# MODIPHY: Multimodal Obscured Detection for IoT using PHantom Convolution-
Enabled Faster YOLO

Shubhabrata Mukherjee, Cory Beard, Zhu Li
School of Science and Engineering, University of Missouri Kansas City
Kansas City, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Low-light conditions and occluded scenarios impede object detection in real-
world Internet of Things (IoT) applications like autonomous vehicles and
security systems. While advanced machine learning models strive for accuracy,
their computational demands clash with the limitations of resource-constrained
devices, hampering real-time performance. In our current research, we tackle
this challenge, by introducing “YOLO Phantom”, one of the smallest YOLO models
ever conceived. YOLO Phantom utilizes the novel Phantom Convolution block,
achieving comparable accuracy to the latest YOLOv8n model while simultaneously
reducing both parameters and model size by 43%, resulting in a significant 19%
reduction in Giga Floating-Point Operations (GFLOPs). YOLO Phantom leverages
transfer learning on our multimodal RGB-infrared dataset to address low-light
and occlusion issues, equipping it with robust vision under adverse
conditions. Its real-world efficacy is demonstrated on an IoT platform with
advanced low-light and RGB cameras, seamlessly connecting to an AWS-based
notification endpoint for efficient real-time object detection. Benchmarks
reveal a substantial boost of 17% and 14% in frames per second (FPS) for
thermal and RGB detection, respectively, compared to the baseline YOLOv8n
model. For community contribution, both the code and the multimodal dataset
are available on GitHub.111The code and dataset are available at
http://tinyurl.com/46y5twku.

###### Index Terms:

Low light object detection, Multimodal fusion, IoT, YOLOv8, Phantom
Convolution

##  I Introduction

Robust object detection under low light and occlusion is crucial for various
applications, including traffic monitoring, public safety, and environmental
monitoring. Instances such as Tesla’s autopilot oversight and GM’s Cruise
pedestrian crash underscore safety concerns in autonomous vehicles [1, 2]. The
challenges extend to facial recognition systems, facing accuracy drops in
inadequate lighting, necessitating advancements in object detection
technologies [3, 4]. AI plays a pivotal role in addressing these low-light
challenges, employing innovative approaches like integrating infrared sensors
to perceive thermal signatures in darkness. The early fusion [5] of thermal
and RGB datasets in multimodal imaging emerge as a powerful solution,
particularly in overcoming occlusion. Fusing thermal, visible light, LiDAR,
and radar distance scanning data through machine learning significantly
improves the performance of object detection and classification systems.

Recent strides in low-light object detection, utilizing deep learning and
fusion techniques [6, 7, 8, 9], exhibit promise. YOLO-based methods, in
particular, have gained acclaim [10, 11, 12]. While deploying neural network
models on edge devices like UAVs presents challenges [13], our work bridges
this gap by enhancing the YOLOv8 nano [14] architecture, giving rise to the
“YOLO Phantom.” Tailored for resource-constrained IoT devices, this adaptation
ensures compatibility with small-scale hardware, achieving substantial speed
improvements for real-time object detection in low-light conditions. Our
presentation delves into these enhancements, scrutinizes their performance
gains, and showcases their effectiveness in pertinent low-light object
detection tasks.

In the context of our current research, we have developed a compact yet
efficient multimodal object detection model optimized for effective low light
object detection. Leveraging the FLIRV2 dataset [15], we crafted a multimodal
dataset to train our own YOLO Phantom model, complemented by integrating a
NoIR camera Module [16] with Raspberry Pi for exceptional performance in low-
light scenarios. The distinctive contributions of this paper include:

  * •

Creation of a compact YOLO model: Designing one of the smallest YOLO models,
by using only 50% of the kernels in three of the layers, demonstrating
superior speed without compromising accuracy.

  * •

YOLO Phantom for multimodal detection: Introducing YOLO Phantom, capable of
detecting multiple modalities such as RGB and thermal. The model underwent
training using various modalities and demonstrates effectiveness in scenarios
characterized by low light conditions and occlusion.

  * •

Innovative convolution block: Proposing the novel “Phantom Convolution” block,
by using Depthwise Separable Convolution [17] and Group Convolution [18]
layers we created a faster and more efficient version of Ghost Convolution
[19].

  * •

Realization of an end-to-end notification system: Realizing the new model
through a practical end-to-end object detection framework. This involves
integrating a Raspberry Pi with RGB and NoIR cameras for simultaneous
detection, along with an AWS cloud-based real-time notification system.

##  II Implementation strategies

Customized detection systems power diverse applications, from self-driving
cars to surveillance, tailoring to specific environments and tasks. Key types
include:

  * •

Aerial Detection System (ADS): ADS is utilized in air defense, public safety,
disaster recovery [20] and vegetation monitoring, with ground-based
installations strategically placed and air-based systems integrated into
manned or unmanned aerial vehicles (UAVs) for enhanced situational awareness.

  * •

Marine Detection System (MDS): MDS is designed to detect and identify threats
to maritime security, using radar, sonar, acoustic sensors, and smart cameras
to track vessels and activities in surrounding waters [21]. AI-based software
and ML algorithms analyze sensor data for threat identification, with examples
like an Automatic Identification System (AIS) for vessel tracking and Long-
Range Identification and Tracking (LRIT) for long-distance identification.

  * •

Ground-based Detection System (GDS): GDS focuses on detecting, classifying, or
tracking objects in terrestrial environments, employing technologies like
ground-based radar, RGB or infrared-sensor-based cameras, and motion sensors
to identify people, vehicles, or object movement [22]. AI-based infrastructure
analyzes collected data for threat identification, including perimeter
intrusion detection systems and object detection with facial recognition using
smart cameras and machine learning.

Figure 1: Detection on a rainy, obscured evening with severe occlusion using a
multimodal YOLO model and a NoIR camera on a Raspberry Pi platform

##  III Role of resource optimization

Limited resources in remote areas often constrain resilient object detection
in IoT systems. Optimized resource usage becomes vital for their extended,
reliable operation in such environments, enhancing lifespan, accuracy, and
overall efficacy. In particular, resource optimization is focused on the
efficient utilization of computation resources such as power, memory, and
temperature control. Limited power in devices like Unmanned Aerial Vehicle
(UAV) or surveillance cars hinders training and running computation-intensive
AI models, compromising efficiency, longevity, and overall performance.
Through significant improvements, particularly the incorporation of our newly
proposed phantom convolution block, we successfully developed one of the most
compact YOLO models to date. As illustrated in Fig. 2, our model boasts the
fewest parameters, minimal GFLOPs, and the smallest size when compared to some
of the most recent compact YOLO models. Performance evaluation revealed
comparable or superior performance in different scenarios compared to other
state-of-the-art (SOTA) smaller models, encompassing both RGB and thermal
modalities. YOLO Phantom’s lightweight design makes it highly suitable for
real-time object detection on edge devices.

Figure 2: Size, Parameters, and GFLOP comparison of smaller YOLO models

##  IV Our approach: YOLO Phantom

We have modified the architecture of the smallest version of the Ultralytics
YOLOv8 model, YOLOv8n, to enable it to detect both RGB and thermal modality
images. Furthermore, we strategically restructured and resized select network
blocks to facilitate expedited model inference. And, finally, we trained it
using transfer learning with a multimodal RGB and thermal image dataset as
described later. This enabled YOLO Phantom to detect multimodal objects much
faster than the Ultralytics YOLOv8, with similar or slightly higher accuracy
(mAP), as seen in Section V.

###  IV-A Ultralytics YOLOv8 architecture

The Ultralytics YOLOv8 architecture is primarily divided into three main
parts: the Backbone, Neck, and Head. The Backbone is responsible for
generating feature maps and incorporates the following components: C2f, a
modified version of Darknet53 with Cross-Stage Partial (CSP) connections for
improved feature re-usability and gradient flow, convolutional layers with
varying kernel sizes. The output of the Backbone is feature maps at multiple
scales, capturing both low-level details and high-level semantic information.
Fig. 3 shows the implementation and our changes in YOLOv8 backbone, which
includes the C2f replaced by improved C2f (“C2fi”) and new Phantom Convolution
blocks. Ultralytics YOLOv8 backbone also consists of SPPF (Spatial Pyramid
Pooling - Fast) which enhances the receptive field and captures multi-scale
features through parallel pooling branches with different kernel sizes and
concatenation of pooled features.

Figure 3: Modified YOLOv8 Backbone [23]

Fig. 4 shows the Ultralytics YOLOv8 implementation of the neck and head, along
with our introduced Phantom Convolution block. Upsample blocks are used to
increase the feature resolution and match with C2fi (C2f with ShortCut False)
blocks using Concat blocks. The outputs from C2fi goes to Convolution blocks
as well as detection heads, used for detecting small, medium, or large
objects. It uses a decoupled head to perform various tasks like detection,
classification, segmentation separately. YOLOv8 also uses SiLU (Swish Linear
Unit) as default activation function, Mosaic augmentation, anchor-free
detection, batch normalization for enriched training, and flexible prediction.
Please note that inside the Ultralytics repository, in the main architecture
file for YOLOV8, the head and neck sections have been collectively referred to
as ‘head’.

###  IV-B Architecture improvement for faster and better inference

In the development of YOLO Phantom, we undertook modifications to accommodate
both RGB and thermal image detection, enhancing the model’s versatility.
Ultralytics YOLOv8 training was conducted on the COCO dataset [24], a
predominantly RGB dataset. It was necessary for us to make adjustments to
architectural elements and parameter scaling by directing our efforts toward
achieving comparable mean average precision (mAP) performance with reduced
model complexity and improved speed. Various novel convolution methods have
been adapted by researchers to improve accuracy performance (better mAP) or
make the model faster (higher FPS) to perform different real-time computer
vision tasks. We experimented with quite a few of them for our training; we
have mentioned our observations from using them in our model architecture.

#### Group Convolution

Group convolution [25] divides filters into groups, each operating on a subset
of input channels. Unlike traditional convolution connecting to all input
channels, group convolution segregates filters, reducing computational costs
and enabling parallelization. It enhances efficiency without compromising
representational power, especially in models with many channels. The group
convolution operation can be expressed as below:

Figure 4: Modified YOLOv8 Neck and Decoupled detection Head [23]

  * •

Input tensor: $X\in\mathbb{R}^{N\times C\times H\times W}$

  * •

Weight tensor: $W\in\mathbb{R}^{G\times\frac{C}{G}\times K\times K}$

  * •

Bias term: $b\in\mathbb{R}^{G\times 1}$

  * •

Output tensor: $Y\in\mathbb{R}^{N\times G\times\frac{C}{G}\times
H^{\prime}\times W^{\prime}}$

$Y_{n,g,i,j}=\sum_{c=1}^{\frac{C}{G}}X_{n,(g-1)\frac{C}{G}+c,i^{\prime},j^{\prime}}\cdot
W_{g,c,k^{\prime},l^{\prime}}+b_{g}$ (1)

#### Depth-wise separable Convolution

Depth-wise Separable convolution [17] is a CNN variant that splits a standard
convolution into two steps: depth-wise convolution, where each input channel
is convolved independently, and point-wise convolution, combining results
across channels with 1x1 filters. This separation significantly reduces
computational complexity and parameters compared to traditional convolutions,
making it widely adopted in mobile and edge devices for efficiency without
compromising expressiveness. It strikes a balance between computational
efficiency and model performance, making it suitable for resource-constrained
environments.

