11institutetext: California Institute of Technology, Pasadena CA 91125, USA
11email<EMAIL_ADDRESS>
# PanDA: Panoptic Data Augmentation

Yang Liu 11 Pietro Perona 11 Markus Meister 11

###### Abstract

The recently proposed panoptic segmentation task presents a significant
challenge of image understanding with computer vision by unifying semantic
segmentation and instance segmentation tasks. In this paper we present an
efficient and novel panoptic data augmentation (PanDA) method which operates
exclusively in pixel space, requires no additional data or training, and is
computationally cheap to implement. By retraining original state-of-the-art
models on PanDA augmented datasets generated with a single frozen set of
parameters, we show robust performance gains in panoptic segmentation,
instance segmentation, as well as detection across models, backbones, dataset
domains, and scales. Finally, the effectiveness of unrealistic-looking
training images synthesized by PanDA suggest that one should rethink the need
for image realism for efficient data augmentation.

###### Keywords:

Data Augmentation, Panoptic Segmentation, Scene Understanding, Image Synthesis

##  1 Introduction

With the rapid development of convolutional neural networks (CNN) and the
availability of large scale annotated datasets like ImageNet [23], modern
computer vision models have reached or surpassed human performance in many
domains of visual recognition [12, 4]. Much of the interest of the community
has since shifted from visual recognition to formulating and solving more
challenging tasks such as object detection, semantic segmentation, and
instance segmentation.

Recently, [15] proposed the panoptic segmentation task that unifies instance
segmentation and semantic segmentation. The task requires a model to assign
each pixel of an image a semantic label and an instance ID. Several panoptic
datasets, such as the Cityscapes [5], Microsoft COCO [17], ADE20K [29], and
Mapillary Vistas [20], have been released. Much of research attention has
focused on developing new models [27, 21, 10]. So far, the best performing
models on the leader boards of various major panoptic challenges are
exclusively CNN based.

In this paper, insead of developing new models, we focus on the data
augmentation aspect of the panoptic segmentation task, and develop a novel
panoptic data augmentation method - PanDA that further improves the
performance of original top performing models from 2019.

We first identify the data deficiency and class imbalance problems and propose
a pixel space data augmentation method for panoptic segmentation datasets that
efficiently generates synthetic datasets from the original dataset. The method
is computationally cheap and fast, and it requires no training or additional
data. To demonstrate the robustness of PanDA, we use a single frozen set of
parameters throughout the paper except for the Ablation Study section.

Next, we experimentally demonstrate that training with PanDA augmented
Cityscapes further improves all performance metrics of original top-performing
UPSNet [27] and Seamless Scene Segmentation [21] models. With PanDA, robust
boosts in panoptic segmentation, instance segmentation, and detection with
UPSNet with ResNet-50/101 backbones and Seamless Scene Segmentation model with
ResNet-50 backbone are reported.

To further demonstrate the generalizability of PanDA across scales and
domains, we apply it to Cityscapes subsets with 10 to 3,000 images, as well as
a 10 times larger COCO subset with 30,000 images. We report performance gains
across scales and datasets. By quantifying the log-linear relationship between
number of training images and final performance metrics, we show that on
average a PanDA generated image is 20-40% as effective as an original image.

Finally, results from the ablation study show that, contrary to common
beliefs, less realistic looking images improve model performance more. It
suggests that the level of image realism is not positively correlated with
data augmentation efficiency.

##  2 Related Work

###  2.1 Panoptic Segmentation

The panoptic segmentation task was first proposed in 2019 by Kirillov et al.
[15] as an attempt to formulate a unified task that bridges the gap between
semantic segmentation and instance segmentation. The task divides objects in
to two super-categories: things and stuff. For classes in things, each pixel
in the image is labeled with a class ID and an instance ID. For classes in
stuff, each pixel is labeled with a class ID only. In addtion to traditional
metrics such as mean intersection over union (mIoU) and average precision of
instance segmentation (AP), the panoptic quality (PQ) is defined as,

$PQ=\underbrace{\frac{\sum_{(p,g)\in{TP}}IoU(p,g)}{|TP|}}_{\text{segmentation
quality
(SQ)}}\times\underbrace{\frac{|TP|}{|TP|+\frac{1}{2}|FP|+\frac{1}{2}|FN|}}_{\text{recognition
quality (RQ)}}$

where true positives (TP) are predicted segments that have strictly greater
than 0.5 overlap with ground truth segments. PQ is the product of segmentation
quality (SQ) and recognition quality (RQ).

###  2.2 Data Augmentation

Despite the efforts towards low-shot learning, modern CNN based models are
still data-hungry in that they have very large model capacity to even memorize
randomly labeled large datasets [28]. One efficient way to regularize models
and promote generalization is data augmentation.

