probability density function (PDF) to assign probability weighting to final
water contents based on the defined uncertainty in the constraints. In this
work, we tested several common statistical functions and identified a gamma
profile to best fit the present-day constraint we adopt. Gamma profiles worked
well since the water mass fractions are restricted to be greater than or equal
to 0 and, consequently, show asymmetric tails, but any continuous probability
distribution may be used here in theory. With a gamma profile, our present-day
water PDFs are described by:

$\large f_{i}(w,a)=\cfrac{\displaystyle
w^{a-1}e^{-w}}{\displaystyle\Gamma(a)}\ ,$ (C1)

where $i$ denotes which planet the PDF is describing (T1-b, c, d, e, f, g, or
h), $w$ is the present-day water content, $a$ is the shape parameter,
$f_{i}(w,a)$ is the relative likelihood of present-day water content $w$ on
planet $i$, and $\Gamma(a)$ is the gamma function evaluated at a particular
shape parameter.

With the PDF describing the present-day water constraints, the next step is to
determine the value of the PDF at the final water content predicted by each
individual simulation across a particular initial water content. We then sum
the PDF value across the individual simulations of a given initial water
content and normalize for the total number of simulations in that suite to
obtain a probability weighting for that particular initial water content. To
normalize the probability weightings per initial water content, we divide by
the number of simulations in the PDF value summation:

$\large\text{Normalized
Likelihood}=\cfrac{\displaystyle\sum_{z=1}^{N}f_{i}(w_{z},a)}{\displaystyle
N}\ ,\ \text{per initial water content},$ (C2)

where $f_{i}(w_{z},a)$ is the PDF for the planet being considered (Equation
C1) evaluated at the final water content of a particular simulation, $w_{z}$,
and $N$ is the total number of simulations at that initial water content.

The initial water content with the highest probability weighting is the one
that best produced the assumed present-day water constraints. If the initial
water content grid is sampled finely enough, we can interpolate across our
distribution of probability weightings to determine where the true most likely
initial water content would be, and what the confidence intervals are on that
prediction. This next step can only be taken if the grid of initial water
contents with simulation data is fine enough that the distribution of
probability weightings appears to be a smooth, continuous curve; in some cases
this may require running more data once the initial peak probability weighting
has been identified. With sufficient data, this interpolation can be
accomplished with a Riemann sum. To perform the Riemann sum, we take the curve
of probability weightings versus initial water content and create area bins
based on the data points, then find at what initial water content is the area
under the curve 50% of the total area – the 50th percentile, or median value
(i.e., the initial water content most likely to produce the present-day
constraints). Then finding the bounds of 68% and 95% area under the curve,
centered at the median value, will yield the 1$\sigma$ and 2$\sigma$
confidence intervals.

In our work, we created bins between adjacent data points with linear lines as
the top of the bins. Thus, the area in a particular bin between two initial
water contents, $IW1$ and $IW2$, is given by:

$\large A_{bin}=\int_{IW1}^{IW2}\left(sx+b\right)dx=\left(\cfrac{\displaystyle
s}{\displaystyle 2}x^{2}+bx\right)\bigg{|}_{IW1}^{IW2}=\cfrac{\displaystyle
s}{\displaystyle 2}\left(IW2^{2}-IW1^{2}\right)+b\left(IW2-IW1\right)\ ,$ (C3)

where $s$ is the slope of the linear fit between the two data points and $b$
is the y-intercept. This integrand is the same equation used to solve for a
true value or confidence interval bound that lies between two data points. In
fitting for the PDF this work, we used SciPy’s stats module (Virtanen et al.,
2020), and NumPy’s polyfit routine (Harris et al., 2020) for the linear fits.

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# Improving 3D Object Detection for Pedestrians with Virtual Multi-View
Synthesis Orientation Estimation

Jason Ku, Alex D. Pon, Sean Walsh, and Steven L. Waslander Jason Ku, Alex D.
Pon, Sean Walsh, and Steven L. Waslander are with the Institute for Aerospace
Studies, University of Toronto, 4925 Dufferin St, North York, ON, Canada.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Accurately estimating the orientation of pedestrians is an important and
challenging task for autonomous driving because this information is essential
for tracking and predicting pedestrian behavior. This paper presents a
flexible Virtual Multi-View Synthesis module that can be adopted into 3D
object detection methods to improve orientation estimation. The module uses a
multi-step process to acquire the fine-grained semantic information required
for accurate orientation estimation. First, the scene’s point cloud is
densified using a structure preserving depth completion algorithm and each
point is colorized using its corresponding RGB pixel. Next, virtual cameras
are placed around each object in the densified point cloud to generate novel
viewpoints, which preserve the object’s appearance. We show that this module
greatly improves the orientation estimation on the challenging pedestrian
class on the KITTI benchmark. When used with the open-source 3D detector AVOD-
FPN, we outperform all other published methods on the pedestrian Orientation,
3D, and Bird’s Eye View benchmarks.

##  I Introduction

Deep neural networks have made remarkable progress on the task of 3D object
detection such that they are robust enough to be deployed on autonomous
vehicles. The KITTI [1] benchmark showcases the success of 3D object detection
methods, especially on the car and cyclist classes, but also highlights areas
needing improvement. The benchmark shows that existing 3D detection methods
[2, 3] are able to very accurately estimate the orientation of cars and
cyclists, with Average Angular Errors (AAE) [4] less than $None$ and $None$,
respectively, whereas the average error for pedestrians is almost $None$.

In this work, we address the task of 3D pose estimation for pedestrians with a
focus on orientation estimation. This task is especially important for
autonomous driving as this information is useful for tracking and is vital for
predicting pedestrian behavior. Furthermore, it is important to incorporate
orientation estimation into the detection pipeline, and not rely on a tracking
method for this estimation, as the orientation of pedestrians waiting to cross
the street must be inferred without the aid of motion cues.

Image based detection methods have access to rich semantic information from
RGB data. To accurately estimate orientation, these methods must extract fine-
grained details of objects. However, extracting semantic information is
challenging because of the varying scale and appearance of objects caused by
the perspective transformation of the 3D scene into an image. Some methods [5,
6] attempt to resolve this issue by using images at multiple scales to extract
features. However, object appearance inconsistencies still exist from ROI
cropping as noted in [4]. In contrast, we propose to learn fine-grained
information by rendering multiple viewpoints of objects through the placement
of virtual cameras within the 3D scene at consistent locations relative to
each object. Using these generated viewpoints retains a more consistent object
appearance, as shown in Fig 1, which facilitates the learning of object
orientation in our neural network.

Figure 1: Virtual Multi-View Synthesis. The core idea of the method is to
generate a set of virtual views for each detected pedestrian, and exploit
these views in both the training and inference procedures to produce an
accurate orientation estimation.

LiDAR methods can take advantage of accurate depth information to achieve
robust localization. For the car and cyclist classes, the ratio of the 3D
object length and width can be leveraged to simplify the orientation
estimation problem [2]. However, LiDAR based methods also face challenges in
extracting fine-grained semantic information of objects. The sparsity of LiDAR
data limits the operational range of these methods, especially for smaller
objects such as pedestrians. At longer distances, fine-grained information is
lost as the sparsity of the LiDAR data becomes so severe that it can become
difficult to distinguish between trees, poles, and pedestrians, leading to
false positives. Fig. 2 shows that even at shorter distances of 20 and 30
meters, with a high density 64 beam HDL-64E LiDAR, it is difficult for humans
to discern meaningful orientation information due to the sparsity of the data.
To solve this sparsity issue, we leverage the task of depth completion to
generate dense point clouds, which allows for a one-to-one pixel-to-point
incorporation of RGB image data. Also, inspired by F-PointNet’s [7] use of a
2D detector for accurate classification, we leverage a 2D detector for false
positive suppression.

Moreover, orientation estimation performance is dependent on the amount of
available training data. The training set must well represent the entire range
of possible orientations, but labelled 3D data is expensive and time consuming
to acquire [8]. The KITTI [1] dataset has only 4500 pedestrian training
instances with 3D labels, making it difficult to train neural networks that
normally require much larger amounts of data to achieve good performance. A
common solution for the scarcity of pose data is to leverage CAD models and
additional annotations for more training data [9, 10]. We, however, do not
require the use of additional data sources or labels. To mitigate the low
amount of data, we develop a virtual multi-view rendering pipeline to generate
novel realistic data from the image and LiDAR inputs. This generated data is
incorporated into our network during both training and inference. Detected
objects are re-rendered from a set of canonical camera views so that objects
maintain a more consistent appearance as compared to using the 2D crops
obtained by the common Region of Interest (ROI) crop-and-resize procedure. At
inference time, these additional viewpoints are used to determine a more
accurate orientation estimate.

To summarize, our contributions in this work are as follows:

  * •

We propose a flexible module that can be adopted into 3D object detection
pipelines to improve orientation estimation.

  * •

We solve the issues of limited pose data and the challenges of obtaining fine-
grain details from images and LiDAR data in a novel pipeline that uses virtual
cameras to generate viewpoints on a colorized depth completed LiDAR point
cloud.