  * •

Input tensor: $X\in\mathbb{R}^{N\times C\times H\times W}$

  * •

Depthwise convolution filter: $D\in\mathbb{R}^{C\times 1\times K_{d}\times
K_{d}}$

  * •

Pointwise convolution filter: $P\in\mathbb{R}^{C^{\prime}\times C\times
1\times 1}$

  * •

Bias term: $b\in\mathbb{R}^{C^{\prime}\times 1}$

  * •

Output tensor: $Y\in\mathbb{R}^{N\times C^{\prime}\times H^{\prime}\times
W^{\prime}}$

Depthwise Convolution:

$Y_{dw}=\text{DepthwiseConv}(X,D)//Y_{dw}\in\mathbb{R}^{N\times C\times
H^{\prime}\times W^{\prime}}$ (2)

Pointwise Convolution:

$Y=\text{PointwiseConv}(Y_{dw},P)+b//Y\in\mathbb{R}^{N\times C^{\prime}\times
H^{\prime}\times W^{\prime}}$ (3)

Now, if we combine above two operations:

$\displaystyle Y_{n,c^{\prime},i,j}=$
$\displaystyle\sum_{c=1}^{C}\sum_{k^{\prime}=1}^{K_{d}}\sum_{l^{\prime}=1}^{K_{d}}X_{n,c,i\times
S+k^{\prime},j\times S+l^{\prime}}\cdot D_{c,1,k^{\prime},l^{\prime}}$
$\displaystyle+\sum_{c=1}^{C}Y_{dw,n,c,i,j}\cdot
P_{c^{\prime},c,1,1}+b_{c^{\prime}}$ (4)

#### Ghost Convolution

Ghost Convolution [19] enhances convolutional neural networks (CNNs) by
introducing a parallel “ghost branch” alongside the main convolutional layers.
The ghost branch, created by subsampling input channels, processes a reduced
input version and combines its output with the main branch. This technique
balances computational efficiency and model expressiveness. By utilizing fewer
channels in the ghost branch, Ghost Convolution achieves resource savings
while retaining essential features. It is particularly valuable in scenarios
with limited computational resources like mobile devices where efficiency is
crucial.

Figure 5: Phantom Convolution and C2fi block architecture

Various strategies, culminating in adopting Ghost Convolution, were employed
to enhance feature extraction with fewer parameters. As seen in Fig. 5, the
Ghost Convolution block underwent re-architecting, with the first layer
transformed into a group convolution (group size 4) for increased operational
speed and larger kernel sizes (5x5) to expand the receptive field. The second
layer utilized Depth-wise Separable Convolution for efficiency, leading to a
compact block with improved computational efficiency, named “Phantom
Convolution”.

The C2f blocks were modified by eliminating forward pass connections,
introducing “C2fi” or “C2f improved” blocks for a smaller, faster, and more
efficient model. These blocks, with shortcut connections set to false,
replaced traditional C2f blocks throughout the architecture. These changes
allowed a reduction in the number of filters in deeper layers, halving the
number of filters in layer 7 and in the head of the backbone. Notably, the
seventh layer incorporated the Phantom Convolution block, and the final
convolution block in the head (layer 19) was replaced with Phantom Convolution
to enhance efficiency and expedite the detection process.

Figure 6: Detection and Notification pipeline

###  IV-C Experimental Setup

In this study, we devised an end-to-end detection and notification system
utilizing IoT-based hardware in conjunction with Amazon Web Services (AWS) for
cloud-based notifications (Fig. 7). Specifically, we employed a Raspberry Pi
[26] version 4, model B (CanaKit Extreme, 128 GB, 8GB, BullsEye OS), as the
hosting and execution platform for the object detection model. An external RGB
camera was interfaced with the USB port of the Raspberry Pi to capture
relevant data. We also connected a NoIR Camera Module V2 (Fig. 6) to the
Raspberry Pi using Camera Serial Interface (CSI). This camera module can see
in low-light without an infrared filter, making it ideal for low-light
detection. To establish connectivity between the Raspberry Pi and the cloud,
we leveraged an AWS IoT Core, thereby constructing an event notification
framework. This integration empowered us to dispatch customized notifications
based on detected events; in our experimentation, we validated this
functionality by triggering notifications upon the detection of a person.
Furthermore, we used AWS Simple Notification Service (SNS), to augment the AWS
IoT Core in disseminating alerts. This feature facilitated the transmission of
notifications to designated email addresses, phone numbers, or mobile
applications, enhancing the versatility and accessibility of our alerting
mechanism.

Figure 7: Detection setup using Raspberry Pi and RGB camera

Figure 8: Various occluded and low-light detection scenario comparison with
smaller YOLO models (left to right) using NoIR or RGB cameras

To bolster the core detection framework, we integrated supplementary safety
controls. A temperature monitoring system for the Raspberry Pi automatically
halts detection when exceeding a conservative 60°C threshold. Additionally, an
audio alert system triggers upon person detection, offering redundancy beyond
visual notifications. These enhancements increase responsiveness and provide
layered alerts for enhanced situational awareness and threat mitigation.

###  IV-D Dataset description and Training

Utilizing the Teledyne FLIR ADAS Thermal Dataset v2 [15], we curated a
multimodal dataset comprising both RGB and thermal imagery. The capability to
perceive thermal infrared radiation offers both complementary and distinct
advantages compared to conventional sensor technologies like visible cameras,
Lidar, and radar systems. Teledyne FLIR thermal sensors excel in detecting and
classifying objects under challenging conditions, including total darkness,
fog, smoke, adverse weather, and glare. The primary dataset comprises 26,442
annotated images spanning 15 categories. From the training and validation
sets, we selected 10,478 thermal and 10,053 RGB images for the training set,
and a combination of 1,128 thermal and 1,069 RGB images for validation to
construct the multimodal dataset for training. Focusing on the four most
represented classes (person, car, traffic light, street sign), we conducted
experiments. Additionally, a separate thermal test dataset of 3,749 images and
an RGB test dataset of 3,493 images were constructed from the main FLIR V2
dataset’s test section for benchmarking purposes.

YOLO Phantom was trained using our multimodal dataset on a CentOS cluster with
two “NVIDIA RTX A6000” GPUs and 64 CPUs for 100 epochs. Images were resized to
640x640 pixels. The “yolov8n.pt” pre-trained weights served as the starting
point for the training process.

##  V Results and Analysis

###  V-A Out of sample testing

This section presents a performance comparison of YOLO Phantom against other
state-of-the-art (SOTA) smaller YOLO models. The evaluation employed a diverse
image set encompassing low-light and occlusion scenarios (thermal imaging,
tunnel interiors, cloudy evenings, foggy mornings, nighttime). Both RGB and
NoIR cameras captured the images. All the detection has been performed with a
non-max-suppression (NMS) 0.5 threshold. Figure 8 showcases YOLO Phantom’s
competitive performance, matching or exceeding smaller models like
YOLOv4-tiny, YOLOv7-tiny, YOLOv8n, and YOLOv5nu, despite having 4-2x fewer
parameters.

###  V-B Performance on different modality data

In this section, we conducted a benchmark analysis of YOLO Phantom against
recent pre-trained models, including YOLOv5nu, Ultralytics YOLOv8n, and
YOLOv8n trained on multimodal data without any architecture modifications.
This comparative assessment was executed on two distinct modality datasets:
RGB and thermal data. The key performance metrics considered for benchmarking
were accuracy (mAP50-95(B)) and frames per second (FPS). The evaluation
encompassed five different platforms: PyTorch, TorchScript, ONNX, OpenVINO,
and ncnn. The benchmarking process was conducted on a M1 MAC OS CPU. As
depicted in Fig. 9, YOLO Phantom demonstrates a significantly superior
performance compared to pre-trained models such as YOLOv5nu and YOLOv8n. It
exhibits a slightly higher accuracy than the YOLOv8n fusion model trained on
our multimodal data. Moreover, YOLO Phantom consistently achieves higher FPS
across various model variants, surpassing both pre-trained smaller models and
the YOLOv8n fusion model. Similar trends were observed in Fig. 10 during the
thermal data benchmarking. YOLO Phantom outperforms its competitors in mAP and
achieves superior speed due to its smaller size.

Figure 9: mAP and FPS performance comparison on RGB data with pretrained
YOLOV5nu, pretrained YOLOv8n, YOLOv8n transfer learned on multimodal data, and
YOLO Phantom

Figure 10: mAP and FPS performance comparison on thermal data with pretrained
YOLOV5nu, pretrained YOLOv8n, YOLOv8n transfer learned on multimodal data, and
YOLO Phantom

###  V-C Cross-modality performance

This section evaluates the performance of multimodal models against unimodal
models using reverse modality data. This approach demonstrably highlights the
added value of multimodal learning. Table I showcases performance (mAP) on RGB
data comparing YOLOv8n trained on thermal, YOLOv8n trained on RGB, and YOLO
Phantom trained multimodally. While the thermal-trained YOLOv8n struggled, the
RGB-trained one excelled. Notably, YOLO Phantom achieved similar or better
performance in most cases, demonstrating its multimodal advantage.

TABLE I: Cross-modality mAP performance on RGB data Format | Thermal Trained | RGB Trained | YOLO Phantom
---|---|---|---
PyTorch | 9.29 | 19.65 | 19.72
TorchScript | 9.19 | 19.57 | 19.51
ONNX | 9.19 | 19.57 | 19.51
OpenVINO | 8.43 | 10.24 | 16.65
ncnn | 5.22 | 11.99 | 11.95

In the next case, we performed a similar comparison, but instead of accuracy
(mAP) we measured speed (FPS). As observed in Table II, in this case, YOLO
Phantom performed similar or better than thermal-trained or RGB-trained
models.

TABLE II: Cross-modality FPS performance on RGB data Format | Thermal Trained | RGB Trained | YOLO Phantom
---|---|---|---
PyTorch | 17.50 | 17.14 | 17.53
TorchScript | 16.35 | 15.77 | 15.50
ONNX | 20.12 | 20.45 | 21.92
OpenVINO | 22.86 | 23.28 | 25.86
ncnn | 23.77 | 23.85 | 27.15

Next, we conducted a cross-modality mAP performance comparison on the thermal
dataset. As seen in Table III, in this case also the reverse modality model
(RGB-trained) performed poorly compared to the same modality (thermal-trained)
YOLOv8n model. However, the thermal-trained model performed marginally better
compared to YOLO Phantom on this occasion.

TABLE III: Cross-modality mAP performance on thermal data Format | RGB Trained | Thermal Trained | YOLO Phantom
---|---|---|---
PyTorch | 7.21 | 25.21 | 24.82
TorchScript | 6.81 | 24.9 | 24.46
ONNX | 6.81 | 24.9 | 24.46
OpenVINO | 2.81 | 24.58 | 23.79
ncnn | 3.87 | 15.26 | 14.94

Finally, we performed a cross-modality FPS comparison on the thermal dataset.
As observed in Table IV, RGB, and thermal modality trained models performed
nearly similarly, but YOLO Phantom shows much better FPS performance compared
to both.

TABLE IV: Cross-modality FPS performance on thermal data Format | RGB Trained | Thermal Trained | YOLO Phantom
---|---|---|---
PyTorch | 17.76 | 16.54 | 16.65
TorchScript | 15.54 | 15.01 | 15.09
ONNX | 19.16 | 18.48 | 19.32
OpenVINO | 22.92 | 22.38 | 24.22
ncnn | 23.98 | 24.03 | 26.96

##  VI Conclusion and future work

YOLO Phantom opens the door to many new real-world object detection tasks,
particularly in resource-constrained environments. This lightweight model
exhibits a remarkable equilibrium between rapid execution, elevated accuracy,
and resource efficiency – the three most critical factors for IoT devices. The
innovative Phantom Convolution block seamlessly integrates into any framework,
unlocking superior feature extraction with reduced computational burden. Its
novel architecture transcends specific tasks, demonstrating exceptional
performance across diverse, resource-scarce detection scenarios. To further
strengthen an implementation’s practical viability for resource-constrained
IoT deployments, user-defined temperature monitoring was added to safeguard
against thermal excursions, while optional audio alerts offer additional
redundancy for enhanced situational awareness and threat mitigation. This
advancement holds promise for broader performance gains across computer
vision, encompassing tasks like classification, segmentation, and even
generative modeling. The impact of YOLO Phantom extends beyond just enhanced
road safety and refined surveillance applications, paving the way for
pioneering applications in environmental monitoring, wildlife tracking, and
beyond.