Many methods still in use for detection and segmentation models [27, 21, 10]
are largely inherited from the ImageNet visual recognition era [16]. These
methods take advantage of object invariances in pixel space: manipulations
such as crop, horizontal flip, resize, color distortion, and noise injection
do not usually change the identity of the object. These methods are simple in
concept and computationally very cheap, but they do not take advantage of the
more informative ground truth labelling that accompanies the harder detection
and segmentation tasks. Some recent methods start to explore the use of
bounding box and segmentation information [18, 1, 8, 24, 7]. InstaBoost [8] is
proposed as a method for instance segmentation where instances are cropped out
and moved to a different location of the same image, then the hole in
background is filled with a classical in-paint network [2]. Shetty et al. [24]
augment images by removing certain instances that provide context and use a
CNN based in-paint model [25] to fill the holes. Dvornik et al. [7] use a CNN
based context model to guide the addition of instances from an “instance-
database” to original images, avoiding making holes in the original images and
the need for in-painting. The Discussion Section offers a detailed comparison
relating [8], [24] and [7] to the present work.

Data simulation is another flavor of data augmentation. Thanks to the
development of graphical simulation engines, one can generate photo-realistic
images together with pixel perfect ground truth. The method is proven to work
for many tasks such as human pose estimate [26], wildlife classification [1],
and object detection [13]. However, this method often requires handcrafted 3D
models and there usually is a significant domain gap between real images and
simulated images.

Recently, with the popularization of generative models such as generative
adversarial networks (GAN), researchers also add images generated by GANs to
the training set [9, 3, 18]. However, as a type of neural network based model,
the GAN itself requires training which in turn is both computationally
expensive and data hungry.

##  3 Panoptic Data Augmentation

Figure 1: Schematics of PanDA. Foreground and background segments are greatly simplified for clear demonstration of the image decomposition and synthesis process. Note that background in the panoptic segmentation task usually takes a small percentage of pixels in an image |  |
---|---|---
(a) | (b) | (c)
|  |
(d) | (e) | (f)
Figure 2: Examples of original cityscapes images (top row) and PanDA generated
images (bottom row)

Annotating images for panoptic segmentation can be costly, since every pixel
in an image has to be semantically labeled and pixels that belong to things
classes must have an additional instance ID associated. One could argue that
it is orders of magnitude harder for a human annotator to generate panoptic
ground truth than to assign a single class ID to an image as needed in visual
recognition. But with great challenges usually come great opportunities, PanDA
takes advantage of the rich information embedded in panoptic annotations and
uses it to synthesize new images with ground truth.

A fundamental feature that distinguishes PanDA from other data augmentation
methods is that PanDA does not aim to create images that look realistic to
human eyes. Many 3D simulation based methods [1, 22, 26] and cut-paste methods
[24, 7, 8] implicitly or explicitly suggest that the level of realism is key
to achieving high performances. Similary, GAN based image synthesis methods
[9, 3, 14] also usually optimize for realism, since the very goal of the
generator network in GANs is to synthesize images realistic enough to fool the
discriminator network into thinking that they are real [11]. In contrast,
PanDA takes more principled approaches to balance classes, breaks local pixel
structures, and increases variations of objects, and the final images
synthesized do not look fully realistic to human eyes.

As shown in Fig 1, PanDA first decomposes a panoptically labelled training
image by using the segmentation ground truth to divide it into foreground and
background segments. It is worth noting that foreground segments not only
include instances in things classes, but also segments in stuff classes which
are considered background in InstaBoost [8] and [7]. Unfilled pixels are
padded with white noise patterns instead of in-painting as seen in [8] and
[24]. Noise patterns are used because they do not belong to any known
categories, making ground truth assignment unambiguous. Foreground segments
are then overlaid on top of the new background image one by one. The same
manipulation is applied to the ground truth segmentation image to generate the
new ground truth segmentation.

For foreground segments, we can perform a series of pixel space operations
such as dropout, resize and shift to control different aspects of each
individual object instance. Dropout is used to remove segments from an image.
It serves to revome well segmented and classfied objects from the image. The
resize operation changes the size of a segment while preserving the original
aspect ratio, it simulates object movement in depth and introduces more size
variance in objects. It resembles zooming and multi-scale training [27] on the
individual object level. Random shifting moves the segment in x and y in pixel
space. It prevents memorization of object locations by the model and breaks
the local pixel relationship between the object and its background, thus
creates new local contexts for the object. Resize and shift together simulate
3 dimensional translation of objects in space, and dropout controls the
frequency of objects. Because the ground truth depth information to recover
the depth ordering of segments is not available, and we do not want larger
segments to occlude smaller ones, we sort the segments by their area and put
largest segments on the bottom layer. It is worth noting that more operations
can be added to the repertoire of the PanDA tool set, and different variants
of the aforementioned operations may also be implemented. In this paper, we
limit the scope to the operations discussed above to allow for in-depth
experiments with PanDA.

##  4 Experiments

###  4.1 UPSNet and Seamless Scene Segmentation Models

The UPSNet model and the Seamless Scene Segmentation (SSS) models are two
independently developed top-performing panoptic segmentation models. To date,
they are also the only two models with code published to reproduce the
results.

The UPSNet model [27] is one of the top performing single models on both
Cityscapes and MS COCO panoptic challenges. It uses a shared CNN feature
extractor backbone and multiple heads on top of it. The backbone is a Mask
R-CNN feature extractor built on a deep residual network (ResNet) with a
feature pyramid network (FPN). UPSNet has an instance head which follows the
Mask R-CNN design and a semantic head that consists of a deformable
convolution based sub-network. The outputs of the two heads are then fed into
the panoptic head which generates the final panoptic prediction.