  * •

At the time of submission, when incorporated into the detection pipeline of
the open-source AVOD-FPN [2] 3D object detector, our method ranks first
compared to all other published methods on the KITTI [1] pedestrian
Orientation, 3D, and Bird’s Eye View benchmarks.

  * •

Compared to other detection methods with comparable orientation estimation
performance, our method runs over 8 times faster

Figure 2: Pedestrian Appearances at 20 m (top row) and 30 m (bottom row). From
left to right: RGB image, LiDAR scan colored by intensity, depth completed
point cloud colored with corresponding RGB pixels. Even for humans, the
classification of objects such as the tree, and the orientations of the
pedestrians are not readily apparent in the LiDAR scan. In our method, rich
semantic image features are preserved and fused directly with the point cloud,
making it much easier to discern this information.

##  II Related Work

Orientation (yaw, pitch, roll) represents one of the components defining the
3D pose of an object. The introduction of 3D pose datasets [1, 11, 12, 13] has
spurred a myriad of 3D pose estimation methods, and as a result many
techniques for orientation estimation have been explored.

Feature Extraction at Multiple Scales. Previous works have recognized that
accurate orientation estimation requires a feature extraction process that
captures fine-grained semantic information of objects. Zhang et al. [14]
identify that standard Faster R-CNN feature maps are too low resolution for
pedestrians and instead use atrous convolutions [15] and pool features from
shallower layers. Pyramid structures including image pyramids [16] and feature
pyramids [17] have also been leveraged to obtain information from multiple
scales. SubCNN [5] use image pyramids to handle scale changes of objects, and
[2] highlight the importance of a pyramid structure for small classes such as
pedestrians. Moreover, [18, 19, 20] have shown the importance in how methods
crop ROI features. Kundu et al. [4] note that the standard ROI crop can warp
the appearance of shape and pose, and propose the use of a virtual ROI camera
to address this. We instead obtain fine-grained details by using multiple
virtual ROI cameras placed canonically around each object’s detected centroid.
As compared to [4], we not only use 2D RGB data, but also use a 3D point cloud
to produce realistic novel viewpoints that maintain consistent object
appearance.

Figure 3: Architecture Diagram. A 3D detector is used to generate 3D
detections, which are passed into the Virtual Multi-View Synthesis Module. The
module places virtual cameras within the scene represented by a colorized
depth completed LiDAR scan to generate $N$ novel viewpoints. Finally, the
Orientation Estimation Module predicts the object orientation from the
generated views.

Keypoints and CAD Models. Using keypoint detections [21, 22, 23, 24] and CAD
models [25, 26] have been shown to be effective in gaining semantic
understanding of objects of interest. The use of 2D keypoint detections to
estimate pose has been well studied as the Perspective-n-Point (PnP) problem
with many proposed solutions [27, 28, 29]. More recently, [25, 26] use 3D CAD
models and convolutional neural networks (CNNs) to detect keypoints to learn
3D pose. Within the autonomous driving context, Deep MANTA [10] predicts
vehicle part coordinates and uses a vehicle CAD model dataset to estimate 3D
poses. CAD models have also been used to create additional ground truth
labels. Su et al. [9] argue the scarcity of training data with viewpoint
annotations hinders viewpoint estimation performance, so they use 3D models to
generate accurate ground truth data. Unlike the above methods, we do not
require additional keypoint labels or external CAD datasets. We propose a
general pipeline that leverages available data, and with our virtual cameras
we generate novel high resolution viewpoints.

Multi-view Learning. Using multiple views has previously been shown to be
effective in allowing neural networks to learn shape and pose information. Su
et al. [30] render multiple views around an object from a CAD dataset, then
predict shape based on each view’s features. Others [31, 32, 33] use multiple
views to ensure a projection consistency to learn shape and pose information.
These methods tend to use CAD models, which segment the object of interest
from the background, contain full $None$ shape information, and allows perfect
generation of data from any viewpoint. However, our application in autonomous
driving only has access to LiDAR scans, which only provides input data from a
single direction. We show that we are still able to exploit this data by
carefully placing the virtual cameras within a certain operating region to
maintain realistically rendered images, as shown in Fig. 4.

Orientation Representation. Most similar to our work are 3D pose estimation
methods designed for autonomous driving scenarios. These methods have mainly
focused on the representation of orientation and designing new loss functions.
Pose-RCNN [34] uses a Biternion representation for orientation as recommended
by [35]. The monocular 3D object detection method Deep3DBox [36] proposes a
angle bin formulation that frames orientation estimation as a hybrid
classification-regression problem. Here, orientation is discretized into
several bins, and the network is tasked to classify the correct bin and to
predict a regression offset. This formulation has been adopted by LiDAR
methods including [7]. [2] identify an ambiguity issue where identical 3D
boxes are created despite having orientation estimates differing by $\pm\pi$
radians. They solve this issue by parameterizing orientation as an angle
vector, while Yan et al. [3] approach this same issue with a sine error loss.
We show in our ablation studies (Sec. V-B) that parameterizing orientation as
an angle vector while using the discrete-continuous angle bin formulation as
an auxiliary loss is most effective.

##  III Pose Estimation Framework

Fig. 3 provides an overview of our pipeline for 3D pedestrian pose estimation.
Given a set of 3D detections, each parameterized by a centroid
$T=(t_{x},t_{y},t_{z})$, dimensions $D=(d_{x},d_{y},d_{z})$, and an Euler
angle based orientation $O=(\theta,\phi,\psi)$, the objective is to provide a
more accurate estimate of the orientation of the object of interest. For the
KITTI benchmark, the pitch and roll are assumed to be zero, while only the
yaw, or heading angle, $\theta_{yaw}$ is considered.

The core idea of our method is to generate realistic novel viewpoints of
objects from a colored densified point cloud, and to use these viewpoints to
extract rich semantic information of objects and thus perform more accurate
orientation estimation. To start, we build upon the performance of existing 3D
detectors by using 3D detections as centroid proposals. These proposals are
processed by the Virtual Multi-View Synthesis module, which renders a set of
novel views using a dense point cloud reconstruction. Importantly, these views
are created in a set of canonical camera viewpoints, by placing virtual
cameras at consistent locations relative to each object’s centroid. The object
views generated from these virtual cameras better preserves the 3D shape and
appearance of the object as compared to taking ROI crops.

For each of the generated novel views, orientation is estimated by passing the
views through a CNN followed by an orientation regression output head. The
final orientation output is produced by merging the orientation estimates.
Lastly, since the 3D detector used in the pipeline is chosen on a basis of
having high recall, we use a robust 2D detector for false positive
suppression.

###  III-A Virtual Multi-View Synthesis

Densified RGB Point Cloud Generation. We first note that a LiDAR point cloud
is simply a sparse representation of the underlying scene, and can be viewed
omnidirectionally, with each view providing a unique visual representation of
the scene. However the sparsity of LiDAR data only provides a low resolution
perspective of the scene. We therefore take the portion of the LiDAR scan
corresponding to the visible portion of the scene in the RGB image, and
process it through a structure preserving depth completion algorithm [37]. In
particular, the LiDAR points are projected into the image using the provided
$3\times 4$ camera projection matrix $P_{cam}$ creating a sparse depth map
$D_{s}$. The depth completion algorithm produces a dense depth map $D_{d}$,
with each pixel representing a depth, that is then re-projected as a 3D point
cloud. As the resulting point cloud comes from the dense depth map which has
the same resolution as the RGB image, we next infuse the semantic information
from the image by coloring each 3D point with its corresponding pixel from the
RGB image. This colorized, dense reconstruction of the scene’s point cloud
resolves the low resolution of the LiDAR scan, and allows for more realistic
novel views to be generated that preserve fine-grained semantic information,
as shown in Fig. 2.

Multi-View Generation. Our goal is to learn fine-grained details of each
object. The most straightforward process is to simply take ROI crops of the
areas of the image corresponding to each object, but as noted by [4] this
common cropping procedure can lead to wildly different appearances of the same
object. The use of object-centric coordinate frames has recently been shown to
be effective for facilitating learning tasks [7, 38], and as such, we design
our module to use canonical viewpoints for estimation.

To create our canonical camera viewpoints, we place virtual cameras in
positions equidistant from the object centroid along $N$ equally spaced angle
divisions. The camera locations are placed level with the center of the object
of interest and between a range of angles, $\rho\in[-\rho_{max},\rho_{max}]$
to the left and right of the horizontal viewing angle $\alpha$, defined by the
ray from the original camera center to the object centroid, as shown in Fig.
4. The minimum and maximum angle of the viewpoints are chosen such that the
generated views do not expose parts of the object that are not visible from
the original camera viewpoint, which would make the object look unrealistic.
$N$ views of the object are generated in a set of canonical viewpoints, which
are evenly spaced along an arc of radius $r$ around the object. Each viewpoint
generates an $H\times W$ ROI image that preserves appearance, as shown in Fig.
1. These rendered outputs densely fuse the point cloud and image information
into a 3 channel RGB format, which allows the use of mature CNN architectures
to be used for orientation estimation.