## References

  * [1] A. Press. (2023, January) Ap: Tesla driver killed after plowing into firetruck on freeway. [Online]. Available: http://tinyurl.com/3bejc9cy
  * [2] Reuters. (2023, November) Gm’s cruise to recall 950 driverless cars after accident involving pedestrian. [Online]. Available: http://tinyurl.com/5fj6mre5
  * [3] R. Al Sobbahi and J. Tekli, “Comparing deep learning models for low-light natural scene image enhancement and their impact on object detection and classification: Overview, empirical evaluation, and challenges,” _Signal Processing: Image Communication_ , p. 116848, 2022.
  * [4] M. Ahmed, K. A. Hashmi, A. Pagani, M. Liwicki, D. Stricker, and M. Z. Afzal, “Survey and performance analysis of deep learning based object detection in challenging environments,” _Sensors_ , vol. 21, no. 15, p. 5116, 2021.
  * [5] R. O. Chavez-Garcia and O. Aycard, “Multiple sensor fusion and classification for moving object detection and tracking,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 17, no. 2, pp. 525–534, 2016.
  * [6] W. Chen and T. Shah, “Exploring low-light object detection techniques,” _arXiv preprint arXiv:2107.14382_ , 2021.
  * [7] R. Kvyetnyy, R. Maslii, V. Harmash, I. Bogach, A. Kotyra, Z. Gradz, A. Zhanpeisova, and N. Askarova, “Object detection in images with low light condition,” in _Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 2017_ , vol. 10445. SPIE, 2017, pp. 250–259.
  * [8] L. Ye and Z. Ma, “Llod: a object detection method under low-light condition by feature enhancement and fusion,” in _2023 4th international seminar on artificial intelligence, networking and information technology (AINIT)_. IEEE, 2023, pp. 659–662.
  * [9] X. Wang, D. Wang, S. Li, S. Li, P. Zeng, and X. Liang, “Low-light traffic objects detection for automated vehicles,” in _2022 6th CAA International Conference on Vehicular Control and Intelligence (CVCI)_. IEEE, 2022, pp. 1–5.
  * [10] W. Liu, G. Ren, R. Yu, S. Guo, J. Zhu, and L. Zhang, “Image-adaptive yolo for object detection in adverse weather conditions,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 36, no. 2, 2022, pp. 1792–1800.
  * [11] X. Yin, Z. Yu, Z. Fei, W. Lv, and X. Gao, “Pe-yolo: Pyramid enhancement network for dark object detection,” in _International Conference on Artificial Neural Networks_. Springer, 2023, pp. 163–174.
  * [12] Z. Wang, Z. Cai, and Y. Wu, “An improved yolox approach for low-light and small object detection: Ppe on tunnel construction sites,” _Journal of Computational Design and Engineering_ , vol. 10, no. 3, pp. 1158–1175, 2023.
  * [13] S. Y. Nikouei, Y. Chen, S. Song, R. Xu, B.-Y. Choi, and T. R. Faughnan, “Real-time human detection as an edge service enabled by a lightweight CNN,” in _2018 IEEE International Conference on Edge Computing (EDGE)_. IEEE, 2018, pp. 125–129.
  * [14] G. Jocher, A. Chaurasia, and J. Qiu, “Ultralytics YOLO,” https://github.com/ultralytics/ultralytics, version 8.0.0, released January 2023. License: AGPL-3.0.
  * [15] T. FLIR, “Free teledyne flir thermal dataset for algorithm training,” 2023, accessed: 2024-01-30. [Online]. Available: https://www.flir.com/oem/adas/adas-dataset-form/
  * [16] Raspberry Pi Foundation. (2024) Raspberry pi camera module 2 noir. [Online]. Available: https://www.raspberrypi.com/products/pi-noir-camera-v2/
  * [17] F. Chollet, “Xception: Deep learning with depthwise separable convolutions,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 1251–1258.
  * [18] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” _Advances in neural information processing systems_ , vol. 25, 2012.
  * [19] K. Han, Y. Wang, Q. Tian, J. Guo, C. Xu, and C. Xu, “Ghostnet: More features from cheap operations,” in _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2020, pp. 1580–1589.
  * [20] S. Mukherjee, O. Coudert, and C. Beard, “Unimodal: Uav-aided infrared imaging based object detection and localization for search and disaster recovery,” in _2022 IEEE International Symposium on Technologies for Homeland Security (HST)_ , 2022, pp. 1–6.
  * [21] M. Zhang, S. Xu, W. Song, Q. He, and Q. Wei, “Lightweight underwater object detection based on yolo v4 and multi-scale attentional feature fusion,” _Remote Sensing_ , vol. 13, no. 22, p. 4706, 2021.
  * [22] X. Tang, Z. Zhang, and Y. Qin, “On-road object detection and tracking based on radar and vision fusion: A review,” _IEEE Intelligent Transportation Systems Magazine_ , vol. 14, no. 5, pp. 103–128, 2022.
  * [23] RangeKing. (2024) Github homepage. [Online]. Available: https://github.com/RangeKing
  * [24] T. Lin, M. Maire, S. J. Belongie, L. D. Bourdev, R. B. Girshick, J. Hays, P. Perona, D. Ramanan, P. Doll’a r, and C. L. Zitnick, “Microsoft COCO: common objects in context,” _CoRR_ , vol. abs/1405.0312, 2014. [Online]. Available: http://arxiv.org/abs/1405.0312
  * [25] Z. Su, L. Fang, W. Kang, D. Hu, M. Pietikäinen, and L. Liu, “Dynamic group convolution for accelerating convolutional neural networks,” in _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part VI 16_. Springer, 2020, pp. 138–155.
  * [26] Raspberry Pi Foundation. (2024) Raspberry pi. [Online]. Available: https://www.raspberrypi.org/


# A note on typicality in random quantum scattering

Michele Avalle  Department of Computer Science, Sapienza University of Rome,
Viale Regina Elena 295, Rome, 00161, Italy  Alessio Serafini
<EMAIL_ADDRESS>Department of Physics & Astronomy, University
College London, Gower Street, London WC1E 6BT, United Kingdom

###### Abstract

We consider scattering processes where a quantum system is comprised of an
inner subsystem and of a boundary, and is subject to Haar-averaged random
unitaries acting on the boundary-environment Hilbert space only. We show that,
regardless of the initial state, a single scattering event will disentangle
the unconditional state (i.e., the scattered state when no information about
the applied unitary is available) across the inner subsystem-boundary
partition. Also, we apply Levy’s lemma to constrain the trace norm
fluctuations around the unconditional state. Finally, we derive analytical
formulae for the mean scattered purity for initial globally pure states, and
provide one with numerical evidence of the reduction of fluctuations around
such mean values with increasing environmental dimension.

Suggested keywords

##  I Typical states under restricted interactions

Typicality arguments, based on Haar averages in the high-dimensional limit,
have been repeatedly advocated as a plausible avenue to justify the second law
and the ubiquity of the thermal state [1, 2, 3, 4]. Such arguments rely on
averages over Haar-distributed unitaries over a composite Hilbert space
$\mathcal{H}_{\sigma\epsilon}=\mathcal{H}_{\sigma}\otimes\mathcal{H}_{\epsilon}$
pertaining to a physical system $\sigma$ interacting with an environment
$\epsilon$. The typicality approach, closely related to the general study of
typical partial entropies [5], does allow one to shed light on the general
thermodynamic behaviour of quantum systems without getting bogged down in
unnecessary detail, and has been more recently extended to other quantifiers
as well, such as quantum coherence [6]. Besides, typical entanglement and
states play an important role in quantum information science, both directly
and through the construction of related $t$-designs, with applications to
quantum cryptography, teleportation thresholds, state estimation, channel
capacities and randomized benchmarking [7, 8, 9, 10, 11, 12, 13].

However, it must be noted that actual physical interactions hardly ever comply
with the notion of typicality set out above, as they do not extend to the
whole system-plus-environment Hilbert space. It is therefore worthwhile to
reconsider the study of typical states and entropies by generalising the
system/environment framework to one where the system is comprised of an inner
part with Hilbert space $\mathcal{H}_{I}$ and of a boundary with Hilbert
$\mathcal{H}_{B}$, and where only the latter interacts with the environment,
with Hilbert space $\mathcal{H}_{E}$, through a Haar-distributed unitary over
the space $\mathcal{H}_{B}\otimes\mathcal{H}_{E}$, as per the sketch below.

$(\stackrel{{\scriptstyle
I}}{{\bullet}}\quad\underbrace{\stackrel{{\scriptstyle
B}}{{\bullet}})\quad\stackrel{{\scriptstyle E}}{{\bullet}}}_{U}\;.$ (1)

Somewhat loosely, we refer to this framework of restricted, ‘partial’ mixing
as ‘random scattering’. This framework will allow us to inquire into how
equilibration [14] and thermalisation play out for many-body systems with only
a partial interface interacting with the environment.

Notably, a set-up similar to the above one plays a central role in the seminal
Hayden-Preskill model for information retrieval from a black hole’s Hawking
radiation. Therein, the internal dynamics of a black hole is modelled through
a random, Haar-distributed unitary acting only on two specific subsystems
(which are maximally entangled with two other reference systems). Building on
previous work [9, 10], Hayden and Preskill make use of an upper bound
constraining the trace distance between the averaged system reference state
and a product state, given in terms of dimensions and inner state purity. Such
structured black hole models, based on the selective application of
‘pseudorandom’ interactions, are currently drawing substantial attention [16,
17, 18, 19]. It is also worthwhile to mention that, in a similar vein,
pseudorandom interactions are finding application in other fundamental
investigations, such as the study of wormhole growth in the AdS/CFT
correspondence [20].

In this work, we determine the average state resulting from a scattering
event, and show that it is always completely uncorrelated across the inner
subsystem-boundary partition; also, by applying a standard argument based on
Levy’s lemma, we exponentially bound the fluctuations around such a mean state
for initial states that are separable across the inner subsystem-boundary
partition, and for any initial state of the environment. Next, we determine
the average purity of the system (inner plus boundary) state under Haar-
averaged scattering interactions for an initial global pure state, as well as
for an initial factorised state of the environment as a function of the
initial inner subsystem-boundary Schmidt coefficients and subsystems’
dimensions alone.

The plan of the paper is as follows: in Section III.1 we define the random
scattering process, set out the basic terminology and introduce some of the
quantities adopted; in Section III we determine the average unconditional
state and study its fluctuations for initially separable states; in Section IV
we determine the mean local purity of initially pure scattered states and, by
addressing numerically GHZ and W states, present evidence that the
fluctuations around such mean values shrink with increasing environmental
dimension; in Section V we draw some conclusions; derivations and preliminary
technical material concerning Haar averages, Levy’s lemma and concentration of
measure are deferred to the appendices.

##  II The scattering map

Let us consider the scattering map $\Phi$ sending the global initial state
$\rho$, defined on the Hilbert space
$\mathcal{H}_{I}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{E}$, into the state
of subsystem $IB$ after a random scattering interaction acting on subsystem
$BE$ alone:

$\Phi(\rho)=\hbox{Tr}_{E}\left[\int_{H}\hbox{dU}_{BE}\left(\mathbb{I}_{I}\otimes
U_{BE}\right)\rho\left(\mathbb{I}_{I}\otimes
U_{BE}\right)^{\dagger}\right]\,,$ (2)

where the Haar average over the unitary group, denoted with
$\int_{H}\hbox{dU}_{BE}$, takes into account the lack of information about the
boundary-environment interaction in a random scattering process. To emphasise
the Haar averaging involved, we shall also refer to the scattered system state
as $\langle\rho_{IB}\rangle=\Phi(\rho)$ in what follows. Basic techniques for
the evaluation of Haar averages are sketched in Appendix A.

As customary in this context, we shall adopt the purity
$\mu=\hbox{Tr}(\rho^{2})$ as an entropic measure. The purity is related to the
so-called linear entropy $S_{L}=(1-\mu)$, which can in turn be derived from
the von Neumann entropy $S_{V}=-\hbox{Tr}(\rho\ln\rho)$ through a Taylor
expansion at lowest order in the quantum state. At variance with the von
Neumann entropy, the purity is not endowed with a direct operational
interpretation but is much more expedient to evaluate and is still very
effective in characterising the mixedness and thermal character of a state,
especially around its extremal values ($1$ for pure states and $1/d$ for
maximally mixed states, $d$ being the system dimension). It is easy to see
directly from its definition that the purity is a convex function on the set
of quantum states.