The SSS model [21], which was developed around the same time early in 2019,
roughly follows the same meta-architecture with a Mask R-CNN instance head and
a Mini Deeplab (MiniDL) semantic head taking input from a common FPN feature
extractor and feeding into a fusion head for panoptic prediction. At the time
of its publication, it became the top performing model in Cityscapes panoptic
challenge with a ResNet-50 backbone [21].

We used all the original hyper-parameters of UPSNet and SSS models for
training and inference. We only scale (1) the number of training iterations to
keep the number of epochs constant across a wide range of dataset sizes and
(2) the learning rate based on the number of GPUs used in the training. All
trainings start with ImageNet pretrained ResNet-50/101 models as reported in
[27] and [21]. The UPSNet models are trained with 8x Nvidia 2080Ti GPUs with
11GB VRAM. The SSS experiments are conducted on Amazon Web Services with 8x
Nvidia Tesla V100 GPUs with 32GB VRAM. For the in-depth experiments, we choose
UPSNet over SSS because it fits in our server’s GPUs.

###  4.2 Datasets

The Cityscapes [5] panoptic dataset has 2,975 training, 500 validation, and
1,525 test images of ego-centric driving scenes in urban environments in many
cities in Europe. Examples are shown in Fig 2 (a-c). The dense annotation
covers 97% of the pixels which consists of 9 things classes and 11 stuff
classes. We choose Cityscapes as the main testbed for PanDA for several
reasons: 1) It is one of the most popular panoptic datasets available and it
has a total of 19 diverse classes that covers a wide range of driving related
scenarios. Many modern models[27, 21, 10] report performance on Cityscapes and
have published code available. 2) Results on a relatively small set are likely
to generalize to other specialized domains where annotated panoptic data is
scarce. 3) The small size also makes both data synthesis and training cost
manageable. As a result, it is suitable for exploratory studies like this.

To demonstrate the generalizability of PanDA, we also performed additional
experiments on the MS COCO dataset, which has 118K training and 5k validation
images including 53 stuff classes and 80 things classes of common objects. It
is both orders of magnitude larger in size and more diverse regarding number
of classes.

###  4.3 Augmenting Cityscapes with PanDA

The fact that multi scale inference and pretraining with additional data
improves performance of the lightweight UPSNet-50 model suggests that model
capacity is not the major limiting factor for the final performance and that
data augmentation may further improve performance. To investigate the
relationship between model performance and size of training dataset, we
trained the UPSNet-50 model from the same ImageNet pretrained backbone on
various subsets of the Cityscapes training set that consist of 10, 100, 1,000,
and 2,975 images. Fig 3 shows near perfect log-linear relationships between
the number of training images and various performance metrics (PQ:
$r^{2}=0.9995$, mean average precision for instance segmentation evaluated at
0.5:0.95 (AP): $r^{2}=0.9744$, mean average precision for detection evaluated
at 0.5:0.95 (AP box): $r^{2}=0.9948$). It suggests that adding training images
is likely to further improve model performance.

Figure 3: Model performance vs training set size on Cityscapes. We train
UPSNet-50 models on various Cityscapes subsets ranging from 10 to 2,975
images. PQ: panoptic quality. AP: mean average precision of instance
segmentation evaluated at 0.5:0.95. AP box: mean average precision of
detection evaluated at 0.5:0.95. Dashed curves are log-linear fits. Panoptic
segmentation, instance segmentation, and instance detection performance are
summarized by PQ, AP, and AP box, respectively

To test whether adding training images indeed helps, we used PanDA to generate
synthetic images from the Cityscapes training set. A key discrepancy between
model training and and the final PQ score is that classes with large areas
provide overwhelming training signals during training and thus are favored
during training, whereas SQ - the first term of the PQ formula - weights small
and large objects equally. To mitigate the issue, we aim to balance average
per class pixel count per image by applying random dropout of segments where
the dropout rate is linearly proportional to the size of the segments. We also
apply resizing and shifting to introduce size and location variance of
objects.

We performed a grid search of the parameters of PanDA to optimize PQ of
UPSNet-50 model on the full Cityscapes dataset. To demonstrate the
generalizability and robustness, we froze PanDA’s parameters and used it
throughout the rest of the paper with the exception of the Ablation Study
section. Specifically, drop out is performed as follows: segments with areas
between 10% and 50% of an image are dropped out with probability linearly
proportional to their sizes. Segments occupying over 50% of an image are
guaranteed to drop out, and segments smaller than 10% of the image frame are
never dropped out. We couple resize and shift with a zooming operation:
enlarged segments are pushed to the periphery and shrunk ones are pulled
towards the center. This simulates an approaching motion of the object. Each
segment is resized in the range of 0.5x to 1.5x, with scales drawn from a
uniform distribution.

In original Cityscapes, road and building together occupy over 50% pixels of
an image on average which supports our observation that some large classes
dominate learning signals. While PanDA removes a large proportion of non-
background pixels on average, it significantly reduces the dominance of common
large classes while largely preserving small and rare classes (Supplementary
Figure 1 and 2).