Figure 4: Virtual Camera Placement. Virtual cameras are placed at positions
equidistant from each object centroid, with viewpoints ranging from $None$ to
$None$ relative to the ray from the original camera center to the object
centroid, shown by the dotted black line. Here, only three of the eleven
camera positions are shown.

###  III-B Orientation Estimation

The rendered ROI images are passed through a CNN to produce the final
orientation estimations. Several methods [36, 7] use a discrete-continuous
loss in the form of $B$ angle bins with regressions within each bin. However,
we hypothesize that this divides the training data as each bin will only have
a fraction of the total number of training samples to learn from. As shown in
Sec. V-B, the network trained in this way does not work as well as when using
the angle vector formulation. This is likely due to the small number of
pedestrians instances available for training. In our network, the orientation
is instead predicted as two values in a vector format,
$(x_{\theta},y_{\theta})=(\cos(\theta),\sin(\theta))$, in the same manner as
[2]. This formulation avoids dividing the training samples and handles angle
wrapping as each angle $\theta$ is represented as a unique unit vector. At
inference time, the ROI images corresponding to each object are processed
through the network as a single batch to produce $N$ orientation estimates.
The final yaw estimation $\theta_{f}$ is calculated as the mean of the angle
vector yaw estimates as:

$\theta_{f}=\tan^{-1}\left(\frac{\frac{1}{N}\sum_{j=1}^{N}y_{\theta_{j}}}{\frac{1}{N}\sum_{j=1}^{N}x_{\theta_{j}}}\right).$
(1)

###  III-C Final Pose Estimation

The final 3D detection is parameterized by the object centroid $T$, dimensions
$D$, and orientation $O$. The centroid and dimension estimates are taken
directly from the 3D detections, while the yaw $\theta$ of the orientation is
set as the output produced by the Orientation Estimation module.

###  III-D False Positive Suppression

Pedestrian LiDAR-based 3D detectors’ average precision (AP) curves on the
KITTI test benchmark show significantly lower precision compared to methods
that incorporate image data. This supports the hypothesis that it is
significantly harder to discern smaller objects like pedestrians from only
LiDAR data, and that false positives are prevalent. To improve detection
performance, the 2D boxes from a robust 2D detector are matched with the 2D
projections of the 3D detections using an intersection over union (IoU)
threshold of 0.4. 3D boxes whose projections do not meet this threshold have
their scores reduced, which acts as false positive suppression. Since the
KITTI Average Orientation Similarity (AOS) evaluation is also dependent on 2D
detection performance, the projection of each 3D box is replaced with its
corresponding 2D box from the 2D detector, which allows AOS to be evaluated.

###  III-E Training Losses

Multi-task training has shown to be effective in improving performance of many
neural networks. Following the discrete-continuous angle bin formulation from
[36], we add an auxiliary output layer that produces $B$ heading bins and $B$
angle regressions. The angle bins are trained with a softmax loss while the
bin regression and angle vector outputs are trained with a smooth L1 loss. The
total loss is computed as:

$L=\alpha L_{ang}+\beta L_{cls}+\gamma L_{reg},$ (2)

where $L_{ang}$ is the angle vector loss, $L_{cls}$ is the angle bin
classification loss, and $L_{reg}$ is the angle bin regression loss. Losses
are weighted such that they converge to values of similar magnitude. We use
values of ${\alpha=50}$, ${\beta=1}$, and ${\gamma=200}$. The output from the
angle bins are not used in the final estimation as they provide lower
performance when used alone. However, incorporating these bins for auxiliary
losses in the multi-task training scheme provides additional orientation
estimation performance as shown in Sec. V-B.

##  IV Implementation Details

###  IV-A Detector Selection

We use the publicly available AVOD-FPN [2] 3D object detector to generate 3D
detections. The selection of this detector is based on the recall analysis in
Sec. V-A, which shows that it achieves the highest recall for pedestrians
compared to other open-source 3D detectors. The open-source 2D detector MS-CNN
[39] is used for false positive suppression. However, our method is compatible
with any 3D or 2D detector.

###  IV-B View Synthesis

For each detected object, the camera viewpoints are set to $N=11$ evenly
spaced locations ranging from $\rho\in[$$,$$]$ relative to the viewing angle
of the object, at a distance $r=4$ m as shown in Fig. 4. These minimum and
maximum angles are empirically chosen to avoid viewing unseen portions of the
pedestrians, which are not captured in the LiDAR scan. The ROI images of size
$H\times W$ are generated at a resolution of $224\times 224$. All 11 images
are efficiently rendered in parallel using the Visualization Toolkit (VTK).

Method | Recall
---|---
| Easy | Moderate | Hard
SECOND[3] | 86.33 | 80.22 | 73.11
F-PointNet[7] | 95.15 | 88.50 | 80.66
AVOD-FPN[2] | 96.47 | 93.90 | 91.16
Table I: 3D Pedestrian Detector Recall Analysis. Recall on moderate pedestrians for top 16 detections at a distance of 0.3 m on the validation split. We determine that AVOD-FPN [2] has the highest recall of the available open-source 3D detectors. Bins | Vector | OS
---|---|---
4 | - | 0.8817
8 | - | 0.8940
12 | - | 0.8892
16 | - | 0.8625
- |  | 0.9060
8 |  | 0.9148
Table II: Angle Regression Ablation. Comparison of Orientation Score (OS) performance for different types of angle regression. Viewpoint | Views | Time (s) | OS
---|---|---|---
ROI Crop | 1 | 0.042 | 0.8909
Virtual Cam | 1 | 0.042 | 0.8692
Virtual Cam | 3 | 0.111 | 0.9001
Virtual Cam | 6 | 0.209 | 0.9074
Virtual Cam | 11 | 0.378 | 0.9148
Virtual Cam | 15 | 0.520 | 0.9061
Table III: Virtual Multi-View Ablation. Effect of ROI versus virtual camera,
the number of multi-views on CNN inference time for 16 objects, and
orientation score (OS) when used for training and inference.

###  IV-C Orientation Estimation Module

For faster training and convergence, the backbone of the network consists of a
modified ResNet-101 [40] architecture, taking only the layers before conv5_1,
pretrained on the KITTI 2D detection dataset. To reduce GPU memory usage, a
1x1 convolutional layer is applied to the final feature map layer to reduce
the depth to 256. The output of this layer is flattened and passed through two
fully connected layers of size $(512,256)$, followed by the orientation
estimation output heads. The network is trained on batches of 32 ROI images,
while inference can run on up to 16 objects (176 ROI images) simultaneously.
The network is trained using an Adam optimizer for 20 epochs with an initial
learning rate of 0.0001, which is decayed to 0.00005 at 10 epochs.

Method | Input | Output | 2D AP | AOS | OS
---|---|---|---|---|---
|  |  | Easy | Moderate | Hard | Easy | Moderate | Hard | Easy | Moderate | Hard
3DOP[41] | RGB+Stereo | 2D | 81.78 | 67.47 | 64.70 | 72.94 | 59.80 | 57.03 | 0.8919 | 0.8863 | 0.8815
SubCNN[5] | RGB | 2D | 83.28 | 71.33 | 66.36 | 78.45 | 66.28 | 61.36 | 0.9420 | 0.9291 | 0.9247
Pose-RCNN[34] | RGB+LiDAR | 2D | 77.53 | 63.40 | 57.49 | 73.95 | 59.90 | 54.27 | 0.9538 | 0.9448 | 0.9440
SECOND[3] | LiDAR | 3D | 65.73 | 55.74 | 49.08 | 51.56 | 43.51 | 38.78 | 0.7844 | 0.7806 | 0.7901
AVOD-FPN[2] | RGB+LiDAR | 3D | 67.32 | 58.42 | 57.44 | 53.36 | 44.92 | 43.77 | 0.7926 | 0.7689 | 0.7620
Ours | RGB+LiDAR | 3D | 81.11 | 70.89 | 67.23 | 78.57 | 67.66 | 63.83 | 0.9687 | 0.9544 | 0.9494
Table IV: Pedestrian Pose Estimation. _$AP_{2D}$_ , _$AOS$_ , and Orientation Similarity ($OS$) on the KITTI _test_ set. $OS$ is $AOS/AP_{2D}$. Our pose estimation outperforms all other methods. Method | BEV AP | 3D AP
---|---|---
| Easy | Moderate | Hard | Easy | Moderate | Hard
SECOND[3] | 55.10 | 46.27 | 44.76 | 51.07 | 42.56 | 37.29
F-PointNet[7] | 58.09 | 50.22 | 47.20 | 51.21 | 44.89 | 40.23
AVOD-FPN[2] | 58.75 | 51.05 | 47.54 | 50.80 | 42.81 | 40.88
Ours | 61.46 | 51.73 | 47.69 | 53.98 | 45.01 | 41.72
Table V: 3D Pedestrian Localization and Detection. _$AP_{BEV}$_ and
_$AP_{3D}$_ AP on the KITTI _test_ set. Our method outperforms the previous
state-of-the-art.

