{Var}(\hat{\alpha}_{1}^{\text{LP}_{\text{NL}}})=0.1202$,
$\text{Var}(\hat{\alpha}_{1}^{\text{QWP}_{\text{NL}}})=0.0135$ and
$\text{Var}(\hat{\alpha}_{1}^{\text{LP}_{\text{L}}})=0.0087$. The variances
for the estimated angles in (b) are
$\text{Var}(\hat{\alpha}_{2}^{\text{LP}_{\text{NL}}})=0.0744$,
$\text{Var}(\hat{\alpha}_{2}^{\text{QWP}_{\text{NL}}})=0.1735$ and
$\text{Var}(\hat{\alpha}_{2}^{\text{LP}_{\text{L}}})=0.4049$.

Although theoretical predictions without considering any source of noise yield
the correct result for the value of the angle, it is evident that the average
value of the estimated angles for all polarimetric procedures and both
estimators, does not equal to the real value of $\alpha=37$. Therefore, the
results suggest that the angles estimated using experimental data contain
statistical bias, reports for which are available in the literature [98, 99].
From Fig. 6(a) we see that indeed non-local polarimetry for LP using entangled
photons gives a better estimate of $\alpha$ as compared to the local case.
Also, we see that the QWP experiment has a far larger error compared with both
of the LP experiments. However, even this is a considerable improvement
compared to the local QWP experiment because in that case QFI is zero and no
angle can be estimated. In Fig. 6(b) we see that applying the estimator
defined by the fidelity maximization procedure, more accurate estimates for
the values of $\alpha$ are obtained. In this estimator, unlike the one given
by Eq. 38, the information from all of the elements of the density matrices
are used and therefore generally it is expected that it will yield a better
estimate. However, the effect of the noise on the estimation of the angle is
highly non-trivial and comparing Fig. 6(a) and Fig. 6(b) we can see that
although the accuracy of non-local polarimetry is higher than local case for
the estimator $\hat{\alpha}_{1}$, the converse is true for $\hat{\alpha}_{2}$
. Additionally it must be stated that the estimate given in Eq. 38 has one
advantage over Eq. 39 and that is the fact that one only needs two density
matrix elements to come up with an angle estimate and full QST is not
required.

Our results demonstrate that although one can use entanglement to enhance
parameter estimation sensitivity, it is also important to consider the effect
of realistic metrological experiments with various possible noise
contributions on the accuracy of the estimated parameters. It is also
important to design estimators which are more robust to loss of accuracy due
to noise. For example, it is shown in Fig. 6 that $\hat{\alpha}_{1}$ is more
susceptible to the noise-induced bias compared with $\hat{\alpha}_{2}$. It is
worth mentioning that for biased estimators the QCRB becomes [92]

$\text{Var}(\alpha)\geq\frac{(1+\partial_{\alpha}b)^{2}}{mF_{Q}},$ (40)

in which $b$ is the bias of the estimator. Our results show that, taking into
account the various sources of experimental noise and uncertainty, estimated
values for the angle will contain bias. To investigate the validity of this
assertion and get more insight, in the next section we will perform QST on a
set of numerically generated data which takes into account several sources of
noise.

##  VIII  Theoretical Model of the Experimental noise

However small it may be, no experiment is immune to noise. It is important to
take into account the effects of noise on the results of the experiment,
quantify it if it is possible, and take measures to mitigate such effects. In
our experiment several sources of error exist that can potentially distort the
data. In our theoretical analysis of the effect of noise we only consider four
sources of noise, namely state preparation noise, dark counts, misalignment of
measurement bases and Poisson noise in photon counting. Other possible sources
of error such as imperfections of the photodetectors and other optical
elements are neglected. We model the state preparation noise as

$\hat{\rho}_{in}=q_{1}\hat{\rho}+(1-q_{1})\hat{\rho}_{r},$ (41)

in which $\hat{\rho}$ is the desired noiseless input state, $\hat{\rho}_{r}$
is a an average of a number of random density matrices and $q_{1}\in[0,1]$ is
a real number quantifying the amount of preparation noise. In addition to the
optical signal, the background noise can also trigger the photodetectors and
induce dark counts. To model this effect, we assume that the quantum state of
the background noise is given by a maximally mixed state and that it also
contributes to the input of the photodetectors [100, 101].

$\hat{\rho}^{\prime}=q_{2}\hat{\rho}_{out}+(1-q_{2})\frac{\hat{I}_{d}}{d}.$
(42)

Here, $q_{2}\in[0,1]$ is a real number quantifying the amount of dark counts
and $\hat{I}_{d}$ is a $d\times d$ identity matrix. $\hat{\rho}_{out}$ is the
density matrix at the output of LP or QWP considering the fact that the input
state is $\hat{\rho}_{in}$. During the experiments, the measurement process is
also subject to noise. In our calculations we model this noise by random
rotations of the projection operators. The rotation operator is given by [102,
103, 100, 101].

