11institutetext: University of Michigan, Ann Arbor MI 48109, USA
11email<EMAIL_ADDRESS>
# ClearPose: Large-scale Transparent Object Dataset and Benchmark

Xiaotong Chen University of Michigan, Ann Arbor MI 48109, USA
11email<EMAIL_ADDRESS>Huijie Zhang University
of Michigan, Ann Arbor MI 48109, USA
11email<EMAIL_ADDRESS>Zeren Yu University of
Michigan, Ann Arbor MI 48109, USA
11email<EMAIL_ADDRESS>Anthony Opipari
University of Michigan, Ann Arbor MI 48109, USA
11email<EMAIL_ADDRESS>
Odest Chadwicke Jenkins University of Michigan, Ann Arbor MI 48109, USA
11email<EMAIL_ADDRESS>
###### Abstract

Transparent objects are ubiquitous in household settings and pose distinct
challenges for visual sensing and perception systems. The optical properties
of transparent objects leave conventional 3D sensors alone unreliable for
object depth and pose estimation. These challenges are highlighted by the
shortage of large-scale RGB-Depth datasets focusing on transparent objects in
real-world settings. In this work, we contribute a large-scale real-world RGB-
Depth transparent object dataset named ClearPose to serve as a benchmark
dataset for segmentation, scene-level depth completion and object-centric pose
estimation tasks. The ClearPose dataset contains over 350K labeled real-world
RGB-Depth frames and 5M instance annotations covering 63 household objects.
The dataset includes object categories commonly used in daily life under
various lighting and occluding conditions as well as challenging test
scenarios such as cases of occlusion by opaque or translucent objects, non-
planar orientations, presence of liquids, etc. We benchmark several state-of-
the-art depth completion and object pose estimation deep neural networks on
ClearPose. The dataset and benchmarking source code is available at
https://github.com/opipari/ClearPose.

###### Keywords:

Transparent Objects. Depth Completion. Pose Estimation. Dataset and Benchmark.

##  1 Introduction

Transparent and translucent objects are prevalent in daily life and household
settings. When compared with opaque and Lambertian objects, they present
additional challenges to visual perception systems. The first challenge is
that transparent objects do not exhibit consistent RGB color features across
varying scenes. Since the visual appearance of these objects depends on a
given scene’s background, lighting, and organization, their visual features
can drastically differ between scenes thereby confounding feature-based
perception systems. The second challenge is the inaccurate depth measurements
among RGB-Depth (RGB-D) cameras on transparent or translucent materials due to
the lack of reliable reflections. This challenge is especially meaningful for
state-of-the-art pose estimation approaches that require accurate depth
measurements as input. To overcome these challenges, computational perception
algorithms have been proposed for a variety of visual tasks, including image
segmentation, depth completion, and object pose estimation. In this paper, our
aim is to complement recent work in transparent object perception by providing
a large-scale, real-world RGB-D transparent object dataset. Furthermore, we
use the new large-scale dataset to perform benchmark analysis of state-of-the-
art perception algorithms on transparent object depth completion and pose
estimation tasks.

Figure 1: Sample images from existing transparent datasets. TOD and
StereOBJ-1M are collected using stereo RGB cameras, and other datasets used
RGB-D cameras (except for Omniverse which is completely synthetic). Table 1:
Comparison between existing transparent object datasets and ClearPose.
∗Trans10K is a transparent segmentation dataset and has no object-centric pose
labels. ∗StereObj1M is not publicly available at the time of submission so the
#frame and #pose annotation are estimated based on the published ratio of
transparent objects in the entire object set [12]

dataset | modality | #obj | #frame | #pose annotation
---|---|---|---|---
TOD [13] | RGB-D | 15 | 48K(real) |  $\sim$0.1M
ClearGrasp [15] | RGB-D | 10 | 50K(syn)+286(real) |  $\sim$0.2M
TODD [22] | RGB-D | 6 | 15K(real) |  $\sim$0.1M
Omniverse [26] | RGB-D | 9 | 60K(syn) |  $\sim$0.2M
TransCG [7] | RGB-D | 51 | 58K(real) |  $\sim$0.2M
Trans10K∗ [20] | RGB | 10K | 15K(real) | seg only
ProLIT [25] | Light-Field | 5 | 75K(syn)+300(real) |  $\sim$0.1M
StereObj1M∗ [12] | Stereo | 7 |  $\sim$150K(real) |  $\sim$0.6M
ClearPose (ours) | RGB-D | 63 | 350K(real) |  $\sim$5M

There are several existing datasets focusing on transparent object perception
with commodity RGB stereo or RGB-D sensors, as summarized in Figure 1. While
these datasets target transparent object perception, most are relatively
small-scale (no more than 50K real-world frames), include few cluttered scenes
(typically with less than 3 objects per image), do not offer diverse
categories of commonplace household objects, and record limited lighting
changes. These limitations motivate the introduction of ClearPose, a large-
scale real-world transparent object dataset labeled with ground truth pose,
depth, instance-level segmentation masks, surface normals, etc. that (1) has a
comparable size with ordinary opaque object pose estimation datasets like YCB-
Video [19]; (2) contains challenging heavy clutter scenes including multiple
layers of occlusion between transparent objects; (3) contains a variety of
commonplace household object categories (bottle, cup, wine, container, bowl,
plate, spoon, knife, fork, and some chemical lab supplies); (4) covers diverse
lighting conditions and multiple adversarial testing scenes. Further details
of ClearPose in relation to existing datasets are included in Table. 1 with
sample images from ClearPose shown in Figure 2.

Figure 2: Sample images from ClearPose dataset. On the left, we show RGB
images taken for different object subsets under various lighting conditions
and backgrounds. On the right, we show examples of different types of testing
scenes, such as covered by a translucent box (top-left), novel background with
opaque distractor objects (top-right), non-planar cases (middle-left), filled
with liquid (middle-right), and heavy clutters of transparent objects
(bottom).

The labeling of such a large-scale dataset requires both efficiency and
accuracy. To achieve these qualities in ClearPose, we take advantage of a
recently published pipeline named ProgressLabeller [5]. The ProgressLabeller
pipeline combines visual SLAM and an interactive graphical interface with
multi-view silhouette matching-based object alignment to enable efficient and
accurate labeling of the transparent object poses from RGB-D videos, which
realizes rapid data annotation and exempts from the broken depth problem by
transparent objects. Given the unique scale and relevance of ClearPose, we
envision it will serve beneficial as a benchmark dataset on transparent
perception tasks. In this current paper, we include benchmark analysis for a
set of state-of-the-art visual perception algorithms on ClearPose. We target
our benchmark analysis on the tasks of depth completion and RGB-D pose
estimation.

##  2 Related Works

###  2.1 Transparent Dataset and Annotation

As mentioned in Table 1, there are several existing datasets and associated
annotation pipelines that focus on transparent objects. With the exception of
works such as Trans10k [20], TransCut [23] and TOM-Net [3] that are focused on
2D image segmentation or matting, most recent transparent perception datasets
are collected in an object-centric 3D setting using RGB-D, stereo or light-
field sensor modalities.

Similar to the case of opaque objects, datasets created in simulation,
supporting photo-realistic rendering with ray-tracing, are more readily
created and can produce very realistic examples of transparent object
appearance. Examples of such simulated datasets include those [15] rendered by
Blender, [25] by Unreal Engine and [26] by Nvidia Omniverse platform. While
simulated datasets are appealing for their ease of creation, they often lack
realistic feature artifacts (e.g. sensor noise, object wear marks, true
lighting etc.) which can impact downstream perception systems that rely on the
synthetic dataset (i.e. the syn-to-real gap).

Among existing real-world datasets, TOD [13] and StereObj1M [12] use RGB
stereo cameras together with AprilTags. First, camera pose transforms are
solved from AprilTag detection, and then several 2D keypoints on the objects
are manually annotated in keyframes that are farthest to each other in the
sequence. The corresponding 3D keypoint positions are solved by multi-view
triangulation from those labeled 2D keypoints, and finally, the object 6D
poses are solved from 3D keypoints as an Orthogonal Procrustes problem and
propagated to all frames. In TOD, the authors also introduced a method to
record ground truth depth images: they record the positions of transparent
objects in the scene, and put their opaque counterparts that share the same
shape at the same pose in separate collects. This pipeline was also used for
real-world data collection in ClearGrasp [15], however, it’s extremely
inefficient to replace transparent objects and their counterparts
repetitively. Moreover, it is unclear how or whether this approach could be
applied to data collection in complex scenes with cluttered objects as is
typical in household settings. In Xu et al. [22], transparent objects are
placed in several fixed locations relative to AprilTag arrays, with pose
distribution diversity achieved by attaching a camera to a robot end-effector.
This method still restricts the relative position between objects and is
inefficient for complex scenes. In Fang et al. [7], all objects are attached
to a large visual IR marker so that an optical tracking algorithm can estimate
the objects’ 6D poses. In this way, the collection can support dynamic scenes.
On the other hand, all the object instances in collected data are accompanied
by visually obscuring external labels which may not exist in natural
environments.

Overall, datasets except for StereObj1M are still not large-scale and require
external efforts on hardware, such as deploying robotic arms, calibration
between multiple sensors, fiducial or optical markers. Instead of using
markers or complex robotic apparatuses, we take advantage of an existing
labeling system, ProgressLabeller [5], that is based on visual SLAM to produce
accurate camera poses efficiently on recorded RGB-D videos. There are two
assumptions in our labeling pipeline, both of which can be easily met: our
pipeline assumes static scenes during video capturing and that scene
backgrounds have adequate RGB features for visual SLAM processing.

###  2.2 Transparent Depth Completion and Object Pose Estimation

Zhang et al. [24] presented early work on the problem of depth completion from
inaccurate depth using deep neural networks. Zhang et al. introduced an
approach to estimate surface normal and boundaries from RGB images that then
solved for the completed depth using optimization. ClearGrasp [15] adapted the
method to work for transparent objects and demonstrated robotic grasping
experiments on transparent objects from completed depth. Tang et al. [16]
integrated the ClearGrasp network structure with adversarial learning to
improve depth completion accuracy. Zhu et al. [26] proposed a framework that
learns local implicit depth functions from the inspiration of neural radiance
fields and performs self-refinement to complete the depth of transparent
objects. Xu et al. [22] proposed to first complete the point cloud by
projecting the original depth using a 3D encoder-decoder U-Net and then re-
project the completed point cloud back to depth, and finally complete this
depth using another encoder-decoder network given the ground truth mask. Fang
et al. [7] also used a U-Net architecture to perform depth completion and
demonstrated robotic grasping capabilities with their approach.