In the following we will consider both the purity
$\hbox{Tr}\left(\Phi(\rho)^{2}\right)$ of the scattered state, as well as the
average of the purity under the Haar measure

$\langle\mu(\rho_{IB})\rangle=\hbox{Tr}_{IB}\left[\int_{H}\hbox{dU}\left(\hbox{Tr}_{E}\left(\mathbb{I}_{I}\otimes
U\right)\rho\left(\mathbb{I}_{I}\otimes U\right)^{\dagger}\right)^{2}\right],$
(3)

Both $\langle\rho_{IB}\rangle$ and $\langle\mu(\rho_{IB})\rangle$ can be
evaluated exactly and, as we will see in the next sections, the fluctuations
around them can be bound by means of Levy’s lemma.

Borrowing from well-established terminology in quantum control theory, we will
refer henceforth to the state $\langle\rho_{IB}\rangle$ as to the
‘unconditional’ state, which is prepared after a scattering event when the
interaction between boundary and environment is completely unknown, and thus
Haar-distributed. The unconditional state is an average of conditional states
$\rho_{IB}$, which would be prepared if information about the unitary
interaction were somehow retrieved. The purity distribution of conditional
states may be of interest too, and will therefore be considered in what
follows.

##  III The unconditional state

Let us first analyse some remarkable properties of the unconditional state,
resulting from the Haar average, and consider their thermodynamical
implications. Throughout the paper, we will use a convention such that the
indices of all matrices and coefficients will always follow the order: inner
system-boundary-environment. Also, we shall set
$d_{X}:=\hbox{dim}(\mathcal{H}_{X})$.

###  III.1 A decoupling theorem

Quite remarkably, the scattering map of Eq.(2) has the effect of totally
suppressing any initial correlation between the boundary and the inner part of
the system.

###### Proposition 1

For any given initial state $\rho$, the expectation value over the Haar
measure of the reduced $IB$ state after the scattering interaction defined in
Eq.(2) is a factorised state:

$\langle\rho_{IB}\rangle=\rho_{I}\otimes\frac{\mathbb{I}_{B}}{d_{B}}\,,$ (4)

with

$\rho_{I}=\hbox{Tr}_{BE}(\rho)\;.$ (5)

The proof of this proposition is deferred to Appendix B.

Notice that the local parts of this unconditional state might have been
predicted by inspection, since the maximally mixed state in the boundary is a
result of the Haar-averaged mixing with the environment, whilst the inner
system is untouched by the interaction, which thus will not alter its local
state. However, the complete destruction of correlations, regardless of the
dimensions involved, is not trivial. This proposition complies with what is
known from the study of Hawking radiation [15], in a form and to a degree that
will be made more explicit in the next subsection.

Before moving on to quantifying the statistical deviation from the
unconditional state, it is worthwhile to spend a few words about its
thermodynamic implications. If one takes the stance that thermalisation can be
justified by localised, yet unknown, surface interactions with an environment,
this statement shows that environmental thermalisation must be entirely
mediated by the surface, i.e. it cannot act on the bulk of a system directly
through previous correlations, regardless of the subsystem sizes at play.

###  III.2 Fluctuations around the unconditional state

Levy’s lemma (see Appendix E for a discussion of its derivation) is a powerful
standard tool to characterise the Haar-generated distribution of conditional
states. Let us first recall the lemma:

###### Lemma 1

(Levy’s lemma) Given a function $f$ : $S^{d}\rightarrow\mathbbm{R}$ defined on
the d-dimensional hypersphere $\mathbbm{S}^{d}$, and a point
$\phi\in\mathbbm{S}^{d}$ chosen at random, the probability $P$ for $f$ to
deviate from its mean value is given by

$\hbox{P}\left[|f(\phi)-\langle f\rangle|\geq\epsilon\right]\leq
2\hbox{exp}\left(-\frac{(d+1)\epsilon^{2}}{9\pi^{3}\eta^{2}}\right)\,,$ (6)

where $\epsilon$ is an arbitrarily small positive constant and $\eta$ is the
Lipshitz constant of $f$, i.e. $\eta$ :
$|f(\phi_{1})-f(\phi_{2})|\leq\eta|\phi_{1}-\phi_{2}|$,
$\forall(\phi_{1},\phi_{2})\in\mathbbm{S}^{d}$.

The above lemma can be applied anytime one deals with pure global quantum
states which, for a Hilbert space of dimension $d$, live on the surface of a
$(2d-1)$ dimensional hypersphere, and is in fact key to thermodynamic
typicality arguments, as in [4]. In our case though, the action of the random,
Haar-averaged unitaries is constrained to the $IB$ subspace, so we cannot
apply the lemma directly.

Yet, let us consider an initial state that is separable across the inner-
boundary sector, i.e., a state of the form

$\displaystyle\rho=$ $\displaystyle\sum_{j}p_{j}\rho_{I,j}\otimes\rho_{BE,j}$
(7) $\displaystyle=$
$\displaystyle\sum_{j,k}p_{j}p_{BE,j,k}\rho_{I,j}\otimes|\psi_{BE,j,k}\rangle\langle\psi_{BE,j,k}|\,,$
(8)

where $p_{j}$ and $p_{BE,j,k}$ are positive probabilities adding up to one
when summed over $j$ and $k$ respectively, the $\rho_{I,j}$’s are quantum
states of the inner sector and $|\psi_{BE,j,k}\rangle$ are unit vectors of the
Hilbert space $\mathcal{H}_{B}\otimes\mathcal{H}_{E}$. The action of the Haar-
averaged unitaries on each element the sum above results in a probability
distribution of pure states in $\mathcal{H}_{B}\otimes\mathcal{H}_{E}$ that
does not depend on $j$ and $k$, and which may be parametrised on the
$(2d_{B}d_{E}-1)$-dimensional hypersphere of pure states on such a space.
Denoting by $\phi$ the variables that parametrise the hypersphere, and by
$|\phi\rangle$ the corresponding pure state of the $BE$ subsystem, one has
that the initial state $\rho$ is mapped into the following conditional state

$\sum_{j,k}p_{j}p_{BE,j,k}\rho_{I,j}\otimes|\phi\rangle\langle\phi|=\rho_{I}\otimes|\phi\rangle\langle\phi|\,,$
(9)

where $\rho_{I}=\sum_{j}p_{j}\rho_{I,j}=\hbox{Tr}_{BE}(\rho)$. Then, as proven
in detail in Appendix F, Levy’s lemma may be applied to obtain the following
characterisation of the distribution of trace distances of the scattered
states

$P\left[\|\rho_{IB,\phi}-\Phi\left(\rho\right)\|_{1}\geq\epsilon+\sqrt{\frac{d_{B}^{2}-1}{d_{E}d_{B}+1}}\right]\leq
2\hbox{e}^{-\frac{d_{B}d_{E}\epsilon^{2}}{18\pi^{3}}}\,.$ (10)

Let us also recall that, by virtue of Helstrom’s theorem [21], the minimum
error probability in discriminating between two quantum states $\rho$ and
$\sigma$, optimised over all POVMs, is given by
$\frac{1}{2}-\frac{1}{4}\|\rho-\sigma\|_{1}$. In this precise sense, states at
vanishing trace distance become operationally indistiguishable, an argument
that was also applied to justify perceived thermodynamic regularities in the
appropriate limit [4].

Inequality (10) only applies to separable inner-boundary initial states (which
also include cases where the environment is initially completely uncorrelated,
although this requirement was not explicitly needed), as Levy’s lemma does not
allow one to make any direct inference for initial entangled states, where the
effect of random unitaries on off-diagonal elements must be taken into
account. Under such a separability assumption, the dimension of the inner
Hilbert space does not play any role in bounding the typical fluctuations. We
will explore more general initial states and address situations where the
inner Hilbert space does play a role, by investigating the purity distribution
of the scattered states in the next section.

Notice also that, as typical in such cases [4], our argument does not
demonstrate an exponential shrinking of the deviation from the average when
the trace distance is arbitrarily small, although this is recovered in the
limit $d_{E}\gg d_{B}$: As one should expect, concentration of measure occurs,
even under partial averaging, as the environment dimension grows, regardless
of the other dimensions involved.

##  IV Mean scattered purity of initial pure states

A scenario which is amenable to an insightful evaluation is that of initial
pure states. Denoting with $\\{|i\rangle\\}$, $\\{|b\rangle\\}$ and
$\\{|e\rangle\\}$ the bases of choice in the spaces $\mathcal{H}_{I}$,
$\mathcal{H}_{B}$ and $\mathcal{H}_{E}$, and with $\psi_{ibe}$ arbitrary state
vector coefficients, one would have for pure states:

$\rho=|\Psi\rangle\\!\langle\Psi|=\sum_{ii^{\prime}bb^{\prime}ee^{\prime}}\psi_{ibe}\psi_{i^{\prime}b^{\prime}e^{\prime}}^{*}|ibe\rangle\\!\langle
i^{\prime}b^{\prime}e^{\prime}|\,.$ (11)

Through a rather lengthy calculation, reported in Appendix (C), one then
obtains the following mean purity

$\langle\mu(\rho_{IB})\rangle=\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{B}d_{E})^{2}-1}\left(\Delta-\Gamma\right)\,,$
(12)

where we have defined

$\displaystyle\Delta$ $\displaystyle:=$
$\displaystyle\sum_{\begin{subarray}{c}i,i^{\prime},(be),(be)^{\prime}\\\
i\neq
i^{\prime},(be)\neq(be)^{\prime}\end{subarray}}|\psi_{i(be)}|^{2}|\psi_{i^{\prime}(be)^{\prime}}|^{2}\,,$
(13) $\displaystyle\Gamma$ $\displaystyle:=$
$\displaystyle\sum_{\begin{subarray}{c}i,i^{\prime},(be),(be)^{\prime}\\\
i\neq
i^{\prime},(be)\neq(be)^{\prime}\end{subarray}}\psi_{i(be)}\psi_{i^{\prime}(be)}^{*}\psi_{i^{\prime}(be)^{\prime}}\psi_{i(be)^{\prime}}^{*}\,.$
(14)

When the initial system state is pure, and the system-environment state is
factorised, the only additional parameters at play, other than the Hilbert
spaces’ dimensions, are the Schmidt coefficients of the $IB$ initial state, on
which the mean purity must necessarily depend. As shown in Appendix D, the
latter then reads

$\langle\mu(\rho_{IB})\rangle=\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{B}d_{E})^{2}-1}\sum_{\begin{subarray}{c}jl\\\
j\neq l\end{subarray}}|c_{j}|^{2}|c_{l}|^{2}\,,$ (15)

where $\\{c_{j}\\}$ are the Schmidt coefficients of the initial $IB$ state.

Clearly, Eq. (15) implies that, when the initial state of the $IB$ system is
also separable, i.e., when
$|\Psi\rangle=|I\rangle\otimes|B\rangle\otimes|E\rangle$, $M=1$, one recovers
the well-known result in the absence of inner system-boundary separation [26,
5]:

$\langle\mu(\rho_{IB})\rangle=\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}\,.$ (16)

This is actually true regardless the initial correlations between boundary and
environment. Indeed, when
$|\Psi\rangle=|I\rangle\otimes|BE\rangle=\sum_{i=1}^{d_{I}}\sum_{j=1}^{d_{B}d_{E}}\gamma_{i}\tau_{j}|\Gamma_{i}\rangle|T_{j}\rangle$:

$\displaystyle\Delta-\Gamma$ $\displaystyle=$
$\displaystyle\sum_{ii^{\prime}jj^{\prime}}\left[|\gamma_{i}\tau_{j}|^{2}|\gamma_{i^{\prime}}\tau_{j^{\prime}}|^{2}-\gamma_{i}\tau_{j}\gamma_{i^{\prime}}^{*}\tau_{j}^{*}\gamma_{i^{\prime}}\tau_{j^{\prime}}\gamma_{i}^{*}\tau_{j^{\prime}}^{*}\right]$
(17) $\displaystyle=$
$\displaystyle\sum_{ii^{\prime}jj^{\prime}}\left[|\gamma_{i}\tau_{j}|^{2}|\gamma_{i^{\prime}}\tau_{j^{\prime}}|^{2}-|\gamma_{i}\tau_{j}|^{2}|\gamma_{i^{\prime}}\tau_{j^{\prime}}|^{2}\right]$
$\displaystyle=$ $\displaystyle 0\,,$

so that Eq. (12) reduces to the standard bipartite system-environment case.