$\hat{U}(\omega_{1},\omega_{2},\omega_{3})=\begin{bmatrix}e^{i\omega_{1}/2}\text{cos}(\omega_{3})&-ie^{i\omega_{2}/2}\text{sin}(\omega_{3})\\\
-ie^{-i\omega_{2}/2}\text{sin}(\omega_{3})&e^{-i\omega_{1}/2}\text{cos}(\omega_{3}),\end{bmatrix}$
(43)

in which $\omega_{1}$, $\omega_{2}$, $\omega_{3}$ are normally distributed
parameters with zero mean and standard deviation $\sigma$. In this framework,
$\sigma$ determines the amount of noise due to random rotations and
misalignment in the measurement bases. The rotated projection operator becomes

$\hat{\Pi}_{i}^{\prime}=(\hat{I}\otimes\hat{U})\hat{\Pi}_{i}(\hat{I}\otimes\hat{U}^{\dagger}).$
(44)

The coincidence counts, including the sources of noise, is given by

$n_{i}^{\prime}=\mathcal{N}_{i}\text{Tr}[\hat{\Pi}_{i}^{\prime}\hat{\rho}^{\prime}],$
(45)

in which $\mathcal{N}_{i}\in\text{Pois}(\mathcal{N})$ is randomly generated
using the Poisson distribution with the mean value of $\mathcal{N}$. Using Eq.
45 we include the effect of Poisson noise in our calculations for photon
counting.

Our goal is to show that the mentioned experimental noise models can induce
bias in our parameter estimation problem. We don’t intend to optimize the
parameters of our noise model to quantify the experimental errors more
accurately. For our calculations we take $\sigma=\pi/720$, $\mathcal{N}=5000$.
The random density matrix in Eq. 41 is averaged out of 20 random density
matrices. Also, to make presentation of our results simpler, we take
$q_{1}=q_{2}=q$. The theoretical results for noisy QST for our chosen set of
parameters is given in Fig. 7. Comparing Fig. 7(a) and Fig. 7(b) it is clear
that compared to $\hat{\alpha}_{1}$ using $\hat{\alpha}_{2}$ as an estimator
will result in a smaller bias. Our calculations show that for
$\hat{\alpha}_{1}$, although there are fluctuations due to Poisson noise, the
bias increases linearly with parameter $q$ as shown in Fig. 7(a). For our
selected set of parameters we see that in Fig. 7(a) that in case of using
$\hat{\alpha}_{1}$ as the estimator $\hat{\alpha}_{1}^{\text{LP}_{\text{NL}}}$
results in the smallest bias. However, using the estimator $\hat{\alpha}_{2}$,
as shown in Fig. 7(b), $\hat{\alpha}_{2}^{\text{LP}_{\text{L}}}$ yields the
smallest value for bias.

Figure 7: The magnitude of bias for the estimator (a) $\hat{\alpha}_{1}$ and
$\hat{\alpha}_{2}$ (b) . In both (a) and (b) the solid green, long-dashed blue
and the dashed red lines are $\hat{\alpha}_{i}^{\text{QWP}_{\text{NL}}}$,
$\hat{\alpha}_{i}^{\text{LP}_{\text{NL}}}$ and
$\hat{\alpha}_{i}^{\text{LP}_{\text{L}}}$ respectively. The expected values
for magnitude of bias are shown by the dash-dotted green line for
$\hat{\alpha}_{i}^{\text{QWP}_{\text{NL}}}$, loosely dash-dotted blue line for
$\hat{\alpha}_{i}^{\text{LP}_{\text{NL}}}$, and dooted red line for
$\hat{\alpha}_{i}^{\text{LP}_{\text{L}}}$.

Our results clearly show that including sources of error in our calculations
will induce bias in the estimated angles. Therefore, to fully utilize the
metrological quantum advantage one can either try to reduce the effect of
noise on the experimental results, or design estimators that are less
susceptible to noise.

##  IX  Conclusion

Our theoretical and experimental framework illustrates that employing nonlocal
polarimetry with an initially entangled state leads to improved precision and
accuracy in extracting information about sample characteristics compared to
the case of local polarimetry. Our results show that using quantum
entanglement can increase the precision and accuracy depending on the chosen
estimators and existing noise channels. We have seen that the experimental
noise induces bias in our estimators and therefore reduces the accuracy of the
estimated results. Therefore, to harness the full potential of quantum states
for metrological tasks, one needs to address the issue of noise-induced bias
as well.

Our investigation reveals the presence of a quantum nonlocality advantage
within the ambit of two-channel entangled state quantum polarimetry. However,
it is pertinent to note that this quantum advantage remains suboptimal. Within
the realm of quantum metrology, the utilization of NOON states, expressed in
the form $(\left|HH\right\rangle+\left|VV\right\rangle)/\sqrt{2}$, could
conceivably offer a higher enhancement in precision compared to local
polarimetry [104]. Furthermore, such states could potentially be harnessed to
attain quadratic advantages with increasing photon numbers. While our current
study identifies the nonlocal advantage, the pursuit of optimization for this
nonlocal scheme remains an open challenge, awaiting exploration in future
investigations.

###### Acknowledgements.

This work has been funded by the European Union’s Horizon 2020 research and
innovation programme (Grant Agreement No. 899580); the German Federal Ministry
of Education and Research (FKZ 13N14877 and 13N15956); and the Cluster of
Excellence “Balance of the Microverse” (EXC 2051-Project No. 390713860). V.B.
thanks for funding of this work also through the ProChance-career program of
the Friedrich Schiller University Jena.

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