KeyPose [13] was proposed for keypoint-based transparent object pose
estimation on stereo images. It outperformed DenseFusion [18], even with
ground truth depth, and achieved high accuracy on the TOD dataset. Chang et
al. [2] proposed a 3D bounding box representation and reported results
comparable to KeyPose in multi-view pose estimation. StereObj1M [12]
benchmarked KeyPose and another RGB-based object pose estimator, PVNet [14],
on more challenging objects and scenes, where both methods achieved lower
accuracy with respect to the ADD-S AUC metric (introduced in [19]) with both
monocular and stereo input. Xu et al. [21] proposed a two-stage pose
estimation framework that performs image segmentation, surface normal
estimation, and plane approximation in the first stage. The second stage then
combines output from the first stage with color and depth RoI features for
input to an RGB-D pose estimator originally designed for opaque objects [17]
to regress 6D object poses. This method also outperformed DenseFusion and [17]
fed with ClearGrasp output depth by a large margin. In this paper, we evaluate
how well state-of-the-art RGB-D methods [9] designed for opaque object pose
estimation can perform on transparent objects compared with [21].

##  3 Dataset

###  3.1 Dataset Objects and Statistics

As shown in Figure 3, there are 63 objects included in the ClearPose dataset.
There are 49 household objects, including 14 water cups, 9 wine cups (with a
thin stem compared with water cups), 5 bottles (with an opening smaller than
the cross-section of the cylindrical body), 6 bowls, 5 containers (with 4
corners while bowls are classified with round shapes), and several other
objects like pitcher, mug, spoon, etc. Moreover, the dataset contains 14
chemical supply objects, including a syringe, a glass stick, 2 reagent
bottles, 3 pans, 2 graduated cylinders, a funnel, a flask, a beaker, and 2
droppers.

Figure 3: Objects included in the ClearPose dataset. On the left, we show the
rendered CAD models for each object. From top to bottom there are bowls and
plates (1st row), containers and bottles (2nd row), wine cups, two mugs and a
pitcher (3rd row), water cups (4th row), and spoons, a fork, knives as well as
chemical supplies (bottom row). On the right, we show real images of household
objects on the top right, and chemical supplies on the bottom right.

All the images are collected using a RealSense L515 RGB-depth camera, with a
raw resolution of 1280$\times$720\. After object pose annotation, the central
part of each image is cropped and reshaped to 640$\times$480 for reduced
storage space and faster CNN training and inference. For the training set, we
separate all 63 objects into 5 separate subsets and collected 4-5 scenes with
different backgrounds for each subset. Each scene is scanned by the hand-held
camera moving around the tabletop scene at 3 different heights with 3
different lighting conditions (bright room light, dim room light, dim room
light with sidelight from a photography lighting board). For the testing set,
as the appearance of transparent objects depends on their context within a
scene, we consider 6 different test cases and collect corresponding scenes as
follows: (1) different backgrounds: novel backgrounds that never appeared in
the training scenes with each object subset. (2) heavy occlusions: cluttered
scenes each with about 25 objects that form multiple layers of occlusion when
viewed from the table’s side. (3) translucent/transparent covers: scenes with
all transparent objects placed inside a translucent box. (4) together with
opaque objects: transparent objects placed together with opaque YCB and HOPE
objects, which did not appear in the training set. (5) filled with liquid:
scenes with transparent objects filled with different colored liquid. (6) non-
planar configuration: scenes with objects placed onto different surfaces with
multiple heights. Sample RGB images from both training and test sets are
included in Figure 2.

We calculate several statistics about the ClearPose dataset. In total there
are 354,481 RGB-D frames captured in 51 scenes, with 5,052,429 object instance
annotations with 6 DoF poses, segmentation masks, surface normals, and ground
truth depth images. The distribution of object instance annotation and camera
viewpoint coverage are shown in Figure 4 colored by object category, where we
see our dataset has roughly even viewpoint coverage for most objects.
Viewpoint coverage is calculated by projecting collected object orientations
onto a unit sphere and counting the covered region percentage over the
sphere’s surface. For symmetrical objects, regions with the same object
appearance are considered together. Some objects like plates, forks, large
bowls can only be placed in certain orientations, so they have reduced
viewpoint coverage. The 2nd water_cup was broken during the data collection
process so it has fewer instance annotations.

Figure 4: Distribution of instance annotations and viewpoint orientation
coverage statistics for every object in the ClearPose dataset. The ‘kitchen’
category includes fork, spoon, knife, mug, plate and pitcher objects.  Figure
5: Transparent object pose annotation pipeline using ProgressLabeller. On top-
left we show one sample of aligned RGB-D images from continuous streams
captured by the camera. The frames are fed into visual SLAM to estimate camera
trajectory, then the objects’ poses are annotated in a multi-view silhouette
alignment interface shown at the bottom-right. Finally, the object poses,
surface normals, and fixed depths are calculated and rendered as output.

###  3.2 Pose Annotation

We use the ProgressLabeller [5] to annotate the 6D poses of transparent
objects and render object-wise segmentation masks, ground truth depths,
surface normals, etc. from the labeled poses. As shown in Figure 5, the first
step of the ProgressLabeller pipeline is to run ORB-SLAM3 [1] on collected
RGB-D video frames to produce camera pose estimates. During data collection,
we notice ORB-SLAM3 sometimes couldn’t estimate camera pose well in case of
extreme transparent object clutter, where background RGB features are heavily
distorted. In these cases, the camera view needs to capture some background
area or other landmark objects that can provide stable features. The next step
is to import the reconstructed scene (cameras and point cloud) into the
Blender workspace, select several camera views from different orientations,
and import the object 3D CAD model into the workspace. Then, the object
silhouettes/boundaries can be directly compared and matched with original RGB
images from multiple views simultaneously when the user drags the object
across the scene to tune its position and orientation. Figure 5 shows an
example case of a matched transparent bottle. Optionally, the user can select
to first locate the object onto the 2D fitting plane of the support table,
etc., as shown on top-left of Figure 5. After labeling all objects’ poses in
the 3D workspace, their poses in every camera frame are calculated by dividing
them with estimated camera poses. The output ground truth depth images (fixed
depth) are generated by overlaying rendered object CAD model depth to the
original depth images. Then the surface normals are calculated from the depth
images. It takes around 30 minutes to finish labeling one scene using this
pipeline, including visual SLAM for camera pose estimation, object pose manual
alignment, and output image rendering.

##  4 Benchmark Experiments

In this section, we provide benchmark results of recent research on depth-
related transparent object perception using deep neural networks, including
scene level depth completion, and both instance-level and category-level RGB-D
pose estimation. As mentioned in Section 3, we test the generalization
capability of such systems on 6 aspects of appearance novelty with transparent
objects: new background, heavy occlusion, translucent cover, opaque distractor
objects, filled with liquid, and non-planar placement. Specifically, around
200K images are selected for training, and for each of 6 test cases, 2K images
from corresponding scenes are randomly sampled to compose the testing set for
evaluation.

###  4.1 Depth Completion

We selected two recent depth completion works that are publicly available,
ImplicitDepth [26] and TransCG [7] as baseline methods for the depth
completion task on transparent object scenes. (ClearGrasp [15] was shown to be
less accurate than both works, and TranspareNet [22] was released around the
date of this submission.) ImplicitDepth is a two-stage method that learns
local implicit depth functions in the first stage through ray-voxel pairs
similar to neural rendering and refines the depth in the second stage. TransCG
is built on DFNet [11] which was initially developed for image completion. We
trained both networks following their original papers’ training iterations and
hyper-parameters. Specifically, ImplicitDepth was trained on a 16G RTX3080 GPU
with a 0.001 learning rate and iterated around 2M frames for each of the two
stages. TransCG was trained on an 8G RTX3070 GPU with a fixed 0.001 learning
rate and iterated around 900K frames in total. Both works use Adam as the
optimizer. Then we evaluated the two works on 6 test sets mentioned in Section
3 with metrics defined in [6]. The results are shown in Table 2. TransCG
surpassed ImplicitDepth in most tests with fewer training iterations, which
implies that methods using DFNet can outperform designs using voxel-based
PointNet for transparent depth completion. Across different tests, both
methods perform poorly in Translucent Cover scenes and achieved the best
performance in New Background scenes. Other scene variations such as Filled
Liquid, Opaque Distractor, and Non Planar do not substantially impact the
methods’ accuracy. Figure 6 shows examples of qualitative results from both
methods compared with the ground truth.

Table 2: Depth completion benchmark results of ImplicitDepth and TransCG on 6
different test scenarios of the ClearPose dataset.

Testset | Metric | RMSE$\downarrow$ | REL$\downarrow$ | MAE$\downarrow$ | $\delta_{1.05}\uparrow$ | $\delta_{1.10}\uparrow$ | $\delta_{1.25}\uparrow$
---|---|---|---|---|---|---|---
New Background | ImplicitDepth | 0.07 | 0.05 | 0.04 | 67.00 | 87.03 | 97.50
TransCG | 0.03 | 0.03 | 0.02 | 86.50 | 97.02 | 99.74
Heavy Occlusion | ImplicitDepth | 0.11 | 0.09 | 0.08 | 41.43 | 66.52 | 91.96
TransCG | 0.06 | 0.04 | 0.04 | 72.03 | 90.61 | 98.73
Translucent Cover | ImplicitDepth | 0.16 | 0.16 | 0.13 | 22.85 | 41.17 | 73.11
TransCG | 0.16 | 0.15 | 0.14 | 23.44 | 39.75 | 67.56
Opaque Distractor | ImplicitDepth | 0.14 | 0.13 | 0.10 | 34.41 | 55.59 | 83.23
TransCG | 0.08 | 0.06 | 0.06 | 52.43 | 75.52 | 97.53
Filled Liquid | ImplicitDepth | 0.14 | 0.13 | 0.11 | 32.84 | 53.44 | 84.84
TransCG | 0.04 | 0.04 | 0.03 | 77.65 | 93.81 | 99.50
Non Planar | ImplicitDepth | 0.18 | 0.16 | 0.15 | 20.34 | 38.57 | 74.02
TransCG | 0.09 | 0.07 | 0.07 | 55.31 | 76.47 | 94.88

Figure 6: Qualitative depth completion results. From left to right, there are
RGB image, raw depth, ground truth depth rendered using object CAD models,
completed depth by TransCG and ImplicitDepth.

