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11institutetext: University of Maryland, College Park, MD 20740, USA
11email: {jiayewu, saeedhd, geng, zwicker<EMAIL_ADDRESS>
22institutetext: University of North Carolina at Chapel Hill, NC, 27599
22email<EMAIL_ADDRESS>
# GaNI: Global and Near Field Illumination Aware Neural Inverse Rendering

Jiaye Wu 11 Saeed Hadadan 11 Geng Lin 11 Matthias Zwicker 11 David Jacobs 11
Roni Sengupta 22

###### Abstract

In this paper, we present GaNI, a Global and Near-field Illumination-aware
neural inverse rendering technique that can reconstruct geometry, albedo, and
roughness parameters from images of a scene captured with co-located light and
camera. Existing inverse rendering techniques with co-located light-camera
focus on single objects only, without modeling global illumination and near-
field lighting more prominent in scenes with multiple objects. We introduce a
system that solves this problem in two stages; we first reconstruct the
geometry powered by neural volumetric rendering NeuS, followed by inverse
neural radiosity NeRad that uses the previously predicted geometry to estimate
albedo and roughness. However, such a naive combination fails and we propose
multiple technical contributions that enable this two-stage approach. We
observe that NeuS fails to handle near-field illumination and strong specular
reflections from the flashlight in a scene. We propose to implicitly model the
effects of near-field illumination and introduce a surface angle loss function
to handle specular reflections. Similarly, we observe that NeRad assumes
constant illumination throughout the capture and cannot handle moving
flashlights during capture. We propose a light position-aware radiance cache
network and additional smoothness priors on roughness to reconstruct
reflectance. Experimental evaluation on synthetic and real data shows that our
method outperforms the existing co-located light-camera-based inverse
rendering techniques. Our approach produces significantly better reflectance
and slightly better geometry than capture strategies that do not require a
dark room.

|  |  | IRON |  | WildLight |  | Ours
---|---|---|---|---|---|---|---
Input Image |  |  Albedo |  |  Roughness |  |  Albedo |  |  Roughness |  |  Albedo |  |  Roughness |
|  |  Geometry |  |  Re-Render |  |  Geometry |  |  Re-Render |  |  Geometry |  |  Re-Render |
Figure 1:  We perform inverse rendering of a scene from multiple images
captured with co-located light and camera. Our method, GaNI, produces better
geometry, albedo, roughness and re-rendering in unseen views than state-of-
the-art approaches, IRON [37] and WildLight [4], that also uses co-located
light-camera.

##  1 Introduction

Decomposing an indoor scene into geometry, material properties, and lighting,
called inverse rendering [1, 2, 15, 37, 4], is a long-standing problem in
computer vision. Inverse rendering has many applications in VR/AR,
computational photography, and robotics perception, e.g. relighting, material
editing, etc. [44, 12, 37, 4]. However, existing methods are often tailored
for individual objects [37, 4, 38, 14] or human faces [29, 11, 26]. For scenes
consisting of multiple objects, recent methods have shown impressive quality
in reconstructing geometry [31, 5], but fail to reconstruct material
reflectance i.e the Bi-directional Reflectance Distribution Function (BRDF),
[28, 12, 43, 35, 45], since it is is highly under-constrained.

To make the inverse rendering problem well-constrained, researchers proposed
capturing images using a co-located flashlight and camera [4, 14, 1, 18, 24,
37]. Such a setup is readily accessible since one can easily capture images by
turning on the flashlight of one’s mobile camera; light and camera location
can be easily calibrated by using Structure-from-motion which imposes
additional constraints on the inverse rendering problem. Recent approaches,
like IRON [37] and WildLight [4], have shown impressive performance in
material prediction of a single object captured using a co-located light and
camera in a dark room and under ambient lighting respectively. However, these
methods fail to generalize to scenes consisting of multiple objects, as noted
in our experimental evaluations 1. We believe this is because the complexity
of scenes with multiple objects surpasses that of individual objects,
particularly with a co-located light-camera setup, which induces a pronounced
near-field effect, where different regions in the scene receive different
intensities of light. Moreover, interactions between objects and surfaces
result in strong inter-reflections or global illumination.

In this paper, we aim to perform inverse rendering of a scene consisting of
multiple objects captured using a co-located light and camera. We present
GaNI, a system that can reconstruct both high-quality geometry and
reflectance, by implicitly modeling both global and near-field illumination,
hence overcoming the shortcomings of existing approaches [37, 4]. We focus on
examples that consist of multiple objects, e.g. coffee table, shoe rack,
table, etc. We believe this is a first step towards developing a system that
can perform accurate inverse rendering of a 360-degree room-scale scene, which
is significantly more challenging.

Our proposed method develops a neural scene representation by optimizing the
parameters for each scene independently. We propose a two-stage approach where
we first solve for geometry followed by reflectance, instead of jointly
optimizing both which is challenging in the presence of complex illumination
effects. We observe that a naive combination of first reconstructing geometry
with neural volume rendering, NeuS [31], followed by reflectance estimation
with InvNeRad [8], an efficient technique to model multi-bounce global
illumination with known geometry, fails due to specularity, global
illumination, and changing light source. We propose several technical
contributions that enable geometry and reflectance estimation using a two-
stage approach involving NeuS [31] and InvNeRad [8] backbones.

In the first stage, we modify NeuS[31] to handle near-field lighting and
strong specular reflections which are more pronounced due to co-located light
and camera. We implicitly model the near-field effect by conditioning NeuS
with a parametrization that encodes the distance between the scene point from
the camera. This approach enables the neural radiance representation to
effectively learn and model spatially varying incident illumination fields,
even in the presence of global illumination. Consequently, we achieve a robust
reconstruction of scene geometry, presenting a significant improvement over
NeuS[31]. To further improve reconstruction at the regions of strong
specularity from the flashlight, we adaptively set weights for each point on
the scene, reducing contributions from those where the light is nearly
perpendicular or parallel to the surface.

In the second stage, we use surface rendering to solve the rendering equation
and extract principled BRDF [3]. Here we observe that Inverse Neural Radiosity
(InvNeRad) [8], which models 2nd-order light bounces using a learnable
radiance cache, expects the scene illumination to remain static across the
capture, which is not true for images captured with a co-located light and
camera. We solve this by training a light position-aware radiance cache
network that can handle moving light sources. While this leads to accurate
albedo prediction, we note that the predicted roughness is often noisy for
regions with faulty geometry, which we address using a total-variation
smoothness prior over roughness.

We perform a detailed quantitative and qualitative analysis of our approach on
3 real and 4 synthetic scenes. First, we show that while co-located light and
camera in a dark room imposes additional constraints on the capture, it
produces significantly better reflectance than alternate capture
possibilities, i.e. natural illumination capture [36] and co-located
flashlight under ambient lighting capture (WildLight) [4]. Next, we show that
in comparison to the prior co-located light and camera-based inverse rendering
algorithm, IRON [37], developed for a single object, our method produces
significantly better geometry and reflectance.

In summary, our contributions are $\bullet$ We propose GaNI, a neural inverse
rendering algorithm that reconstructs the geometry and reflectance of a scene
consisting of multiple objects captured with a co-located light and camera.
$\bullet$ We build upon the neural volume rendering technique NeuS [31] and
the neural global illumination modeling technique InvNeRad [8] by proposing
key technical contributions such as: (i) implicitly model near-field lighting
(ii) surface angle loss to downweight regions with strong specular reflections
(iii) learning radiance cache with varying light source to model global
illumination (iv) total-variation smoothness prior on roughness. $\bullet$ Our
evaluations show that the proposed approach reconstructs significantly better
geometry and reflectance compared to the state-of-the-art object-centric
inverse rendering technique IRON [37]; and predicts better reflectance and
similar geometry compared to alternate capture setup, such as natural
illumination [36] and WildLight [4].

Our code and data will be available upon acceptance.

Figure 2: Overview of our pipeline. Our system consists of two stages. In the
first stage, we reconstruct geometry with volume rendering under near-field
and global illumination. In the second stage, we extract accurate material
properties with surface rendering while accounting for multi-bounce global
illumination while solving the rendering equation by minimizing the
radiometric prior. Our output is principled BRDF[3] (albedo and roughness)
represented by two separate neural networks and a neural signed distance
field.

##  2 Related Works

####  2.0.1 Role of capture setup.

Researchers have explored different capture configurations with specific
advantages that can prove beneficial depending on the intended applications.
For example, some works [28, 12] attempt to solve inverse rendering from
unconstrained images captured in-the-wild but produce low-quality results. On
the other hand, researchers have used Light Stage [26, 7, 39], a spherical
gantry consisting of well-calibrated lights and cameras, to reconstruct very
high-quality geometry and reflectance but can rarely be used in-the-wild.

Recently researchers have focused on simpler, ‘at-home’ configurations, that
can be easily replicated in the wild but also provide sufficient constraints
to generate better reconstructions than unconstrained approaches [28, 12].
These ‘at-home’ capture setups can be under unconstraint natural illumination
[40, 41, 32, 17, 38, 23, 9], a co-located light and camera [4, 1, 18, 24, 37,
27] or a moving camera and a flashlight [15, 42]. Additionally, WildLight [4]
has explored inverse rendering with co-located light and camera in the
presence of ambient illumination. Recent researches have shown that the choice
of illumination condition can heavily impact the quality of BRDF recovery. A
natural illumination capture setup is most popular due to its simplicity, as
it only requires multi-view images. However, such methods often cannot extract
accurate reflectance, especially in multi-object scenes with complicated
geometry due to fundamental ambiguity between geometry, lighting and
reflectance. On the other hand, a co-located setup often gives high quality
inverse rendering results, but requires a darkroom. WildLight combines the
best of the two worlds, but requires roughly equal ambient illumination energy
and flashlight energy, which is difficult to satisfy in multi-object scenes.
Our work is focused on co-located light and camera setup, which allows
accurate inverse rendering results. We show that the state-of-the-art inverse
rendering technique with co-located light and camera fails to model near-field
and global illumination effects that are more visible in multi-object scenes
than objects. Our approach is the first to enable co-located light-camera-
based inverse rendering in scenes by implicitly modeling near-field and global
illumination effects.

####  2.0.2 Geometry Reconstruction.

Recently, NeuS [31] and VolSDF [33] have gained popularity since they achieve
high-accuracy geometry by adopting a volumetric approach to optimize a neural
implicit surface representation as a Signed Distance Field (SDF). We adopt
NeuS [31] for reconstructing the geometry. However, NeuS cannot handle near-
field illumination and strong specularity. Therefore, we propose to model
near-field illumination and use surface angle loss to handle this scenario.

####  2.0.3 Material Estimation Under Co-located Light and Camera.

Most previous inverse rendering algorithms, such as IRON [37], WildLight [4],
for Co-located Light and Camera do not attempt to handle global illumination.
While global illumination is often negligible for objects, such inter-
reflection is much more prominent for multi-object scenes. Some previous
methods [41, 36] targeted at natural illumination estimate global illumination
by leveraging a radiance field fitted to image observations. However such
approaches are problematic for co-located light and camera setup as the
radiance of only a subset of the scene can be observed.

To properly define radiance for the entire scene requires solving the
rendering equation. Such a solution typically requires solving path integrals,
i.e. integration over the path space and building paths that connect the
camera to the light source. The most naive approach that serves this purpose
is _differentiable path tracing_ , which is known to have an intractable
memory footprint in complex scenes and linear time complexity in the path
length. To alleviate the time and memory requirements, a recent work, InvNeRad
[8], adopted radiance caching techniques and only computes the primary ray
intersection, and queries the cache for the contribution of the rest of the
path, thus accounting for global illumination. In particular, InvNeRad [8]
uses a neural network to represent the radiance cache and solve the rendering
equation in self-supervised fashion. It shows significant improvements in
terms of memory and time compared to previous work for inverse rendering.

However, InvNeRad assumes static illumination, and cannot be directly applied
to co-located light and camera setup. Therefore, we propose a light source
conditioned neural radiance cache to avoid creating hundreds of radiance
caches and allow sharing computed radiance across different light positions.
On real data, we also found small errors in geometry often lead to errors in
roughness. Therefore, we propose a roughness regularization to reduce the
error.

##  3 Method

Capture Setup. Similar to [37], we capture images with a smartphone with a
flashlight turned on in a dark room. Such a setup provides multi-illumination
images, which helps resolve ambiguity between illumination and material
properties that occur in a natural illumination setup, allowing our method to
obtain significantly better material estimation than capture under natural
illumination. While this means our method can only be used at night time, it
produces significantly better BRDF without requiring any sophisticated
hardware equipment or calibration.