Figure 1: Purity of the state resulting from a random scattering boundary-
environment interaction for (see Eqs.(IV,23)) (a) $\rho^{W}$, (b)
$\rho^{GHZ}$, (c) $\tilde{\rho}^{W}$, (d) $\tilde{\rho}^{GHZ}$ initial states.
Each value on the $x$ axis labels a different $U_{BE}$ random extraction. The
straight horizontal lines in the plots represent the expected values of the
purity, computed with Eqs.(IV), 23, where in (a),(b) $d_{I}=d_{B}=2$, whereas
in (c),(d) $d_{I}=2$, $d_{B}=4$.

If, on the other hand, the initial inner system-boundary state is maximally
entangled, assuming, as is reasonable, $d_{B}\leq d_{I}$, one can insert
$c_{j}=1/\sqrt{d_{B}}$ $\forall$ $j$ to obtain, from Eq. (15), the mean purity

$\langle\mu(\rho_{IB})\rangle=\frac{(d_{B}^{2}-1)d_{E}+d_{E}^{2}-1}{(d_{B}^{2}d_{E}^{2}-1)}\;,$
(18)

which, in the limit $d_{E}\rightarrow\infty$, yields
$\langle\mu(\rho_{IB})\rangle=1/d_{B}^{2}$: the boundary’s dimension
constrains the equilibrium mean purity to a value higher than the minimum
value $1/(d_{I}d_{B})$.

This corresponds to the fact that full mixing (and, in settings where the
total energy is set, full thermalisation) cannot be achieved via interaction
with a boundary, unless further boundary-inner subsystem interactions are also
taken into account.

The fundamental difference between our scattering model and the conventional
scenario in which the whole system interacts with an environment lies in the
fact that in the latter case the expectation value of the purity is
independent of the initial state, whilst in the former case, in general, this
is clearly not true. This is the case even in the limit of infinite dimension
of the environment:

$\lim_{d_{E}\to\infty}\langle\mu(\rho_{IB})\rangle=\frac{1}{d_{B}}\left(1-\Delta+\Gamma\right)\,.$
(19)

Here, Levy’s lemma cannot be applied directly to bound the fluctuations around
these mean values, as the random unitary transformations do not in general
result in a state distribution that can be parametrised on a hypersphere.
However, we have checked numerically the validity of Eqs. (12) foro initial
GHZ and W states. For $N$ qubits, these two classes of states, which are
important for the study of multipartite entanglement [23], are defined as:

$\displaystyle|GHZ\rangle_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|0\rangle^{\otimes N}+|1\rangle^{\otimes
N}\right)\,,$ (20) $\displaystyle|W\rangle_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\left(|10\dots 0\rangle+|01\dots
0\rangle+\dots+|00\dots 1\rangle\right).$

Defining $\rho^{GHZ}=|GHZ\rangle\\!\langle GHZ|_{N}$ and
$\rho^{W}=|W\rangle\\!\langle W|_{N}$, a direct calculation leads to the
following mean scattered purities:

$\displaystyle\langle\mu(\rho_{IB}^{GHZ})\rangle$ $\displaystyle=$
$\displaystyle\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{B}d_{E})^{2}-1}\frac{1}{2}$
$\displaystyle\langle\mu(\rho_{IB}^{W})\rangle$ $\displaystyle=$
$\displaystyle\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{B}d_{E})^{2}-1}\frac{2N_{I}(N-N_{I})}{N^{2}}\;,$

where $D=d_{I}d_{B}d_{E}=2^{N}$ (here, the $X$ subsystem comprises
$N_{X}=\log_{2}(d_{X})$ qubits). Interestingly, the average scattered purity
of W states is affected by the inner subsystem dimension. Also, it can be seen
that the average purity of initial W states is larger than the GHZ one (for
given $d_{B}$ and $d_{E}$) as long as $N_{I}<N/2$, i.e., as long as the
boundary interaction Hilbert space is not too thin compared with the system
bulk.

We have also considered the case where three-qubit GHZ and W states interact
with an $N_{E}=N-3$ qubit environment prepared in a fiducial state
$\tilde{\rho}^{GHZ}:=|GHZ\rangle\\!\langle GHZ|_{3}\otimes|0\rangle\\!\langle
0|_{E}^{\otimes(N-3)}$, $\tilde{\rho}^{W}:=|W\rangle\\!\langle
W|_{3}\otimes|0\rangle\\!\langle 0|_{E}^{\otimes(N-3)}$, finding

$\displaystyle\langle\mu(\tilde{\rho}_{IB}^{GHZ})\rangle$ $\displaystyle=$
$\displaystyle\langle\mu(\rho_{IB}^{GHZ})\rangle\,,$ (23)
$\displaystyle\langle\mu(\tilde{\rho}_{IB}^{W})\rangle$ $\displaystyle=$
$\displaystyle\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{B}d_{E})^{2}-1}\frac{4}{9}\,.$
(24)

Note that these formulae apply to both $N_{I}=1$, $N_{B}=2$ and to $N_{I}=2$,
$N_{B}=1$, since the Schmidt coefficients across the inner boundary partition
are the same in both set-ups.

As can be appreciated in Fig.(1), the fluctuations around the expected value
of $\mu$ are quite large for an environment composed of $N_{E}=1$, $N_{E}=2$
qubits, but are already strongly suppressed for $N_{E}\approx 10$ and almost
completely damped for $N_{E}\approx 100$.

##  V Conclusions

Summing up, we have considered a scattering scenario where random unitary
transformations act jointly on an environment and a boundary subsystem rather
than on the whole system, which comprises an inner subsystem too, and derived
the following:

  * •

A proof that any single, Haar-distributed scattering event is able to
disentangle completely boundary and inner subsystems.

  * •

A proof that, for initially separable inner-boundary states, the conditional
scattered states concentrate exponentially in trace norm around the average
conditional state, by virtue of Levy’s lemma.

  * •

An analytical formula for the mean purity of initially pure scattered states
in terms of the subsystems’ dimensions and initial state’s coefficients.

  * •

An analytical formula for the mean scattered purity of initially globally pure
states uncorrelated with the environment, in terms of the Schmidt coefficients
of the initial state.

Besides, by addressing initial GHZ and W states, we have provided numerical
evidence for the concentration of measure of scattered purities with
increasing environmental dimension.

Our approach and findings are a step towards a generalised study of
thermalisation and equilibration for structured systems and restricted
interactions, and may be of interest to quantum thermodynamics approaches
based on typicality and other fundamental investigations, such as the study of
black hole entropy.

###### Acknowledgements.

We wish to thank the Madonna del Carmelo for xer unrelenting support.

## References

  * [1] E. Lubkin, J. Math. Phys. 19, 1028 (1978).
  * [2] J. Gemmer, A. Otte, and G. Mahler, Phys. Rev. Lett. 86, 1927 (2001).
  * [3] J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics (Springer, Dordrecht, 2009).
  * [4] S. Popescu, A.J. Short, and A. Winter, Nat. Phys. 2, 754 (2006).
  * [5] O.C.O. Dahlsten, C. Lupo, S. Mancini, and A. Serafini, J. Phys. A: Math. Theor. 47, 363001 (2014).
  * [6] U. Singh, L. Zhang, and A.K. Pati, Phys. Rev. A 93, 032125 (2016).
  * [7] A. Hayashi, T. Hashimoto, and M. Horibe, Phys. Rev. A 72, 032325 (2005).
  * [8] S. Iblisdir and J. Roland, Phys. Lett. A, 358, 368 (2006).
  * [9] P. Hayden, M. Horodecki, A Winter, and J. Yard, Open Syst. Inf. Dyn. 15, 7 (2008).
  * [10] A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, Proc. R. Soc. A 465, 2537 (2009).
  * [11] J. Emerson, R. Alicki, and K. Życzkowski, J. Opt. B: Quantum Semiclass. Opt. 7, S347 (2005).
  * [12] E. Knill, D. Leibfried, R. Reichle, J. Britton, R.B. Blakestad, J.D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D.J. Wineland, Phys. Rev. A 77, 012307 (2008).
  * [13] E. Magesan, J.M. Gambetta, and J. Emerson, Phys. Rev. A 85, 042311 (2012).
  * [14] N. Linden, S. Popescu, A.J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009).
  * [15] P. Hayden and J. Preskill, J. High Energy Phys. 2007, 120 (2007).
  * [16] B. Yoshida, Phys. Rev. D 100, 086001 (2019).
  * [17] L. Piroli, C. Sünderhauf, and X.-L. Qi, JHEP 2020, 63 (2020)
  * [18] C. Akers, N. Engelhardt, D. Harlow, G. Penington, and S. Vardhan, arXiv:2207.06536.
  * [19] I.H. Kim and J. Preskill, JHEP 2023, 233 (2023).
  * [20] A. Bouland, B. Fefferman, and U. Vazirani, arXiv:1910.14646.
  * [21] C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
  * [22] A.S. Holevo, Theor. Prob. Appl. 23, 411 (1978).
  * [23] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 864 (2009).
  * [24] S. Aubert and C.S. Lam, J. Math. Phys. 44, 6112 (2003).
  * [25] W. Fulton and J. Harris, Representation Theory (Springer, 2004).
  * [26] M. Müller, Lecture Notes on Selected Advanced Topics in Quantum Information, (University of Waterloo, 2012).
  * [27] M. Ledoux,The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs 89 (American Mathematical Society, 2001).
  * [28] M. Gerken, Selected Topics in Mathematical Physics: Quantum Information, (2013).

##  Appendix A Integrals over the unitary group

Here we just summarise the results of [24] which are relevant for the purposes
of our model.

The maps Eqs.(2,3) involve the calculation of integrals of the form

$\int_{H}\hbox{dU}\left[U^{*}_{i,j}U_{k,l}\right]\,,\quad\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{2},j_{2}}U_{k_{1},l_{1}}U_{k_{2},l_{2}}\right]\,,$
(25)

where $\int_{H}$ means integrating over the Haar measure. Integrals of this
kind can be tackled using Schur’s lemma (see e.g. [25, 26]), but here we
follow the, somewhat easier, approach described in [24].

In general, for some degrees of the polynomials $p$, $q$ in $U^{*}$ and $U$,
one wants to compute

$\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}\dots
U^{*}_{i_{p},j_{p}}U_{k_{1},l_{1}}\dots
U_{k_{q},l_{q}}\right]=\int_{H}\hbox{dU}\left[\prod_{a=1}^{p}U^{*}_{i_{a},j_{a}}\prod_{b=1}^{q}U_{k_{b},l_{b}}\right]\equiv\int_{H}\hbox{dU}\left[U^{*}_{I_{p}J_{p}}U_{K_{q}L_{q}}\right]\equiv\langle
I_{p},J_{p}|K_{q},L_{q}\rangle\,,$ (26)

where we have defined $X_{p}=(x_{1},x_{2},\dots x_{p})$. It is shown in [24]
that the only non-zero integrals are the ones in which the degrees are such
that $p=q$ (thus we will drop this index), $K=I$ and $L=J_{Q}$, where $J_{Q}$
is any permutation of the indices in the set $J$:

$\langle I_{p},J_{p}|K_{q},L_{q}\rangle=\langle I,J|I,J_{Q}\rangle\,.$ (27)

When $p=1$

$\int_{H}\hbox{dU}\left[U^{*}_{i,j}U_{k,l}\right]=\int_{H}\hbox{dU}\left[U^{*}_{i,j}U_{i,j}\right]=\langle
i,j|i,j\rangle=\frac{1}{d}\,,$ (28)

where $d$ is the dimension of $U$.