##  V Experiments and Results

We validate our method on the challenging KITTI Object Detection benchmark,
where we evaluate our results on the Orientation, 3D, and Bird’s Eye View
(BEV) object detection tasks. We compare with current state-of-the-art methods
across all tasks, and show performance improvement across all categories. We
follow the common training and validation split used by other methods [2, 7,
41, 42], which split the training data at a roughly 1:1 training to validation
ratio. As with [2, 7], a separate training split is used for test submission
for better generalization. For training, ROI images are generated from the
ground truth centroid annotations, and are augmented with contrast,
brightness, and left-right flipping. To simulate 3D localization errors from
the 3D detection network, 2D translation is sampled from a uniform
distribution up to 10 pixels in any direction. In our ablation studies, we
validate our method using renders centered around the top 16 scoring
pedestrian centroids per scene generated by AVOD-FPN [2]. Experiments are
compared using Orientation Score [36] (${OS=AOS/AP_{2D}}$), which is the AOS
normalized by detection AP. For inference on the test data, we also use the 3D
detections from AVOD-FPN. We follow the standard KITTI evaluation difficulty
categories of easy, moderate, and hard to evaluate our estimations.

###  V-A Object Recall Analysis

For our method, the selection of the 3D detector is based on the need for high
recall as a robust 2D detector can provide strong false positive rejection.
Recall analysis for pedestrians based on 3D box IoU can be affected greatly by
the initial pose estimation since the bounding boxes are very small. Since we
only care about the centroid location and we note that errors along the
vertical direction are usually minimal, or less than 8 cm on average, we use
an alternative recall metric that evaluates the centroid distance at a
threshold of 0.3 m along the horizontal and depth directions, $x$ and $z$, to
evaluate whether an object has been detected.

---

---

Figure 5: Qualitative Results. We show a comparison of AVOD-FPN detections
with and without our orientation module. From left to right: AVOD-FPN [2],
Ours, Ground Truth. AVOD-FPN detects all pedestrian instances, but the
orientation estimation is poor for several objects and also includes false
positives in the detections. Our method estimates orientations that more
closely match the ground truth, while also removing false positives.

In Tab. I, we evaluate the recall of three pre-trained open-source 3D
detection methods using the top 16 scoring pedestrian detections per scene on
the validation split. We find that the LiDAR only method, SECOND, misses a
larger portion of instances, with a recall of 80.22% on the moderate
pedestrian instances, while the fusion based AVOD-FPN detector provides the
highest recall of 93.90% on moderate pedestrians. F-PointNet [7] also provides
a similarly high recall of 88.50% on moderate, but the 2D detector used in
their method is not publicly available.

###  V-B Angle Regression

Tab. II investigates the effect of using each type of orientation
representation. The table shows that the angle bin estimation provides better
results at 8 bins compared to 12 or 16, which aligns with the intuition that
splitting the training data for each bin results in worse performance.
However, using only 4 bins requires the network to learn larger regression
values, which does not work as well. The angle vector formulation outperforms
the angle bin method, and the final row of the table shows that using the
multi-task loss formulation with the angle bins as an auxiliary loss provides
the best result.

###  V-C Effect of Rendered Views

Tab. III shows the influence of the rendered views on the CNN inference time
and orientation estimation performance. The basic ROI crop provides better
results than a single virtual camera, likely due to the fact that the novel
views are being rendered from sparser data. However, with only 3 views, the
multi-view formulation outperforms the ROI crop. Performance increases with
the number of rendered views, but the improvement comes at the cost of
increased inference time on the GPU. However, at 15 views performance no
longer improves likely due to the fact the novel views are rendered too close
together resulting in repeated data. The final version of the pipeline uses 11
views, which provides the best OS performance. The rendering of 11 views only
takes an additional 5 ms over a single view, but each additional view takes 34
ms to inference in the network.

###  V-D Comparison with State-of-the-Art

We apply our method on the unseen data of the test samples, and submit the
results to the online server for evaluation. In Tab. IV, we compare our method
with existing methods on the tasks of 2D detection and orientation estimation.
On the easy and moderate difficulties, our proposed method achieves comparable
2D AP performance to other 2D methods that also provide orientation
estimation, while performing better on hard instances due to the high recall
of our selected 3D detector. On the task of orientation estimation, evaluated
as AOS on the test server, our method outperforms all other methods on the
benchmark across all difficulties, and produces the highest OS. We also
compare our results to other 3D detection methods that provide orientation
estimates and show significant improvements in orientation estimation
performance.

In Tab. V, we compare our results with the existing state-of-the-art 3D
detections methods on the tasks of 3D localization $AP_{BEV}$ and 3D detection
$AP_{3D}$. As expected, our performance improves upon the base AVOD-FPN
detector due to false positive rejection from MS-CNN, as well as the more
accurate orientation estimation which increases 3D and BEV IoU due to better
3D box alignment.

###  V-E Timing

We assess the run time of each component in our proposed method. For depth
completion, we use the multi-scale version of [37] which takes 33 ms per LiDAR
scan. The View Synthesis module takes 40 ms to render 11 views for 16 objects
with VTK. On the KITTI dataset, the CNN inference takes 72 ms per frame on
average on a Titan Xp GPU using the AVOD-FPN 3D detections, which equates to
24 ms per object. The AVOD-FPN 3D detector runs at 100 ms per frame. In total,
the average time per frame is 245 ms. However, it is important to note that
our pipeline does not affect the initial object detection time, which depends
on the 3D object detector used. Even so, the full estimation is still 8x
faster than other methods with comparable pose estimation performance; SubCNN
[5] and Pose-RCNN [34] take 2 s, and 3DOP [41] takes 3 s per frame.

##  VI Qualitative Results

Fig. 5 shows a qualitative comparison of pose estimation results on the KITTI
dataset. AVOD-FPN detects all pedestrian instances within the scene. However,
orientations are poorly estimated for several instances and there are also
false positives. Our module successfully recovers more accurate orientation
estimations for the pedestrians within the scene, and suppresses several false
positives.

##  VII Conclusion

This work presents a method to generate virtual viewpoints on a depth
completed point cloud fused with RGB data that addresses the issues of
learning semantic information from sparse LiDAR data, and learning from small
amounts of pose training data. The generation of virtual views in canonical
viewpoints also preserves object appearance. The accurate estimation of
pedestrian pose is especially important for safe autonomous driving as the
behavior of pedestrians can be more easily predicted. Our module is designed
to be incorporated into existing 3D detectors to provide accurate pose
estimation, and sets new state-of-the-art results on the KITTI object
detection benchmark.

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$U_{\textrm{all}}$ with one group of rows deleted:

$\displaystyle U_{\textrm{all}}$
$\displaystyle=\begin{pmatrix}U_{X,\alpha}&0&0&0\\\
U_{Y,\alpha}&0&U_{Y,\eta}&0\end{pmatrix}$

and compute the same formulas with only $\boldsymbol{X}$ and $\boldsymbol{Y}$
as the available data.

##  S10 Product

The predicted clusters, oxygen, temperature, and salinity values, as well as
their uncertainties, are given and described in the Google Drive folder
https://drive.google.com/drive/folders/1_rG5woV45UHsLpX7sWl3C-lWRaNvZysX?usp=sharing.

In Figure S5, we plot the predicted clusters for each of the 12 months.

Figure S5: Cluster membership information for each month of the year.

##  S11 Additional Data Analysis Plots

We show additional data analysis plots of interest in a variety of figures. In
Figure S6, we plot our AIC-based model selection criterion for different
number of principal components. In Figures S7 and S8, we demonstrate the
seasonality dependence of the cluster memberships, showing fairly different
clusters for January versus July. In Figures S9, S10, and S11, we plot
predictions, predictive variances, and means for temperature and salinity,
respectively, which are obtained naturally from our analysis of temperature,
salinity, and oxygen. Similarly, we plot the first principal component of each
estimated in Figure S12: principal components of temperature and salinity,
linear transformation functions, and principal components for $\mathcal{E}$.
In Figure S13, we break down the oxygen variability by the cluster. We plot
the length of uncertainty estimates in Figure S14 in the leave-out
experiments, showing that, as expected, predicted uncertainties for leave-
float-out are higher than for leave-profile-out. In Figure S15, we give an
account of the proportion of variance explained in oxygen by location, and in
Figure S16 we consider the two contributions to the oxygen uncertainty in (10)
and (11). In Figure S16 we can identify areas where the cluster membership is
uncertain. In Figure S17, we plot predicted functions over varying latitude
with fixed longitude. In Figure S18, we plot the estimated log-likelihood (as
calculated in Section 3.4), as a function of the EM iteration, showing that
our estimation strategy estimates parameters that increase the log-likelihood.
In Figure S19, we plot predictions and variances for nitrate.

Figure S6: Akaike Information Criterion values for varying number of principal
components for the covariates and the response. In general, the AIC seems to
be decreasing as a function of both principal components, though it increases
from 12 to 13 response PCs.

Figure S7: Marginal membership information for January (left, summer) and July
(right, winter). This demonstrates seasonal differences in oceanographic
fronts.  Figure S8: Cluster membership of BGC Argo profiles, in a histogram
summarized by the day of the year (seasonality).

Figure S9: Temperature (left) and salinity (right) predictions for January
15th, 2020 at 150 decibar.

Figure S10: Temperature (left) and salinity (right) predicted standard
deviations for January 15th, 2020 at 150 decibar.