###  4.2 Instance-Level Object Pose Estimation

There is one recent work of Xu et al. [21] focusing on transparent object pose
estimation using raw RGB and depth images. This work doesn’t have source code
publicly available, so we re-implemented the proposed method following the
original paper for inclusion in our benchmark analysis. This method is
implemented as a two-stage pipeline, for which we trained Mask R-CNN [8] for
instance-level segmentation and DeepLabv3 [4] for surface normal estimation,
and with an XYZ 3D coordinate map of the supporting plane feature in the first
stage. The second stage replicates most of the architecture and loss functions
described in [17] to ultimately regress dense pixel-wise 3D translation, 3D
rotation delta from a set of fixed rotation anchors, and confidence scores. In
practice, we trained the networks on an RTX2080-SUPER GPU. Mask R-CNN has
trained 5 epochs with SGD optimizer, batch size of 5, and learning rate of
0.005. DeepLabv3 was trained 2 epochs with Adam optimizer, batch size of 4 and
learning rate of 0.0001, and second stage network was trained around 180K
iterations with Adam optimizer, batch size of 4, and learning rate of 0.0005.
We compare this method [21] with a state-of-the-art RGB-D pose estimator that
was originally designed for opaque objects, FFB6D [9]. The FFB6D estimator
follows a two-stream RGB and point cloud encoder-decoder architecture with
fusion between blocks. FFB6D is trained on a 16G RTX3080 GPU for 5 epochs with
batch size 6. All the hyper-parameters follow the default value from the
original implementation. For our analysis of FFB6D pose estimation
performance, we run experiments with different depth options in training and
testing: with raw, ground truth, and completed depth from TransCG, as detailed
in Table 3.

Table 3: Pose estimation accuracy comparison on different test sets of the
ClearPose dataset. $\textrm{FFB6D}_{r/r}$ refers to train and test FFB6D model
both on raw depth. Similarly, $\textrm{FFB6D}_{g/c}$, $\textrm{FFB6D}_{g/g}$
refer to train on ground truth, test on completed depth from TransCG, and
train and test both on ground truth, respectively. The values are averaged
across all objects in the dataset.

Testset | Metric | Xu et al. | $\textrm{FFB6D}_{r/r}$ | $\textrm{FFB6D}_{g/c}$ | $\textrm{FFB6D}_{g/g}$
---|---|---|---|---|---
New Background | Accuracy | 50.958 | 44.264 | 49.517 | 59.694
ADD(-S) | 45.233 | 43.452 | 47.691 | 58.224
Heavy Occlusion | Accuracy | 24.193 | 14.723 | 15.160 | 26.331
ADD(-S) | 22.953 | 17.869 | 17.862 | 31.875
Translucent Cover | Accuracy | 14.353 | 5.5617 | 4.5345 | 13.433
ADD(-S) | 14.311 | 7.5983 | 5.8054 | 17.620
Opaque Distractor | Accuracy | 42.630 | 0.4618 | 1.3331 | 2.3525
ADD(-S) | 39.036 | 0.7628 | 1.5516 | 3.0685
Filled Liquid | Accuracy | 34.500 | 7.6908 | 9.0584 | 16.228
ADD(-S) | 32.251 | 11.153 | 10.828 | 18.583
Non Planar | Accuracy | 21.024 | 7.4924 | 7.5843 | 15.567
ADD(-S) | 18.411 | 7.8021 | 6.7339 | 16.986

For evaluation, we use two metrics based on Average-point-Distance (ADD and
ADD-S) from [19]. ADD is calculated as the average Euclidean distance of
corresponding point pairs from two object point clouds separately at the
ground truth pose and predicted pose. ADD-S is calculated as the minimum
distance of every point from the predicted point cloud to the ground truth
point cloud. In Table 3, ‘Accuracy’ is calculated as the percentage of all
pose estimates on the test set with ADD error less than 10cm. ‘ADD(-S)’ is
calculated as Accuracy-Under-Curve area integrated from 0-10cm error, which is
then scaled from 1 to 100 as the percentage.

Figure 7: Visualization of pose estimation in ClearPose dataset. From left to
right, they are raw image, ground truth object poses, pose estimation results
of method from Xu et al., FFB6Dg/g, FFB6Dg/c, FFB6Dr/r. From top to bottom,
results are shown in different test scenes in Table 3. Objects are projected
to color masks based on their pose estimates, with their 6DoF poses marked as
the red-green-blue coordinate frame.

As shown in Table 3, from the comparison between different training and
testing combination within FFB6D, the upper bound performance appears when the
network are both trained and tested on ground truth depth. When both the
training and testing data come to raw, the metric drops a lot. Obviously,
inaccurate depth would be the difficulty for transparent object pose
estimation. It should be mentioned that training on ground truth depth,
testing on completed depth (from TransCG) almost display the same accuracy.
Although TransCG is good at depth completion, the disparity between ground
truth and depth completion would make the network in vain. Generally, the
easiest test case is New Background, and the accuracy drops a lot in the other
5 scenarios. When we compare the accuracy of Xu et al. with variants of FFB6D,
we find they are comparable in New Background, Heavy Occlusion, Translucent
Cover, and Non Planar scenes, while Xu et al. is much better in Opaque
Distractor and Filled Liquid scenes. One possible reason is that there are
some unseen colors mixing in the transparent objects, adding remarkable noise
to object keypoint regression during the FFB6D inference process, which is not
used by Xu et al. Overall, the pose estimation accuracy of current methods is
still much worse than that on opaque objects with RGB features (with ADD-S
around 90 on public datasets [10]). Some qualitative examples of pose
estimates are shown in Figure 7.

##  5 Discussions

There are some common classes of objects with transparency/translucency not
included in our dataset, for example, those with colored
transparent/translucent materials, with markers or labels, together with
opaque parts, etc. Instead, our focus in the ClearPose dataset is to
investigate pure transparency that exhibits relatively few features for
perception. On the other hand, we anticipate the open-source ProgressLabeller
[5] will facilitate more large-scale customized transparent datasets in the
future.

Figure 8: Examples of multi-layer transparent object appearance. In the left
image, the annotated bounding boxes show large overlap between object pairs,
where the objects behind are still perceivable in some cases with less light
distortion. In the right image, objects behind the translucent surface are
still detectable as well.

As for benchmarking perception models, we didn’t include a complete list of
recent state-of-the-art approaches due to resource constraints (i.e. compute
and time limitations). Based on our current dataset and benchmark results,
there are several possible extensions: (1) Comparison of RGB-only pose
estimators with RGB-D methods that are free of transparent object broken depth
issues. (2) Category-level pose estimation for transparent objects, for which
the ClearPose dataset has categories of bowls, bottles, wine_cups, etc. that
are with similar 3D shape and topology. (3) Neural rendering on transparent
objects considering environment contexts, such as varied lighting and
occlusions. (4) Transparent object grasping and manipulation experiment in
practical scenes, including the 6 test scenarios mentioned in the benchmark.

Besides, an especially interesting problem emerging from our heavy cluttered,
and translucent covered test scenes is the multi-layer appearance of
transparent objects. As shown in Figure 8, because of
transparency/translucency, some image pixels could belong to more than one
object. New detection and segmentation annotation rules, such as bounding box
non-maximum suppression threshold, or segmentation mask format over the image,
could be proposed and explored based on our dataset as future work.

##  6 Conclusions

In this paper, we described the contribution of ClearPose, a new large-scale
RGB-D transparent object dataset with annotated poses, masks, and associated
labels created using a recently proposed pipeline. We performed a set of
benchmarking experiments on depth completion and object pose estimation tasks
using state-of-the-art methods over 6 different generalization test cases that
are common in practical scenarios. Results from our experiments demonstrate
that there is still much room for improvement in some cases, such as heavy
clutter, transparent objects filled with liquid, or being covered by other
translucent surfaces. The dataset and benchmark code implementations will be
made public with the intention to support further research progress in
transparent RGB-D visual perception.

Acknowledgement. We thank greatly the support from Dr. Peter Gaskell and
Weishu Wu at the University of Michigan, who provided devices and objects for
dataset collection.

## References

  * [1] Campos, C., Elvira, R., Rodríguez, J.J.G., Montiel, J.M., Tardós, J.D.: Orb-slam3: An accurate open-source library for visual, visual–inertial, and multimap slam. IEEE Transactions on Robotics 37(6), 1874–1890 (2021)
  * [2] Chang, J., Kim, M., Kang, S., Han, H., Hong, S., Jang, K., Kang, S.: Ghostpose*: Multi-view pose estimation of transparent objects for robot hand grasping. In: 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). pp. 5749–5755. IEEE (2021)
  * [3] Chen, G., Han, K., Wong, K.Y.K.: Tom-net: Learning transparent object matting from a single image. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 9233–9241 (2018)
  * [4] Chen, L.C., Papandreou, G., Schroff, F., Adam, H.: Rethinking atrous convolution for semantic image segmentation. arXiv preprint arXiv:1706.05587 (2017)
  * [5] Chen, X., Zhang, H., Yu, Z., Lewis, S., Jenkins, O.C.: Progresslabeller: Visual data stream annotation for training object-centric 3d perception. arXiv preprint arXiv:2203.00283 (2022)
  * [6] Eigen, D., Puhrsch, C., Fergus, R.: Depth map prediction from a single image using a multi-scale deep network. Advances in neural information processing systems 27 (2014)
  * [7] Fang, H., Fang, H.S., Xu, S., Lu, C.: Transcg: A large-scale real-world dataset for transparent object depth completion and grasping. arXiv preprint arXiv:2202.08471 (2022)
  * [8] He, K., Gkioxari, G., Dollár, P., Girshick, R.: Mask r-cnn. In: Proceedings of the IEEE international conference on computer vision. pp. 2961–2969 (2017)
  * [9] He, Y., Huang, H., Fan, H., Chen, Q., Sun, J.: Ffb6d: A full flow bidirectional fusion network for 6d pose estimation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 3003–3013 (2021)
  * [10] Hodaň, T., Sundermeyer, M., Drost, B., Labbé, Y., Brachmann, E., Michel, F., Rother, C., Matas, J.: Bop challenge 2020 on 6d object localization. In: European Conference on Computer Vision. pp. 577–594. Springer (2020)
  * [11] Hong, X., Xiong, P., Ji, R., Fan, H.: Deep fusion network for image completion. In: Proceedings of the 27th ACM international conference on multimedia. pp. 2033–2042 (2019)
  * [12] Liu, X., Iwase, S., Kitani, K.M.: Stereobj-1m: Large-scale stereo image dataset for 6d object pose estimation. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 10870–10879 (2021)
  * [13] Liu, X., Jonschkowski, R., Angelova, A., Konolige, K.: Keypose: Multi-view 3d labeling and keypoint estimation for transparent objects. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 11602–11610 (2020)
  * [14] Peng, S., Liu, Y., Huang, Q., Zhou, X., Bao, H.: Pvnet: Pixel-wise voting network for 6dof pose estimation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 4561–4570 (2019)
  * [15] Sajjan, S., Moore, M., Pan, M., Nagaraja, G., Lee, J., Zeng, A., Song, S.: Clear grasp: 3d shape estimation of transparent objects for manipulation. In: 2020 IEEE International Conference on Robotics and Automation (ICRA). pp. 3634–3642. IEEE (2020)
  * [16] Tang, Y., Chen, J., Yang, Z., Lin, Z., Li, Q., Liu, W.: Depthgrasp: Depth completion of transparent objects using self-attentive adversarial network with spectral residual for grasping. In: 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). pp. 5710–5716. IEEE (2021)
  * [17] Tian, M., Pan, L., Ang, M.H., Lee, G.H.: Robust 6d object pose estimation by learning rgb-d features. In: 2020 IEEE International Conference on Robotics and Automation (ICRA). pp. 6218–6224. IEEE (2020)
  * [18] Wang, C., Xu, D., Zhu, Y., Martín-Martín, R., Lu, C., Fei-Fei, L., Savarese, S.: Densefusion: 6d object pose estimation by iterative dense fusion. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 3343–3352 (2019)
  * [19] Xiang, Y., Schmidt, T., Narayanan, V., Fox, D.: Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes. Robotics Science and Systems (2018)
  * [20] Xie, E., Wang, W., Wang, W., Ding, M., Shen, C., Luo, P.: Segmenting transparent objects in the wild. In: European conference on computer vision. pp. 696–711. Springer (2020)
  * [21] Xu, C., Chen, J., Yao, M., Zhou, J., Zhang, L., Liu, Y.: 6dof pose estimation of transparent object from a single rgb-d image. Sensors 20(23), 6790 (2020)
  * [22] Xu, H., Wang, Y.R., Eppel, S., Aspuru-Guzik, A., Shkurti, F., Garg, A.: Seeing glass: Joint point cloud and depth completion for transparent objects. arXiv preprint arXiv:2110.00087 (2021)
  * [23] Xu, Y., Nagahara, H., Shimada, A., Taniguchi, R.i.: Transcut: Transparent object segmentation from a light-field image. In: Proceedings of the IEEE International Conference on Computer Vision. pp. 3442–3450 (2015)
  * [24] Zhang, Y., Funkhouser, T.: Deep depth completion of a single rgb-d image. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 175–185 (2018)
  * [25] Zhou, Z., Chen, X., Jenkins, O.C.: Lit: Light-field inference of transparency for refractive object localization. IEEE Robotics and Automation Letters 5(3), 4548–4555 (2020)
  * [26] Zhu, L., Mousavian, A., Xiang, Y., Mazhar, H., van Eenbergen, J., Debnath, S., Fox, D.: Rgb-d local implicit function for depth completion of transparent objects. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 4649–4658 (2021)