Overview. We observe that existing one-stage colocated inverse rendering
systems, such as WildLight [4], which represents radiance as principled BRDF
(Bidirectional Reflectance Distribution Function) parameters under a co-
located flashlight, are often not robust against global illumination effects,
such as interreflection between objects, and fails for real geometry
reconstruction. Therefore, our reconstruction process consists of two
optimization stages, as depicted in Figure 2. The first stage (Section A3)
performs volume rendering by improving NeuS [31] to reconstruct the geometry
with implicit near-field illumination and handle strong specular reflection
with a surface angle loss. The second stage (Section A4) builds on InvNeRAD
[8] and uses the predicted geometry to solve the rendering equation at all
moving light positions and extracts principled BRDF parameters[3] with a
smoothness prior on roughness.

###  3.1 Stage 1: Geometry Recovery with Volume Rendering

We build our volume rendering procedure on top of NeuS[31], and we similarly
represent the scene by two neural networks.

Geometry Network
$\operatorname{S_{\Theta_{S}}}(\mathbf{x})\rightarrow\\{s,\mathbf{f}\\}$,
which maps a 3D position $\mathbf{x}$ to its signed distance to the closest
surface, and a feature vector $\mathbf{f}$. The network represents the
geometry of the scene.

Color Network
$\operatorname{C_{\Theta_{C}}}(\mathbf{x},\mathbf{n},\mathbf{v},\mathbf{f})\rightarrow\mathbf{c}$,
where $\mathbf{x}$ is the 3D position of the point, $\mathbf{n}$ is the normal
at position $\mathbf{x}$, $\mathbf{v}$ is the view direction of the camera and
$\mathbf{f}$ is the feature vector output by the geometry network. The network
represents the radiance of the scene.

The images are rendered with volume rendering as defined by NeuS[31]. Using
their unbiased and occlusion-aware weight $\operatorname{w}(\cdot)$, with
$\mathbf{o}$ as camera position, $\mathbf{v}$ as view direction, t as distance
along camera ray, and $\operatorname{p}(t)$ as 3D positions along camera ray,
we have the following equation.

$\operatorname{C}(\mathbf{o},\mathbf{v})=\int_{0}^{+\infty}w(t)\operatorname{C_{\Theta_{C}}}(\operatorname{p}(t),\mathbf{v})dt$
(1)

For a larger scene, the distance from co-located light to different parts of
the scene will be different. However, as shown previously, the color network
of NeuS does not model such differences. As shown in table 1 and Figure 4.0.1,
previous inverse rendering methods such as IRON [37], which directly use
NeuS[31] often fails in our multi-object dataset.

We model the co-located flashlight as a point light source. To handle the
change of radiance, we additionally parameterize the color network with the
flashlight position $\mathbf{x_{i}}$:

$\operatorname{C_{\Theta_{C}}}(\mathbf{x},\mathbf{n},\mathbf{v},\mathbf{f},\mathbf{x_{i}})\rightarrow\mathbf{c}$
(2)

Since the camera is co-located with the flashlight,
$\mathbf{x_{i}}=\mathbf{x}+t\mathbf{v}$, where $t$ is the distance between the
flashlight and queried point. We can uniquely identify a
$(\mathbf{x_{i}},\mathbf{x},\mathbf{v})$ combination by just
$(t,\mathbf{x},\mathbf{v})$. Additionally, since the flashlight is a point
light source, the irradiance follows the inverse quadratic rule
$\frac{1}{t^{2}}$ as distance increases from the flashlight. As such, we
propose the following parametrization:

$\operatorname{C_{\Theta_{C}}}(\mathbf{x},\mathbf{n},\mathbf{v},\mathbf{f},\frac{1}{t^{2}})\rightarrow\mathbf{c}$
(3)

By encouraging but not forcing the light conditioned radiance network to model
near field light effects from a point light source, we can robustly and
accurately reconstruct geometry even in the presence of global illumination.

Similar to previous methods, we encode view direction with spherical
harmonics[22] and every other input with positional encoding[21].

We supervise the pixels with ground-truth image pixel values via
reconstruction loss. However, as our color network models the illumination
change, we supervise the network in linear RGB space to bypass the non-
linearity introduced by gamma function. Together, we use linearized log loss
introduced by RawNeRF [20] to better learn radiance in dark regions. Denote
$y_{i}$ as the i-th pixel of groundtruth.

$\tilde{L}_{\text{recons}}(\hat{y},y)=\sum_{i}\left(\frac{\operatorname{C}(\mathbf{o},\mathbf{v})-y_{i}}{sg(\operatorname{C}(\mathbf{o},\mathbf{v}))+\epsilon}\right)^{2}$
(4)

Surface Angle Weighting We observe that strong reflections from the flashlight
heavily deteriorate geometry prediction. While several works, such as neural-
pbir[30] or Ref-NeuS, have explored loss weighting schemes to improve geometry
reconstruction for specular objects, we found these methods often tend to
degrade the geometry for concave scenes.

Our weighting scheme is based on the fact that the light position is known,
and co-located with the camera, so we can calculate the angle the light ray
forms with the surface normal. Our intuition is to downweight very small
angles (perpendicular direction) that produce strong specular reflections and
very large angles (grazing direction) that cause the diffuse component to
become minimal, while specular inter-reflection become even stronger due to
fresnel effects.

Since we do not have ground truth geometry, we use the neural sdf during
training as proxy geometry. We render the surface normal similar to color:

$\operatorname{N}(\mathbf{o})=\int_{0}^{+\infty}w(t)\nabla\operatorname{S_{\Theta_{S}}}(\mathbf{x})dt$
(5)

Given angle $\theta=\operatorname{arccos}(N\cdot L)$ between normal $N$ and
light ray orientation $L$, we propose to weight the reconstruction loss
$L_{\text{recons}}$ of each pixels by:

$W_{a,b}(\theta)=\left\\{\begin{array}[]{ll}\operatorname{max}(\operatorname{cos}(a(\theta-\frac{\pi}{4})),0)&x\leq\frac{\pi}{4}\\\
\operatorname{max}(\operatorname{cos}(b(\theta-\frac{\pi}{4})),0)&x>\frac{\pi}{4}\\\
\end{array}\right.$ (6)

Our choice of function is motivated by the following requirements: decrease
slowly in the middle near $\frac{\pi}{4}$, where neither specularity nor
fresnel effects are prominent, and decrease rapidly on the two extremes where
those effects quickly dominate observation; the function needs to be
asymmetrical, as retroreflective specular highlights are only observable from
a very narrow range of angles w.r.t surface normal, but specular inter-
reflection is prominent over a significantly larger range of angles. We use
$a=2$, $b=3$ for all our experiments.

Structure-from-Motion (SfM) pointcloud supervision on real data. Similar to
other neural reconstruction techniques, we require camera poses to be known
ahead of time before reconstruction, which is commonly estimated from
structure-from-motion (SfM). Similar to prior works such as Geo-NeuS[5],
Neilf++[36], instead of discarding the point cloud estimated in SfM, our
system can optionally use the point cloud to supervise geometry by requiring
the neural signed distance field to be zero at point cloud locations
$X_{\text{sfm}}$. We use SfM point cloud supervision on real data.

###  3.2 Stage 2: BRDF Estimation with Surface Rendering

In the second stage, we assume a surface based rendering model, and by
properly modeling global illumination effects, we recover material properties
represented by popular principled BRDF [3], which allows intuitive human
editing for downstream applications. Specifically, we optimize for spatially
varying albedo and roughness. We represent them using separate neural
networks, $\phi_{a}(x)$ and $\phi_{r}(x)$, respectively, where $x$ is any 3D
position.

With a co-located light and camera capture setup, illumination changes in
every image, and we observe a subset of the radiance field. Hence, we cannot
compute global illumination directly from image observations similar to some
previous inverse rendering methods[41]. To recover unobserved radiance, we
leverage an efficient global illumination inverse rendering technique InvNeRad
[8] that solves the rendering equation directly on geometry defined by the
signed distance field.

####  3.2.1 Background: Inverse Neural Radiosity.

InvNeRad [8] is an inverse rendering method that efficiently accounts for
global illumination without the need to explicitly simulate multiple
scattering events through path tracing. It uses a radiance cache $L_{\theta}$
represented as a neural network with parameter set $\theta$, and after
bouncing the ray only once, it queries the cache to collect the contribution
of the rest of the path. More formally, the incident radiance at pixel $k$,
denoted as $I_{k}$, is determined by the measurement equation,

$I_{k}=\int_{\mathcal{A}}\int_{\mathcal{H}^{2}}W_{k}(x,\omega).(E+\mathcal{T}(L_{\theta})(x,\omega))dxd\omega^{\perp}$
(7)

where $W_{k}(x,\omega)$ models the response of a sensor pixel to incident
radiance over its area $\mathcal{A}$ and the hemisphere of directions
$\mathcal{H}^{2}$, $E$ is the emitted radiance distribution $E(x,\omega_{o})$,
$\mathcal{T}$ is the transport operator.

To optimize the scene parameters $\phi$ from the ground truth images, we
denote the image rendered using $\phi$ as $I(\phi)$, then we can define a
reconstruction loss as

$\mathcal{L}_{\mathrm{recons}}(I(\phi))=\|I(\phi)-I^{\mathrm{GT}}\|.$ (8)

To train the radiance cache, InvNeRad [8] introduces a radiometric prior

$\mathcal{L}_{\mathrm{prior}}(\theta)=\|L_{\theta}(x,\omega_{o})-(E(x,\omega_{o})+\mathcal{T}(L_{\theta})(x,\omega_{o}))\|.$
(9)

Following InvNeRad [8], we additionally constrain the neural radiance cache
with groundtruth radiance from images by directly rendering the neural
radiance cache into an image $I^{LHS}_{k}$. Then we define an additional loss
term to match the image to groundtruth.

$\displaystyle\mathcal{L}_{\mathrm{LHS}}(\theta)=\left\|I^{LHS}(\theta)-I^{\mathrm{GT}}\right\|^{2}.$
(10)

####  3.2.2 Co-located Light Inverse Neural Radiosity.

InvNeRad assumes illumination is constant across the input views. With our co-
located light capture setup, the lighting changes with each view. A naive
approach is to have a radiance cache for every flashlight $l$, and formulate
the radiometric prior as:

$\mathcal{L}_{\mathrm{prior}}(\theta)=\|L_{l,\theta}(x,\omega_{o})-(E_{l}(x,\omega_{o})+\mathcal{T}(L_{l,\theta})(x,\omega_{o}))\|$
(11)

where for each captured image with flashlight $l$, $L_{l,\theta}$ requires a
separate network. This is computationally prohibitive as we could easily have
close to 1000 different light positions, thus 1000 different networks.
Instead, we propose to use a single radiance cache network to represent
radiance at all flashlight positions, which will take additional parameters
$x_{l}$ and $d_{l}$, the position of the flashlight and orientation, as input.
We also noticed that under a co-located light scenario, direct illumination
often has strong discontinuities that change with light positions, making it
hard for the network to model. However, it can be computed easily. Therefore,
we additionally condition the neural radiance network on direct illumination
$L_{\text{direct}}$. Denote visibility of surface from flashlight as
$\mathbbm{1}_{\text{vis}}$, flashlight radiant intensity as
$E_{\text{flash}}$, and distance between flashlight and surface as $d$.

$L_{\text{direct}}=\mathbbm{1}_{\text{vis}}\frac{E_{\text{flash}}}{d^{2}}$
(12)

####  3.2.3 Roughness Regularization.

Unlike the original InvNeRad setup, we do not have access to groundtruth
geometry. We found small imperfections will often lead to significant errors
in roughness. Therefore, we additionally incorporate a total variation loss
for roughness, and we set $\lambda=0.02$ in all our experiments.

$L_{rough}=\lambda|\nabla\phi_{r}|$ (13)

Finally, the optimization task is

$\phi^{*},\theta^{*}=\operatorname*{arg\,min}_{\phi,\theta}\mathcal{L}_{\mathrm{recons}}(I(\phi))+\mathcal{L}_{\mathrm{prior}}(\theta)+\mathcal{L}_{\mathrm{rough}}(\phi)+\mathcal{L}_{\mathrm{LHS}}(\theta)$
(14)

####  3.2.4 Implementation Details

Similar to InvNeRad, we implement our second stage on top of Mitsuba 3 [10],
and use hashgrid encoding to encode the 3D position inputs to the reflectance
fields $\phi_{a}$ and $\phi_{r}$. We also use the hashgrid encoding for
flashlight positions, and spherical harmonics for flashlight orientation to
encourage smoothness between contiguous flashlight positions but still allow
high frequency changes.

To further speedup global illumination computation, we extract the neural sdf
at $2048^{3}$ resolution and store as a mesh. We found the resolution
sufficient to represent the details of the scenes we tested.