Examples of original and synthetic image pairs are shown in Fig 2 (and
Supplementary Fig 5): some of the commonly seen classes with large area such
as road and building are more frequently dropped out and smaller instances
such as person and pole are often kept. It has caught our attention that the
synthetic images are no longer realistic and coherent scenes but rather
nonsensical to human eyes. However we show in later sections that adding these
synthetic images improves model performance.

Table 1: Results on Cityscapes. PanDA augmented Cityscapes datasets generated with a single frozen set of parameters improve all performance of UPSNet-50, UPSNet-101, SSS-50. PQ: panoptic quality. SQ: segmentation quality. RQ: recognition quality. PQth: PQ for things classes. PQst: PQ for stuff classes. mIoU: mean intersection over union. AP: mean average precision of instance segmentation evaluated at 0.5:0.95. APbox: mean average precision of detection evaluated at 0.5:0.95 Model | PanDA | PQ | SQ | RQ | PQTh | PQSt | mIoU | AP | APBox
---|---|---|---|---|---|---|---|---|---
UPSNet-50 |  | 58.8 | 79.5 | 72.6 | 54.5 | 62.0 | 75.1 | 33.5 | 38.7
✓ | 59.9 | 79.9 | 73.8 | 55.4 | 63.3 | 76.2 | 34.6 | 39.6
UPSNet-101 |  | 59.8 | 80.0 | 73.5 | 55.6 | 62.9 | 76.7 | 33.0 | 38.2
✓ | 60.5 | 80.2 | 74.1 | 56.3 | 63.6 | 77.1 | 35.0 | 40.6
SSS-50 |  | 60.3 | – | – | 55.5 | 63.8 | 77.4 | 32.4 | 36.3
✓ | 60.9 | – | – | 56.4 | 64.2 | 78.0 | 33.8 | 37.0

To demonstrate the usefulness of PanDA, we first trained original UPSNet and
SSS models from ImageNet pre-trained ResNet backbones to establish baseline
performance metrics (Table 1). The reproduced results largely recapitulate
those published in the original papers (For details, see Supplementary Table
1). Then we generated synthetic training sets with PanDA and trained original
UPSNet and SSS models on the augmented training sets. We first used PanDA to
double the Cityscapes training set and trained the original UPSNet-50 model on
the PanDA augmented Cityscapes dataset with a total of 3,000 images. The model
(UPSNet-50 with PanDA in Table 1 and Supplementary Fig 3) outperforms the
baseline model (as well as the one reported in [27]) in all metrics. In
summary, UPS-50 trained on PanDA augmented images improves on the baseline by
1.1 PQ, 0.9 AP, 1.1 AP box, evaluated at 0.5:0.95) and 1.1 mean Intersection
over Union (mIoU) (0.7 PQ, 1.0 AP, 1.3 AP box, and 1.0 mIoU improvements over
[27]) on the validation set. Additionally, overall and per class performance
in both instance segmentation task (Supplementary Table 2 and detection task
(Supplementary Table 3) improve upon baseline.

One commonly used method to improve model performance at the cost of increased
computation is to use a larger and deeper backbone. We trained a UPSNet-101
model which uses the deeper ResNet-101 and observed performance gain for all
metrics except for AP and AP box. We froze PanDA parameters and trained
UPSNet-101 model with PanDA augmented images. Table 1 shows that the PanDA
enhanced UPSNet-101 model in turn outperforms all other UPSNet models. Two
observations can be made from the additional experiments with UPSNet-101.
First, it is remarkable that with a half-sized back bone, the PanDA enhanced
UPSNet-50 model outperforms the baseline UPSNet-101 model in five out of eight
metrics: PQ, RQ, PQ stuff, AP and AP box. Second, the fact that PanDA further
improves UPSNet-101 suggests that using a deeper backbone and training on
PanDA augmented images help the model in complementary ways. In other words,
it demonstrates that PanDA generalizes across backbones.

In addition to UPSNet, we also conducted experiments on SSS, which is the only
other model with code available to date. We trained the SSS model on PanDA
augmented images with the same frozen parameters. The last row in Table 1
shows all-around improvements of the SSS model trained with PanDA. Together,
the set of experiments with UPSNet-50/101 and SSS shows that PanDA is
effective across models and backbones.