When $p=2$, the non-zero integrals are:

$\displaystyle\bullet\,(i_{1}\neq i_{2},j_{1}\neq j_{2})$ $\displaystyle:$
$\displaystyle\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{2},j_{2}}U_{i_{1},j_{1}}U_{i_{2},j_{2}}\right]=\frac{1}{d^{2}-1}\,;$
(29) $\displaystyle\bullet\,(i_{1}\neq i_{2},j_{1}\neq j_{2})$
$\displaystyle:$
$\displaystyle\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{2},j_{2}}U_{i_{1},j_{2}}U_{i_{2},j_{1}}\right]=-\frac{1}{d(d^{2}-1)}\,;$
(30) $\displaystyle\bullet\,(i_{1}=i_{2},j_{1}\neq j_{2})$ $\displaystyle:$
$\displaystyle\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{1},j_{2}}U_{i_{1},j_{1}}U_{i_{1},j_{2}}\right]=\frac{1}{d(d+1)}\,,$
(31)
$\displaystyle\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{1},j_{2}}U_{i_{1},j_{2}}U_{i_{1},j_{1}}\right]=\frac{1}{d(d+1)}\,;$
$\displaystyle\bullet\,(i_{1}\neq i_{2},j_{1}=j_{2})$ $\displaystyle:$
$\displaystyle\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{2},j_{1}}U_{i_{1},j_{1}}U_{i_{2},j_{1}}\right]=\frac{1}{d(d+1)}\,;$
(32) $\displaystyle\bullet\,(i_{1}=i_{2},j_{1}=j_{2})$ $\displaystyle:$
$\displaystyle\int_{H}\hbox{dU}\left[U^{*}_{i_{1},j_{1}}U^{*}_{i_{1},j_{1}}U_{i_{1},j_{1}}U_{i_{1},j_{1}}\right]=\frac{2}{d(d+1)}\,.$
(33)

Knowing how to deal with this kind of integrals allows one to compute relevant
quantities such as the mean reduced state $\overline{\rho_{S}}$ of a bipartite
system-environment ($SE$) density matrix:

$\displaystyle\overline{\rho_{S}}$ $\displaystyle=$
$\displaystyle\hbox{Tr}_{E}(\langle\rho_{SE}\rangle_{H})=\hbox{Tr}_{E}\left\\{\int_{H}\hbox{dU}\left[U\rho_{SE}U^{\dagger}\right]\right\\}$
(34) $\displaystyle=$
$\displaystyle\hbox{Tr}_{E}\left\\{\int_{H}\hbox{dU}\sum_{acge=1}^{d_{S}}\sum_{ik=1}^{d_{S}}\sum_{bdhf=1}^{d_{E}}\sum_{jl=1}^{d_{E}}U_{(ab)(cd)}U^{*}_{(ef)(gh)}\psi_{ij}\psi^{*}_{kl}|ab\rangle\\!\langle
cd|\cdot|ij\rangle\\!\langle kl|\cdot|gh\rangle\\!\langle ef|\right\\}$
$\displaystyle=$
$\displaystyle\hbox{Tr}_{E}\left\\{\sum_{ikae=1}^{d_{S}}\sum_{jlbf=1}^{d_{E}}\psi_{ij}\psi^{*}_{kl}\underbrace{\int_{H}\hbox{dU}\left[U_{(ab)(ij)}U_{(ef)(kl)}^{*}\right]}_{=\frac{1}{d_{S}d_{E}}\delta^{ab}_{ef}\delta^{ij}_{kl}}\right\\}$
$\displaystyle=$
$\displaystyle\hbox{Tr}_{E}\left\\{\frac{1}{d_{S}d_{E}}\sum_{ij}|\psi_{ij}|^{2}\sum_{a=1}^{d_{S}}\sum_{b=1}^{d_{E}}|ab\rangle\\!\langle
ab|\right\\}=\frac{1}{d_{S}d_{E}}\sum_{\xi=1}^{d_{E}}\langle\xi|\sum_{a=1}^{d_{S}}\sum_{b=1}^{d_{E}}|ab\rangle\\!\langle
ab|\cdot|\xi\rangle=\frac{1}{d_{S}}\sum_{a=1}^{d_{S}}|a\rangle\\!\langle
a|\,.$

##  Appendix B Proof of proposition 1

Proof: Due to the linearity of averaging and partial tracing, it will suffice
to prove this statement for initially pure states. So, substituting Eq.(11)
into Eq.(2) and expressing the scattering unitaries in terms of their
components in the $BE$ basis,
$U_{BE}=\sum_{i,k=1}^{d_{B}}\sum_{j,l=1}^{d_{E}}U_{(ij)(kl)}|ij\rangle\\!\langle
kl|$, yields

$\displaystyle\Phi(\rho_{IBE})$ $\displaystyle=$
$\displaystyle\hbox{Tr}_{E}\left[\int_{H}\hbox{dU}\left(\mathbb{I}_{I}\otimes
U_{BE}\right)\rho_{IBE}\left(\mathbb{I}_{I}\otimes
U_{BE}\right)^{\dagger}\right]$ (35) $\displaystyle=$
$\displaystyle\sum_{\xi=1}^{d_{E}}\langle\xi|\int_{H}\hbox{dU}\sum_{\begin{subarray}{c}jklm\\\
mnop\end{subarray}}\sum_{\begin{subarray}{c}sbe\\\
s^{\prime}b^{\prime}e^{\prime}\end{subarray}}U_{(ij)(kl)}\psi_{sbe}|ij\rangle\langle{kl}|sbe\rangle\langle{s^{\prime}b^{\prime}e^{\prime}}|mn\rangle\\!\langle
op|\psi_{s^{\prime}b^{\prime}e^{\prime}}^{*}U_{(op)(mn)}^{*}|\xi\rangle$
$\displaystyle=$
$\displaystyle\sum_{\xi=1}^{d_{E}}\sum_{\begin{subarray}{c}jklm\\\
mnop\end{subarray}}\sum_{\begin{subarray}{c}sbe\\\
s^{\prime}b^{\prime}e^{\prime}\end{subarray}}\psi_{sbe}\psi_{s^{\prime}b^{\prime}e^{\prime}}^{*}\underbrace{\langle\xi|{ij}\rangle\langle{kl}|sbe\rangle\\!\langle
s^{\prime}b^{\prime}e^{\prime}|{mn}\rangle\langle{op}|\xi\rangle}_{\delta_{j}^{\xi}\delta^{k}_{b}\delta_{e}^{l}\delta_{b^{\prime}}^{m}\delta_{e^{\prime}}^{m}\delta_{o}^{\xi}}\int_{H}\hbox{dU}\left[U_{(op)(mn)}^{*}U_{(ij)(kl)}\right]$
$\displaystyle=$
$\displaystyle\sum_{\xi=1}^{d_{E}}\sum_{\begin{subarray}{c}sbe\\\
s^{\prime}b^{\prime}e^{\prime}\end{subarray}}\sum_{io}\psi_{sbe}\psi_{s^{\prime}b^{\prime}e^{\prime}}^{*}|si\rangle\\!\langle
s^{\prime}o|\int_{H}\hbox{dU}\left[U_{(o\xi)(b^{\prime}e^{\prime})}^{*}U_{(i\xi)(be)}\right]=\sum_{\xi=1}^{d_{E}}\sum_{\begin{subarray}{c}sbe\\\
s^{\prime}b^{\prime}e^{\prime}\end{subarray}}\sum_{io}\psi_{sbe}\psi_{s^{\prime}b^{\prime}e^{\prime}}^{*}|si\rangle\\!\langle
s^{\prime}o|\frac{1}{d_{B}d_{E}}\delta_{i}^{o}\delta_{b^{\prime}e^{\prime}}^{be}$
$\displaystyle=$ $\displaystyle\frac{1}{d_{B}d_{E}}\sum_{\xi
ss^{\prime}bei}\psi_{sbe}\psi_{s^{\prime}be}^{*}|si\rangle\\!\langle
s^{\prime}i|=\sum_{ss^{\prime}be}\psi_{sbe}\psi_{s^{\prime}be}^{*}|s\rangle\\!\langle
s^{\prime}|\otimes\frac{1}{d_{B}}\sum_{i=1}^{d_{B}}|i\rangle\\!\langle
i|=\rho_{I}\otimes\overline{\rho}_{B}\,,$

where for the fourth equality we have used the results of Appendix A.

$\square$

##  Appendix C Expectation value of the purity

Here we explicitely show how to get to Eq.(12). Substituting Eq.(11) into
Eq.(3) and expressing the scattering unitaries in terms of their components in
the $BE$ basis,
$U_{BE}=\sum_{i,k=1}^{d_{B}}\sum_{j,l=1}^{d_{E}}U_{(ij)(kl)}|ij\rangle\\!\langle
kl|$:

$\displaystyle\langle\mu(\rho_{IB})\rangle$ $\displaystyle=$
$\displaystyle\hbox{Tr}_{IB}\left\\{\int_{H}\hbox{dU}\left[\hbox{Tr}_{E}\left(\left(\mathbb{I}_{I}\otimes
U_{BE}\right)\rho_{IBE}\left(\mathbb{I}_{I}\otimes
U_{BE}\right)^{\dagger}\right)\right]^{2}\right\\}$ (36) $\displaystyle=$
$\displaystyle\int_{H}\hbox{dU}\sum_{\gamma\gamma^{\prime}\beta\beta^{\prime}\xi\xi^{\prime}}\langle(\gamma\beta)\xi|\sum_{\begin{subarray}{c}ibe\\\
i^{\prime}b^{\prime}e^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}jklm\\\
nopq\end{subarray}}\psi_{ibe}\psi_{i^{\prime}b^{\prime}e^{\prime}}^{*}U_{(jk)(lm)}U^{*}_{(pq)(no)}|jk\rangle\langle{lm}|ibe\rangle\langle{i^{\prime}b^{\prime}e^{\prime}}|no\rangle\langle{pq}|(\gamma^{\prime}\beta^{\prime})\xi\rangle\\!\langle(\gamma^{\prime}\beta^{\prime})\xi^{\prime}|$
$\displaystyle\times$
$\displaystyle\sum_{\begin{subarray}{c}i^{2}b^{2}e^{2}\\\
i^{3}b^{3}e^{3}\end{subarray}}\sum_{\begin{subarray}{c}rstu\\\
vwxy\end{subarray}}\psi_{i^{2}b^{2}e^{2}}\psi_{i^{3}b^{3}e^{3}}^{*}U_{(rs)(tu)}U^{*}_{(xy)(vw)}|rs\rangle\langle{tu}|i^{2}b^{2}e^{2}\rangle\langle{i^{3}b^{3}e^{3}}|vw\rangle\langle{xy}|(\gamma^{\prime}\beta^{\prime})\xi^{\prime}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\gamma\gamma^{\prime}\beta\beta^{\prime}\xi\xi^{\prime}}\sum_{\begin{subarray}{c}beb^{\prime}e^{\prime}\\\
b^{2}e^{2}b^{3}e^{3}\end{subarray}}\psi_{\gamma
be}\psi^{*}_{\gamma^{\prime}b^{\prime}e^{\prime}}\psi_{\gamma^{\prime}b^{2}e^{2}}\psi^{*}_{\gamma
b^{3}e^{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta^{\prime}\xi)(b^{\prime}e^{\prime})}U^{*}_{(\beta\xi^{\prime})(b^{3}e^{3})}U_{(\beta\xi)(be)}U_{(\beta^{\prime}\xi^{\prime})(b^{2}e^{2})}\right]$
$\displaystyle\equiv$ $\displaystyle I+J\,,$

where we have split the sum into two components

$\displaystyle I$ $\displaystyle:=$
$\displaystyle\sum_{\gamma\beta\beta^{\prime}\xi\xi^{\prime}}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma n_{1}}\psi_{\gamma n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta^{\prime}\xi)n_{1}}U^{*}_{(\beta\xi^{\prime})n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta^{\prime}\xi^{\prime})n_{2}}\right]\,,$
$\displaystyle J$ $\displaystyle:=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\beta\beta^{\prime}\xi\xi^{\prime}\\\
(\gamma\neq\gamma^{\prime})\end{subarray}}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma^{\prime}n_{1}}\psi_{\gamma^{\prime}n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta^{\prime}\xi)n_{1}}U^{*}_{(\beta\xi^{\prime})n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta^{\prime}\xi^{\prime})n_{2}}\right]\,.$

In the above equalities we have also shortened the notation, merging the
boundary and environment indices pertaining to both the coefficients of
$|\Psi\rangle$ and $U$ into a single one. Let us take care of $I$ first.