Supplementary Material Xiaotong ChenHuijie ZhangZeren YuAnthony Opipari
Odest Chadwicke Jenkins

##  Appendix 0.A Dataset Collection Details

The 3D CAD models of objects in the ClearPose dataset are manually created in
Blender, with geometry size measurement of real objects. Then during the
labeling process of aligning object models to RGB images, the object
dimensions are further verified and corrected. Finally, the object models
could align well with RGB images in multiple scene sequences.

We used a series of table cloth with different textures as background when
collecting the dataset. For each scene, after we randomly put objects in
place, a collector will hold the camera and move steadily around the scene
with normal room light from relative low, middle, high altitude view angles,
for 1 round for each view angle, then another 3 rounds with dim room light and
3 rounds with dim room light plus side light from the lighting board mounted
on a tripod on the side. The RealSense camera records aligned RGB-depth image
pairs around 30Hz with resolution 1280$\times$720 in scenes with normal room
light, and 10-20Hz in other scenes. Most device mentioned above are shown in
Figure 9.

A video with detailed annotation steps and results of depth completion and
object pose estimation is available at YouTube

Figure 9: On the left there are lighting board, tripod, and Intel RealSense
L515 camera put on the background round table. On the right there are most of
background table cloth used in the dataset.

##  Appendix 0.B Object Pose Estimation

More qualitative results with visualization on depth completion and object
pose estimation are included in the accompanied video.

Specific to FFB6D, eight 3D sample keypoints on each object are required for
the deep neural network to regress. We manually specify keypoints as shown
below Figure 10-14. For axial-symmetric objects, all of eight keypoints are
generated with equal interval along the symmetric axis. For other objects,
keypoints are manually selected to be at feature-rich areas, such as corners
or edges.

Figure 10: Object names and labeled keypoints in set 1 of 5. Figure 11: Object
names and labeled keypoints in set 2 of 5. Figure 12: Object names and labeled
keypoints in set 3 of 5. Figure 13: Object names and labeled keypoints in set
4 of 5. Figure 14: Object names and labeled keypoints in set 5 of 5.


# Spread Complexity of High Energy Neutrino Propagation over Astrophysical
Distances

Khushboo Dixit  Centre for Astro-Particle Physics (CAPP) and Department of
Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South
Africa  S. Shajidul Haque  High Energy Physics, Cosmology & Astrophysics
Theory Group and The Laboratory for Quantum Gravity & Strings, Department of
Mathematics & Applied Mathematics, University of Cape Town, Cape Town, South
Africa  Department of Mathematics and Natural Sciences, Brac University,
Dhaka, Bangladesh  National Institute for Theoretical and Computational
Sciences (NITheCS), Private Bag X1, Matieland, South Africa  Soebur Razzaque
Centre for Astro-Particle Physics (CAPP) and Department of Physics, University
of Johannesburg, PO Box 524, Auckland Park 2006, South Africa  Department of
Physics, The George Washington University, Washington, DC 20052, USA  National
Institute for Theoretical and Computational Sciences (NITheCS), Private Bag
X1, Matieland, South Africa

###### Abstract

Spread complexity measures the minimized spread of quantum states over all
choices of basis. It generalizes Krylov operator complexity to quantum states
under continuous Hamiltonian evolution. In this paper, we study spread
complexity in the context of high-energy astrophysical neutrinos and propose a
new flavor ratio based on complexity. Our findings indicate that our proposal
might favor an initial ratio of fluxes as
$\phi_{\nu_{e}}^{0}:\phi_{\nu_{\mu}}^{0}:\phi_{\nu_{\tau}}^{0}=1:0:0$ over a
more generally expected ratio of $1:2:0$, when the IceCube neutrino
observatory achieves its projected sensitivity to discriminate between
flavors. Additionally, complexity-based definitions of flavor ratios exhibit a
slight but nonzero sensitivity to the neutrino mass ordering, which
traditional flavor ratios cannot capture.

##  I Introduction

Neutrinos have long been a subject of profound interest within the field of
particle physics. The discovery of neutrino oscillations, which signifies the
massive nature of neutrinos, has significantly impacted the scientific
community, prompting an intensified quest for physics beyond the Standard
Model (BSM) [1, 2, 3, 4, 5, 6, 7, 8]. Presently, numerous long-baseline
experiments, both established and planned, are dedicated to scrutinizing
oscillation parameters with heightened precision and resolution [9, 10, 11,
12]. Additionally, some experiments such as the IceCube neutrino observatory,
situated at the South Pole, are actively engaged in the pursuit of high-energy
astrophysical neutrinos. This pursuit is motivated by the potential for these
neutrinos and their flavor composition to unveil crucial insights into their
production mechanisms within celestial entities [13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26].

In addition, various analyses have demonstrated that neutrinos exhibit
intriguing properties such as entanglement and several other nonlocal
correlations [27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. These characteristics
underscore their significant potential for executing diverse tasks pertinent
to Quantum Information Processing (QIP). One of the important aspects related
to quantum information and computation theory is known as quantum complexity,
which was proposed as a useful probe [37] for the interior of a black hole by
using the anti-de Sitter/conformal field theory (AdS/CFT) duality. Holographic
complexity, proposed to be equivalent to the ”volume” or ”action,” elucidates
the physics behind the horizon of an eternal AdS black hole [37, 38, 39, 40,
41]. However, in recent days, complexity has proven to be extremely useful for
understanding various aspects of quantum systems such as quantum chaos [42,
43, 44, 45, 46, 47], quantum phase transitions [48], quantum decoherence [49,
50], and more. It has also been applied to investigate cosmological
perturbation models and the evolution of the universe [51, 52, 53].

Complexity can be described in two ways when dealing with quantum mechanical
systems. First, according to Nielsen’s definition, complexity is the minimal
distance in the space of the unitaries between the identity and the time
evolution operator $U(t)$ of the system [54, 55, 56, 57, 58]. The second
approach defines complexity as minimizing the spread of the state over a
specified basis during the time evolution [59, 60]. The complexity of the
spread of states has recently been studied in the context of neutrino
oscillations in accelerator neutrino-based long baseline experiments [61].
This study not only indicated that complexity could provide more information
on the CP-violating phase compared to the oscillation probability, but it also
provided a perspective on the preferred value of the CP-violating phase angle.
Motivated by this, in this paper, we examine the complexity of astrophysical
neutrinos by considering three different initial flavor ratios based on their
possible production mechanisms. IceCube Neutrino Observatory, currently
operating in the Antarctica has been collecting data on astrophysical
neutrinos and has potential to measure their flavor ratios [24], especially
after the recent detection of astrophysical tau neutrinos [26]. Our findings
suggest that the modified flavor ratios, in terms of complexities, support the
idea that neutron-decay (with an initial flux ratio of
$\phi_{\nu_{e}}^{0}:\phi_{\nu_{\mu}}^{0}:\phi_{\nu_{\tau}}^{0}=1:0:0$) might
be favored over $\pi$-decay (with an initial flux ratio of $1:2:0$) as the
mechanism responsible for the production of neutrinos in celestial bodies. We
delve into the details of neutrino oscillations and flavor ratios in the
context of astrophysical neutrinos in sec. II.1, followed by defining spread
complexity in sec. II.2. We then provide the analytical expressions of flavor
ratios modified in terms of complexities in sec. II.3. Finally, we present our
results and discussions in sec. III and IV, respectively.

##  II Neutrino Oscillations and Quantum Complexity

In this section, we discuss neutrino oscillations in the context of
astrophysical neutrinos and recall the complexity embedded in a state
evolution. This is followed by a discussion on the complexities for each
initial neutrino flavor modified in the context of astrophysical neutrinos.
Additionally, we propose the redefined flavor ratios of neutrinos in terms of
complexities.