###  4.4 PanDA Across Scales and Datasets

Table 2: PanDA generalization results. We used the same set of parameters for all experiments in this section. As shown in the table, PanDA generalizes well not only across scales of Cityscapes subsets, but also to COCO subsets that are 10 times larger than the original Cityscapes dataset Dataset | # images | PanDA | PQ | SQ | RQ | PQTh | PQSt | mIoU | AP | APBox
---|---|---|---|---|---|---|---|---|---|---
Cityscapes | 10 |  | 23.0 | 58.1 | 29.6 | 9.2 | 33.1 | 36.3 | 2.9 | 5.5
10 | ✓ | 23.5 | 54.9 | 30.3 | 12.5 | 30.3 | 34.3 | 4.6 | 7.6
100 |  | 37.7 | 72.5 | 48.4 | 26.4 | 45.9 | 52.4 | 10.5 | 16.7
100 | ✓ | 40.1 | 70.6 | 51.0 | 29.5 | 47.7 | 54.1 | 12.7 | 19.1
1,000 |  | 52.7 | 77.9 | 65.8 | 45.5 | 57.9 | 68.7 | 25.8 | 31.1
1,000 | ✓ | 55.0 | 78.4 | 68.5 | 49.1 | 59.2 | 70.1 | 27.3 | 32.5
3,000 |  | 59.3 | 79.7 | 73.0 | 54.6 | 62.7 | 75.2 | 33.3 | 39.1
3,000 | ✓ | 59.9 | 79.9 | 73.8 | 55.4 | 63.3 | 76.2 | 34.6 | 39.6
COCO | 30,000 |  | 36.5 | 76.2 | 45.7 | 41.6 | 28.8 | 45.3 | 26.5 | 29.0
30,000 | ✓ | 37.4 | 76.9 | 46.6 | 43.2 | 28.7 | 45.9 | 28.0 | 31.2

To test how well PanDA generalizes across scales, we applied it to smaller
subsets of Cityscapes. Such generalization would be particularly useful if one
develops a new panoptic task in a new domain where annotated data is expensive
and scarce. In Table 2 (and Supplementary Table 4), we show the consistent
performance improvement on UPSNet models trained with PanDA augmented
Cityscapes subsets across scales of 10, 100, 1,000, and 3,000 images.

We then ask whether the improvement with PanDA is specific to the Cityscapes
dataset. Arguably, the ego-centric driving scenarios in urban environments in
Cityscapes are a specialized domain which may raise concerns of
generalizability across domains. In addition, Cityscapes is one of the smaller
panoptic datasets available. Although previous experiments demonstrate that
PanDA performs well when the dataset scales down, it remains unknown how well
it performs when scaled up. To address the concerns, we applied PanDA to a
30,000 image subset of COCO dataset which is obtained by going through a list
of the original 118K training set with step size 4. The COCO 30K subset will
not only test whether PanDA generalizes to a different domain (examples in
Supplementary Fig 6), but also breaks the 3,000-image upper limit of
Cityscapes. The bottom two rows of Table 2 show that PanDA indeed works on the
10x larger COCO dataset, which further supports the claim that PanDA
generalizes well across scales and domains.

###  4.5 Data Efficiency

In this section, we explore the data efficiency of PanDA generated images. As
shown in Fig 3 (and Supplementary Fig 4), PQ, AP, and AP box scales linearly
with the logarithm of number of training images. The log-linear regression
functions are,

$\displaystyle PQ=6.3255ln(n)+8.5413,(r^{2}=0.9995)$ (1) $\displaystyle
AP=5.4384ln(n)-11.494,(r^{2}=0.9744)$ (2) $\displaystyle
AP^{box}=5.8271ln(n)-8.7893,(r^{2}=0.9948)$ (3)

where $n$ is the number of original images used to train the model, PQ, AP,
and AP box are respective model performance. PQ, AP, AP box values are plugged
into the regression functions to in turn estimate effective training set size
$N$ for experiments with either original images only or original plus PanDA
images. We define data efficiency (DE) as,

$\displaystyle DE=\frac{N^{aug}-N^{orig.}}{N^{orig.}}\times 100\%$ (4)

where Norig. is the estimated effective training set size of models trained on
original images only, and Naug is that of models trained on original and PanDA
images. The definition is specific to the case where the ratio of original and
synthetic images is 1. A higher DE suggests higher per image data efficiency,
and one would expect a DE of 100% if a synthetic image is as informative as an
original image.

Table 3: Data efficiency. Three sets of estimates of effective training set sizes are made per experiment with PQ, AP, and AP box, respectively. # original images: number of original images used in training. $DE$: data efficiency in percent. $N$: number of effective training images estimated by model performance. Superscripts orig. and aug denote model trained with original images only or original and PanDA images. Subscripts denote which performance metric is used to estimate $N$ or $DE$ # orig. img | $N_{PQ}^{orig.}$ | $N_{PQ}^{aug}$ | $DE_{PQ}$ | $N_{AP}^{orig.}$ | $N_{AP}^{aug}$ | $DE_{AP}$ | $N_{APbox}^{orig.}$ | $N_{APbox}^{aug}$ | $DE_{APbox}$
---|---|---|---|---|---|---|---|---|---
10 | 9.8 | 10.6 | 8.2 | 14.1 | 19.3 | 36.7 | 11.6 | 16.7 | 43.4
100 | 100.5 | 146.8 | 46.1 | 57.1 | 85.5 | 49.9 | 79.4 | 119.8 | 51.0
1000 | 1076.1 | 1547.9 | 43.9 | 951.0 | 1253.1 | 31.8 | 939.6 | 1194.8 | 27.2
3000 | 2822.6 | 3358.7 | 19.0 | 3918.2 | 4796.6 | 22.4 | 3462.4 | 4040.7 | 16.7

Two conclusions can be drawn from results in Table 3. First, across scales and
metrics, PanDA images are not as efficient as original images because none of
DEs are near 100%. However, this is expected since we only reuse the object
instances in the original images, and the synthetic images have only 40% non-
background pixels per image on average. Second, the fact that a synthetic
image can be half as efficient as a real one suggests that there is
significant amount of information embedded in the original images that is not
learned by current models. One can expect superior models to capture this
additional information without the help of data augmentation.