Figure S11: Temperature (left) and salinity (right) means for each cluster.

Figure S12: The first estimated functions for each group, for a) temperature
PCs, b) salinity PCs, c) linear transformation functions, d) principal
components of $\mathcal{E}$. Figure S13:  Breakdown of oxygen variability by
cluster based on the model). Measurement error standard deviation refers to
$\sigma_{Y}$, standard deviation where temperature and salinity is known is
$\sqrt{\sigma_{Y}^{2}+B(p)\Theta_{\mathcal{Y},g}\mathbb{V}\textrm{ar}(\eta_{g})\Theta_{\mathcal{Y},g}^{\top}B(p)^{\top}}$,
while the total standard deviation is
$\sqrt{\sigma_{Y}^{2}+B(p)\Lambda_{g}\mathbb{V}\textrm{ar}(\alpha_{g})\Lambda_{g}^{\top}B(p)^{\top}+B(p)\Theta_{\mathcal{Y},g}\mathbb{V}\textrm{ar}(\eta_{g})\Theta_{\mathcal{Y},g}^{\top}B(p)^{\top}}$.
Figure S14: The median predicted standard deviations for leave-out
experiments, by pressure. Figure S15: Proportion of oxygen variance explained
by the covariates and spatial correlation in prediction, January 15, 2020.

Figure S16: (Left) Model-based uncertainty of oxygen based on the scores in
term (10) for January 15th, 2020 at 150 decibars. (Right) Model-uncertainty
based on the clustering in term (11). The combination of this uncertainty
represents the overall uncertainty in the oxygen estimates.

Figure S17: Estimated oxygen (left) and oxygen standard deviation (right) for
longitude -90∘ for varying latitude and pressure. As one moves from south
(dark) to north (light), the predicted oxygen and their uncertainties take
different shapes.  Figure S18: Log-likelihood
$\widehat{f(\boldsymbol{X},\boldsymbol{Y})}$ for each MC iteration when
considering $G=5$, $Q_{1}=Q_{2}=12$.

Figure S19: (Far Left) Nitrate prediction for January 15th, 2020 at 300
decibars, based on 5 clusters. (Left) Nitrate prediction standard deviation
for the same quantities. (Right) Cluster membership map for January 15, 2020.
(Far Right) Mean nitrate functions for each cluster.


11institutetext: Laboratory of Information Technologies, Joint Institute for
Nuclear Research, Dubna, Russia22institutetext: Peoples’ Friendship University
of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian
Federation 33institutetext: A. Razmadze Mathematical Institute, Iv.
Javakhishvili Tbilisi State University, Tbilisi, Georgia44institutetext:
Institute of Quantum Physics and Engineering Technologies, Georgian Technical
University, Tbilisi, Georgia
44email<EMAIL_ADDRESS>
# On overall measure of non-classicality of $N$-level quantum system and its
universality in the large $N$ limit

V. Abgaryan 1122 0000-0001-6713-4095 A. Khvedelidze 113344 0000-0002-5953-0140
I. Rogojin 11 0000-0001-6440-5451

###### Abstract

In this report we are aiming at introducing a global measure of non-
classicality of the state space of $N$-level quantum systems and estimating it
in the limit of large $N$. For this purpose we employ the Wigner function
negativity as a non-classicality criteria. Thus, the specific volume of the
support of negative values of Wigner function is treated as a measure of non-
classicality of an individual state. Assuming that the states of an $N$-level
quantum system are distributed by Hilbert-Schmidt measure (Hilbert-Schmidt
ensemble), we define the global measure as the average non-classicality of the
individual states over the Hilbert-Schmidt ensemble. We present the numerical
estimate of this quantity as a result of random generation of states, and
prove a proposition claiming its exact value in the limit of $N\to\infty$.

###### Keywords:

Wigner function phase space formalism non-classicality Hilbert-Schmidt measure
.

##  1 Introduction

With the rise of quantum information and computation paradigms alongside the
adjacent fields, one time and again encounters the characterization ”non-
classical” when describing the quantum states involved. It must be pointed out
that the notion of non-classicality of quantum states is not well defined.
Under this label, we usually understand the effects predicted by quantum
mechanics which are incomprehensible from the standpoint of classical
intuition. These include everything spanning from quantum entanglement and
other purely quantum correlations to sub-poissonian statistics and squeezing
of electromagnetic fields. Quite often these become resources for new powerful
techniques, as is the case, for example, with quantum entanglement. A question
of quantitative description of the degree of non-classicality and hence of the
resource itself arises here. Obviously, due to the wideness and vagueness of
the question, it would be naive to assume the existence of a universal measure
encompassing the intensities of all the quantum effects. However, it seems
that the central object of quantum mechanics on the phase space, the Wigner
function, somehow encodes the crucial information about the non-classical
features of the state through the property of having negative values. Indeed,
to name just a few examples: it has been shown that quantum circuits where the
initial state together with the quantum operations is representable with
positive Wigner functions can be classically efficiently simulated [2];
s-waves are entangled if and only if corresponding Wigner function has
negative domains [1]; the negativity volume of the Wigner function is an
entanglement indicator for hybrid qubit-bosonic states if certain conditions
are met [3], etc.

Elaborating on the property of Wigner function to have negative values,
several measures of non-classicality have been introduced (see [4] and
references therein). Here, we generalize these well-established ideas from the
level of individual states to the whole state space.

The article is organized as follows. In the next section we introduce the
necessary basics about the Wigner function. In section 3 we define the main
quantities which will be used in the rest of paper. Section 4.1 contains
results on the global measure of non-classicality of density matrices from the
Hilbert-Schmidt ensemble. Finally, in section 4.2 the analysis of behavior of
the introduced measure of non-classicality for large $N$ is given.

##  2 The Wigner function of a density operator of $N$-level system

Wigner function [5] was introduced in an attempt of phase space description of
quantum mechanics. For quantum states represented by a density operator
$\hat{\rho}\in\mathcal{D}(L^{2}(\mathbb{R}^{n}))$ acting on the Hilbert space
$\mathcal{H}=L^{2}(\mathbb{R^{n}})$ the Wigner function of $\hat{\rho}$ is
defined over a phase space $(\mathbb{R}^{2n}\,,w)$ with the standard
symplectic 2-form $w:=\sum_{j}^{n}\,dp_{j}\wedge dq_{j}$ and is given by the
so-called Wigner transform:

$W_{\hat{\rho}}(\boldsymbol{q},\boldsymbol{p})=\left(\frac{1}{2\pi\hbar}\right)^{n}\int_{\mathbb{R}^{n}}\,d{\boldsymbol{\eta}}\,\left\langle\boldsymbol{q}-\frac{\boldsymbol{\eta}}{2}\middle|\,\hat{\rho}\,\middle|{\boldsymbol{q}}+\frac{\boldsymbol{\eta}}{2}\right\rangle
e^{\displaystyle{\frac{i}{\hbar}\boldsymbol{p}\boldsymbol{\eta}}}.$ (1)

According to the Weyl-Wigner formalism one can establish an invertible map
between the self-adjoint semipositive definite operator $\hat{\rho}$ and its
Weyl symbol $(2\pi\hbar)^{n}W_{\hat{\rho}}(\boldsymbol{q},\boldsymbol{p})\,$
in (1)

$\hat{\varrho}\ \rightleftarrows\ W_{\rho}(\boldsymbol{q},\boldsymbol{p})\,.$
(2)

Generalizing the Weyl-Wigner mapping (2) to the case of an arbitrary self-
adjoint operator,

$\hat{A}\rightleftarrows W_{A}(\boldsymbol{q},\boldsymbol{p})\,,$ (3)

the quantum mechanical prediction of the operator, i.e.,
$\mathbb{E}[\hat{A}]=\mbox{tr}[\hat{\rho}\hat{A}]$ is expressible in the form
of conventional ensemble average in classical mechanics defined as the mean of
an operator symbol $A(\boldsymbol{q},\boldsymbol{p})$ over the phase-space
with the distribution $W_{\varrho}:$

$\mathbb{E}[\hat{A}]=\overline{A}\,\qquad\overline{A}:=\int_{\mathbb{R}^{2n}}\,d\boldsymbol{p}\,d\boldsymbol{q}\,W_{A}(\boldsymbol{q},\boldsymbol{p})\,W_{\rho}(\boldsymbol{p},\,\boldsymbol{q})\,.$
(4)

However, the similarity between the quantum and classical expressions is
somewhat illusive. Though, the marginal distributions of momenta on one side
and of coordinates on the other are true probability density functions, due to
the limitation of simultaneous measurements of coordinates and momenta in
quantum mechanics by Heisenberg uncertainty principle, Wigner function is not
free from ”faults”. Namely, it may be shown that there are states for which
Wigner function has negative values. Hence it can’t be considered as a true
probability density function, and is usually called a quasiprobability density
function.