Table 1: Quantitative comparison of geometry on synthetic data. Our method produces significantly better geometry than IRON [37] and slightly better geometry than WildLight [4] with images captured with co-located lighting and camera in a darkroom.  |  Chamfer ($\times 100$) | Depth Map L1
---|---|---
| Bedroom | Shelf | Counter | Bedroom | Shelf | Counter
IRON | 1.012 | 0.194 | 1.672 | 0.124 | 0.076 | 0.174
WildLight | 0.000 | 0.013 | 0.076 | 0.004 | 0.026 | 0.024
Ours | 0.032 | 0.011 | 0.023 | 0.012 | 0.014 | 0.015

##  4 Evaluation

We perform qualitative evaluation on real data and quantitative and
qualitative evaluation on synthetic data.

####  4.0.1 Synthetic data.

We created three scenes, bedroom, shelf, and kitchen counter based on openly
available Blender and Mitsuba scenes. We render the dataset with a moving co-
located point light and camera as HDR images with the Mitsuba 3 renderer. Each
synthetic scene contains 1000 training images and 50 validation images with
randomly generated camera poses.

Real data. We captured four real scenes using co-located light and camera,
window sill, table, shoe rack, and coffee table that contain 1006, 704, 564,
and 650 images respectively. We reserve 5% of the images of each scene as the
validation split.

| IRON | WildLight | Ours | GT
---|---|---|---|---
Bedroom |  |  |  |
Shelf |  |  |  |
Counter |  |  |  |
Shoe Rack |  |  |  |
Table |  |  |  |
Sill |  |  |  |
Coffee Table |  |  |  |

(a) Qualitative evaluation of geometry. Our method produces significantly
better geometry than IRON [37] on synthetic and real data. Compared to
WildLight [4] we produce slightly better geometry on synthetic data and
significantly better on real data.

(b) Qualitative comparison of re-rendering. We present re-rendering in
validation views for the synthetic scenes (row 1-3) and real scenes (row 4-7).
Our method produces better re-rendering w.r.t. IRON [37] due to our ability to
better model near-field and global illumination. (c) Qualitative comparison of
geometry and re-rendering.

Table 2: Qualitative evaluation of reflectance on synthetic data (Stage 2). We
present MSE ($\times 10$) of re-rendering in unseen views, albedo, and
roughness on the validation set of synthetic data. Best is shown in bold and
second places underlined.

Figure 3:  Qualitative comparison of reflectance estimation. We present
estimated albedo, and roughness in validation views for the synthetic scenes
(row 1-3) and real scenes (row 4-7). Our method produces significantly better
albedo, roughness and re-rendering w.r.t. IRON [37] due to our ability to
better model near-field and global illumination. (Please zoom in for better
visualization)

####  4.0.2 Comparison.

We compare our approach with that of WildLight [4], which does not handle
global illumination, and IRON [37], which additionally does not model near-
field illumination. We test all compared methods in a darkroom. While
WildLight is designed for capture under ambient ilumination, we found such
design has trouble converging correctly for multi-object scenes under ambient
illumination. We will show the results in supplementary. The original
implementation of IRON uses Mitsuba roughplastic BRDF. Since all of our
synthetic scenes use principled BRDF, we modified IRON to use principled BRDF
for a fair comparison. We also compare with Nelif++ [36] which performs
inverse rendering under natural illumination without requiring a dark room.

We reconstruct better geometry and reflectance than IRON [37] and WildLight
[4] (see Fig. 4.0.1, 4.0.1 and Tab. 1, 4.0.1). Compared to other capture
setups that do not require dark room, Nelif++ [36] we improve the geometry
slightly but significantly improve BRDF prediction.

###  4.1 Comparison with Inverse Rendering Techniques that use Co-located
Light & Camera

####  4.1.1 Evaluation of Geometry.

We compare the geometry reconstructed by our method with state of art inverse
rendering methods using a co-located capture setup. We compare to IRON [37]
and WildLight[4] quantitatively on synthetic data in Tab. 1 and qualitatively
on both synthetic and real data in Fig. 4.0.1.

As the mesh models of our synthetic scene contain hidden parts not visible in
any camera views, we cannot naively compute the distance between reconstructed
geometry and ground-truth meshes. Therefore, we rendered a depth map of
ground-truth mesh and reconstructed mesh on validation views, and compute the
L1 distance between the two depth maps as our depth map score. Additionally,
we can convert the depth map of ground-truth mesh and of reconstructed mesh as
two-point clouds. Then we compute the Chamfer distance from the ground-truth
point cloud to the point cloud of the reconstructed mesh.

IRON [37] does not attempt to model near-field illumination and performs
poorly in multi-object scenes. WildLight [4] assumes a point light source
model which does not model global illumination. Therefore, while WildLight
performed well on an open scene without much inter-reflection, such as
‘bedroom’, the performance is worse on scenes with strong concavity such as
‘shelf’, ‘kitchen counter’ and complex geometry, e.g. the sign in ‘shelf’
scene, the teddy bear in the back of the kitchen ‘counter’, or the left-most
shoe on the second row of the ‘shoe rack’. Additionally, WildLight often
catastrophically fails on many real scenes with prominent inter-reflection,
such as the ‘table’, ‘window sill’, or ‘coffee table’.

In comparison, our method is more robust in the presence of global
illumination. However, since we do not strictly enforce a physically based
BRDF model but implicitly regularize the radiance with a neural network, we
perform slightly worse in textureless areas such as the wall.

Figure 4:  Comparison of our method with Neilf++ [36], a state-of-the-art
inverse rendering algorithm for natural illumination. We captured the same
scene under both natural and co-located illuminations with similar number of
images and camera poses. We found that our method significantly outperforms
Neilf++, epsecially in albedo.

####  4.1.2 Quantitative Evaluation of Reflectance on Synthetic data

We compare the reflectance estimation of our method with that of IRON [37]
quantitatively on synthetic data in 4.0.1 and qualitatively on both synthetic
and real data in Figure 4.0.1 on held-out validation set. Along with albedo
and roughness estimation, we also re-render the scene in validation views,
unseen during training, to visualize the overall effect of inverse rendering.
We compute Mean Squared Error (MSE) for re-rendering in validation views and
prediction of albedo and roughness. During re-rendering in validation views,
we clip the range of both ground-truth and prediction to $[0,1]$ to prevent
high-intensity pixels from dominating the error.

For quantitative evaluation, we optimize the point flashlight energy during
training for all methods. Therefore, we rescale the predicted albedo of each
method by the flashlight energy. For qualitative evaluation, we mask out
pixels that do not have the corresponding geometry reconstructed.

Overall, our method significantly outperforms IRON [37] on average for albedo
estimation (MSE 0.066 vs 0.552), roughness estimation (MSE: 0.846 vs 1.792),
and re-rendering in validation views (MSE: 0.014 vs 0.129). Similarly on real
data, we significantly outperform IRON [37]. Re-rendering images of IRON [37]
contain many artifacts due to poorly reconstructed geometry, e.g. Cabinet gets
fused with in wall in Bedroom; two layers of ‘shelf’ are reconstructed as one
solid block. On the other hand, our method has very minimal errors in re-
rendering of novel views, and hence predicted geometry and reflectance.

###  4.2 Comparison with Natural Illumination and Hybrid Illumination

Both our approach and IRON expects dark room for capture, which means it can
be only performed at nighttime. Recent methods like Neilf++[36] can perform
inverse rendering in the presence of natural illumination. that our method
outperforms Neilf++[36], which can capture scenes under natural illumination.

We captured the same scene with similar number of images and poses under both
natural and co-located illumination. We captured 750 images under both
illuminations for the first example shoe rack, and 649 and 686 for co-located
and natural illumination respectively for the second example coffee table.
However, in Fig. 4.0.1 we show prominent errors exist on the right side of the
white shoe under the hat, as incorrect estimation of local illumination
results in very high intensity albedo. Shading also gets baked into albedo of
the hat. Similarly shading are baked into the albedo of the table for the
second example. The difference is more obvious for reflectance estimation, and
hence generating re-rendering views, highlighting the importance of using co-
located capture for high-quality reflectance estimation.

Ours | w/o our weighting
---|---
|

Figure 5:  Ablation of our proposed surface angle weighting. Red box
highlights areas where specular inter-reflection causes artifacts in geometry
without surface angle weighting. Such error become significantly less
pronounced in our full variant.

###  4.3 Ablation studies

We perform ablation of individual component of our system, surface angle
weighting, and our second stage to show their effectiveness.

####  4.3.1 Ablation study of Surface Angle Weighting

In Figure 5, we conduct ablation study of our proposed weighting scheme in the
first stage. The artifacts on the table highlighted in red are present in the
ablated version which is caused by interreflection. Such artifacts become
significantly less pronounced in the full version as we put less weight on
pixels at parallel angle.

####  4.3.2 Ablation study of Ours Second Stage

To separate the effect of geometry and reflectance, we perform an ablation
study where we replace the second stage of our system with IRON’s second
stage, which we denote as Ours-Abl. Ours-Abl uses the same reconstructed
geometry, only differing from ours in material properties extraction. As shown
in Figure 4.0.1, even with good geometry initialization, Ours-Abl often leaves
strong artifacts in the albedo, such as on the side of cabinet of kitchen
counter or the top right corner of the pillow in table. Without proper
handling global illumination, Ours-Abl often heavily bakes global illumination
effects into albedo. Moreover, Ours-Abl often has trouble recovering roughness
in regions where global illumination is prominent, such as the side of
cabinet, or the inside of shelf.

##  5 Conclusion

We introduce a novel system for inverse rendering for scenes using co-located
light and camera. By modeling changing point light source near field
illumination, we are able to obtain robust and accurate reconstruction of
geometry under co-located flashlight. Then we propose light position
conditioned radiance cache which extends InvNeRad to co-located light and
extracts accurate material properties under global illumination. Our
experiments show strong performance against state of the art methods both in
geometry reconstruction and reflectance recovery. We believe our work will be
an important step toward accurate inverse rendering of scenes.

Limitations Similar to previous co-located light and camera approaches, our
approach requires capturing at nightime with lights off. Our current approach
is focused on reconstruction of multi-object scenes rather than room scale
scenes. One potential future direction is to extend our approach to full room
scale scenes.