###  II.1 Astrophysical neutrino oscillations and flavor ratios

It is now well known that neutrinos can change their flavor while traveling to
a distant place. Depending on their energy as well as their sensitivity to the
mass squared difference associated to their mass eigenstates one can calculate
the oscillation length for such flavor transitions. Neutrino oscillations
occur as the flavor states of neutrino are not the mass eigenstates, but they
mix via a unitary matrix to form these mass eigenstates. In the case of three
flavor oscillations, the form of this unitary matrix can be given as

$U=\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\ 0&-s_{23}&c_{23}\\\
\end{pmatrix}\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta}\\\ 0&1&0\\\
-s_{13}e^{i\delta}&0&c_{13}\\\ \end{pmatrix}\begin{pmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\ \end{pmatrix},$

where, $c_{ij}$ and $s_{ij}$ are $\cos\theta_{ij}$ and $\sin\theta_{ij}$ with
mixing angles $\theta_{ij}$ ($i,j=1,2,3$). The probability of survival or
oscillation after the time evolution of the neutrino with a specific flavor
can be given by

$\displaystyle P_{\alpha\beta}=\delta_{\alpha\beta}$
$\displaystyle-4\sum_{j>k}Re\\{U_{\alpha j}^{\ast}U_{\beta j}U_{\alpha
k}U_{\beta k}^{\ast}\\}\sin^{2}\left(\frac{\Delta m^{2}_{ij}L}{4E}\right)$
$\displaystyle-2\sum_{j>k}Im\\{U_{\alpha j}^{\ast}U_{\beta j}U_{\alpha
k}U_{\beta k}^{\ast}\\}\sin\left(\frac{\Delta m^{2}_{ij}L}{2E}\right)$

In the case of astrophysical neutrinos where neutrinos travel through the
distances of Mpc or larger order, the sine and cosine terms carrying the phase
factor get averaged out and the probabilities take the following form

$P{\alpha\beta}=\delta_{\alpha\beta}-2\sum_{j>k}Re\\{U_{\alpha
j}^{\ast}U_{\beta j}U_{\alpha k}U_{\beta k}^{\ast}\\}$

The quantity of interest in the experiments searching for high energy
astrophysical neutrinos is the so-called flavor ratio. The fraction of each
neutrino flavor observed in astrophysical neutrino experiments can be defined
using the flavor oscillation probabilities and the initial flavor ratios. The
initial $\nu_{\tau}$ flavor is supposed to be zero as the neutrino production
mechanism in these high energy astrophysical objects like blazars or AGNs are
considered to be either neutron beta decay, $n\to p+e^{-}+{\bar{\nu}}_{e}$,
which induces initial flavor ratio of
$\phi_{e}^{0}:\phi_{\mu}^{0}:\phi_{\tau}^{0}=1:0:0$, or pion decay,
$\pi^{+}\to e^{+}+\nu_{e}+{\bar{\nu}}_{\mu}+\nu_{\mu}$, with initial flavor
ratio of $1:2:0$. In the case of $\mu$-damped $\pi$-decay, when the $\mu$
loses substantial energy before decaying [62], the initial ratio of $0:1:0$ is
also possible. The fractions of each neutrino flavor arriving on the Earth can
be expressed as follows

$\displaystyle
X_{e}\equiv\frac{\phi_{\nu_{e}}}{\phi_{\nu_{e}}+\phi_{\nu_{\mu}}+\phi_{\nu_{\tau}}}=\frac{P_{ee}\phi_{\nu_{e}}^{0}+P_{\mu
e}\phi_{\nu_{\mu}}^{0}}{\phi_{\nu_{e}}^{0}+\phi_{\nu_{\mu}}^{0}+\phi_{\nu_{\tau}}^{0}}$
$\displaystyle
X_{\mu}\equiv\frac{\phi_{\nu_{\mu}}}{\phi_{\nu_{e}}+\phi_{\nu_{\mu}}+\phi_{\nu_{\tau}}}=\frac{P_{e\mu}\phi_{\nu_{e}}^{0}+P_{\mu\mu}\phi_{\nu_{\mu}}^{0}}{\phi_{\nu_{e}}^{0}+\phi_{\nu_{\mu}}^{0}+\phi_{\nu_{\tau}}^{0}}$
$\displaystyle
X_{\tau}\equiv\frac{\phi_{\nu_{\tau}}}{\phi_{\nu_{e}}+\phi_{\nu_{\mu}}+\phi_{\nu_{\tau}}}=\frac{P_{e\tau}\phi_{\nu_{e}}^{0}+P_{\mu\tau}\phi_{\nu_{\mu}}^{0}}{\phi_{\nu_{e}}^{0}+\phi_{\nu_{\mu}}^{0}+\phi_{\nu_{\tau}}^{0}}$
(1)

###  II.2 Complexity of spread of states

In this section, we recall the formulation of the complexity of the spread of
states as introduced in [59, 60]. The evolution of a state $\ket{\psi(0)}$ is
governed by the Schrodinger equation as follows

$i\partial_{t}\ket{\psi(t)}=H\ket{\psi(t)},$

where, $H$ is the time independent Hamiltonian of the system and the solution
to this equation gives us the time evolved state

$\displaystyle\ket{\psi(t)}$ $\displaystyle=e^{-iHt}\ket{\psi(0)},$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-iHt)^{n}}{n!}\ket{\psi(0)},$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-it)^{n}}{n!}\ket{\psi_{n}},$

where, $\ket{\psi_{n}}=H^{n}\ket{\psi}$. We then apply the Gram-Schmidt
procedure to obtain an ordered orthonormal (Krylov) basis using the set of
infinite states $\\{\ket{\psi_{n}}\\}$. The number of states spanning such a
basis can be equal to or less than the dynamics of the system, however, it has
been seen that in case of neutrinos, the number of basis states is exactly
equal to the dimension of the system [61]. We expect more complex spread of a
state if the dynamics involved is chaotic. To formulate the degree of
complexity a cost function can be defined as

$\chi=\sum_{n}|\langle K_{n}|\psi(t)\rangle|^{2}$ (2)

It has been shown in [59] that the above cost function achieves its minimum
for the Krylov basis, hence, this minimized cost function defined in Eq. (2)
can be treated as the complexity embedded in the evolution of the state of the
system.

###  II.3 Spread complexity in flavor oscillations of astrophysical neutrinos

Spread complexity introduced in Eq. (2) has been formulated for the general
case of three flavor neutrino oscillations in [61]. The complexity for
neutrino starting as $\nu_{e}$ flavor at the initial point can be expressed
below

$\displaystyle\chi_{e}=$ $\displaystyle
P_{e\mu}(t)\left[N_{1e}^{2}|a_{1}|^{2}+2N_{2e}^{2}|b_{1}|^{2}\right]+P_{e\tau}(t)\left[N_{1e}^{2}|a_{2}|^{2}+2N_{2e}^{2}|b_{2}|^{2}\right]$
$\displaystyle+2\Re\left[N_{1e}^{2}a_{1}^{\ast}a_{2}A_{e\mu}(t)A_{e\tau}(t)^{\ast}\right]+4\Re\left[N_{2e}^{2}b_{1}^{\ast}b_{2}A_{e\mu}(t)A_{e\tau}(t)^{\ast}\right].$
(3)

Here, $P_{e\mu}$ and $P_{e\tau}$ are the oscillation probabilities for
$\nu_{e}\rightarrow\nu_{\mu}$ and $\nu_{e}\rightarrow\nu_{\tau}$ transitions,
respectively and $A_{e\mu}$ and $A_{e\tau}$ are the corresponding oscillation
amplitudes. In the case of astrophysical neutrinos where the travel distance
is significantly large (of the order of Mpc or larger), the sin and cos terms
carrying the phase factor depending on the mass squared differences
$m_{j}^{2}-m_{i}^{2}$ ($j>i$ and $i,j=1,2,3$) corresponding to the mass
eigenstates driving flavor oscillations get averaged out. Therefore, these
probabilities and amplitudes take the following form

$\displaystyle P_{e\mu}=$ $\displaystyle-2\left(\Re(U_{e2}^{\ast}U_{\mu
2}U_{e1}U_{\mu 1}^{\ast})+\Re(U_{e3}^{\ast}U_{\mu 3}U_{e1}U_{\mu
1}^{\ast})+\Re(U_{e3}^{\ast}U_{\mu 3}U_{e2}U_{\mu 2}^{\ast})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{8}\sin\theta_{12}\cos\theta_{12}\cos^{2}\theta_{13}\left(4\cos\delta\cos
2\theta_{12}\sin\theta_{13}\sin 2\theta_{23}\right.$ $\displaystyle\left.+\sin
2\theta_{12}\left(2\cos 2\theta_{13}\sin^{2}\theta_{23}+3\cos
2\theta_{23}+1\right)\right),$ $\displaystyle P_{e\tau}=$
$\displaystyle-2\left(\Re(U_{e2}^{\ast}U_{\tau 2}U_{e1}U_{\tau
1}^{\ast})+\Re(U_{e3}^{\ast}U_{\tau 3}U_{e1}U_{\tau
1}^{\ast})+\Re(U_{e3}^{\ast}U_{\tau 3}U_{e2}U_{\tau 2}^{\ast})\right)$
$\displaystyle=$
$\displaystyle-\cos^{2}\theta_{13}\left(\frac{1}{4}\sin\theta_{13}\left(\cos\delta(4\sin
2\theta_{12}+\sin 4\theta_{12})\sin 2\theta_{23}\right.\right.$
$\displaystyle\left.\left.+8\sin^{2}\theta_{12}\sin\theta_{13}\cos^{2}\theta_{23}\right)-2\cos^{4}\theta_{12}\sin^{2}\theta_{13}\cos^{2}\theta_{23}\right.$
$\displaystyle\left.-2\sin^{2}\theta_{12}\cos^{2}\theta_{12}\sin^{2}\theta_{23}\right),$
$\displaystyle A_{e\mu}A_{e\tau}^{\ast}=$ $\displaystyle|U_{e1}|^{2}U_{\mu
1}^{\ast}U_{\tau 1}+|U_{e2}|^{2}U_{\mu 2}^{\ast}U_{\tau 2}+|U_{e3}|^{2}U_{\mu
3}^{\ast}U_{\tau 3}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\cos^{2}\theta_{13}(\sin 2\theta_{23}(-(\cos
4\theta_{12}+7)\cos 2\theta_{13}+3\cos 4\theta_{12}+5)$ $\displaystyle+4\sin
4\theta_{12}\sin\theta_{13}(\cos\delta\cos 2\theta_{23}+i\sin\delta)).$ (4)

The effect of averaging out the sine and cosine terms will be visible in
probabilities $P_{e\mu}$ & $P_{e\tau}$ and the cross term
$A_{e\mu}A_{e\tau}^{\ast}$ only. Other terms such as $|a_{1}|^{2}$,
$|a_{2}|^{2}$, $|b_{1}|^{2}$, $|b_{2}|^{2}$, $a_{1}^{\ast}a_{2}$ and
$b_{1}^{\ast}b_{2}$ are given in the appendix.

One important point should also be noted that since the oscillation terms
carrying the phase factor in terms of $\Delta m^{2}$ get averaged out, the
discrimination between the normal ordering (NO) and inverted ordering (IO) of
neutrino mass eigenstates is impossible through probabilities. However,
complexities have slight but non-vanishing discrimination in the effects of NO
and IO as can be seen in the expressions of $|a_{1}|^{2}$, $|a_{2}|^{2}$,
$|b_{1}|^{2}$, $|b_{2}|^{2}$ and so on in Eq. (IV). Similarly, the
complexities associated with the evolution of neutrino states can be obtained
in case of initial $\nu_{\mu}$ and $\nu_{\tau}$ flavors as well.

---
Figure 1: Complexities $\chi_{\alpha}$ (normal ordering (NO) and inverted
ordering (IO) of neutrino masses in blue and red curves, respectively) and
$1-P_{\alpha\alpha}$ (black curve) plotted with respect to the $CP$-violating
phase $\delta$. Here $\alpha=e,\mu,\tau$ depends on the initial neutrino
flavor. All other parameters are set to their best-fit values provided in a
recent analysis [63].