###  4.6 Ablation Study

Table 4: Ablation study on Cityscapes dataset. Training on the original dataset for more iterations does not improve model performance. Best performance is achieved by combining dropout, resize, and shift Extra Iterations | Dropout | Resize | Shift | PQ | SQ | RQ | PQTh | PQSt | mIoU | AP | APBox
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | 58.8 | 79.5 | 72.6 | 54.5 | 62.0 | 75.1 | 33.5 | 38.7
✓ |  |  |  | 58.7 | 80.1 | 71.9 | 53.5 | 62.4 | 75.6 | 33.2 | 38.1
✓ | ✓ |  |  | 59.3 | 79.6 | 73.1 | 54.1 | 63.1 | 75.8 | 33.4 | 39.2
✓ | ✓ | ✓ |  | 59.0 | 79.8 | 72.6 | 55.3 | 61.7 | 75.8 | 33.4 | 39.3
✓ | ✓ |  | ✓ | 59.3 | 79.9 | 72.8 | 53.9 | 63.2 | 76.0 | 34.1 | 39.4
✓ | ✓ | ✓ | ✓ | 59.9 | 79.9 | 73.8 | 55.4 | 63.3 | 76.2 | 34.6 | 39.6

We conducted an ablation study to evaluate the effectiveness of the individual
operations of PanDA. We use the reproduced UPSNet-50 model trained on the
original Cityscapes training set as the baseline (first row in Table 4). For
each experiment group, certain subsets of PanDA image operations are ablated.
We optimized PanDA’s parameters under the ablation constraint and report the
best performance obtained. Experiment groups are trained on their respective
PanDA 1x augmented Cityscapes dataset. Doubling the number of training epochs
without any augmented data (second row in Table 4) does not improve model
performance which suggests that the original UPSNet training parameters are
near optimal. PanDA with dropout operation only improves model performance,
presumably by mitigating the data imbalance issue (Supplementary Fig 1 and 2).
Without resize or shifting, original object relationships are preserved and
thus still realistic (Fig 4(b)). As more operations are included in PanDA, the
level of realism of synthesized images decreases (Fig 4(c)). However, the fact
that best performance is achieved with the least realistic image set suggest
that, contrary to popular belief, the level of realism of synthetic images is
not necessarily correlated with data augmentation performance.

|  |
---|---|---
(a) Original | (b) Dropout only | (c) Dropout+Resize+Shift
Figure 4: Examples of image with decreasing levels realism (a) Original image.
(b) Image with dropout only. The dominant segment - road is dropped out to
balance per class pixel count. The composition of the picture is largely
preserved and many objects still appear in realistic contexts. (c) Image with
dropout, resize, and shift. In addition to dropout as seen in (b), segments
are resized and moved to create new contexts. The additional operations make
the new image unrealistic

##  5 Discussion

Compared with GAN based and 3D simulation based data augmentation methods,
PanDA has several advantages. First of all, PanDA does not require training.
GAN based methods need to be trained to generate realistic images in the
desired domain first before the generative network can be used for data
augmentation. Second, no additional data is needed for PanDA. As mentioned
before, training GANs not only takes time but also usually requires a large
unlabelled dataset from the same domain. 3D simulation based methods almost
always require hand crafted 3D models of classes of interest. Finally, it is
computationally very cheap to use PanDA since it operates exclusively in pixel
space. Many image operations such as cropping and resizing could be further
parallelized and optimized which allows for real time image synthesis at
training time in an active learning fashion.

As mentioned in the Related Work section, InstaBoost [8] relies on two
operations - shift of instance segments and in-paint of background - to boost
the performance of instance segmentation and detection. There are several key
differences between PanDA and InstaBoost. First, in the panoptic segmentation
task, a large part of the “background” in the instance segmentation task now
belongs to the stuff superclass. The “background” in the panoptic segmentation
images usually takes up less than 15% of the image, making it infeasible to
use the same in-paint process. Secondly, the motivation behind in-painting the
background is not explicitly explained and presumably to maintain the image’s
level of realism. In contrast, PanDA boosts panoptic segmentation performance
despite losing image realism. Finally, in addition to the shift operation in
InstaBoost, PanDA has resize and drop out operations which are shown to be
essential for its effectiveness by the ablation study (Table 4). The method
proposed by Dvornik et al. [7] adds new instances to original images and
avoids holes in the background from removing instances as seen in InstaBoost.
However, the pasting of new instances has to be guided by a CNN-based context
model to determine the location and class of instances to add, and that model
has to be trained which requires data and time. Similar to InstaBoost, the
method only applies to things classes which takes less than 50% of pixels in
an image on average. Shetty et al. [24] propose removal of objects to break
context information and help deep models to generalize in classification and
semantic segmentation. This approach relies heavily on a trainable CNN based
in-paint network to fill the holes in the background. Data and time are needed
to train the in-paint CNN network in this method. Additionally, experiments in
the original paper show that removing large objects such as mountains makes it
hard for the in-paint network to fill the hole left behind and indeed hurts
model performance. In contrast, PanDA removes the large objects like road and
buildings with high probabilities and we show that is essential for
performance gain.