## References

  * [1] Bi, S., Xu, Z., Sunkavalli, K., Kriegman, D., Ramamoorthi, R.: Deep 3d capture: Geometry and reflectance from sparse multi-view images. In: 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 5959–5968. IEEE Computer Society, Los Alamitos, CA, USA (jun 2020). https://doi.org/10.1109/CVPR42600.2020.00600, https://doi.ieeecomputersociety.org/10.1109/CVPR42600.2020.00600
  * [2] Bi, S., Xu, Z., Sunkavalli, K., Hašan, M., Hold-Geoffroy, Y., Kriegman, D., Ramamoorthi, R.: Deep reflectance volumes: Relightable reconstructions from multi-view photometric images. In: Computer Vision – ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part III. p. 294–311. Springer-Verlag, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-030-58580-8_18, https://doi.org/10.1007/978-3-030-58580-8_18
  * [3] Burley, B., Studios, W.D.A.: Physically-based shading at disney. In: Acm Siggraph. vol. 2012, pp. 1–7. vol. 2012 (2012)
  * [4] Cheng, Z., Li, J., Li, H.: Wildlight: In-the-wild inverse rendering with a flashlight. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4305–4314 (June 2023)
  * [5] Fu, Q., Xu, Q., Ong, Y.S., Tao, W.: Geo-neus: Geometry-consistent neural implicit surfaces learning for multi-view reconstruction. In: Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., Oh, A. (eds.) Advances in Neural Information Processing Systems. vol. 35, pp. 3403–3416. Curran Associates, Inc. (2022), https://proceedings.neurips.cc/paper_files/paper/2022/file/16415eed5a0a121bfce79924db05d3fe-Paper-Conference.pdf
  * [6] Goldman, D.B.: Vignette and exposure calibration and compensation. IEEE Transactions on Pattern Analysis and Machine Intelligence 32(12), 2276–2288 (2010). https://doi.org/10.1109/TPAMI.2010.55
  * [7] Guo, K., Lincoln, P., Davidson, P., Busch, J., Yu, X., Whalen, M., Harvey, G., Orts-Escolano, S., Pandey, R., Dourgarian, J., Tang, D., Tkach, A., Kowdle, A., Cooper, E., Dou, M., Fanello, S., Fyffe, G., Rhemann, C., Taylor, J., Debevec, P., Izadi, S.: The relightables: Volumetric performance capture of humans with realistic relighting. ACM Trans. Graph. 38(6) (nov 2019). https://doi.org/10.1145/3355089.3356571, https://doi.org/10.1145/3355089.3356571
  * [8] Hadadan, S., Lin, G., Novák, J., Rousselle, F., Zwicker, M.: Inverse global illumination using a neural radiometric prior. In: ACM SIGGRAPH 2023 Conference Proceedings. SIGGRAPH ’23, Association for Computing Machinery, New York, NY, USA (2023). https://doi.org/10.1145/3588432.3591553, https://doi.org/10.1145/3588432.3591553
  * [9] Hasselgren, J., Hofmann, N., Munkberg, J.: Shape, Light, and Material Decomposition from Images using Monte Carlo Rendering and Denoising. arXiv:2206.03380 (2022)
  * [10] Jakob, W., Speierer, S., Roussel, N., Nimier-David, M., Vicini, D., Zeltner, T., Nicolet, B., Crespo, M., Leroy, V., Zhang, Z.: Mitsuba 3 renderer (2022), https://mitsuba-renderer.org
  * [11] Kim, H., Zollöfer, M., Tewari, A., Thies, J., Richardt, C., Theobalt, C.: Inversefacenet: Deep single-shot inverse face rendering from a single image. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2018)
  * [12] Li, Z., Shafiei, M., Ramamoorthi, R., Sunkavalli, K., Chandraker, M.: Inverse rendering for complex indoor scenes: Shape, spatially-varying lighting and svbrdf from a single image. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (June 2020)
  * [13] LibRaw LLC: Libraw. https://github.com/LibRaw/LibRaw
  * [14] Lichy, D., Sengupta, S., Jacobs, D.W.: Fast light-weight near-field photometric stereo. In: 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 12602–12611. IEEE Computer Society, Los Alamitos, CA, USA (jun 2022). https://doi.org/10.1109/CVPR52688.2022.01228, https://doi.ieeecomputersociety.org/10.1109/CVPR52688.2022.01228
  * [15] Lichy, D., Wu, J., Sengupta, S., Jacobs, D.W.: Shape and material capture at home. In: 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6119–6129. IEEE Computer Society, Los Alamitos, CA, USA (jun 2021). https://doi.org/10.1109/CVPR46437.2021.00606, https://doi.ieeecomputersociety.org/10.1109/CVPR46437.2021.00606
  * [16] Lindenberger, P., Sarlin, P.E., Larsson, V., Pollefeys, M.: Pixel-Perfect Structure-from-Motion with Featuremetric Refinement. In: ICCV (2021)
  * [17] Liu, Y., Wang, P., Lin, C., Long, X., Wang, J., Liu, L., Komura, T., Wang, W.: Nero: Neural geometry and brdf reconstruction of reflective objects from multiview images (2023)
  * [18] Luan, F., Zhao, S., Bala, K., Dong, Z.: Unified shape and svbrdf recovery using differentiable monte carlo rendering. Computer Graphics Forum 40 (2021), https://api.semanticscholar.org/CorpusID:232404668
  * [19] Maik Riechert: rawpy. https://github.com/letmaik/rawpy (2022)
  * [20] Mildenhall, B., Hedman, P., Martin-Brualla, R., Srinivasan, P.P., Barron, J.T.: NeRF in the dark: High dynamic range view synthesis from noisy raw images. CVPR (2022)
  * [21] Mildenhall, B., Srinivasan, P.P., Tancik, M., Barron, J.T., Ramamoorthi, R., Ng, R.: Nerf: Representing scenes as neural radiance fields for view synthesis. Commun. ACM 65(1), 99–106 (dec 2021). https://doi.org/10.1145/3503250, https://doi.org/10.1145/3503250
  * [22] Müller, T., Evans, A., Schied, C., Keller, A.: Instant neural graphics primitives with a multiresolution hash encoding. ACM Trans. Graph. 41(4) (jul 2022). https://doi.org/10.1145/3528223.3530127, https://doi.org/10.1145/3528223.3530127
  * [23] Munkberg, J., Hasselgren, J., Shen, T., Gao, J., Chen, W., Evans, A., Müller, T., Fidler, S.: Extracting Triangular 3D Models, Materials, and Lighting From Images. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 8280–8290 (June 2022)
  * [24] Nam, G., Lee, J.H., Gutierrez, D., Kim, M.H.: Practical svbrdf acquisition of 3d objects with unstructured flash photography. ACM Trans. Graph. 37(6) (dec 2018). https://doi.org/10.1145/3272127.3275017, https://doi.org/10.1145/3272127.3275017
  * [25] NVIDIA: Nvidia optix ray tracing engine, https://developer.nvidia.com/rtx/ray-tracing/optix
  * [26] Pandey, R., Orts-Escolano, S., LeGendre, C., Haene, C., Bouaziz, S., Rhemann, C., Debevec, P., Fanello, S.: Total relighting: Learning to relight portraits for background replacement. vol. 40 (August 2021). https://doi.org/10.1145/3450626.3459872
  * [27] Schmitt, C., Donne, S., Riegler, G., Koltun, V., Geiger, A.: On joint estimation of pose, geometry and svbrdf from a handheld scanner. In: Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) (2020)
  * [28] Sengupta, S., Gu, J., Kim, K., Liu, G., Jacobs, D.W., Kautz, J.: Neural inverse rendering of an indoor scene from a single image. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 8598–8607 (2019)
  * [29] Sengupta, S., Kanazawa, A., Castillo, C.D., Jacobs, D.W.: Sfsnet: Learning shape, reflectance and illuminance of facesin the wild’. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 6296–6305 (2018)
  * [30] Sun, C., Cai, G., Li, Z., Yan, K., Zhang, C., Marshall, C., Huang, J., Zhao, S., Dong, Z.: Neural-pbir reconstruction of shape, material, and illumination. In: 2023 IEEE/CVF International Conference on Computer Vision (ICCV). pp. 18000–18010. IEEE Computer Society, Los Alamitos, CA, USA (oct 2023). https://doi.org/10.1109/ICCV51070.2023.01654, https://doi.ieeecomputersociety.org/10.1109/ICCV51070.2023.01654
  * [31] Wang, P., Liu, L., Liu, Y., Theobalt, C., Komura, T., Wang, W.: Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. In: Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P., Vaughan, J.W. (eds.) Advances in Neural Information Processing Systems. vol. 34, pp. 27171–27183. Curran Associates, Inc. (2021), https://proceedings.neurips.cc/paper_files/paper/2021/file/e41e164f7485ec4a28741a2d0ea41c74-Paper.pdf
  * [32] Yao, Y., Zhang, J., Liu, J., Qu, Y., Fang, T., McKinnon, D., Tsin, Y., Quan, L.: Neilf: Neural incident light field for physically-based material estimation. In: European Conference on Computer Vision (ECCV) (2022)
  * [33] Yariv, L., Gu, J., Kasten, Y., Lipman, Y.: Volume rendering of neural implicit surfaces. In: Thirty-Fifth Conference on Neural Information Processing Systems (2021)
  * [34] Yariv, L., Kasten, Y., Moran, D., Galun, M., Atzmon, M., Ronen, B., Lipman, Y.: Multiview neural surface reconstruction by disentangling geometry and appearance. Advances in Neural Information Processing Systems 33 (2020)
  * [35] Yu, Y., Smith, W.A.: Inverserendernet: Learning single image inverse rendering. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2019)
  * [36] Zhang, J., Yao, Y., Li, S., Liu, J., Fang, T., McKinnon, D., Tsin, Y., Quan, L.: Neilf++: Inter-reflectable light fields for geometry and material estimation. In: International Conference on Computer Vision (ICCV) (2023)
  * [37] Zhang, K., Luan, F., Li, Z., Snavely, N.: Iron: Inverse rendering by optimizing neural sdfs and materials from photometric images. In: 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 5555–5564 (2022). https://doi.org/10.1109/CVPR52688.2022.00548
  * [38] Zhang, K., Luan, F., Wang, Q., Bala, K., Snavely, N.: Physg: Inverse rendering with spherical gaussians for physics-based material editing and relighting. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 5453–5462 (June 2021)
  * [39] Zhang, X., Fanello, S., Tsai, Y.T., Sun, T., Xue, T., Pandey, R., Orts-Escolano, S., Davidson, P., Rhemann, C., Debevec, P., Barron, J.T., Ramamoorthi, R., Freeman, W.T.: Neural light transport for relighting and view synthesis. ACM Trans. Graph. 40(1) (jan 2021). https://doi.org/10.1145/3446328, https://doi.org/10.1145/3446328
  * [40] Zhang, X., Srinivasan, P.P., Deng, B., Debevec, P., Freeman, W.T., Barron, J.T.: Nerfactor: Neural factorization of shape and reflectance under an unknown illumination. ACM Trans. Graph. 40(6) (dec 2021). https://doi.org/10.1145/3478513.3480496, https://doi.org/10.1145/3478513.3480496
  * [41] Zhang, Y., Sun, J., He, X., Fu, H., Jia, R., Zhou, X.: Modeling indirect illumination for inverse rendering (2022). https://doi.org/10.48550/ARXIV.2204.06837, https://arxiv.org/abs/2204.06837
  * [42] Zhao, D., Lichy, D., Perrin, P.N., Frahm, J.M., Sengupta, S.: Mvpsnet: Fast generalizable multi-view photometric stereo. arXiv preprint arXiv:2305.11167 (2023)
  * [43] Zhu, J., Huo, Y., Ye, Q., Luan, F., Li, J., Xi, D., Wang, L., Tang, R., Hua, W., Bao, H., Wang, R.: I2-sdf: Intrinsic indoor scene reconstruction and editing via raytracing in neural sdfs. In: CVPR (2023)
  * [44] Zhu, J., Luan, F., Huo, Y., Lin, Z., Zhong, Z., Xi, D., Wang, R., Bao, H., Zheng, J., Tang, R.: Learning-based inverse rendering of complex indoor scenes with differentiable monte carlo raytracing. In: SIGGRAPH Asia 2022 Conference Papers. SA ’22, Association for Computing Machinery, New York, NY, USA (2022). https://doi.org/10.1145/3550469.3555407, https://doi.org/10.1145/3550469.3555407
  * [45] Zhu, R., Li, Z., Matai, J., Porikli, F., Chandraker, M.: Irisformer: Dense vision transformers for single-image inverse rendering in indoor scenes. In: 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2812–2821. IEEE Computer Society, Los Alamitos, CA, USA (jun 2022). https://doi.org/10.1109/CVPR52688.2022.00284, https://doi.ieeecomputersociety.org/10.1109/CVPR52688.2022.00284

##  A1 Summary

Here we summarize our supplementary.

  * •

We render all our scenes under natural lighting conditions with our recovered
geometry and material parameters using path tracing renderer mitsuba [10], and
denoise with its built-in NVIDIA OptiX denoiser [25]. The videos feature
ambient lighting with moving point light sources and cameras to demonstrate
the practical applications of our method. They are included as separate mp4
files.

  * •

In Figure 4.0.1, we show qualitative results of WildLight on WildLight style
capture setup (Co-Located Light and Camera under ambient natural
illumination). We found WildLight often fail to converge on our multi-object
scenes.

  * •

In section A2, we provide additional details regarding how we capture and
process our real data.

  * •

In section A3, we provide additional details for our stage 1 architecture and
training.

  * •

In section A4, we provide additional details for our stage 2 training.

  * •

In Figure 4.0.1, Figure 4.0.1, Figure 4.0.1, we provide additional
visualization of qualitative comparisons on synthetic data. In Figure 4.0.1,
Figure 4.0.1, Figure 4.0.1, Figure 4.0.1, we provide additional visualization
of qualitative comparisons on real data. Additional details of the comparison
are discussed in section A5.

Figure 6:  Qualitative results of WildLight on Co-located Light and Camera
under ambient natural illumination.

##  A2 Real Data Capture Setup and Post-processing

We capture all of our real data using an iPhone XS Max and an iPhone 11 Pro.
We capture all the image with ProCamera app on iOS as raw dng file. During
capture, we keep manual and fixed white balance, focus and exposure. For co-
located capture, we also keep flashlight constantly on through the capture
session. We process the raw files with RawPy [19], which is a python interface
around libraw [13]. We perform structure-from-motion reconstruction using
pixel perfect sfm [16]. We apply camera undistortion parameters estimated by
pixel perfect sfm to our captured images. We found that both iPhone XS Max and
iPhone 11 Pro experience significant vignetting. To calibrate for vignetting,
we use a piece of white paper on a sunny day under direct sunlight as the
calibration target. We model the vignetting as 6-th degree even order
polynomial [6], and apply vignetting correction accordingly. We store all
final processed images as 16-bit unsigned png images with linear response
curve without any gamma curve applied, which are used for all following
experiments.

##  A3 Additional Details of Stage 1 Architecture and Training

In section 3.1 of the main paper, we described some changes to original NeuS
architecture, including introducing light position conditioned radiance field.
Here we described additional details of implementation.