###  II.4 Redefining Flavor Ratios Using Complexity

In this section, we will propose a new flavor ratio based on complexity. We
can consider the initial flavor ratio to be
$\phi_{\nu_{e}}^{0}:\phi_{\nu_{\mu}}^{0}:\phi_{\nu_{\tau}}^{0}=(x:1-x:0)\phi_{\nu_{e}}^{0}$,
where $x=0,1,1/3$ which denotes flavor ratios
$\phi_{\nu_{e}}^{0}:\phi_{\nu_{\mu}}^{0}:\phi_{\nu_{\tau}}^{0}=(0:1:0),~{}(1:0:0),~{}(1:2:0)$,
respectively. Here,
$\phi_{\nu_{e}}+\phi_{\nu_{\mu}}+\phi_{\nu_{\tau}}=\phi_{\nu_{e}}^{0}+\phi_{\nu_{\mu}}^{0}+\phi_{\nu_{\tau}}^{0}$
as a consequence of probability conservation. Hence, fractions of distinct
flavors defined in Eq. (II.1) can be revised as

$\displaystyle X_{e}$ $\displaystyle=xP_{ee}+(1-x)P_{\mu e}$ $\displaystyle
X_{\mu}$ $\displaystyle=xP_{\mu e}+(1-x)P_{\mu\mu}$ $\displaystyle X_{\tau}$
$\displaystyle=(2x-1)P_{e\tau}+(1-x)(1-P_{\tau\tau}).$ (5)

Then we introduce here the analogous equation to the Eq. (II.4) in terms of
complexities as

$\displaystyle X^{\prime}_{e}$ $\displaystyle=x(1-\chi_{e})+(1-x)P_{\mu e}.$
$\displaystyle X^{\prime}_{\mu}$ $\displaystyle=xP_{\mu
e}+(1-x)(1-\chi_{\mu})$ $\displaystyle X^{\prime}_{\tau}$
$\displaystyle=(2x-1)P_{e\tau}+(1-x)\chi_{\tau}.$ (6)

Here, we have replaced $1-P_{\alpha\alpha}$ with corresponding
$\chi_{\alpha}$, for flavor $\alpha=e,\mu,\tau$. We take inspiration from the
findings in [61] and Fig. 1 to make this replacement, noting that
$\chi_{\alpha}$ and $1-P_{\alpha\alpha}$ share similar characteristics. The
flavor ratios outlined in Eq. (II.4) represent a combination of complexities
and oscillation probabilities. Despite this, they can provide valuable
insights, as complexities are known to encapsulate a greater amount of
pertinent information about the system.

##  III Results

In this section, we initially present our results concerning flavor ratios
using traditional probability-based definitions, followed by a modification of
these definitions to incorporate complexities for each neutrino flavor. In
[61], a significant characteristic of complexities has been identified,
indicating that the complexities $\chi_{\alpha}$, $(\alpha=e,\mu,\tau)$
emulate the traits of their corresponding total oscillation probabilities,
i.e., $1-P_{\alpha\alpha}$, although $\chi_{\alpha}$ provides more
information. A similar observation is evident in the context of astrophysical
neutrinos, as illustrated in Fig. 1. It becomes apparent that both
complexities $\chi_{e}$ (depicted as blue(solid) and red(dashed) curves for
normal and inverted mass ordering, respectively) and the total oscillation
probability $1-P_{ee}$ (denoted by the brown(dot-dashed) curve) are not
impacted by the CP-phase $\delta$. However, $\chi_{\mu}$, $\chi_{\tau}$, as
well as $1-P_{\mu\mu}$ and $1-P_{\tau\tau}$, exhibit variations in relation to
the $\delta$ phase. Furthermore, these quantities display symmetric
characteristics around $\delta=0$, signifying their CP-even nature, where CP-
even denotes $f(\delta)=f(-\delta)$, with $f$ encompassing $\chi_{\alpha}$ and
$1-P_{\alpha\alpha}$. Notably, complexities $\chi_{\mu}$ and $\chi_{\tau}$
demonstrate enhanced sensitivity to the CP-phase. Additionally, Fig. 1
highlights a subtle sensitivity of complexities $\chi_{\alpha}$ to the normal
and inverted mass ordering, a feature absent in the case of
$1-P_{\alpha\alpha}$.

Figure 2: The flavor ratios $X_{\alpha}$ ($X^{\prime}_{\alpha}$) are shown on
a ternary plot representation. Dark-gray and light-gray regions represent the
68% and 95% CL of Asimov sensitivity to measure flavor composition using the
current GlobalFit sample [63], while the dark-pink and light-pink regions show
the same sensitivity with the inclusion of 12 years of HESE sample as well
[64]. Blue and red colors depict the cases of relative fractions of
$X^{\prime}_{e}$, $X^{\prime}_{\mu}$ and $X^{\prime}_{\tau}$ corresponding to
the normal ordering (NO) and inverted ordering (IO), respectively. The purple
points represent the fraction of traditionally defined flavor ratios, $i.e.,$
$X_{e}$, $X_{\mu}$ and $X_{\tau}$. Solid lines (purple, blue and red)
represents the flavor ratios with varying $x$ in the range [0,1]. The cases of
initial flavor ratios ($1:0:0$), ($0:1:0$) and ($1:2:0$) corresponding to
$x=1,0,1/3$, $i.e.,$ respectively are shown in different styles of points as
mentioned in the legend. Figure 3: $X^{\prime}_{e}$, $X^{\prime}_{\mu}$ and
$X^{\prime}_{\tau}$ defined in Eq. (II.4) are plotted with respect to $x$. The
solid and dashed curves represent $X^{\prime}_{\alpha}$ with NO and
$X^{\prime}_{\alpha}$ with IO, while the associated initial neutrino flavor
$\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$ are represented by blue, orange and
black color, respectively. Dot-dashed curves represent traditional ratios
$X_{\alpha}$ defined in Eq. (II.4), respectively.

In Fig. 2, the ternary plot shows the representation of flavor ratios. The
traditional flavor ratios $X_{\alpha}$ are depicted in purple, while their
analogous quantities $X^{\prime}_{\alpha}$ are shown in blue (for NO) and red
(for IO). The dark-gray and light-gray regions represent the 68% and 95% C.L.
of Asimov sensitivity for measuring flavor composition using the current
GlobalFit sample [64]. The dark-pink and light-pink regions show the same
sensitivity when the 12 years of HESE sample is included, with 68% and 95%
C.L., respectively [64].

It is interesting that in contrast with the traditional flavor ratios, the
complexity-based flavor ratios favor the initial ratio of neutrino flavors to
be $1:0:0$. In general, the most anticipated initial neutrino flavor ratio is
$1:2:0$, expected to be observed as a $1:1:1$ ratio at Earth. However, there
is insufficient evidence supporting this assumption. The findings in reference
[20] indicate that the best fit to IceCube data at the time is achieved with a
$1:0:0$ ratio at Earth, which does not align with any of these proposed
initial flavor ratios, namely ($1:0:0$), ($0:1:0$), ($1:2:0$), or any other
arbitrary choice as shown in Fig. 3 where we vary the initial fraction via the
parameter $x$ between [0,1].

Apart from the above observations, the following points are important to be
mentioned:

  * •

Firstly, the analogous flavor ratios $X^{\prime}_{\alpha}$ defined in terms of
complexities $\chi_{e}$, $\chi_{\mu}$ and $\chi_{\tau}$ can have more spread
with respect to the variation in $CP$-phase $\delta$ both in the case of NO
and IO in comparison to the traditional quantities $X_{\alpha}$ as the
complexities in Fig. (1) are shown to have more variations with respect to
$\delta$ than the total oscillation probabilities. That means these newly
defined flavor ratios can provide better resolution for the $CP$-violating
phase value. Therefore, complexity-based flavor ratios can also offer the
possibility to constrain or exclude a significant range of the $\delta$-phase,
in principle.

  * •

Secondly, in Fig. (2), the newly defined flavor ratios in terms of
complexities also show small but non-zero sensitivity to the neutrino mass
ordering (slight difference between the blue and red lines). Specifically, the
initial flavor ratio
$\phi_{\nu_{e}}^{0}:\phi_{\nu_{\mu}}^{0}:\phi_{\nu_{\tau}}^{0}=1:0:0$ favors
this condition the most. This sensitivity is absent in the case of traditional
flavor ratios (purple). The point to be noticed here is that complexity-based
ratios also favor the case of $1:0:0$ over $1:2:0$ as initial flavor ratios of
the astrophysical neutrinos, while compared with the sensitivity contours
shown in the plot.

The above-mentioned advantages of the redefined flavor ratios,
$X^{\prime}_{\alpha}$ in Eq. (II.4) in terms of the complexities over the
traditional ones, $X_{\alpha}$, in Eq. (II.4) can be attributed to the cross
terms present in the complexities defined in Eq. (3) along with the terms
proportional to the oscillation probabilities. We also provide individual
fractions of neutrino flavors varying with respect to arbitrary values of $x$
(the parameter defining the initial neutrino flavor ratio) in Fig. (3), which
may correspond to non-traditional neutrino production mechanism.

##  IV Discussion

In this work, we have attempted to redefine the neutrino flavor ratios
arriving on the Earth in terms of the complexities of neutrino propagation due
to the motivation that these complexities and the total oscillation
probabilities of neutrino flavor states depict equivalent features, whereas
the former convey more information. Comparing these complexity-based flavor
ratios with the traditional ones, we find that the former favors the initial
flavor ratio of $1:0:0$ in place of $1:2:0$ might be indicating the production
mechanism of neutrinos to be $n-decay$ in astrophysical objects. Furthermore,
these quantities are found to be sensitive to the neutrino mass ordering,
unlike the case of traditional flavor ratios. Also, notably the complexity-
based flavor ratios manifest a greater degree of variability in the
$CP$-violating phase. These features might be tested in the future with
observational data from the neutrino telescopes.

###### Acknowledgements.

This work was partially supported by grants from the National Research
Foundation (NRF), South Africa, through the National Institute of Theoretical
and Computational Sciences (NITheCS) and from the University of Johannesburg
Research Council.