There are certain limitations within the current implementation of PanDA that
leave doors open for more exploration. First, the functions used in background
padding, dropout, resize, and shift can be further optimized or complemented
with more optimized functions. For example, resizing can be drawn from a
normal instead of uniform distribution. Operations such as rotation and
warping can be added. Further optimizations can be performed in an automated
way [6]. Second, since the 3,000 training images in Cityscapes can be divided
into 78,000 segments, if we allow the creation of hybrid images where segments
in the new image can come from different original images, we could further
expand the image variation combinatorially. Third, it remains unclear how many
useful synthetic images can be generated with PanDA per original image. In
principle, one could create an infinite number of synthetic images from the
original dataset. Due to the high computation cost of training with very large
datasets, this paper limits the scope to doubling the number of training
images. It will be an interesting direction to investigate the limit of the
number of useful synthetic images per original image.

It is worth noting that for challenges on ego-centric driving datasets like
Cityscapes, it is also possible to improve model performance by pretraining on
a larger dataset from a related domain. For example, better performing models
can be obtained by pretraining on MS COCO and fine tuning with Cityscapes[27].
However, in a new and specialized domain, there might not exist a dataset
available to pretrain on. In addition, for a large dataset like MS COCO,
pretraining on smaller datasets is unlikely to help.

Finally, although PanDA operations are well justified from the standpoint of
statistics, we were surprised that the best performing PanDA augmented
datasets do not look natural or realistic to human eyes at all. Many PanDA
generated images look qualitatively similar to the ones shown in Fig 2 where
positioning and occlusion of objects are not realistic. Many objects in the
synthetic images appear to be “floating” on top of the noise background out of
the original context (Fig 2 (d), Fig 4 (c)); sometimes they cluster and
overlap each other in the wrong depth order (Fig 2 (f)). It is known that
contextual information helps human visual detection[19]. We suspect that by
taking objects out of their original context, PanDA presents harder challenges
to the model and therefore forces the model to pay more attention to the
pixels within the object foreground. Secondly, despite the fact that PanDA
drastically reduced the total non-background pixels per synthetic image, the
augmented datasets are more balanced. It suggests that a balanced but small
dataset might be more helpful than a large but unbalanced dataset.

##  6 Conclusions

We present a simple and efficient method for data augmentation of annotated
panoptic images to improve panoptic segmentation performance. PanDA is
computationally cheap, and requires no training or additional data. After
training with PanDA augmented datasets, top performing panoptic segmentation
models further improve performance on two popular datasets, Cityscapes and MS
COCO. To the best of our knowledge, PanDA is the first pixel-space data
augmentation method with demonstrated performance gain for leading models on
panoptic segmentation tasks. Further improvement is possible with fine-tuned
parameters. The effectiveness of unrealistic images suggests that we should
reconsider maximizing realism in image synthesis for data augmentation.
Finally, PanDA opens new opportunities for exploring efficient pixel space
data augmentation methods for detection and segmentation datasets, and we
believe the community would benefit from these explorations in data
augmentation. Code will be available on GitHub upon publication.

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PanDA: Panoptic Data Augmentation - Appendix