With our synthetic dataset, the images can contain “background pixels” where
the primary ray from the camera does not intersect with any scene geometry
during rendering. Since we are not using environment map for our scenes, the
values of these pixels are undefined, and we use a per image binary mask to
ignore these pixels during training. Consequently, we do not put any
supervision in the background region. To prevent the network from producing
arbitrary values for the background, we adopt mask loss commonly used by prior
works [31, 34]. NeuS [31] defines unbiased weights $w_{k,i}$ along the k-th
camera ray based on the underlying signed distance field. Denote
$\hat{O}_{k}=\sum_{i=1}^{n}w_{i}$ as the the sum of weights along the k-th
camera ray, $M_{k}=\\{0,1\\}$ as the value of the binary mask on the k-th
pixel, and $\operatorname{BCE}$ as the binary cross entropy loss, we have the
following equation.

$L_{\text{mask}}=\operatorname{BCE}(M_{k},\hat{O}_{k})$ (15)

Such mask loss is only used for synthetic data, and not used for real data.

Stage 1 is trained with batch size of 512 rays, learning rate of $5\times
10^{-4}$ for 500K steps on synthetic data, and 1M steps on real data.

##  A4 Additional Details of Stage 2 Architecture and Training

Here we provide additional details regarding the second stage of our system.
We use Principled BRDF [3] in stage 2. Similar to InvNeRad[8], we only
optimize for albedo and roughness, while keeping other parameters fixed. We
set all fixed parameters to zeros, except for “specular” (which sometimes
called “specular albedo”), which we set to 0.6 for synthetic scenes and 0.5
for real scenes. For fairness of comparison, we also fix “specular” to the
same values for IRON[37].

We train our stage 2 for $48000$ iterations at learning rate of $5\times
10^{-4}$ with batch size of 16384 for synthetic data and 15625 for real data.

##  A5 Additional Visualization of Qualitative Results

In Figure 4.0.1, Figure 4.0.1, Figure 4.0.1, we provide additional
visualization of qualitative results on synthetic data. In Figure 4.0.1,
Figure 4.0.1, Figure 4.0.1, Figure 4.0.1, we provide additional visualization
of qualitative results on real data.

Figure 7:  Qualitative comparison on synthetic scene bedroom.

Figure 8:  Qualitative comparison on synthetic scene coffee table.

Figure 9:  Qualitative comparison on synthetic scene shelf.

Figure 10:  Qualitative comparison on real scene shoe rack.

Figure 11:  Qualitative comparison on real scene table.

Figure 12:  Qualitative comparison on real scene window sill.

| IRON | Ours-Abl | WildLight | Ours | GT
---|---|---|---|---|---
Render |  |  |  |  |
Albedo |  |  |  |  |
Roughness |  |  |  |  |
Render |  |  |  |  |
Albedo |  |  |  |  |
Roughness |  |  |  |  |

Figure 13:  Qualitative comparison on real scene coffee table.


#  BRiSTOL \- a Band-limited RMS Stationarity Test Tool for Gravitational Wave
Data

F. DI RENZO  (∗) F. FIDECARO  M.RAZZANO and N. SORRENTINO  Università di Pisa,
Dipartimento di Fisica, I-56127 Pisa, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy
${}^{(*)}\mathrm{Corresponding\ author}$<EMAIL_ADDRESS>
###### Abstract

Common techniques in Gravitational Wave data analysis assume, to some extent,
the stationarity and Gaussianity of the detector noise. These assumptions are
not always satisfied because of the presence of short duration transients,
namely glitches, and other slower variations in the statistical properties of
the noise, which might be related to malfunctioning subsystems. We present
here a new technique to test the stationarity hypothesis with minimal
assumptions on the data, exploiting the band-limited root mean square and the
two-samples Kolmogorov-Smirnov test. The outcome is a time-frequency map
showing where the hypothesis is to be rejected. This technique was used as
part of the event validation procedure for assessing the quality of the LIGO
and Virgo data during O3. We also report on the applications of the test to
both simulated and real data, highlighting its sensitivity to various kinds of
non-stationarities.

##  1 Introduction

Gravitational Waves (GWs) are propagating “ripples” in the fabric of space-
time, produced by the coalescence of compact binary star systems (CBCs) or
other violent phenomena in the Universe [1]. Interferometric GW detectors,
like Advanced LIGO [2] and Advanced Virgo [3], measure their transit from the
differential strain induced on the detector arms. Many sources of _noise_ ,
either of environmental or instrumental origin, can produce a similar effect.
This noise is conveniently described as a stochastic (or random) process, and
our ability to extract the information on the astrophysical signal is
determined by how good we know the statistical properties of the noise [4].

Common GW data analysis techniques assume that the noise process can be
modeled as a stationary and Gaussian one [5]. Stationarity implies that its
statistical properties do not vary over time, and can then be estimated from a
single, sufficiently long, realization of the process. Gaussianity ensures
that the second order distribution moments are sufficient to fully
characterize it. In particular, the power spectral density (PSD) of the
Gaussian process is the sole quantity needed to fully represent its
statistical properties.

When these assumptions cease to be valid, the noise characterization gets
complicated, the techniques used within the stationary and Gaussian assumption
are no more optimal [6], and the corresponding estimates of the GW source
properties are altered [7]. Moreover, noise non-stationarities are often the
manifestation of some subsystem malfunction in the detector [8]. For these
reasons, it is important to identify them, both for the assessment of the data
quality in correspondence of a candidate event and from the point of view of
the characterization and improvement of the detector.

##  2 A Stationarity Test Based on the Signal Band-Limited RMS

We present here a new test of stationarity based on the signal Band-Limited
RMS (BLRMS) that we have called BRiSTOL: Band-RMS Stationarity Test toOL [9].

###  2.1 Estimation of the Signal BLRMS

The BLRMS is the average power of a signal in a limited frequency band
$[f^{\min},f^{\max}]$:

$\mathit{BLRMS}\big{(}t;[f^{\min},f^{\max}]\big{)}=\sqrt{\frac{1}{f^{\max}-f^{\min}}\int_{f^{\min}}^{f^{\max}}\hat{S}(f;t)df}$
(1)

where $\hat{S}(f;t)$ is an estimate of the signal PSD referred to the time
$t$.

Using this quantity provides two advantages with respect to standard analyses
based on the PSD variability. Firstly, the highly correlated GW detector noise
has regions of its spectrum characterized by very diverse behaviors. There are
parts of it with characteristic features, like _spectral lines_ , bumps and
other structures where one would need a better frequency resolution, and
regions where the spectrum is _flatter_ , described for example by some
characteristic power law. Averaging over the latter frequency regions has a
variance reduction effect similar to that obtained with the Welch method [10]
in the time domain.

Secondly, many noise sources affect specific frequency bands only. If one
knows a band division of the spectrum able to bound them, these bands can be
used for the BLRMS estimation. Then, once a non-stationary noise component
stands out in a specific band, this can be immediately associated to a known
possible source.

Figure 1: Left: effect of spectral line removal on the Amplitude Spectral
Density (ASD) of Virgo. Right: glitch removal from a Virgo BLRMS time series,
where a glitch at about $200$ seconds from the start is removed by the
thresholding MAD (orange band). These plots have been created with O3a open
data of April 5, 2019 [13].

###  2.2 Modified BLRMS Algorithm

The algorithm to compute the BLRMS from Eq. 1 can be improved with two
additions to make it more effective for the purpose of GW detector noise
studies. Firstly, if a spectral line is contained in the band where we are
estimating the BLRMS, this is going to dominate the result. This is sometimes
undesirable, especially if one is mostly interested in the non-stationarities
of the noise floor. For this reason, we have included in the method for
computing the BLRMS the possibility to automatically identify spectral lines
and remove them with a technique similar to the one described in [11].

Secondly, GW data often presents _glitches_. These are rapid excess of noise,
with typical duration shorter than one second. This feature makes them similar
to _burst_ GW signals. The best algorithms to identify them are the so called
_Event Trigger Generators_ (ETGs) [12]. Then, to focus on slower non-
stationarities, usually not targeted by these ETGs, we may want to exclude the
glitches from our stationarity test. For this reason, we added the possibility
to identify and remove them with an algorithm that computes the moving median
of the BLRMS’ time series and applies a threshold based on their Median
Absolute Deviation (MAD).

Examples of spectral line and glitch removal on Virgo O3a data are shown in
Fig. 1.

###  2.3 BLRMS-based Stationarity Test

We have based our stationarity test on the modified BLRMS computation
described above. Firstly, for every frequency band, we divide the BLRMS time
series in consecutive chunks containing a statistically significant number of
points. Then, we make use of specific _metrics_ to test the stationarity
hypothesis comparing the BLRMS estimates in consecutive chunks.

A particularly balanced metric, useful for having an algorithm sensitive to a
large variety of non-stationarities, is the one provided by the _two-samples
Kolmogorov-Smirnov test_ [14], based on the maximum absolute difference
between the BLRMS empirical c.d.f.s in neighboring chunks, respectively with
$n$ and $n^{\prime}$ data points, $\\{x_{i}\\}_{i=1,...,n^{(\prime)}}$:

$\hat{F}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}I(x_{i}\leq x)$ (2)

where $I(\ldots)$ is the _indicator_ or _characteristic function_ , equals to
$1$ if its argument is true, zero otherwise. The previous quantity counts the
proportion of the sample points below level $x$. The test statistic is
therefore:

$\mathit{KS}_{nn^{\prime}}=\sqrt{\frac{nn^{\prime}}{n+n^{\prime}}}\sup_{x}\left|\hat{F}_{n}(x)-\hat{F}_{n^{\prime}}(x)\right|.$
(3)

With this normalization, the test statistic asymptotically approaches a
_Kolmogorov distribution_ , which is independent on the distribution of the
data (_Kolmogorov Theorem_). Also, $p$-values, the probabilities of obtaining
more extreme values from the same distribution, are available for this
statistic and can be reported in a time-frequency map where to verify the
stationarity hypothesis. An example of this representation for the test
outcome is shown in Fig. 2.

Figure 2: Time-frequency map of the $p$-values for the null hypothesis of
stationarity obtained with BRiSTOL. Red regions are times and frequency bands
where this hypothesis should be rejected at a significance level $\alpha=1\%$.
Left: application to real Virgo O3a open data of April 5, 2019. Right:
simulated colored Gaussian Virgo data, randomly generated from the ASD in Fig.
1 with the addition of some noise non-stationarities.

##  3 Applications and Conclusions

In Fig. 2 we reported two examples of application of BRiSTOL. The time-
frequency maps show the $p$-values for the stationarity hypothesis in various
time chunks and frequency bands. Red regions are where stationarity should be
rejected at a significance level $\alpha=1\%$, that is, the $p$-value there is
smaller than this threshold. On the left, we reported the application to real
O3a Virgo data, while on the right we generated a (stationary) colored
Gaussian noise from the ASD in Fig. 1, to which we added some noise features:
a sinusoid at frequency $90$ Hz from time $500$ to $1000$ sec,111Notice that a
sinusoidal signal is stationary. The only features highlighted by the colormap
in Fig. 2 are indeed its onset and offset, respectively at time $500$ and
$1000$ seconds from the start. a moving line with exponentially varying
frequencies between $10$ and $1000$ Hz from $1500$ to $2000$ sec, a “ramp” of
white noise with increasing amplitude from $2500$ and $3000$ sec, and lastly a
$5$ seconds “burst” (covering all frequency bands) at time $3250$ sec. All
these features have amplitude (the maximum one for the ramp and the burst)
equals to $10^{-22}$ in strain units.

BRiSTOL has been used for the validation of candidate events during O3 and for
detector characterization studies. Thanks to the minimal assumptions on the
data, namely that the possible non-stationarities can show up in the second
order moments,222A subsequent Gaussianity test can also reveal if this is
sufficient to exclude non-stationarities at higher order. this test can be
used as a first step for assessing data quality. If non-stationary regions are
identified, further analyses are triggered to investigate the possible origin
of them or, in the case of candidate event validation, to check whether they
can be detrimental for the estimations on the GW source properties.