As it was mentioned above the Wigner transform is well adapted to the case of
a quantum mechanical system associated to the Hilbert space
$\mathcal{H}=L^{2}(\mathbb{R^{n}})\,.$ The natural question arises how to deal
with other quantum systems whose Hilbert space $\mathcal{H}$ is different from
$L^{2}(\mathbb{R^{n}})\,?$ In 1957, based on the Weyl-Wigner approach,
R.L.Stratonovich formulated [6] general principles of constructing the mapping
(3), which should be satisfied for any quantum system associated to some
Hilbert space. These principles, later on, received the name of Stratonovich-
Weyl (SW) correspondence. Since in the present note we are interested in
quantification of “quantumness” in systems whose Hilbert space is
$\mathcal{H}=\mathbb{C}^{N}$, below basics of SW correspondence are reproduced
in a form adapted to the case of finite-dimensional quantum systems.

The basic idea of realisation of mapping (3) is to use the kernel operator
$\Delta(\Omega_{N})$ defined over the symplectic manifold $\Omega_{N}$ endowed
with some symplectic 2-form. The mapping is given by formulae

$\displaystyle W_{A}(\Omega_{N})=\mbox{tr}\left(A\Delta(\Omega_{N})\right)\,,$
(5)
$\displaystyle\hat{A}=\int_{\Omega_{N}}\mathrm{d}\Omega_{N}\,\Delta(\Omega_{N})W_{A}(\Omega_{N})\,;$
(6)

Here the kernel $\Delta(\Omega_{N})$ is self-dual, in sense that the same
kernel defines as direct as well an inverse mapping (5), and it is the so-
called Staratonovich-Weyl kernel. According to the Stratonovich-Weyl
principles in order to have a correct phase-space formulation of quantum
theory SW kernel should provide fulfilment of the following compulsory
requirements:

  * •

the kernel must be Hermitian,
$\Delta(\Omega_{N})^{\dagger}=\Delta(\Omega_{N})$ guaranteeing the reality of
symbols;

  * •

the kernel must be the trace class operator, i.e.,
$\int_{\Omega_{N}}d\Omega_{N}\,\Delta(\Omega_{N})=1,$ ensuring completeness of
quantum states as well as classical ones;

  * •

the unitary symmetry of states $\rho^{\prime}=g\,\rho g^{\dagger}\,,\ g\in
SU(N)$ induces the adjoint transformation of SW kernel,
$\Delta(\Omega_{N}^{\prime})=g^{\dagger}\Delta(\Omega_{N})g$ where
$\Omega_{N}^{\prime}$ is an image of point $\Omega_{N}$ under the action of
$g\,.$

It has been shown [7] that for $N$-level quantum system the Stratonovich-Weyl
correspondence clauses admit simple formulation in the form of algebraic
equations on spectrum of SW kernel:

$\mbox{tr}[\Delta(\Omega_{N})]=1\quad{\text{and}}\quad\mbox{tr}[\Delta(\Omega_{N})^{2}]=N\,.$
(7)

These equations leave $N-2$ parametric freedom of choice of the spectrum of SW
kernel. Taking into account this ambiguity we can write the SVD decomposition
of SW kernel

$\Delta(\Omega_{N}|\boldsymbol{\nu})=U(\Omega_{N})\,P(\boldsymbol{\nu})\,U^{\dagger}(\Omega_{N})\,,$
(8)

where $P(\boldsymbol{\nu})$ is a diagonal matrix whose elements are
specifically ordered eigenvalues of the SW kernel,
$\boldsymbol{spec}(\Delta)=\\{\pi_{1}(\boldsymbol{\nu}),\pi_{2}(\boldsymbol{\nu}),\dots,\pi_{N}(\boldsymbol{\nu})\\}\,$.
Eigenvalues $\pi(\boldsymbol{\nu})$ are functions of a real $(N-2)$\- tuple
$\boldsymbol{\nu}=\left(\nu_{1},\,\cdots,\,\nu_{N-2}\right)$ parameterising
the moduli space of solutions to (7). Hereafter, dealing with the Wigner
function of density matrix $\rho$ we will point at this ambiguity by
explicitly writing dependence of SW kernel on the moduli space parameters
$\boldsymbol{\nu}$:

$W_{\rho}^{(\boldsymbol{\nu})}(\Omega_{N})=\mbox{tr}[\rho\,\Delta(\Omega_{N}|\boldsymbol{\nu})]\,.$
(9)

See more on the moduli space of parameters in [8].

Finally, a few remarks on symplectic space $\Omega_{N}\,$ are in order. From
the SVD decomposition (8) it follows that its structure, particularly its
dimension depends on the choice of kernel. Now, assuming that its isotropy
group $H\in U(N)$ is of the form

$H_{\boldsymbol{k}}={U(k_{1})\times U(k_{2})\times U(k_{s+1})}\,,$

then the corresponding phase-space $\Omega_{N}$ can be identified with a
complex flag variety $\mathbb{F}^{N}_{d_{1},d_{2},\dots,d_{s}}={U(N)}/{H}\,,$
where $(d_{1},d_{2},\dots,d_{s})$ are positive integers with sum $N$, such
that $k_{1}=d_{1}$ and $k_{i+1}=d_{i+1}-d_{i}$ with $d_{s+1}=N\,.$ Therefore,
each SW kernel is in one-to one correspondence with a point of moduli space
(with $\boldsymbol{\nu}$\- being the corresponding coordinate) and it is
defined over the phase $\Omega_{N,\boldsymbol{k}}$, member of the finite
family of flag varieties labeled by an integer $(s+1)$-tuple
$\boldsymbol{k}=(k_{1},\dots,k_{s+1})\,.$ The volume form on
$\Omega_{N,\boldsymbol{k}}$ is determined by the bi-invariant normalised Haar
measure $d\mu_{SU(N)}$ on $SU(N)$ group [7]:

$d\Omega_{N,\boldsymbol{k}}=N\,\mbox{Vol}(H_{\boldsymbol{k}})\,\frac{d\mu_{SU(N)}}{d\mu_{H_{\boldsymbol{k}}}}\,,$
(10)

where $d\mu_{H_{\boldsymbol{k}}}$ is the bi-invariant measure over the
isotropy group $H_{\boldsymbol{k}}$.

##  3 Measures of non-classicality of state and overall quantum system

Before introducing the main quantity we are interested in, it is worth to
remind a few auxiliary notions. We begin with the definition of the state
space $\mathfrak{P}_{N}$ of an $N$-level quantum system.

###### Definition 1

The state space $\mathfrak{P}_{N}$ is a $N^{2}-1$ dimensional subset in the
space of $N\times N$ complex matrices $M_{N}(\mathbb{C})$ given by following
conditions:

$\mathfrak{P}_{N}=\\{X\in M_{N}(\mathbb{C})\ |\ X=X^{\dagger}\,,\quad X\geq
0\,,\quad\mathrm{tr}\left(X\right)=1\\}\,.$ (11)

Let, $\Delta(\Omega_{N}\,|\,\boldsymbol{\nu})\,$ be the Stratonovich-Weyl (SW)
kernel with moduli parameter $\boldsymbol{\nu}\,.$ Due to possible symmetries
of state $\rho$ and SW kernel the corresponding WF function has domain of
definition not over the whole $\Omega_{N}$, but is restricted to its certain
subset. Having in mind this fact we introduce two additional definitions.

###### Definition 2

$\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]\in\Omega_{N}$ represents a support of
WF associated to a given state $\rho\in\mathfrak{P}_{N}\,$ and SW kernel.

###### Definition 3

We call $\Omega_{N}^{(-)}[\rho\,|\,{\boldsymbol{\nu}}]$ the negative support
of the Wigner function associated to a given SW kernel and state
$\rho\in\mathfrak{P}_{N}\,,$

$\Omega_{N}^{(-)}[\rho\,|\,{\boldsymbol{\nu}}]=\left\\{\omega\in\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]\,\,|\,\,W_{\rho}^{\boldsymbol{\nu}}(\omega)<0\right\\}.$
(12)

Associating non classicality with the discrepancy between positivity
requirement on classical probability distribution and a property of the Wigner
function to attain negative values, we introduce a measure quantifying
quantumness of state via a relative volume of the subset of phase space where
this discrepancy occurs. The next definitions give formalization of this idea.

###### Definition 4

For a state $\rho$ of an $N$-dimensional quantum system we define its non-
classicality measure (or quantumness)
$\mathfrak{Q}_{N}[\rho\,,\boldsymbol{\nu}]$ as

$\mathfrak{Q}_{N}[\rho\,,{\boldsymbol{\nu}}]=\frac{\mathrm{Vol}(\Omega_{N}^{(-)}[\rho\,|\,{\boldsymbol{\nu}}])}{\mathrm{Vol}\left(\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]\right)}\,.$
(13)

It is necessary to note that in definition (13) it is assumed that the volume
is evaluated using the symplectic volume form which is a projection of the
corresponding volume form on the phase space $\Omega_{N}$ to the subset
$\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]$.

###### Definition 5

We call the following unions,

$\Omega_{N}[\boldsymbol{\nu}]=\bigcup_{\rho\in\mathfrak{P}_{N}}\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]\,,\quad\mathrm{and}\quad\Omega^{(-)}_{N}[\boldsymbol{\nu}]=\bigcup_{\rho\in\mathfrak{P}_{N}}\Omega_{N}^{(-)}[\rho\,|\,{\boldsymbol{\nu}}]$
(14)

as the “symplectic superspace” and the collection of supports of negativity of
the Wigner function will be called correspondingly as “negativity
supersupport”.