$\displaystyle|a_{1}|^{2}=$ $\displaystyle(\Delta
m_{21}^{2})^{2}|U_{e2}|^{2}|U_{\mu 2}|^{2}+(\Delta
m_{31}^{2})^{2}|U_{e3}|^{2}|U_{\mu 3}|^{2}+2(\Delta m_{21}^{2})(\Delta
m_{31}^{2})\Re(U_{e2}^{\ast}U_{\mu 2}U_{e3}U_{\mu 3}^{\ast})$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{13}\left((\Delta
m^{2}_{21})^{2}\sin^{2}\theta_{12}\left(\sin^{2}\delta\sin^{2}\theta_{12}\sin^{2}\theta_{13}\sin^{2}\theta_{23}\right.\right.$
$\displaystyle\left.\left.+\left(\cos\theta_{12}\cos\theta_{23}-\cos\delta\sin\theta_{12}\sin\theta_{13}\sin\theta_{23}\right)^{2}\right)+\Delta
m^{2}_{31}\sin\theta_{13}\sin\theta_{23}\right.$
$\displaystyle\left.\left(\Delta m^{2}_{21}\cos\delta\sin
2\theta_{12}\cos\theta_{23}+\sin\theta_{13}\sin\theta_{23}\left(\Delta
m^{2}_{31}-2\Delta m^{2}_{21}\sin^{2}\theta_{12}\right)\right)\right)$
$\displaystyle|a_{2}|^{2}=$ $\displaystyle(\Delta
m_{21}^{2})^{2}|U_{e2}|^{2}|U_{\tau 2}|^{2}+(\Delta
m_{31}^{2})^{2}|U_{e3}|^{2}|U_{\tau 3}|^{2}+2(\Delta m_{21}^{2})(\Delta
m_{31}^{2})\Re(U_{e2}^{\ast}U_{\tau 2}U_{e3}U_{\tau 3}^{\ast})$
$\displaystyle=$ $\displaystyle\cos^{2}\theta_{13}\left(-\frac{1}{4}\Delta
m^{2}_{21}\cos\delta\sin 2\theta_{12}\sin\theta_{13}\sin 2\theta_{23}(\Delta
m_{21}^{2}\cos 2\theta_{12}-\Delta m_{21}^{2}+2\Delta m_{31}^{2})\right.$
$\displaystyle\left.+(\Delta
m_{21}^{2})^{2}\sin^{2}\theta_{12}\cos^{2}\theta_{12}\sin^{2}\theta_{23}\right.$
$\displaystyle\left.+\frac{1}{4}\sin^{2}\theta_{13}\cos^{2}\theta_{23}(\Delta
m_{21}^{2}(-\cos 2\theta_{12})+\Delta m_{21}^{2}-2\Delta
m_{31}^{2})^{2}\right)$ $\displaystyle|b_{1}|^{2}=$ $\displaystyle(\Delta
m_{21}^{2})^{2}(\Delta m_{21}^{2}-2E_{\nu}A_{e})^{2}|U_{e2}|^{2}|U_{\mu
2}|^{2}+(\Delta m_{31}^{2})^{2}(\Delta
m_{31}^{2}-2E_{\nu}A_{e})^{2}|U_{e3}|^{2}|U_{\mu 3}|^{2}$
$\displaystyle+2(\Delta m_{21}^{2})(\Delta m_{31}^{2})(\Delta
m_{21}^{2}-2E_{\nu}A_{e})(\Delta
m_{31}^{2}-2E_{\nu}A_{e})\Re(U_{e2}^{\ast}U_{\mu 2}U_{e3}U_{\mu 3}^{\ast})$
$\displaystyle|b_{2}|^{2}=$ $\displaystyle(\Delta m_{21}^{2})^{2}(\Delta
m_{21}^{2}-2E_{\nu}A_{e})^{2}|U_{e2}|^{2}|U_{\tau 2}|^{2}+(\Delta
m_{31}^{2})^{2}(\Delta m_{31}^{2}-2E_{\nu}A_{e})^{2}|U_{e3}|^{2}|U_{\tau
3}|^{2}$ $\displaystyle+2(\Delta m_{21}^{2})(\Delta m_{31}^{2})(\Delta
m_{21}^{2}-2E_{\nu}A_{e})(\Delta
m_{31}^{2}-2E_{\nu}A_{e})\Re(U_{e2}^{\ast}U_{\tau 2}U_{e3}U_{\tau 3}^{\ast})$
$\displaystyle a_{1}^{\ast}a_{2}=$ $\displaystyle(\Delta
m_{21}^{2})^{2}|U_{e2}|^{2}U_{\mu 2}^{\ast}U_{\tau 2}+(\Delta
m_{31}^{2})^{2}|U_{e3}|^{2}U_{\mu 3}^{\ast}U_{\tau 3}$ $\displaystyle+(\Delta
m_{21}^{2})(\Delta m_{31}^{2})[U_{e2}U_{\mu 2}^{\ast}U_{e3}^{\ast}U_{\tau
3}+U_{e3}U_{\mu 3}^{\ast}U_{e2}^{\ast}U_{\tau 2}]$ $\displaystyle=$
$\displaystyle\frac{1}{8}\cos^{2}\theta_{13}\left(-8e^{-i\delta}\Delta
m^{2}_{21}\sin\theta_{12}\cos\theta_{12}\sin\theta_{13}\left(-\sin^{2}\theta_{23}+e^{2i\delta}\cos^{2}\theta_{23}\right)\right.$
$\displaystyle\left.\left(\Delta m^{2}_{21}\sin^{2}\theta_{12}-\Delta
m^{2}_{31}\right)-(\Delta m^{2}_{21})^{2}\sin^{2}2\theta_{12}\sin
2\theta_{23}\right.$ $\displaystyle\left.+\sin^{2}\theta_{13}\sin
2\theta_{23}(2\Delta m^{2}_{21}\sin^{2}\theta_{12}-2\Delta
m^{2}_{31})^{2}\right),$ $\displaystyle b_{1}^{\ast}b_{2}=$
$\displaystyle(\Delta m_{21}^{2})^{2}(\Delta
m_{21}^{2}-2E_{\nu}A_{e})^{2}|U_{e2}|^{2}U_{\mu 2}^{\ast}U_{\tau 2}+(\Delta
m_{31}^{2})^{2}(\Delta m_{31}^{2}-2E_{\nu}A_{e})^{2}|U_{e3}|^{2}U_{\mu
3}^{\ast}U_{\tau 3}$ $\displaystyle+(\Delta m_{21}^{2})(\Delta
m_{31}^{2})(\Delta m_{21}^{2}-2E_{\nu}A_{e})(\Delta
m_{31}^{2}-2E_{\nu}A_{e})[U_{e2}U_{\mu 2}^{\ast}U_{e3}^{\ast}U_{\tau
3}+U_{e3}U_{\mu 3}^{\ast}U_{e2}^{\ast}U_{\tau 2}]$ $\displaystyle=$
$\displaystyle\left[e^{-i\delta}(\Delta m^{2}_{21})^{2}(\Delta
m^{2}_{31})^{2}(\Delta m^{2}_{21}-\Delta
m^{2}_{31})^{2}\sin^{2}\theta_{12}\cos^{2}\theta_{12}\sin^{2}\theta_{13}\cos^{2}\theta_{13}\left(\Delta
m^{2}_{21}\sin\theta_{12}\right.\right.$ (7)
$\displaystyle\left.\left.\cos\theta_{12}\sin\theta_{23}-e^{i\delta}\sin\theta_{13}\cos\theta_{23}\left(\Delta
m^{2}_{31}-\Delta m^{2}_{21}\sin^{2}\theta_{12}\right)\right)\right.$
$\displaystyle\left.\left(\sin\theta_{13}\sin\theta_{23}\left(\Delta
m^{2}_{31}-\Delta m^{2}_{21}\sin^{2}\theta_{12}\right)+e^{i\delta}\Delta
m^{2}_{21}\sin\theta_{12}\cos\theta_{12}\cos\theta_{23}\right)\right]$
$\displaystyle\times\left[\left(-(\Delta
m^{2}_{21})^{2}\sin^{4}\theta_{12}\cos^{2}\theta_{13}+\Delta
m^{2}_{21}\sin^{2}\theta_{12}\right.\right.$ $\displaystyle\left.\left.(\Delta
m^{2}_{21}+\Delta m^{2}_{31}\cos 2\theta_{13}-\Delta m^{2}_{31})+(\Delta
m^{2}_{31})^{2}\sin^{2}\theta_{13}\right)^{2}\right].$

The normalization constants can be spelled as

$\displaystyle N_{1e}^{2}=$ $\displaystyle\left[(\Delta
m_{21}^{2})^{2}|U_{e2}|^{2}(1-|U_{e2}|^{2})+(\Delta
m_{31}^{2})^{2}|U_{e3}|^{2}(1-|U_{e3}|^{2})\right.$
$\displaystyle\left.-2(\Delta m_{21}^{2})(\Delta
m_{31}^{2})|U_{e2}|^{2}|U_{e3}|^{2}\right]^{-1}$ (8) $\displaystyle
N_{2e}^{2}=$ $\displaystyle\left[(\Delta m_{21}^{2})^{2}(\Delta
m_{21}^{2}-2E_{\nu}A_{e})^{2}|U_{e2}|^{2}(1-|U_{e2}|^{2})+(\Delta
m_{31}^{2})^{2}(\Delta m_{31}^{2}-2E_{\nu}A_{e})^{2}|U_{e3}|^{2}\right.$
$\displaystyle\left.(1-|U_{e3}|^{2})-2(\Delta m_{21}^{2})(\Delta
m_{31}^{2})(\Delta m_{21}^{2}-2E_{\nu}A_{e})(\Delta
m_{31}^{2}-2E_{\nu}A_{e})|U_{e2}|^{2}|U_{e3}|^{2}\right]^{-1}$ (9)

where,

$\displaystyle A_{e}=\frac{1}{2E}$ $\displaystyle\left[(\Delta
m_{21}^{2})^{3}|U_{e2}|^{2}(1-|U_{e2}|^{2})+(\Delta
m_{31}^{2})^{3}|U_{e3}|^{2}(1-|U_{e3}|^{2})-\Delta m_{21}^{2}\Delta
m_{31}^{2}|U_{e2}|^{2}|U_{e3}|^{2}\right.$ $\displaystyle\left.(\Delta
m_{21}^{2}+\Delta m_{31}^{2})\right]\left[(\Delta
m_{21}^{2})^{3}|U_{e2}|^{2}(1-|U_{e2}|^{2})+(\Delta
m_{31}^{2})^{3}|U_{e3}|^{2}(1-|U_{e3}|^{2})\right.$
$\displaystyle\left.-2(\Delta m_{21}^{2})(\Delta
m_{31}^{2})|U_{e2}|^{2}|U_{e3}|^{2}\right]^{-1}$