## References

## References

  * [1] B.S. Sathyaprakash and B.F. Schutz, Living Rev. Rel. 12, 2 (2009).
  * [2] J. Aasi et al. (LIGO Scientific Collaboration), Class. Quant. Grav. 32, 7-074001 (2015).
  * [3] F. Acernese et al. (Virgo Collaboration), Class. Quant. Grav. 32, 2-024001 (2015).
  * [4] B.P. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration), Class. Quant. Grav. 37, 5-055002 (2020).
  * [5] P. Jaranowski and A. Krolak, Living Rev. Rel. 8 3 (2005).
  * [6] B. Zackay et al. , arXiv:1908.05644.
  * [7] O. Edy, A. Lundgren and L.K. Nuttall, arXiv:2101.07743.
  * [8] J. Aasi et al. (Virgo Collaboration), Class. Quant. Grav. 29, 155002 (2012).
  * [9] F. Di Renzo, PhD Thesis, University of Pisa, etd-06192020-081853.
  * [10] P. Welch, IEEE Transactions on Audio and Electroacoustics 15 2, 70-73 (1967).
  * [11] F. Acernese et al. (Virgo Collaboration), Class. Quant. Grav. 22, S1189-S1196 (2005).
  * [12] F. Robinet et al. , SoftwareX 12 100620 (2020).
  * [13] R. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), _SoftwareX_ 13 100658 (2021).
  * [14] A. Kolmogorov, The Annals of Mathematical Statistics 12 4, 461-463 (1941).


# Entanglement polygon inequalities for pure states in qudit systems

Xian Shi<EMAIL_ADDRESS>College of Information Science and Technology,
Beijing University of Chemical Technology, Beijing 100029, China

###### Abstract

Entanglement is one of the important resources in quantum tasks. Recently,
Yang $et$ $al.$ [arXiv:2205.08801] proposed an entanglement polygon
inequalities (EPI) in terms of some entanglement measures for $n$-qudit pure
states. Here we continue to consider the entanglement polygon inequalities.
Specifially, we show that the EPI is valid for $n$-qudit pure states in terms
of geometrical entanglement measure (GEM), then we study the residual
entanglement in terms of GEM for pure states in three-qubit systems. At last,
we present counterexamples showing that the EPI is invalid for higher
dimensional systems in terms of negativity, we also present a class of states
beyond qubits satisfy the EPI in terms of negativity.

###### pacs:

03.65.Ud, 03.67.Mn

##  I introduction

Quantum entanglement is an essential feature of quantum mechanics. It plays an
important role in quantum informa- tion and quantum computation theory
horodecki2009quantum , such as superdense coding bennett1992communication ,
teleportation bennett1993teleporting , and the speedup of quantum algorithms
shimoni2005entangled .
One of the important properties of multipartite entanglement is that
entanglement cannot be freely shared. For a tripartite entangled state
$\rho_{ABC},$ there exists six different bipartite entanglements
$E_{A|B},E_{A|C},E_{B|C},$$E_{A|BC},E_{B|AC}$ and $E_{C|AB}$. Here $E$ is an
entanglement measure for bipartite systems, $E_{A|BC}$ can be seen as a $one$
$to$ $group$ entanglement, and $E_{A|B}$ can be seen as a $one$ $to$ $one$
entanglement. In 2000, Coffman, Kundu and Wootters showed a famous inequality
for a three qubit system
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$
coffman2000distributed ,

$\displaystyle E_{A|BC}\geq E_{A|B}+E_{A|C},$ (1)

This inequality means that the entanglement between a singled out qubit to a
group of qubits is bounded by the sum of the entanglement between the singled
out qubit to each qubit in the group. Later, this inequality is generalized to
multiqubit systems in terms of the squashed entanglent measure
christandl2004squashed , the squared entanglement of formation bai2014general
, the squared Tsallis-$q$ entanglement measure luo2016general , and the
squared Renyi entanglement measure song2016general . As (1) is not valid for
multipartite higher dimensional systems in terms of generic bipartite
entanglement measures, other types of monogamy of entanglement were presented
de2014monogamy ; gour2018monogamy ; shi2021multilinear .

Recently, entanglement polygon inequality (EPI) that is on the $one$ $to$
$group$ entanglement was proposed for multipartite entangled pure states
$\otimes_{i}\mathcal{H}_{i}$ qian2018entanglement ; yang2022entanglement ,

$\displaystyle E_{j|\overline{j}}\leq\sum_{k\neq j}E_{k|\overline{k}}.$ (2)

This inequality was proved valid for pure states in multi-qubit systems in
terms of bipartite entanglement measures qian2018entanglement  and in
arbitrary dimensional systems in terms of the $q$-concurrence
yang2021parametrized  when $q\geq 2$ and the unified-$(q,s)$ entangled measure
when $q\geq 1,s\geq 0$ yang2022entanglement . However, whether the EPI in
valid for pure states in higher dimensional systems in terms of concurrence or
negativity is unknown.

This paper is organized as follows. In Sec. II, we present the preliminary
knowledge needed here, In Sec. III, we present our main results, first we
present that the EPI is valid in terms of GEM for pure states in arbitrary
dimensional systems, we also consider the entanglement indicators based the
EPI. At last, We present a class of pure states that doesnot satisfy the EPI
in terms of negativity. Moreover, we present a class of pure states in higher
dimensional systems satisfy the EPI in terms of negativity, we also present
that the EPI is valid in terms of concurrence for pure states in arbitrary
dimensional systems. In Sec. IV, we end with a summary.

##  II PRELIMINARIES

An $n$-partite pure state $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ is full
product if it can be written as

$\displaystyle|\psi\rangle_{A_{1}A_{2}\cdots
A_{n}}=|\phi_{1}\rangle_{A_{1}}|\phi_{2}\rangle_{A_{2}}\cdots|\phi_{n}\rangle_{A_{n}},$
(3)

otherwise, it is entangled. A multipartite pure state is called genuinely
entangled if

$\displaystyle|\psi\rangle_{A_{1}A_{2}\cdots
A_{n}}\neq|\phi\rangle_{S}|\zeta\rangle_{\overline{S}},$ (4)

for any partite $S|\overline{S}$, here $S$ is a subset of
$\boldsymbol{A}=\\{A_{1},A_{2},\cdots,A_{n}\\}$, and
$\overline{S}=\boldsymbol{A}-S$.

Next we recall some entanglement measures for a bipartite state $\rho_{AB}.$ A
bipartite pure state $|\psi\rangle_{AB}$ can be always written as

$\displaystyle|\psi\rangle_{AB}=\sum_{i=0}^{d-1}\sqrt{\lambda_{i}}|i\rangle_{A}|i\rangle_{B},$

where $\lambda_{i}\geq\lambda_{i+1}\geq 0,$ $i=0,1,\cdots,d-2,$ and
$\sum_{i}\lambda_{i}=1$. The geometric entanglement measure for a pure state
$|\phi\rangle_{AB}=\sum_{i}\sqrt{\lambda_{i}}|ii\rangle$ is defined as

$\displaystyle G(|\phi\rangle):=$ $\displaystyle 1-\mathop{\rm
max}_{|\psi\rangle=|a_{1}\rangle|a_{2}\rangle}\langle\psi|\rho|\psi\rangle$
(5)

The geometric entanglement measure for a mixed state $\rho_{AB}$ is generally
defined as wei2003geometric ,

$\displaystyle G(\rho_{AB})=\mathop{\rm min}\sum_{i}p_{i}G(|\phi_{i}\rangle),$
(6)

where the minimum takes over all the decompositions of $\rho_{AB}$.

Negativity is an important entanglement measure for bipartite systems. It is
defined as follows for a pure state $|\phi\rangle_{AB}$ vidal2002computable ,

$\displaystyle
N(|\phi\rangle_{AB})=\frac{\lVert|\psi\rangle\langle\psi|^{T_{A}}\rVert_{1}-1}{2},$
(7)

here $\langle ij|\rho_{AB}^{T_{A}}|kl\rangle=\langle kj|\rho_{AB}|il\rangle$.

Concurrence is the other entanglement measure for bipartite mixed states
$\rho_{AB}$. The concurrence of a pure state $|\psi\rangle_{AB}$ is defined as

$\displaystyle C(|\psi\rangle_{AB})=\sqrt{2(1-\mathop{\rm Tr}\rho_{A}^{2})}.$
(8)

Here $\rho_{A}=\mathop{\rm Tr}_{B}|\psi\rangle_{AB}\langle\psi|$. For a mixed
state $\rho_{AB},$ it is defined as

$\displaystyle C(\rho_{AB})=\mathop{\rm
min}_{\\{p_{i},|\phi_{i}\rangle_{AB}\\}}\sum_{i}p_{i}C(|\phi_{i}\rangle_{AB}),$
(9)

where the minimum takes over all the decompositions of
$\rho_{AB}=\sum_{i}p_{i}|\phi_{i}\rangle_{AB}\langle\phi_{i}|$ with $p_{i}\geq
0$ and $\sum p_{i}=1.$
For a two-qubit mixed state $\rho_{AB},$ Wootters derived an analytical
formula wootters1998entanglement :

$\displaystyle C_{AB}=$ $\displaystyle\mathop{\rm
max}\\{\sqrt{\mu_{1}}-\sqrt{\mu_{2}}-\sqrt{\mu_{3}}-\sqrt{\mu_{4}},0\\},$ (10)

where $\mu_{1},\mu_{2},\mu_{3},$ and $\mu_{4},$ are the eigenvalues of the
matrix
$\rho_{AB}(\sigma_{y}\otimes\sigma_{y})\rho_{AB}^{*}(\sigma_{y}\otimes\sigma_{y})$
with nonincreasing order.

Recently, a parametrized entanglement measure, the $q$-concurrence, was
proposed in yang2021parametrized . The $q$-concurrence for a pure state
$|\psi\rangle_{AB}$ is defined as

$\displaystyle C_{q}(|\psi\rangle_{AB})=1-Tr\rho_{A}^{q}.$ (11)

In yang2022entanglement , the authors showed that the $q$-concurrence
satisfies the EPI for pure states when $q\geq 2$.

###### Lemma 1

yang2022entanglement  For any $n$-qudit pure entangled state $|\psi\rangle$ in
$\otimes_{i}\mathcal{H}_{i}.$ When $q\geq 2$, the following inequality holds,

$\displaystyle C_{q}^{j|\overline{j}}\leq\sum_{k\neq j,\forall
k}C_{q}^{k|\overline{k}}(|\psi\rangle),$ (12)

##  III Main Results

In this section, we first recall the EPI proposed in qian2018entanglement ;
yang2022entanglement , then we present that the geometric entanglement measure
(GEM) also satisfies the EPI for three qudit pure states, and we also present
an application based on the EPI in terms of GEM. At last, we present a class
of counterexamples which shows that negativity is invalid for three qudit pure
states, we also present a class of examples that satisfies the EPI in terms of
negativity.

###  III.1 EPI IN TERMS OF GEOMETRIC ENTANGLEMENT MEASURE

Assume $|\psi\rangle_{AB}=\sum_{i=0}^{d-1}\sqrt{\lambda_{i}}|ii\rangle$ is a
pure state with $\lambda_{i}\geq\lambda_{i+1}\geq 0,$ $i=0,1,\cdots,d-2,$ and
$\sum_{i}\lambda_{i}=1$. By the formula (5), we have

$\displaystyle G(|\psi\rangle)=1-\lambda_{0},$ (13)

Next we recall the Schatten $p$-norm of a bounded operator $M$ on the Hilber
space $\mathcal{H}$ with finite dimensions,

$\displaystyle\lVert M\rVert_{p}:=$ $\displaystyle[\mathop{\rm
Tr}(|M|^{p})]^{\frac{1}{p}},$ $\displaystyle=$
$\displaystyle(\sum_{i=0}^{d-1}s_{i}^{p}(M))^{\frac{1}{p}}.$ (14)

here $s_{i}(M)$ are the sigular values of $M$ with decreasing order, $i.$$e.$
the eigenvalues of $\sqrt{M^{\dagger}M}$. And when $p\rightarrow\infty,$
$\lVert M\rVert_{p}\rightarrow s_{0}(M).$ Next we recall the following lemma.

###### Lemma 2

audenaert2007subadditivity  For any bipartite state $\rho$ on a Hilbert space
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, the inequality

$\displaystyle 1+\lVert\rho\rVert_{q}\geq\lVert Tr_{1}\rho\rVert_{q}+\lVert
Tr_{2}\rho\rVert_{1}$ (15)

holds for $q>1$.

Clearly, the inequality $(\ref{sq})$ can be written as

$\displaystyle\sum_{i=1}^{2}1-\lVert Tr_{i}\rho\rVert_{q}\geq
1-\lVert\rho\rVert_{q},\hskip 14.22636pt\textit{when $q>1.$}$ (16)

Then we have the following result.

###### Theorem 3

Assume $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ is a pure state in the
following system $\otimes_{i=1}^{n}\mathcal{H}_{i}$, then we have the
following inequality,

$\displaystyle G(|\psi\rangle_{P_{i}|\overline{P_{i}}})\leq\sum_{j\neq
i}G(|\psi\rangle_{P_{j}|\overline{P_{j}}}),$ (17)

here $\\{P_{1},P_{2},\cdots,P_{k}\\}$ is a partition of the set
$\\{A_{1},A_{2},\cdots,A_{n}\\}$ with $k\leq n$.