This definitions are in given in a sense of the famous Wheeler’s superspace
notion in General Relativity (see [9]). Basically
$\Omega_{N}[\boldsymbol{\nu}]$ is the collection of the supports of the WF of
all possible states of $N$-level quantum system with fixed SW kernel.
Following the same logic as before one can introduce the measure of
quantumness on the “symplectic superspace” as well.

###### Definition 6

For a given SW kernel, the global non-classicality measure
$\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ of $N$-level quantum system is

$\mathfrak{Q}_{N}[\boldsymbol{\nu}]=\frac{\mathrm{Vol}_{\mathrm{g}}(\Omega_{N}^{(-)}[\boldsymbol{\nu}])}{\mathrm{Vol}_{\mathrm{g}}\left(\Omega_{N}[\boldsymbol{\nu}]\right)}\,.$
(15)

In the definition (15) under the volume of the “symplectic superspace” we
assume a result of an average of the symplectic volume of
$\mathrm{Vol}(\Omega_{N}[\rho,\boldsymbol{\nu}])$ over all possible states
distributed in accordance with the measure
$\mathrm{d}\mathfrak{m}_{\mathrm{g}}[\rho]\,,$ associated to a certain
Riemannian metric $\mathrm{g}$ on $\mathfrak{P}_{N}$:

$\mathrm{Vol}_{\mathrm{g}}\left(\Omega_{N}[\boldsymbol{\nu}]\right)=\int_{\mathfrak{P}_{N}}\mathrm{d}\mathfrak{m}_{\mathrm{g}}[\rho]\,\mathrm{Vol}(\Omega_{N}[\rho,\boldsymbol{\nu}])$
(16)

Below we introduce notions allowing us to relate the definition of the global
indicator of system quantumness $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ given in
terms of the “symplectic superspace” with the corresponding notion formulated
in terms of the state space $\mathfrak{P}_{N}\,.$

###### Definition 7

For an arbitrary point $\omega\in\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]$, the
subspace
$\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}\,|\,\omega]\subset\mathfrak{P}_{N}$
of state space is defined as

$\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}\,|\,\omega]=\left\\{\rho\in\mathfrak{P}_{N}\,|\,\omega\in\Omega_{N}[\rho\,|\,\,{\boldsymbol{\nu}}]\,,\,W_{\rho}^{{\boldsymbol{\nu}}}(\omega)<0\,\right\\}\,.$
(17)

###### Proposition 1

The volume of $\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}\,|\,\omega]$ evaluated
with respect to the unitary invariant measure on $\mathfrak{P}_{N}$ is
independent of $\omega\,,$

$\frac{\mathrm{d}}{\mathrm{d}\omega}\,\mathrm{Vol}_{g}\left(\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}\,|\,\omega]\right)=0\,.$
(18)

###### Proof

Indeed, let us write down the volume integral (18) over negativity domain via
the Heaviside step function $\theta[-W_{\rho}^{{\boldsymbol{\nu}}}(\omega)]$
and use SVD decomposition (8) for SW kernel $\Delta(\omega|\boldsymbol{\nu})$
of the Wigner function

$\mathrm{Vol}_{g}\left(\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}\,|\,\omega]\right)=\int_{\mathfrak{P}_{N}}\mathrm{d}\mathfrak{m}_{\mathrm{g}}[\rho]\,\theta\left[-\mathrm{tr}(U(\omega)P^{\boldsymbol{\nu}}U(\omega)^{\dagger}\rho)\right]=\int_{\mathfrak{P}_{N}}\mathrm{d}\mathfrak{m}_{\mathrm{g}}[\rho^{\prime}]\,\theta\left[-\mathrm{tr}(P^{\boldsymbol{\nu}}\rho^{\prime})\right]\,,$
(19)

In the last line of (19) we perform transformation
$\rho^{\prime}=U(\omega)^{\dagger}\rho U(\omega)$. Noting that the state space
$\mathfrak{P}_{N}$ is $SU(N)$ invariant space endowed with the invariant
measure we get convinced that
$\mathrm{Vol}\left(\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}\,|\,\omega]\right)$
is the same for all $\omega\in\Omega_{N}[\rho\,|\,{\boldsymbol{\nu}}]\,.$

Based on this observation, afterwards we choose $\omega$ corresponding to the
diagonal SW kernels, i.e., $\omega=0\,$ and simplify notation of the
negativity subset, $\mathfrak{P}_{N}^{(-)}[\boldsymbol{\nu}]$.

Now we are in position to formulate the Proposition which interrelates two
ways of interpretation of the global measure of quantumness.

###### Proposition 2

The global non-classicality measure $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ can
be expressed as the relative volume of the subset
$\mathfrak{P}_{N}^{(-)}[\boldsymbol{\nu}]$ with respect to total volume of
state space $\mathfrak{P}_{N}$:

$\mathfrak{Q}_{N}[\boldsymbol{\nu}]=\frac{\mathrm{Vol}_{\mathrm{g}}(\mathfrak{P}_{N}^{(-)}[\boldsymbol{\nu}])}{\mathrm{Vol}_{\mathrm{g}}(\mathfrak{P}_{N}[\boldsymbol{\nu}])}\,.$
(20)

where the volume of state space is evaluated with respect to the metric $g$
generating the measure $\mathrm{d}\mathfrak{m}_{\mathrm{g}}[\rho]$ in
definition (15).

###### Proof

At first let us note that contribution to (20) from components of “symplectic
superspace” associated to non-generic states (degenerate and non-maximal rank
density matrices) is zero owing to zero integration measure of this states.

Hence, the integration effectively projects only to the components of
“symplectic superspace” corresponding to the stratum of the generic states,
whose isotropy group is conjugated to subgroup $H=U(1)^{N}\,.$ Therefore the
structure of $\Omega_{N}[\rho,\boldsymbol{\nu}]$ is solely determined by the
SW kernel and does not depend on the $\rho$, which means that,

$\mathrm{Vol}_{g}(\Omega_{N}[\boldsymbol{\nu}])=\mathrm{Vol}_{g}(\mathfrak{P}_{N})\mathrm{Vol}\left(\frac{U(N)}{U(1)^{N}}\right)\,,$
(21)

The same argumentation lead the relation

$\mathrm{Vol}_{g}(\Omega_{N}^{(-)}[\boldsymbol{\nu}]=\mathrm{Vol}_{g}{}(\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}])\mathrm{Vol}\left(\frac{U(N)}{U(1)^{N}}\right)\,,$
(22)

thus proving the Proposition.

##  4 Global measure of quantumness as geometric probability

In this section we outline an interpretation of the above introduced measure
of nonclassicality $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ as a certain geometric
probability. Indeed, according to the representation (20) the global non-
classicality measure $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ can be expressed as
the relative volume of the subset
$\mathfrak{P}^{(-)}_{N}[\boldsymbol{\nu}]\in\mathfrak{P}_{N}\,,$ consisting
out of states $\rho$ whose Wigner functions
$W_{\rho}(\omega\,|\,\boldsymbol{\nu})$ evaluated at some fixed point of phase
space, say $\omega=0$ are negative. Therefore, in consent to the Theory of
Geometric Probability, this relative volume can be identified with the
probability of finding of states with the negative WF among a certain random
ensemble of states:

$\mathcal{P}^{(-)}:=\frac{Number~{}of~{}states~{}with~{}negative~{}WF}{Total~{}number~{}of~{}generated~{}states}$
(23)

Note, that this identification of $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ and
$\mathcal{P}^{(-)}$ is correct if random states are distributed in ensemble
according to the measure $\mathrm{d}\mathfrak{m}_{\mathrm{g}}[\rho]$ in
definition (15).

Following this identification of $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ and
probability $\mathcal{P}^{(-)}$, we will generate random ensemble of the
Hilbert-Schmidt states of $N$-level quantum system and then construct the
corresponding Wigner functions with different kernels and test them on
negativity.

###  4.1 Quantumness of Hilbert-Schmidt states for different SW kernels

There is an elegant method of generation of random density matrices from the
Hilbert-Schmidt ensemble . Its starting point is is the generation of the so-
called Ginibre ensemble, i.e., the set of complex matrices whose elements have
real and imaginary parts distributed as independent normal random variables.
Considering a square $N\times N$ complex random matrix $z$ from the Ginibre
ensemble, one can construct the density matrix from the Hilbert-Schmidt
ensemble as

$\rho_{{}_{\mathrm{HS}}}=\frac{z^{\dagger}z}{\mbox{tr}(z^{\dagger}z)},$ (24)

We have generated such set of density matrices we evaluate
$\mathfrak{Q}_{N}(\boldsymbol{\nu})$ according to (23), for the next families
of SW kernels:

  * •

Kernels whose isotropy group is $H=SU(N-1)$, i.e., $(N-1)$\- fold degenerate
eigenvalues $\frac{1+\sqrt{1+N}}{N}$ and one smallest eigenvalue,
$\frac{1+(1-N)\sqrt{1+N}}{N}$;

  * •

Kernels whose isotropy is $H=SU(N-2)\times SU(2)$, i.e., $(N-2)$\- fold
degenerate eigenvalues $\frac{2-N-\sqrt{2}\sqrt{(N-2)(N-1)(1+N)}}{(N-2)N}$ and
double degenerate eigenvalues $\frac{2-\sqrt{2}\sqrt{(N-2)(N-1)(1+N)}}{2N}$;

  * •

Kernels whose isotropy group is $H=SU(N-3)\times SU(3)$, , i.e., $(N-3)$\-
fold degenerate eigenvalues
$\frac{3-N-\sqrt{3}\sqrt{3-N-3N^{2}+N^{3}}}{(N-3)N}$ and triple of degenerate
eigenvalues, $\frac{3-\sqrt{3}\sqrt{3-N-3N^{2}+N^{3}}}{3N}$;

  * •

Random kernels, which almost always are generic.