$\displaystyle I$ $\displaystyle=$
$\displaystyle\underbrace{\sum_{\begin{subarray}{c}\gamma\beta\xi\xi^{\prime}\\\
\xi\neq\xi^{\prime}\end{subarray}}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma n_{1}}\psi_{\gamma n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta\xi)n_{1}}U^{*}_{(\beta\xi^{\prime})n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta\xi^{\prime})n_{2}}\right]}_{=:\hbox{{I}}(\xi\neq\xi^{\prime})}+$
$\displaystyle\underbrace{\sum_{\begin{subarray}{c}\gamma\beta\beta^{\prime}\xi\\\
\beta\neq\beta^{\prime}\end{subarray}}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma n_{1}}\psi_{\gamma n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta^{\prime}\xi)n_{1}}U^{*}_{(\beta\xi)n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta^{\prime}\xi)n_{2}}\right]}_{=:\hbox{{II}}(\beta\neq\beta^{\prime})}+$
$\displaystyle\underbrace{\sum_{\gamma\beta\xi}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma n_{1}}\psi_{\gamma n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta\xi)n_{1}}U^{*}_{(\beta\xi)n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta\xi)n_{2}}\right]}_{=:\hbox{{III}}(\beta\xi=\beta^{\prime}\xi^{\prime})}\,.$

We should now further decompose the sums above to get to a sum of integrals
like Eq.s(29-33) of Appendix A. Writing it down explicitly would be rather
unmanageable, though. Arguably the best way to work it out is to group the
decomposition into a table.

$\sum_{n_{0}n_{1}n_{2}n_{3}}$ | I$(\xi\neq\xi^{\prime})$ | II$(\beta\neq\beta^{\prime})$ | III$(\beta\xi=\beta^{\prime}\xi^{\prime})$
---|---|---|---
$n_{0}=n_{1}\neq n_{2}=n_{3}$ |  $d_{B}d_{E}(d_{E}-1)\cdot$(29) |  $d_{B}d_{E}(d_{B}-1)\cdot$(30) |  $d_{B}d_{E}\cdot$(31)
$n_{0}=n_{3}\neq n_{1}=n_{2}$ |  $d_{B}d_{E}(d_{E}-1)\cdot$(30) |  $d_{B}d_{E}(d_{B}-1)\cdot$(29) |  $d_{B}d_{E}\cdot$(31)
$n_{0}=n_{1}=n_{2}=n_{3}$ |  $d_{B}d_{E}(d_{E}-1)\cdot$(32) |  $d_{B}d_{E}(d_{B}-1)\cdot$(32) |  $d_{B}d_{E}\cdot$(33)
Table 1: Decomposition of $I$ into a sum of the non-zero integrals
Eqs.(29-33). In the second to fourth columns, each of these factors is
multiplied by their multiplicity.

Using Tab.(1), after a little bookkeeping we obtain

$\displaystyle I$ $\displaystyle=$
$\displaystyle\left(d(d_{E}-1)+d(d_{B}-1)\right)\left(\sum_{\begin{subarray}{c}n_{0}n_{2}\gamma\\\
n_{0}\neq n_{2}\end{subarray}}\frac{|\psi_{\gamma n_{0}}|^{2}|\psi_{\gamma
n_{2}}|^{2}}{(d^{2}-1)}+\sum_{n_{0}\gamma}\frac{|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma
n_{0}}|^{2}}{d(d+1)}-\sum_{\begin{subarray}{c}n_{0}n_{2}\gamma\\\ n_{0}\neq
n_{2}\end{subarray}}\frac{|\psi_{\gamma n_{0}}|^{2}|\psi_{\gamma
n_{2}}|^{2}}{d(d^{2}-1)}\right)$ (39) $\displaystyle+$ $\displaystyle
2d\left(\sum_{\begin{subarray}{c}n_{0}n_{2}\gamma\\\ n_{0}\neq
n_{2}\end{subarray}}\frac{|\psi_{\gamma n_{0}}|^{2}|\psi_{\gamma
n_{2}}|^{2}}{d(d+1)}+\sum_{n_{0}\gamma}\frac{|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma n_{0}}|^{2}}{d(d+1)}\right)\,$

with $d=d_{E}d_{B}$.

We can do exactly the same thing with $J$. First, decompose the sum as in
Eq.(C)

$\displaystyle J$ $\displaystyle=$
$\displaystyle\underbrace{\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}\beta\xi\xi^{\prime}\\\
\xi\neq\xi^{\prime}\end{subarray}}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma^{\prime}n_{1}}\psi_{\gamma^{\prime}n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta\xi)n_{1}}U^{*}_{(\beta\xi^{\prime})n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta\xi^{\prime})n_{2}}\right]}_{\hbox{{I}}(\xi\neq\xi^{\prime})}+$
$\displaystyle\underbrace{\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}\beta\beta^{\prime}\xi\\\
\beta\neq\beta^{\prime}\end{subarray}}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma^{\prime}n_{1}}\psi_{\gamma^{\prime}n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta^{\prime}\xi)n_{1}}U^{*}_{(\beta\xi)n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta^{\prime}\xi)n_{2}}\right]}_{\hbox{{II}}(\beta\neq\beta^{\prime})}+$
$\displaystyle\underbrace{\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\beta\xi}\sum_{n_{0}n_{1}n_{2}n_{3}}\psi_{\gamma
n_{0}}\psi^{*}_{\gamma^{\prime}n_{1}}\psi_{\gamma^{\prime}n_{2}}\psi^{*}_{\gamma
n_{3}}\int_{H}\hbox{dU}\left[U^{*}_{(\beta\xi)n_{1}}U^{*}_{(\beta\xi)n_{3}}U_{(\beta\xi)n_{0}}U_{(\beta\xi)n_{2}}\right]}_{\hbox{{III}}(\beta\xi=\beta^{\prime}\xi^{\prime})}\,,$

and then use Tab.(1) [the integrals are the same of Eq.(C)] to obtain, after
some algebra,

$\displaystyle J$ $\displaystyle=$ $\displaystyle
d(d_{E}-1)\left(\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\frac{\psi_{\gamma
n_{0}}\psi_{\gamma^{\prime}n_{0}}^{*}\psi_{\gamma^{\prime}n_{2}}\psi_{\gamma
n_{2}}^{*}}{(d^{2}-1)}-\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\frac{|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{2}}|^{2}}{d(d^{2}-1)}+\sum_{\begin{subarray}{c}n_{0}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\frac{|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{0}}|^{2}}{d(d+1)}\right)$ (41)
$\displaystyle+$ $\displaystyle
d(d_{B}-1)\left(\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\frac{|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{2}}|^{2}}{(d^{2}-1)}-\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\frac{\psi_{\gamma
n_{0}}\psi_{\gamma^{\prime}n_{0}}^{*}\psi_{\gamma^{\prime}n_{2}}\psi_{\gamma
n_{2}}^{*}}{d(d^{2}-1)}+\sum_{\begin{subarray}{c}n_{0}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\frac{|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{0}}|^{2}}{d(d+1)}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{(d+1)}\left(\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\psi_{\gamma
n_{0}}\psi_{\gamma^{\prime}n_{0}}^{*}\psi_{\gamma^{\prime}n_{2}}\psi_{\gamma
n_{2}}^{*}\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{2}}|^{2}+2\sum_{\begin{subarray}{c}n_{0}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{0}}|^{2}\right).$

Putting it all together and rearranging a bit:

$\displaystyle I+J$ $\displaystyle=$
$\displaystyle\frac{d_{E}+d_{B}}{d+1}\left(\sum_{\begin{subarray}{c}\gamma
n_{0}n_{2}\\\ n_{0}\neq n_{2}\end{subarray}}|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma n_{2}}|^{2}+\sum_{\gamma n_{0}}|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma
n_{0}}|^{2}+\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}n_{0}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{0}}|^{2}\right)$ (42) $\displaystyle=$
$\displaystyle\left[\frac{dd_{B}-d_{E}}{(d-1)(d+1)}\right]\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{2}}|^{2}+\left[\frac{dd_{E}-d_{B}}{(d-1)(d+1)}\right]\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\psi_{\gamma
n_{0}}\psi_{\gamma^{\prime}n_{0}}^{*}\psi_{\gamma^{\prime}n_{2}}\psi_{\gamma
n_{2}}^{*}.$

The calculation to get to the final expression of the average purity is rather
tedious, but quite trivial. It only involves some algebra and some care in
grouping the right terms to form the trace of the global state. Ultimately,
this leads to

$\langle\mu(\rho_{IB})\rangle=\frac{d_{B}+d_{E}}{d_{E}d_{B}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{E}d_{B})^{2}-1}\sum_{\begin{subarray}{c}\gamma\gamma^{\prime}\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}n_{0}n_{2}\\\
n_{0}\neq n_{2}\end{subarray}}\left(|\psi_{\gamma
n_{0}}|^{2}|\psi_{\gamma^{\prime}n_{2}}|^{2}-\psi_{\gamma
n_{0}}\psi_{\gamma^{\prime}n_{0}}^{*}\psi_{\gamma^{\prime}n_{2}}\psi_{\gamma
n_{2}}^{*}\right)\,.$ (43)

$\square$

##  Appendix D Mean purity of initially uncorrelated pure states

Proof: The initial state $\rho_{IBE}=|\Psi\rangle\\!\langle\Psi|$ is such that

$|\Psi\rangle=|\Psi_{IB}\rangle\otimes|\Psi_{E}\rangle=\left(\sum_{i}^{M}c_{i}|I_{i}\rangle\otimes|B_{i}\rangle\right)\otimes\left(\sum_{j}\xi_{j}|E_{j}\rangle\right)\,,$
(44)

where $M=\hbox{min}(\left(d_{B},d_{I}\right)$. By plugging the coefficients of
the above equation into Eq.(12), we have that the last term in the sum must
($\Gamma$) be null (this is due to the fact that
$\psi_{i(be)}\propto\delta_{ib}$, because of the Schmidt decomposition). So we
have:

$\displaystyle\Delta-\Gamma$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i,i^{\prime},(be),(be)^{\prime}\\\
i\neq
i^{\prime},(be)\neq(be)^{\prime}\end{subarray}}\left[|\psi_{i(be)}|^{2}|\psi_{i^{\prime}(be)^{\prime}}|^{2}-\psi_{i(be)}\psi_{i^{\prime}(be)}^{*}\psi_{i^{\prime}(be)^{\prime}}\psi_{i(be)^{\prime}}^{*}\right]=\sum_{\begin{subarray}{c}jklm\\\
j\neq l\end{subarray}}|c_{j}\xi_{k}|^{2}|c_{l}\xi_{m}|^{2}$ (45)
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}jl\\\ j\neq
l\end{subarray}}|c_{j}|^{2}|c_{l}|^{2}\sum_{k}|\xi_{k}|^{2}\sum_{m}|\xi_{m}|^{2}=\sum_{\begin{subarray}{c}jl\\\
j\neq l\end{subarray}}|c_{j}|^{2}|c_{l}|^{2}\,,$

where the last equality follows from the fact that the reduced state of the
environment is trace-one. Hence when the initial state is separable such as is
Eq.(44), the resulting purity is:

$\langle\mu(\rho_{IB})\rangle=\frac{d_{B}+d_{E}}{d_{B}d_{E}+1}+\frac{d_{B}(1-d_{E}^{2})}{(d_{B}d_{E})^{2}-1}\sum_{\begin{subarray}{c}jl\\\
j\neq l\end{subarray}}|c_{j}|^{2}|c_{l}|^{2}\,.$ (46)

$\square$

##  Appendix E Measure concentration and Levy’s lemma

$d$-dimensional pure quantum states can be described as points on the surface
of a $(2d-1)$-dimensional unit sphere. This can be realized by expressing a
generic state in complex coordinates $|\psi\rangle=(z_{1},z_{2},\dots,z_{d})$,
where $z_{j}\in\mathbb{C}$, for $j=1,2,\dots,d$, with
$\sum_{j}^{d}|z_{j}|^{2}=1$ and writing the coordinates in real components
$z_{j}=x_{j}+iy_{j}$, so that $\sum_{j}^{d}x_{j}^{2}+\sum_{j}^{d}y_{j}^{2}=1$.