###### Proof.

As

$\displaystyle
G(|\psi\rangle_{AB})=1-s_{0}(\rho_{A})=\lim_{p\rightarrow\infty}[1-\lVert\rho_{A}\rVert_{p}],$
(18)

then we have

$\displaystyle G(|\psi\rangle_{P_{i}|\overline{P_{i}}})=$
$\displaystyle\lim_{p\rightarrow\infty}[1-\lVert\rho_{P_{i}}\rVert_{p}]$
$\displaystyle=$
$\displaystyle\lim_{p\rightarrow\infty}[1-\lVert\rho_{\cup_{j\neq
i}P_{j}}\rVert_{p}]$ $\displaystyle\leq$
$\displaystyle\lim_{p\rightarrow\infty}\sum_{j\neq
i}[1-\lVert\rho_{P_{j}}\rVert_{q}]$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}G(|\psi\rangle_{P_{j}|\overline{P_{j}}}).$ (19)

Here the first equality is due to $(\ref{ge}),$ the first inequality is due to
the inequality $(\ref{sq1})$, and the last inequality is due to the equality
$(\ref{ge}).$ $\sqcap$$\sqcup$

Next we propose a generalized EPI for $n$-qudit pure states in terms of the
GEM.

###### Lemma 4

When $a,b,c\in(0,1]$, and $a+b\geq c$, then

$\displaystyle a^{\alpha}+b^{\alpha}\geq c^{\alpha},$

when $\alpha\in(0,1]$.

###### Proof.

Here we can always assume $a\geq b$, and when $0<\alpha\leq 1,$
$(\frac{a}{b}+1)^{\alpha}-(\frac{a}{b})^{\alpha}\leq 1,$ then we have

$\displaystyle a^{\alpha}+b^{\alpha}\geq(a+b)^{\alpha}\geq c^{\alpha}.$

$\sqcap$$\sqcup$

###### Corollary 5

Assume $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ is a pure state in the
following system $\otimes_{i=1}^{n}\mathcal{H}_{i}$, then we have the
following inequality,

$\displaystyle
G(|\psi\rangle_{P_{i}|\overline{P_{i}}})^{\alpha}\leq\sum_{j\neq
i}G(|\psi\rangle_{P_{j}|\overline{P_{j}}})^{\alpha},$ (20)

here $\\{P_{1},P_{2},\cdots,P_{k}\\}$ is a partition of the set
$\\{A_{1},A_{2},\cdots,A_{n}\\}$ with $k\leq n$, and $\alpha\in(0,1]$.

The above corollary is due to the Theorem 3 and Lemma 4. Then we present an
example for the above corollary.

###### Example 6

$\displaystyle|\psi\rangle_{ABC}=\frac{3}{5}|102\rangle+\frac{2\sqrt{2}}{5}|200\rangle+\frac{2}{5}|010\rangle+\frac{\sqrt{2}}{5}|020\rangle+\frac{\sqrt{2}}{5}|001\rangle,$

through computation, we have

$\displaystyle
G(|\psi\rangle_{A|BC})=\frac{9}{25},G(|\psi\rangle_{B|AC})=\frac{19}{25},G(|\psi\rangle_{C|AB})=\frac{14}{25},$
(21)

then the inequality $(\ref{gm2})$ can be written as

$\displaystyle g=$ $\displaystyle
G(|\psi\rangle_{A|BC})^{\alpha}+G(|\psi\rangle_{C|BA})^{\alpha}-G(|\psi\rangle_{B|AC})^{\alpha}$
$\displaystyle=$
$\displaystyle(\frac{9}{25})^{\alpha}+(\frac{14}{25})^{\alpha}-(\frac{19}{25})^{\alpha},$
(22)

we plot $(\ref{e1})$ in Fig. 1, from the figure, we see that the inequality
$(\ref{e1})$ is bigger than 0 when $\alpha\in(0,1).$

Figure 1: In this figure, we present the inequality (22).

###  III.2 APPLICATIONS OF THE EPI IN TERMS OF GEM

Assume $|\psi\rangle_{A_{1}A_{2}\cdots A_{n}}$ is a pure state in
$\otimes_{i}\mathcal{H}_{i}$, then we consider an entanglement indicator
defined as zhu2014entanglement ; yang2022entanglement ,

$\displaystyle\delta_{\alpha}^{G}(|\psi\rangle)=$ $\displaystyle\mathop{\rm
min}_{i}\tau_{\alpha}^{i}(|\psi\rangle),$
$\displaystyle\tau_{\alpha}^{i}(|\psi\rangle)=$ $\displaystyle\sum_{j\neq
i}G^{\alpha}(|\psi\rangle_{A_{j}|\overline{A_{j}}})-G^{\alpha}(|\psi\rangle_{A_{i}|\overline{A_{i}}}).$
(23)

By the Corollary 5, we have that $\delta_{\alpha}^{G}\geq 0,$
$\alpha\in(0,1)$. When considering $|\psi\rangle$ is a three-qubit pure state,
we have the following theorem.

###### Theorem 7

Assume $|\psi\rangle_{A_{1}A_{2}A_{3}}$ is a three-qubit pure state, then
$\delta_{\alpha}^{G}=0,$ $\forall\alpha\in(0,1)$ if and only if
$|\psi\rangle_{A_{1}A_{2}A_{3}}$ is a biseparable state.

###### Proof.

$\Rightarrow:$ When $\alpha\in(0,0.5)$, as $2\alpha\in(0,1),$ we have

$\displaystyle
G^{2\alpha}(|\psi\rangle_{i|\overline{i}})-G^{2\alpha}(|\psi\rangle_{j|\overline{j}})-G^{2\alpha}(|\psi\rangle_{k|\overline{k}})$
$\displaystyle=$
$\displaystyle(G^{\alpha}(|\psi\rangle_{j|\overline{j}})+G^{\alpha}(|\psi\rangle_{k|\overline{k}}))^{2}-G^{2\alpha}(|\psi\rangle_{j|\overline{j}})-G^{2\alpha}(|\psi\rangle_{k|\overline{k}})$
$\displaystyle=$ $\displaystyle
2G^{\alpha}(|\psi\rangle_{j|\overline{j}})G^{\alpha}(|\psi\rangle_{k|\overline{k}})=0.$

Here $j\neq k\neq i.$ And From the above, we have
$G(|\psi\rangle_{j|\overline{j}})=0$ or $G(|\psi\rangle_{k|\overline{k}})=0.$

A three-qubit pure state $|\psi\rangle_{A_{1}A_{2}A_{3}}$ can be written in
the generalized Schmidt decomposition acin2000generalized :

$\displaystyle|\psi\rangle=l_{0}|000\rangle+l_{1}e^{i\theta}|100\rangle+l_{2}|101\rangle+l_{3}|110\rangle+l_{4}|111\rangle,$
(24)

where $\theta\in[0,\pi)$, $l_{i}\geq 0$ $(i=0,1,2,3,4),$ and
$\sum_{i=0}^{4}l_{i}^{2}=1.$ From simple computation, we have the Schmidt
coefficients of $|\psi\rangle$ in terms of $A|BC$ is

$Sch(|\psi\rangle_{A|BC})=(l_{0}^{2},1-l_{0}^{2}),$

the Schmidt coefficients of $|\psi\rangle$ in terms of $B|AC$ is

$Sch(|\psi\rangle_{B|AC})=(\frac{1+\sqrt{1-4\Delta_{0}}}{2},\frac{1-\sqrt{1-4\Delta_{0}}}{2}),$

here
$\Delta_{0}=l_{0}^{2}l_{3}^{2}+l_{0}^{2}l_{4}^{2}+l_{1}^{2}l_{4}^{2}+l_{2}^{2}l_{3}^{2}-2l_{1}l_{2}l_{3}l_{4}\cos\theta,$
the Schmidt coefficients of $|\psi\rangle$ in terms of $C|AB$ is

$\displaystyle
Sch(|\psi\rangle_{C|AB})=(\frac{1+\sqrt{1-4\Delta_{1}}}{2},\frac{1-\sqrt{1-4\Delta_{1}}}{2}),$

here
$\Delta_{1}=l_{0}^{2}l_{2}^{2}+l_{0}^{2}l_{4}^{2}+l_{1}^{2}l_{4}^{2}+l_{2}^{2}l_{3}^{2}-2l_{1}l_{2}l_{3}l_{4}\cos\theta.$

When $G(|\psi\rangle_{A|BC})=0,$ $l_{0}=0$ or $l_{0}=1,$ then
$|\psi\rangle_{ABC}$ is biseparable. When $G(|\psi\rangle_{B|AC})=0,$ that is,
$\Delta_{0}=0,$ as
$\Delta_{0}=l_{0}^{2}(l_{3}^{2}+l_{4}^{2})+(l_{1}l_{4}-l_{2}l_{3})^{2}+4l_{1}l_{2}l_{3}l_{4}\sin^{2}\frac{\theta}{2}=0,$
then $l_{3}=l_{4}=0,$ or $l_{0}=0,$ then $|\psi\rangle_{ABC}$ is biseparable.
The case $G(|\psi\rangle_{C|AB})=0$ is similar to $G(|\psi\rangle_{B|AC})=0$.

$\Leftarrow:$ Assume $|\psi\rangle=|\phi\rangle_{i}|\varphi\rangle_{jk},$
$i\neq j\neq k,$ then $G(|\psi\rangle_{i|jk})=0,$
$G(|\psi\rangle_{j|ki})=G(|\psi\rangle_{k|ji}),$ then $\delta_{\alpha}^{G}=0,$
$\forall\alpha\in(0,1).$ $\sqcap$$\sqcup$

###  III.3 EPI IN TERM OF NEGATIVITY AND CONCURRENCE

In this subsection, we present a class of counterexamples that does not
satisfy the EPI in terms of negativity and concurrence, which answers the
problem proposed in yang2022entanglement . We also propose a class of
multipartite pure states satisfying the EPI in terms of negativity.

First we present an example.

###### Example 8

$\displaystyle|\psi\rangle_{ABC}=\frac{|000\rangle+|101\rangle+|202\rangle+|310\rangle+|411\rangle+|512\rangle+|620\rangle+|721\rangle+|822\rangle}{3},$
(25)

According to the definition of Negativity (7), we have that
$N(|\psi\rangle_{A|BC})=4,$ $N(|\psi\rangle_{B|AC})=1$,
$N(|\psi\rangle_{C|AB})=1$. Clearly, $N(|\psi\rangle_{A|BC})\geq
N(|\psi\rangle_{B|AC})+N(|\psi\rangle_{C|AB})$, that is, the EPI is invalid
for $|\psi\rangle_{ABC}$ in terms of negativity.

###### Remark 9

In the above example, we have that $\rho_{BC}=\frac{1}{9}I_{3}\otimes I_{3}$,
that is, $|\psi\rangle_{ABC}$ is a purification of a product state
$\rho_{BC}$.

Next we present a class of states which doesnot satisfy the EPI in terms of
negativity.

###### Theorem 10

Assume $\rho$ and $\sigma$ are two states on the system $\mathcal{H}_{d}$,
$\rho=\sum_{i}a_{i}|i\rangle\langle i|,$ $\sigma=\sum_{j}b_{j}|j\rangle\langle
j|$, $\mathop{\rm rank}(\rho),\mathop{\rm rank}(\sigma)>1,$ and
$\\{|i\rangle\\}$ and $\\{|j\rangle\\}$ are the orthonormal bases of
$\mathcal{H}_{d}.$ Let
$|\psi\rangle_{ABC}=\sum_{ij}\sqrt{a_{i}b_{j}}|ij\rangle_{AB}|ij\rangle_{C}$
be a purification of $\rho_{A}\otimes\sigma_{B}$, then we have

$\displaystyle N(|\psi\rangle_{C|AB})\geq
N(|\psi\rangle_{A|BC})+N(|\psi\rangle_{BAC}).$ (26)

###### Proof.

Assume $|\phi\rangle_{AB}=\sum_{i}\sqrt{\lambda_{i}}|ii\rangle,$ then

$\displaystyle N(|\phi\rangle_{AB})=$
$\displaystyle\frac{(\sum_{i}\sqrt{\lambda_{i}})^{2}-1}{2},$ (27)

Due to (27), we have

$\displaystyle
N(|\psi\rangle_{C|AB})-N(|\psi\rangle_{A|BC})-N(|\psi\rangle_{BAC})$
$\displaystyle=$
$\displaystyle\frac{1}{2}[(\sum_{ij}\sqrt{a_{i}b_{j}})^{2}-1-(\sum_{i}\sqrt{a_{i}})^{2}+1-(\sum_{j}\sqrt{b_{j}})^{2}+1]$
$\displaystyle=$
$\displaystyle\frac{1}{2}[(1-(\sum_{i}\sqrt{a_{i}})^{2})(1-(\sum_{j}\sqrt{b_{j}})^{2})].$
(28)

As $\sum_{i}a_{i}=\sum_{j}b_{j}=1,$ and $\mathop{\rm rank}\rho,\mathop{\rm
rank}\sigma>1,$ then $(\ref{nep})>0.$ Then we finish the proof.
$\sqcap$$\sqcup$

At last, we present a class of pure states, the generalized W class (GW)
states, satisfy the EPI in terms of negativity. These states were first
studied on the problem of the monogamy relations in terms of concurrence for
higher dimensional systems san2008generalized . Recently, the general monogamy
relations of this class of states attracts much attention of the revelent
researchers shi2020monogamy ; lai2021tighter .