Figure 1: Dependence of $\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ on number of
levels $N$ for different types SW kernels described in the text. For systems
with number of levels greater than $N=2^{8}$ the outputs of all but one plots
are suppressed, since the difference of values of
$\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ for different kernels are within the
statistical error.

In Fig. 1 we have plotted $\mathfrak{Q}_{N}(\boldsymbol{\nu})$ depending on
$N$, for different SW kernels. Approximately $\sim 10^{8}$ matrices have been
generated and tested on the Wigner function negativity for each $N$. This plot
shows that with growing number of levels the quantumness of system becomes
independent of SW kernel and tends to a certain value. In the next section we
will give argumentation of this universality of
$\mathfrak{Q}_{N}(\boldsymbol{\nu})$ for the Hilbert-Schmidt ensemble.

###  4.2 The large $N$ limit of global non-classicality

###### Proposition 3

In the limit $N\to\infty$ the global non-classicality measure
$\mathfrak{Q}_{N}[\boldsymbol{\nu}]$ of the Hilbert-Schmidt ensemble does not
depend on the choice of SW kernel. Furthermore, for the infinite level system
the quantumness measure is

$\lim_{N\to\infty}\mathfrak{Q}_{N}(\boldsymbol{\nu})=\mathrm{erfc}\left(\frac{1}{\sqrt{2}}\right)$
(25)

###### Proof

Suppose that SW kernel $\Delta=UP(\boldsymbol{\nu})U^{\dagger}$ is given by
$P(\boldsymbol{\nu})=\mbox{diag}||\pi_{1},\pi_{2},\cdots,\,\pi_{N}||$, where
the eigenvalues are presented in a decreasing order and that only $m$ of them
are non negative. We assume the following notations

$Z_{1}=\sum_{i=1}^{m}\pi_{i},\quad Z_{2}=\sum_{i=m+1}^{N}|\pi_{i}|,\quad
M_{1}=\sum_{i=1}^{m}\pi_{i}^{2},\quad M_{2}=\sum_{i=m+1}^{N}\pi_{i}^{2}\,.$
(26)

In this notations the equations SW kernel obeys may be rewritten as

$\displaystyle Z_{1}-Z_{2}=1$ (27) $\displaystyle M_{1}+M_{2}=N.$ (28)

Remembering that the matrices generated by the procedure
$\rho=\frac{zz^{\dagger}}{tr(zz^{\dagger})}$, with normally distributed real
and imaginary parts of $\varrho$, uniformly cover the set of density matrices
with respect to Hilbert-Schmidt measure, we observe that
$\mbox{Prob}[\mbox{tr}[P\rho]<0]$ is equal to the probability of event

$\sum_{i=1}^{N}P_{i\,i}\,\rho_{i\,i}<0\,,$ (29)

or alternatively, due to the positivity of ${\mbox{tr}}(zz^{\dagger})$ to the
probability of

$\sum_{i=1}^{N}P_{i\,i}\,(zz^{\dagger})_{i\,i}<0\,.$ (30)

Equivalently rewriting this event we get

$\displaystyle\sum_{i=1}^{m}\pi_{i}\,(zz^{\dagger})_{i\,i}<\sum_{i=m+1}^{N}|\pi_{i}|\,(zz^{\dagger})_{i\,i}\,.$
(31)

Further we denote $\xi_{j}^{(i)}=z_{i,j}z_{i,j}^{*}$, so that
$(zz^{\dagger})_{i\,i}=\sum_{j=1}^{N}\xi_{j}^{(i)}$, and
$\beta_{j}^{(i)}=|\pi_{i}|\,\xi_{j}^{(i)}$. In this terms 31 may be rewritten

$\sum_{i=1}^{m}\sum_{j=1}^{N}\beta_{i}^{(j)}<\sum_{i=m+1}^{N}\sum_{j=1}^{N}\beta_{i}^{(j)}\,.$
(32)

Since $Re(z_{i,j})$ and $Im(z_{i,j})$ are distributed by normal distribution
with zero mean and unit variance, then $\xi$’s are distributed with
$\chi^{2}_{2}$, distribution while $\beta_{j}^{(i)}$’s are distributed with
mean $\mathbb{E}(\beta_{j}^{(i)})=2\,|\pi_{i}|$ and variance
$\mbox{var}(\beta_{j}^{(i)})=4\pi_{i}^{2}$. Now according to central limit
theorem

$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{m}\sum_{j=1}^{N}\beta_{j}^{(i)}-\sum_{i=1}^{m}\sum_{j=1}^{N}\mathbb{E}\left(\beta_{j}^{(i)}\right)}{\left(\sum_{i=1}^{m}\sum_{j=1}^{N}\mbox{var}\left(\beta_{j}^{(i)}\right)\right)^{{\frac{1}{2}}}}=$
(34)
$\displaystyle\frac{\sum_{i=1}^{m}\sum_{j=1}^{N}\beta_{j}^{(i)}-2NZ_{1}}{\sqrt{4NM_{1}}}$

as well as

$\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=m+1}^{N}\sum_{j=1}^{N}\beta_{j}^{(i)}-\sum_{i=m+1}^{N}\sum_{j=1}^{N}\mathbb{E}\left(\beta_{j}^{(i)}\right)}{\left(\sum_{i=m+1}^{N}\sum_{j=1}^{N}\mbox{var}\left(\beta_{j}^{(i)}\right)\right)^{\frac{1}{2}}}=$
(36)
$\displaystyle\frac{\sum_{i=m+1}^{m}\sum_{j=1}^{N}\beta_{j}^{(i)}-2NZ_{2}}{\sqrt{4NM_{2}}}\,,$

are distributed normally with zero mean and unit variance. Transforming eq. 32
by subtracting from both sides $2N(Z_{1}+Z_{2})$ and dividing by
$\sqrt{4NM_{1}M_{2}}$ we get

$\frac{x}{\sqrt{M_{2}}}-\frac{NZ_{2}}{\sqrt{NM_{1}M_{2}}}<\frac{y}{\sqrt{M_{1}}}-\frac{NZ_{1}}{\sqrt{NM_{1}M_{2}}}\,,$
(37)

or taking into account the equations for the Statonovich-Weyl kernel we get

$x<y\sqrt{\frac{N-M_{1}}{M_{1}}}-\sqrt{\frac{N}{M_{1}}}.$ (38)

Now, let us denote $t=\sqrt{\frac{N-M_{1}}{M_{1}}}$ so that the initial
probability is equal to the probability of

$x<y\,t-\sqrt{t^{2}+1}.$ (39)

Since $x$ and $y$ are distributed normally the probability of the event
described by equation (39) will be

$\mathcal{P}(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dy\int_{-\infty}^{yt-\sqrt{t^{2}+1}}dx\,e^{-\frac{y^{2}}{2}}e^{-\frac{x^{2}}{2}}.$
(40)

It may be checked that

$\frac{d\mathcal{P}(t)}{dt}=\frac{1}{2\pi}\int_{-\infty}^{\infty}dy\,\frac{e^{-\frac{y^{2}}{2}-\frac{1}{2}\left(\sqrt{1+t^{2}}-ty\right)^{2}}\left(-t+\sqrt{1+t^{2}}y\right)}{\sqrt{t^{2}+1}}=0$
(41)

Hence,
$\mathcal{P}(t)=\mathcal{P}(0)=\frac{1}{2}\,\mbox{erfc}\left(\frac{1}{\sqrt{2}}\right)=0.158655$,
where erfc is the complimentary error function. Which proves the preposition.

##  5 Conclusions

Summarizing, in this work we have introduced a global measure of non-
classicality of the state space of $N$-level quantum systems. By computer
simulations the measure was computed for several SW kernel families depending
on the number of levels. It was proven that for large $N$ the measure of non-
classicality does not depend on the choice of the SW kernel, thus it is an
invariant over the moduli space of SW kernels.

However, it must be noted, that it is unreasonable to suppose that a single
measure might capture all the non-classical aspects of quantum states, let
alone of the whole state space. Hence, there must be different ways of
defining non-classicality measures underlining this or that features of
quantum behaviour (see for comparison [10]).

##  6 Acknowledgments

The publication has been prepared with the support of the “RUDN University
Program 5-100” (recipient V.A.).

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