Heuristically, the phenomenon of measure concentration on a unit sphere
$S^{(2d-1)}$ in $\mathbb{R}^{2d}$ translates to the fact that almost all
surface measure of the sphere is concentrated around the equator, for any
equator. That is, for any random choice of a coordinate $x_{j}$, consider an
equator of width $\epsilon$

$E_{\epsilon}:=\\{x_{j}\in
S^{(2d-1)}\,|\,d(x_{j},0)\leq\frac{\epsilon}{2}\\}\,,$ (47)

where $d(x,y)=\hbox{arccos}\langle x,y\rangle$ $\forall x,y\in S^{(2d-1)}$ is
the angular distance. Provided a normalised surface measure
$\xi(S^{(2d-1)})=1$, it can be shown that

$\xi(E_{\epsilon})\geq 1-e^{(-kd\epsilon^{2})}\,,$ (48)

where $k>0$ is some constant.

Measure concentration is at the basis of Levy’s lemma (Sec.(IV)), as we show
in the following. The kind of Levy’s lemma we will sketch the derivation of
here is slightly different from the one applied in the next section but the
two formulations are strictly related. For space reasons, not all the details
of the calculations will be shown; the interested reader can find them in
[27].

To proceed, we first need to define two quantities:

###### Definition 1

(Median): Let $X$ be a metric space and $f\,:\,X\,\rightarrow\,\mathbb{R}$ a
continuous function. A median $M_{f}$ is defined by:

$\xi\\{x\in X\,|\,f(x)\leq M_{f}\\}=\frac{1}{2}\,.$ (49)

###### Definition 2

(Concentration function): Let $X$ be a metric space and $S$ a subset of it,
with $\xi(S)=\frac{1}{2}$. For any $\epsilon>0$, the concentration function is
defined as:

$\alpha_{X}(\epsilon):=sup\\{\xi(X\setminus N_{\epsilon}(S))\\}\,,$ (50)

where $N_{\epsilon}(S)$ is the $\epsilon$-neighborhood of $S$:

$N_{\epsilon}(S):=\\{x\in X\,|\,\exists s\in S\,:\,d(s,x)<\epsilon\\}\,.$ (51)

These definitions allow one to formulate the following lemma:

###### Lemma 2

Let $X$ be a metric space and $f\,:\,X\,\rightarrow\,\mathbb{R}$ a Lipschitz-
continuous function with constant 1, then

$\xi\\{x\in X\,|\,f(x)\geq M_{f}+\epsilon\\}\leq\alpha_{X}(\epsilon)\;.$ (52)

Proof: Take $S\,:\,\\{x\,|\,f(x)\leq M_{f}\\}$ so that $\xi(S)=1/2$ and
consider a subset $B\subseteq X$ such that $f(b)\geq M_{f}+\epsilon$, $\forall
b\in B$. Because $f$ is Lipschitz continuous, all points $x\in
N_{\epsilon}(S)$ satisfy $f(x)<M_{f}+\epsilon$, so it must be $b\notin
N_{\epsilon}(S)$, $\forall b\in B$. That means $B$ is a subset of $X$:
$\\{b\in X\,|\,f(b)\geq M_{f}+\epsilon\\}\subseteq X\setminus N_{\epsilon}(S)$
and thus $\xi\\{x\in X\,|\,f(x)\geq M_{f}+\epsilon\\}\leq\xi(X\setminus
N_{\epsilon}(S))\leq\alpha_{X}(\epsilon)$.

$\square$

In terms of probabilities, and by rescaling of the $\epsilon$ to
$\epsilon\rightarrow\epsilon^{\prime}=\eta\epsilon$ for Lipschitz functions
such that $|f(x)-f(y)|\leq\eta\|x-y\|\leq\eta\epsilon$, the above lemma reads:

$\hbox{Prob}(f(x)\geq
M_{f}+\epsilon^{\prime})\leq\alpha_{X}\left(\frac{\epsilon^{\prime}}{\eta}\right)\,.$
(53)

In order to calculate the value of the concentration function
$\alpha_{S^{(2d-1)}}$, one needs to invoke the isoperimetric inequality for
the sphere (see e.g. [27]):

###### Lemma 3

(Isoperimetric inequality for the sphere): Let $A\subseteq S^{(2d-1)}$ be a
closed subset of the sphere and let $C(a,r):=\\{x\in X\,|\,d(a,x)\leq
r\\}\subset S^{(2d-1)}$ a spherical cap around any point $a\in S^{(2d-1)}$,
with the radius r chosen such that $\xi(C(a,r))=\xi(A)$. Then

$\xi(N_{\epsilon}(A))\geq\xi(N_{\epsilon}(C(a,r)))\,.$ (54)

Therefore we have

$\alpha_{S^{(2d-1)}}(\epsilon)=sup\\{\xi(S^{(2d-1)}\setminus
N_{\epsilon}(S))\\}=\xi(S^{(2d-1)})-\hbox{inf}\\{\xi(N_{\epsilon}(S))\\}=1-\hbox{inf}\\{\xi(N_{\epsilon}(S))\\}=1-\xi(C(a,\frac{\pi}{2}+\epsilon))\leq
e^{-d\epsilon^{2}}\,,$ (55)

where the details of the calculation leading to the last inequality can be
found in [28].

So far then, for functions $f$ with Lipschitz constant $\eta\leq 1$:

$\xi\\{f(x)\geq M_{f}+\epsilon\\}\leq\alpha_{S^{(2d-1)}}(\epsilon)\leq
e^{-d\epsilon^{2}}\,.$ (56)

Applying Lemma 2 to the function $g(x)=-f(x)$, one gets $\xi\\{g(x)\geq
M_{f}-\epsilon\\}\leq\alpha_{S^{(2d-1)}}(\epsilon)$, thus

$\xi\\{|f(x)-M_{f}|\geq\epsilon\\}\leq 2\alpha_{S^{(2d-1)}}(\epsilon)\,.$ (57)

By rescaling $\epsilon\rightarrow\epsilon\eta$ for functions with $\eta\geq 1$
and interpreting the relative measure above as a probability, we get to

$\hbox{Prob}\\{|f(x)-M_{f}|\geq\epsilon\\}\leq
2\hbox{e}^{-d\frac{\epsilon^{2}}{\eta^{2}}}\,.$ (58)

Finally, an inequality can be shown which relates median and expectation value
of $f$, bringing the missing factors in the exponential which appear in the
version of the Levy’s lemma we made use of in Eq. (6).

##  Appendix F Bound to state fluctuations

Let us now make use of Levy’s lemma to bound the fluctuations around the
average, unconditional state
$\langle\rho_{IB}\rangle=\rho_{I}\otimes\mathbb{I}/d_{B}$. To this aim, we
intend to apply the lemma to the trace distance between states, which
quantifies their operational distinguishability [21, 22], in particular
between the conditional state $\rho_{I}\otimes|\phi\rangle\langle\phi|$ under
the assumption of separability between the initial inner and boundary systems
[see Eq. (9)], and the unconditional, average state:
$f(\phi)=\|\rho_{I}\otimes\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\rho_{I}\otimes\mathbb{I}/d_{B}\|_{1}=\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}$.
Levy’s lemma applied to $f(\phi)$ reads:

$\hbox{P}\big{[}|\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}-\langle\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}\rangle\bigr{\rvert}\geq\epsilon\big{]}\leq
2\hbox{exp}\left(-\frac{2d_{B}d_{E}\epsilon^{2}}{9\pi^{3}\eta^{2}}\right)\,.$
(59)

It is convenient to rearrange Eq.(59) such that we get to an expression of the
form:

$\hbox{P}\left[\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}\geq\gamma\right]\leq\gamma^{\prime}\,,$
(60)

where

$\gamma=\epsilon+\langle\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}\rangle\,,\quad\gamma^{\prime}=2\hbox{exp}\left(-\frac{2d_{B}d_{E}\epsilon^{2}}{9\pi^{3}\eta^{2}}\right)\,.$
(61)

So in order to estimate the fluctuations around the mean state we need to
bound
$\langle\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}\rangle$.
Following an argument presented in [4], it is now convenient to turn to the
more accessible Hilbert-Schmidt norm
$\|M\|_{2}=\sqrt{\hbox{Tr}(M^{\dagger}M)}$ by exploiting its relationship with
the trace norm $\|M\|_{1}=\hbox{Tr}\sqrt{M^{\dagger}M}$, which satisfies, for
any $n\times n$ matrix $M$, the relation $\|M\|_{1}^{2}\leq n\|M\|_{2}^{2}$.
We thus have:

$\displaystyle\langle\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{1}\rangle$
$\displaystyle\leq$
$\displaystyle\sqrt{d_{B}}\langle\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{2}\rangle\leq\sqrt{d_{B}\langle\|\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\|_{2}^{2}\rangle}$
$\displaystyle=$
$\displaystyle\sqrt{d_{B}\langle\hbox{Tr}\left[\left(\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)-\mathbb{I}/d_{B}\right)^{2}\right]\rangle}=\sqrt{d_{B}\langle\hbox{Tr}_{B}\left[\left(\hbox{Tr}_{E}(|\phi\rangle\langle\phi|)\right)^{2}\right]\rangle-1}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{d_{B}^{2}-1}{d_{B}d_{E}+1}}\,,$

where we inserted the average local purity $(d_{B}+d_{E})/(d_{B}d_{E}+1)$ in
the absence of initial $IB$ correlations [see Eq. (43)].

To complete the Levy’s bound we are looking for, we make use of the following
lemma [4]:

###### Lemma 4

The Lipschitz constant $\eta$ of the function
$f(\phi)=\|\hbox{Tr}_{E}\left[|\phi\rangle\langle\phi|-\right]-\mathbb{I}/d_{B}\|_{1}$
satisfies $\eta\leq 2$.

Proof: One has

$\displaystyle|f(\phi_{1})-f(\phi_{2})|^{2}$
$\displaystyle=|\|\hbox{Tr}_{E}\left[|\phi_{1}\rangle\langle\phi_{1}|\right]-\mathbb{I}/d_{B}\|_{1}-\|\hbox{Tr}_{E}\left[|\phi_{2}\rangle\langle\phi_{2}|\right]-\mathbb{I}/d_{B}\|_{1}|^{2}\leq\|\hbox{Tr}_{E}\left[|\phi_{1}\rangle\langle\phi_{1}|-|\phi_{2}\rangle\langle\phi_{2}|\right]\|_{1}^{2}$
$\displaystyle\leq$
$\displaystyle\||\phi_{1}\rangle\langle\phi_{1}|-|\phi_{2}\rangle\langle\phi_{2}|\|_{1}=4\left(1-|\langle\phi_{1}|\phi_{2}\rangle|^{2}\right)\leq
4||\phi_{1}\rangle-|\phi_{2}\rangle|^{2}\;,$ (63)

where we used the non-increase of the trace norm under partial tracing as well
as the reverse triangle inequality. The last inequality is equivalent to
$\eta\leq 2$.

$\square$

The lemma above allows one to upper bound the quantity $\gamma^{\prime}$ in
Eq.(60) and thus obtain Inequality (10) (where
$\rho_{IB,\phi}\equiv\rho_{I}\otimes\hbox{Tr}[|\phi\rangle\langle\phi|]$):

$\hbox{P}\left[\|\rho_{I}\otimes\hbox{Tr}[|\phi\rangle\langle\phi|]-\Phi\left(\rho_{\phi}\right)\|_{1}\geq\epsilon+\sqrt{\frac{d_{B}^{2}-1}{d_{B}d_{E}+1}}\right]\leq
2\hbox{exp}\left(-\frac{d_{B}d_{E}\epsilon^{2}}{18\pi^{3}}\right)\,,$ (64)

which bounds the fluctuations around the average state of Eq. (4).