## References

  * [1] K. S. Hirata et al.,  Real time, directional measurement of B-8 solar neutrinos in the Kamiokande-II detector.  Phys. Rev. D, 44:2241, 1991.  [Erratum: Phys.Rev.D 45, 2170 (1992)].
  * [2] B. T. Cleveland, Timothy Daily, Raymond Davis, Jr., James R. Distel, Kenneth Lande, C. K. Lee, Paul S. Wildenhain, and Jack Ullman.  Measurement of the solar electron neutrino flux with the Homestake chlorine detector.  Astrophys. J., 496:505–526, 1998.
  * [3] Q. R. Ahmad et al.,  Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory.  Phys. Rev. Lett., 89:011301, 2002.
  * [4] Y. Fukuda et al.  Atmospheric muon-neutrino / electron-neutrino ratio in the multiGeV energy range.  Phys. Lett. B, 335:237–245, 1994.
  * [5] Y. Fukuda et al.  Evidence for oscillation of atmospheric neutrinos.  Phys. Rev. Lett., 81:1562–1567, 1998.
  * [6] Y. Abe et al.  Indication for the disappearance of reactor electron antineutrinos in the Double Chooz experiment.  Phys.Rev.Lett., 108:131801, 2012.
  * [7] D. Adey et al.  Measurement of the Electron Antineutrino Oscillation with 1958 Days of Operation at Daya Bay.  Phys. Rev. Lett., 121(24):241805, 2018.
  * [8] G. Bak et al.  Measurement of Reactor Antineutrino Oscillation Amplitude and Frequency at RENO.  Phys. Rev. Lett., 121(20):201801, 2018.
  * [9] B. Abi et al. [DUNE], Eur. Phys. J. C 80 (2020) no.10, 978 [arXiv:2006.16043 [hep-ex]].
  * [10] J. Zhang [JUNO], J. Phys. Conf. Ser. 2156 (2021) no.1, 012110 [arXiv:2111.10112 [physics.ins-det]].
  * [11] M. A. Acero et al. [NOvA], Phys. Rev. D 106 (2022) no.3, 032004 [arXiv:2108.08219 [hep-ex]].
  * [12] K. Abe et al. [T2K], Eur. Phys. J. C 83 (2023) no.9, 782 [arXiv:2303.03222 [hep-ex]].
  * [13] G. Barenboim and C. Quigg, Phys. Rev. D 67, 073024 (2003) [arXiv:hep-ph/0301220 [hep-ph]].
  * [14] Z. Z. Xing and S. Zhou, Phys. Rev. D 74, 013010 (2006) [arXiv:astro-ph/0603781 [astro-ph]].
  * [15] P. Lipari, M. Lusignoli and D. Meloni, Phys. Rev. D 75, 123005 (2007) [arXiv:0704.0718 [astro-ph]].
  * [16] S. Pakvasa, W. Rodejohann and T. J. Weiler, JHEP 02, 005 (2008) [arXiv:0711.4517 [hep-ph]].
  * [17] A. Esmaili and Y. Farzan, Nucl. Phys. B 821, 197-214 (2009) [arXiv:0905.0259 [hep-ph]].
  * [18] K. C. Lai, G. L. Lin and T. C. Liu, Phys. Rev. D 80, 103005 (2009) [arXiv:0905.4003 [hep-ph]].
  * [19] S. Choubey and W. Rodejohann, Phys. Rev. D 80, 113006 (2009) [arXiv:0909.1219 [hep-ph]].
  * [20] O. Mena, S. Palomares-Ruiz and A. C. Vincent, Flavor Composition of the High-Energy Neutrino Events in IceCube, Phys. Rev. Lett. 113 (2014), 091103 [arXiv:1404.0017 [astro-ph.HE]].
  * [21] A. Palladino, G. Pagliaroli, F. L. Villante and F. Vissani, Phys. Rev. Lett. 114, no.17, 171101 (2015) [arXiv:1502.02923 [astro-ph.HE]].
  * [22] M. G. Aartsen et al. [IceCube], Phys. Rev. Lett. 114, no.17, 171102 (2015) [arXiv:1502.03376 [astro-ph.HE]].
  * [23] M. Bustamante, J. F. Beacom and W. Winter, Phys. Rev. Lett. 115, no.16, 161302 (2015) [arXiv:1506.02645 [astro-ph.HE]].
  * [24] M. G. Aartsen et al. [IceCube], Astrophys. J. 846 (2017) no.2, 136 doi:10.3847/1538-4357/aa8508 [arXiv:1705.02383 [astro-ph.HE]].
  * [25] N. Song, S. W. Li, C. A. Argüelles, M. Bustamante and A. C. Vincent, JCAP 04, 054 (2021) [arXiv:2012.12893 [hep-ph]].
  * [26] R. Abbasi et al. [IceCube], Eur. Phys. J. C 82, no.11, 1031 (2022) [arXiv:2011.03561 [hep-ex]].
  * [27] M. Blasone, F. Dell’Anno, S. De Siena, M. Di Mauro, and F. Illuminati.  Multipartite entangled states in particle mixing.  Phys. Rev. D, 77:096002, 2008.
  * [28] M. Blasone, F. Dell’Anno, S. De Siena, and F. Illuminati,  Entanglement in neutrino oscillations.  EPL, 85:50002, 2009.
  * [29] D. Gangopadhyay, D. Home, and A. Sinha Roy.  Probing the Leggett-Garg Inequality for Oscillating Neutral Kaons and Neutrinos.  Phys. Rev. A, 88(2):022115, 2013.
  * [30] A. K. Alok, S. Banerjee, and S. Uma Sankar.  Quantum correlations in terms of neutrino oscillation probabilities.  Nucl. Phys. B, 909:65–72, 2016.
  * [31] S. Banerjee, A. K. Alok, R. Srikanth, and B. C. Hiesmayr.  A quantum information theoretic analysis of three flavor neutrino oscillations.  Eur. Phys. J. C, 75(10):487, 2015.
  * [32] J. A. Formaggio, D. I. Kaiser, M. M. Murskyj, and T. E. Weiss.  Violation of the Leggett-Garg Inequality in Neutrino Oscillations.  Phys. Rev. Lett., 117(5):050402, 2016.
  * [33] Q. Fu and X. Chen.  Testing violation of the Leggett–Garg-type inequality in neutrino oscillations of the Daya Bay experiment.  Eur. Phys. J. C, 77(11):775, 2017.
  * [34] X. K. Song, Y. Huang, J. Ling and M. H. Yung, Phys. Rev. A 98 (2018) no.5, 050302 [arXiv:1806.00715 [hep-ph]].
  * [35] M. M. Ettefaghi, Z. S. Tabatabaei Lotfi, and R. Ramezani Arani.  Quantum correlations in neutrino oscillation: Coherence and entanglement.  EPL, 132(3):31002, 2020.
  * [36] S. Shafaq and P. Mehta, J. Phys. G 48 (2021) no.8, 085002 [arXiv:2009.12328 [hep-ph]].
  * [37] L. Susskind, Fortsch. Phys. 64 (2016), 49-71 [arXiv:1411.0690 [hep-th]].
  * [38] L. Susskind, Fortsch. Phys. 64 (2016), 24-43 [arXiv:1403.5695 [hep-th]].
  * [39] D. Stanford and L. Susskind, Phys. Rev. D 90 (2014) no.12, 126007 [arXiv:1406.2678 [hep-th]].
  * [40] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Phys. Rev. Lett. 116 (2016) no.19, 191301 [arXiv:1509.07876 [hep-th]].
  * [41] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Phys. Rev. D 93 (2016) no.8, 086006 [arXiv:1512.04993 [hep-th]].
  * [42] T. Ali, A. Bhattacharyya, S. S. Haque, E. H. Kim, N. Moynihan and J. Murugan, Phys. Rev. D 101 (2020) no.2, 026021 [arXiv:1905.13534 [hep-th]].
  * [43] A. Bhattacharyya, W. Chemissany, S. Shajidul Haque and B. Yan, Eur. Phys. J. C 82 (2022) no.1, 87 [arXiv:1909.01894 [hep-th]].
  * [44] A. Bhattacharyya, S. S. Haque and E. H. Kim, JHEP 10 (2021), 028 [arXiv:2011.04705 [hep-th]].
  * [45] V. Balasubramanian, M. Decross, A. Kar and O. Parrikar, JHEP 01 (2020), 134 [arXiv:1905.05765 [hep-th]].
  * [46] A. Bhattacharyya, W. Chemissany, S. S. Haque, J. Murugan and B. Yan, SciPost Phys. Core 4 (2021), 002 [arXiv:2007.01232 [hep-th]].
  * [47] V. Balasubramanian, M. DeCross, A. Kar, Y. Li and O. Parrikar, JHEP 07 (2021), 011 [arXiv:2101.02209 [hep-th]].
  * [48] T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim and N. Moynihan, Phys. Lett. B 811 (2020), 135919 [arXiv:1811.05985 [hep-th]].
  * [49] A. Bhattacharyya, T. Hanif, S. S. Haque and A. Paul, Phys. Rev. D 107 (2023) no.10, 106007 [arXiv:2210.09268 [hep-th]].
  * [50] A. Bhattacharyya, T. Hanif, S. S. Haque and M. K. Rahman, Phys. Rev. D 105 (2022) no.4, 046011 [arXiv:2112.03955 [hep-th]].
  * [51] A. Bhattacharyya, S. Das, S. Shajidul Haque and B. Underwood, Phys. Rev. D 101 (2020) no.10, 106020 [arXiv:2001.08664 [hep-th]].
  * [52] A. Bhattacharyya, S. Das, S. S. Haque and B. Underwood, Phys. Rev. Res. 2 (2020) no.3, 033273 [arXiv:2005.10854 [hep-th]].
  * [53] S. S. Haque, C. Jana and B. Underwood, Phys. Rev. D 106 (2022) no.6, 063510 [arXiv:2110.08356 [hep-th]].
  * [54] M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Science 311 (2006) no.5764, 1133-1135 [arXiv:quant-ph/0603161 [quant-ph]].
  * [55] M. A. Nielsen, Quant. Inf. Comput. 6 (2006) no.3, 213-262 [arXiv:quant-ph/0502070 [quant-ph]].
  * [56] M. R. Dowling and M. A. Nielsen, Quant. Inf. Comput. 8 (2008) no.10, 0861-0899 [arXiv:quant-ph/0701004 [quant-ph]].
  * [57] R. Jefferson and R. C. Myers, JHEP 10 (2017), 107 [arXiv:1707.08570 [hep-th]].
  * [58] T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim and N. Moynihan, JHEP 04 (2019), 087 [arXiv:1810.02734 [hep-th]].
  * [59] V. Balasubramanian, P. Caputa, J. M. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106 (2022) no.4, 046007 [arXiv:2202.06957 [hep-th]].
  * [60] P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106 (2022) no.19, 195125 [arXiv:2205.05688 [hep-th]].
  * [61] K. Dixit, S. S. Haque and S. Razzaque, Quantum spread complexity in neutrino oscillations, Eur. Phys. J. C 84 (2024) no.3, 260 [arXiv:2305.17025 [hep-ph]].
  * [62] J. P. Rachen and P. Meszaros, Phys. Rev. D 58, 123005 (1998) [arXiv:astro-ph/9802280 [astro-ph]].
  * [63] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020), 178 [arXiv:2007.14792 [hep-ph]].
  * [64] R. Abbasi et al. [IceCube], PoS ICRC2023 (2023), 1122 [arXiv:2308.15213 [astro-ph.HE]].