Now let us recall the definition of the GW states $|W_{n}^{d}\rangle$,

$\displaystyle|W_{n}^{d}\rangle_{A_{1}\cdots
A_{n}}=\sum_{i=1}^{d}(a_{1i}|i0\cdots 0\rangle+\cdots+a_{ni}|00\cdots
i\rangle),$ (29)

where we assume $\sum_{i=1}^{d}\sum_{j=1}^{n}|a_{ji}|^{2}=1.$ Next we present
a lemma and then we show the main result of this subsection.

###### Lemma 11

san2008generalized  Assume $|\psi\rangle_{AB_{1}B_{2}\cdots B_{n-1}}$ is a GW
state, then for an arbitrary partition $\\{P_{1},P_{2},\cdots,P_{m}\\}$ of the
set $S=\\{A,B_{1}B_{2},\cdots,B_{n-1}\\},$ the state
$|\psi\rangle_{P_{1}P_{2}\cdots P_{m}}$ is also a GW state, here we assume
$P_{i}\cap P_{j}=\emptyset$ $(i\neq j)$ and $\cup_{i}P_{i}=S,m\leq n$.

###### Theorem 12

Assume $|\psi\rangle_{A_{1}\cdots A_{n}}$ is a GW state, and here we denote
$\\{P_{1},P_{2},P_{3}\\}$ is a partition of the set
$\\{A_{1},A_{2},\cdots,A_{n}\\},$ $n\geq 3,$ then we have

$\displaystyle N(|\psi\rangle_{P_{1}|P_{2}P_{3}})\leq
N(|\psi\rangle_{P_{2}|P_{1}P_{3}})+N(|\psi\rangle_{P_{3}|P_{2}P_{1}}).$ (30)

###### Proof.

Due to the above lemma, we have that $|\psi\rangle_{P_{1}P_{2}P_{3}}$ is also
a GW state, that is, $|\psi\rangle_{P_{1}P_{2}P_{3}}$ can be written as

$\displaystyle|\psi\rangle_{P_{1}P_{2}P_{3}}=\sum_{i=1}^{d}(a_{1i}|i00\rangle+a_{2i}|0i0\rangle+a_{3i}|00i\rangle),$
(31)

through computation, we have that

$\displaystyle
N(|\psi\rangle_{P_{1}|P_{2}P_{3}})=\sqrt{\sum_{i}|a_{1i}|^{2}}\times\sqrt{\sum_{i}(|a_{2i}|^{2}+|a_{3i}|^{2})},$
$\displaystyle
N(|\psi\rangle_{P_{2}|P_{1}P_{3}})=\sqrt{\sum_{i}|a_{2i}|^{2}}\times\sqrt{\sum_{i}(|a_{1i}|^{2}+|a_{3i}|^{2})},$
$\displaystyle
N(|\psi\rangle_{P_{3}|P_{1}P_{3}})=\sqrt{\sum_{i}|a_{3i}|^{2}}\times\sqrt{\sum_{i}(|a_{2i}|^{2}+|a_{1i}|^{2})},$
(32)

let $a={\sum_{i}|a_{1i}|^{2}},$ $b={\sum_{i}|a_{2i}|^{2}},$
$c={\sum_{i}|a_{3i}|^{2}}$, as

$\displaystyle\sqrt{a(b+c)}\leq$
$\displaystyle\sqrt{ab+ac+2bc+2\sqrt{bc(a+c)(a+b)}},$ $\displaystyle=$
$\displaystyle\sqrt{[\sqrt{b(a+c)}+\sqrt{c(a+b)}]^{2}}.$

Then we finish the proof.  $\sqcap$$\sqcup$

Similar to the proof of Theorem 12 and Corollary 5, we can generalize the
Theorem 12 to the following.

###### Corollary 13

Assume $|\psi\rangle_{A_{1}\cdots A_{n}}$ is a GW state, and here we denote
$\\{P_{1},P_{2},\cdots,P_{k}\\}$ is a partition of the set
$\\{A_{1},A_{2},\cdots,A_{n}\\},$ $n\geq 3,$ then we have the following
inequality,

$\displaystyle
N(|\psi\rangle_{P_{i}|\overline{P_{i}}})^{\alpha}\leq\sum_{j\neq
i}N(|\psi\rangle_{P_{j}|\overline{P_{j}}})^{\alpha},$ (33)

here $\\{P_{1},P_{2},\cdots,P_{k}\\}$ is a partition of the set
$\\{A_{1},A_{2},\cdots,A_{n}\\}$ with $k\leq n$, and $\alpha\in(0,1]$.

Then we present an example to show the meaning of Theorem $(\ref{gw}).$

###### Example 14

Here we consider a tripartite pure state,

$\displaystyle|\psi\rangle_{ABCD}=0.3|0001\rangle+0.4|0020\rangle+0.5|0100\rangle+\sqrt{0.5}|1000\rangle,$

here we take $P_{1}=\\{A\\},P_{2}=\\{BC\\},$ $P_{3}=\\{D\\},$ then

$\displaystyle N(|\psi\rangle_{P_{1}|P_{2}P_{3}})=0.5,$ $\displaystyle
N(|\psi\rangle_{P_{2}|P_{1}P_{3}})=\sqrt{0.2419},$ $\displaystyle
N(|\psi\rangle_{P_{3}|P_{1}P_{2}})=$ $\displaystyle\sqrt{0.0819},$

then the inequality $(\ref{n3})$ can be written as

$\displaystyle h=$ $\displaystyle
N(|\psi\rangle_{P_{2}|P_{1}P_{3}})^{\alpha}+N(|\psi\rangle_{P_{3}|P_{1}P_{2}})^{\alpha}-N(|\psi\rangle_{P_{1}|P_{2}P_{3}})^{\alpha}$
$\displaystyle=$ $\displaystyle
0.2419^{\frac{\alpha}{2}}+0.0819^{\frac{\alpha}{2}}-0.5^{\alpha},$ (34)

Figure 2: In this figure, we present the inequality (34).

we plot $(\ref{e2})$ in Fig. 2, from the figure, we see that the inequality
$(\ref{e1})$ is bigger than 0 when $\alpha\in(0,1).$

At last, we present that the EPI is valid in terms of concurrence for pure
states in arbitrary dimensional systems.

Comparing the $(\ref{qc})$ with $(\ref{c})$, we have when $|\psi\rangle_{AB}$
a bipartite pure state,
$C(|\psi\rangle_{AB})=\sqrt{2C_{2}(|\psi\rangle_{AB})}.$ Then combing the
Lemma 1 and similar proof of Corollary 5, we have the following result,

###### Theorem 15

For any $n$-qudit pure entangled state $|\psi\rangle$ in
$\otimes_{i}\mathcal{H}_{i},$ we have

$\displaystyle C_{j|\overline{j}}\leq\sum_{k\neq j,\forall
k}C_{k|\overline{k}}(|\psi\rangle),$ (35)

##  IV CONCLUSION

In this article, we have investigated the EPI for pure states in multipartite
systems. First we present the EPI is valid in terms of GEM for pure states in
arbitrary dimensional systems. Based on the inequalities, we have presented
the entanglement indicators in terms of GEM. At last, we have presented a
class of pure states that does not satisfy the EPI in terms of negativity, we
have also presented a class of pure states satisfying the EPI in terms of
negativity. Moreover, we have presented the EPI is valid in terms of
concurrence for pure states in arbitrary dimensional systems. Due to the
importance of the study of the higher-dimensional multipartite entanglement
systems, our results can provide a reference for future work on the study of
multiparty quantum entanglement.

## References

  * (1) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” _Reviews of modern physics_ , vol. 81, no. 2, p. 865, 2009\.
  * (2) C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on einstein-podolsky-rosen states,” _Physical review letters_ , vol. 69, no. 20, p. 2881, 1992.
  * (3) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” _Physical review letters_ , vol. 70, no. 13, p. 1895, 1993.
  * (4) Y. Shimoni, D. Shapira, and O. Biham, “Entangled quantum states generated by shor’s factoring algorithm,” _Physical Review A_ , vol. 72, no. 6, p. 062308, 2005.
  * (5) V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” _Physical Review A_ , vol. 61, no. 5, p. 052306, 2000.
  * (6) M. Christandl and A. Winter, ““squashed entanglement”: an additive entanglement measure,” _Journal of mathematical physics_ , vol. 45, no. 3, pp. 829–840, 2004.
  * (7) Y.-K. Bai, Y.-F. Xu, and Z. Wang, “General monogamy relation for the entanglement of formation in multiqubit systems,” _Physical review letters_ , vol. 113, no. 10, p. 100503, 2014.
  * (8) Y. Luo, T. Tian, L.-H. Shao, and Y. Li, “General monogamy of tsallis q-entropy entanglement in multiqubit systems,” _Physical Review A_ , vol. 93, no. 6, p. 062340, 2016.
  * (9) W. Song, Y.-K. Bai, M. Yang, M. Yang, and Z.-L. Cao, “General monogamy relation of multiqubit systems in terms of aared rényi-$\alpha$ entanglement,” _Physical Review A_ , vol. 93, no. 2, p. 022306, 2016.
  * (10) T. R. De Oliveira, M. F. Cornelio, and F. F. Fanchini, “Monogamy of entanglement of formation,” _Physical Review A_ , vol. 89, no. 3, p. 034303, 2014.
  * (11) G. Gour and Y. Guo, “Monogamy of entanglement without inequalities,” _Quantum_ , vol. 2, p. 81, 2018.
  * (12) X. Shi, L. Chen, and M. Hu, “Multilinear monogamy relations for multiqubit states,” _Physical Review A_ , vol. 104, no. 1, p. 012426, 2021.
  * (13) X.-F. Qian, M. A. Alonso, and J. H. Eberly, “Entanglement polygon inequality in qubit systems,” _New Journal of Physics_ , vol. 20, no. 6, p. 063012, 2018\.
  * (14) X. Yang, Y.-H. Yang, and M.-X. Luo, “Entanglement polygon inequality in qudit systems,” _arXiv preprint arXiv:2205.08801_ , 2022.
  * (15) T.-C. Wei and P. M. Goldbart, “Geometric measure of entanglement and applications to bipartite and multipartite quantum states,” _Physical Review A_ , vol. 68, no. 4, p. 042307, 2003.
  * (16) G. Vidal and R. F. Werner, “Computable measure of entanglement,” _Physical Review A_ , vol. 65, no. 3, p. 032314, 2002.
  * (17) W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” _Physical Review Letters_ , vol. 80, no. 10, p. 2245, 1998.
  * (18) X. Yang, M.-X. Luo, Y.-H. Yang, and S.-M. Fei, “Parametrized entanglement monotone,” _Physical Review A_ , vol. 103, no. 5, p. 052423, 2021.
  * (19) K. M. Audenaert, “Subadditivity of q-entropies for q¿ 1,” _Journal of Mathematical Physics_ , vol. 48, no. 8, p. 083507, 2007.
  * (20) X.-N. Zhu and S.-M. Fei, “Entanglement monogamy relations of qubit systems,” _Physical Review A_ , vol. 90, no. 2, p. 024304, 2014.
  * (21) A. Acin, A. Andrianov, L. Costa, E. Jane, J. Latorre, and R. Tarrach, “Generalized schmidt decomposition and classification of three-quantum-bit states,” _Physical Review Letters_ , vol. 85, no. 7, p. 1560, 2000.
  * (22) J. San Kim and B. C. Sanders, “Generalized w-class state and its monogamy relation,” _Journal of Physics A: Mathematical and Theoretical_ , vol. 41, no. 49, p. 495301, 2008.
  * (23) X. Shi and L. Chen, “Monogamy relations for the generalized w-class states beyond qubits,” _Physical Review A_ , vol. 101, no. 3, p. 032344, 2020.
  * (24) L.-M. Lai, S.-M. Fei, and Z.-X. Wang, “Tighter monogamy and polygamy relations for a superposition of the generalized w-class state and vacuum,” _Journal of Physics A: Mathematical and Theoretical_ , vol. 54, no. 42, p. 425301, 2021.