Original Cityscapes

PanDA Cityscapes

Figure 1: Treemap views of average pixel count by class per image of Cityscapes training set. Class 20 is the background which is considered out of area of interest. Classes 1 and 3 together occupy the majority of the non-background pixels of an image in the original Cityscapes, and make up less than half the non-background pixels.  Figure 2: Per class pixel percentage per image. Bar graph of average per class pixel percentage of non-background classes per image. Each bar is computed by dividing the average number of pixels of a given class by the sum of the average number of non-background pixels. Pixel percentages of common classes are reduced and those of rare classes are increased, making the synthetic images more class-balanced. Figure 3: Results on Cityscapes In each case data augmentation with PanDA improves performance. PQ: panoptic quality. PQ things: PQ for things classes. PQ stuff: PQ for stuff classes. mIoU: mean intersection over union. AP: segmentation average precision evaluated at 0.5:0.95. APbox: bounding box average precision evaluated at 0.5:0.95 Table 1: Extended Results on Cityscapes. Despite the fact that our baseline performance (best of 5 runs) is lower than that reported in the original UPSNet paper [27], our models trained on PanDA augmented datasets outperform the original UPSNet-50 model without COCO pretraining in all metrics. PanDA 1x consists of 3,000 original Cityscapes images plus 3,000 PanDA synthetic images, PanDA 2x consists of 3,000 original Cityscapes images plus 6,000 PanDA images. PQ: panoptic quality. SQ: segmentation quality. RQ: recognition quality. PQth: PQ for things classes. PQst: PQ for stuff classes. mIoU: mean intersection over union. AP: segmentation average precision evaluated at 0.5:0.95. APbox: bounding box average precision evaluated at 0.5:0.95 Model | PQ | SQ | RQ | PQTh | PQSt | mIoU | AP | APBox
---|---|---|---|---|---|---|---|---
UPSNet-50 Xiong et al.[27] | 59.3 | 79.7 | 73.0 | 54.6 | 62.7 | 75.2 | 33.3 | 39.1
UPSNet-50 baseline | 58.8 | 79.5 | 72.6 | 54.5 | 62.0 | 75.1 | 33.5 | 38.7
UPSNet-50 w/ PanDA 1x | 59.9 | 79.9 | 73.8 | 55.4 | 63.3 | 76.2 | 34.6 | 39.6
UPSNet-50 w/ PanDA 2x | 60.0 | 80.3 | 73.5 | 55.8 | 63.1 | 76.2 | 35.6 | 40.5
UPSNet-101 baseline | 59.8 | 80.0 | 73.5 | 55.6 | 62.9 | 76.7 | 33.0 | 38.2
UPSNet-101 w/ PanDA 1x | 60.5 | 80.2 | 74.1 | 56.3 | 63.6 | 77.1 | 35.0 | 40.6
SSS-50 Porzi et al. [21] | 60.3 | – | – | 56.1 | 63.3 | 77.5 | 33.5 | –
SSS-50 baseline | 60.3 | – | – | 55.5 | 63.8 | 77.4 | 32.4 | 36.3
SSS-50 w/ PanDA 1x | 60.9 | – | – | 56.4 | 64.2 | 78.0 | 33.8 | 37.0
Table 2: Per class instance segmentation results on Cityscapes. Segmentation AP are reported. We observe not only large relative improvement on rare classes such as train and bicycle (18.4% and 13.1%, respectively), but also small gains on common classes such as car and person (3.5% and 3.5%, respectively) Models | person | rider | car | truck | bus | train | motorcycle | bicycle
---|---|---|---|---|---|---|---|---
UPSNet-50 Xiong et al.[27] | 31.2 | 25.1 | 50.9 | 33.6 | 55.0 | 33.2 | 19.2 | 18.3
UPSNet-50 our basline | 31.1 | 24.8 | 51.0 | 33.2 | 53.5 | 35.8 | 19.4 | 18.9
UPSNet-50 w/ PanDA 1x | 31.5 | 25.7 | 52.2 | 34.3 | 54.4 | 37.9 | 20.8 | 20.3
UPSNet-50 w/ PanDA 2x | 32.3 | 25.6 | 52.7 | 36.9 | 56.6 | 39.3 | 20.6 | 20.7
Table 3: Per class detection results on Cityscapes. AP box are reported Models | person | rider | car | truck | bus | train | motorcycle | bicycle
---|---|---|---|---|---|---|---|---
UPSNet-50 Xiong et al.[27] | 38.6 | 42.1 | 56.6 | 33.4 | 56.3 | 27.2 | 28.4 | 30.1
UPSNet-50 our basline | 38.5 | 41.6 | 56.3 | 33.6 | 55.4 | 26.0 | 27.1 | 31.2
UPSNet-50 w/ PanDA 1x | 39.0 | 43.4 | 57.8 | 33.6 | 55.8 | 27.2 | 28.4 | 31.2
UPSNet-50 w/ PanDA 2x | 40.1 | 43.4 | 58.6 | 36.3 | 57.5 | 29.3 | 27.4 | 31.7
Figure 4: Model performance vs training set size on Cityscapes. We train UPSNet-50 models on various Cityscapes subsets ranging from 10 to 2,975 images. PQ: panoptic quality. AP: instance segmentation average precision evaluated at 0.5:0.95. AP box: bounding box average precision evaluated at 0.5:0.95. Dashed curves are log-linear fits of baseline experiments, solid curves are fits of PanDA experiments. Panoptic segmentation, instance segmentation, and instance detection performance are summarized by PQ, AP, and AP box, respectively. PanDA enhanced models consistently outperform original models across scales in all metrics. Data efficiency (DE) corresponds to the amount of right shift of PanDA curves to account for the improved performance with PanDA  Original | Original annotation | PanDA | PanDA annotation
---|---|---|---
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|  |  |
Figure 5: Examples of original and PanDA Cityscapes images with ground truth annotation. Left two columns: original Cityscapes images and ground truth annotation images. Right two columns: PanDA generated images and ground truth annotation images. Original | Original annotation | PanDA | PanDA annotation
---|---|---|---
|  |  |
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Figure 6: Examples of original and PanDA COCO images with ground truth
annotation. Left two columns: original COCO images and ground truth annotation
images. Right two columns: PanDA generated images and ground truth annotation
images.


# Distribution of Fidelity in Quantum State Transfer Protocols

Salvatore Lorenzo  Università degli Studi di Palermo, Dipartimento di Fisica e
Chimica - Emilio Segrè, via Archirafi 36, I-90123 Palermo, Italy  Francesco
Plastina  Dipartimento di Fisica, Università della Calabria, 87036 Arcavacata
di Rende (CS), Italy  INFN, gruppo collegato di Cosenza  Tony J. G. Apollaro
Department of Physics, University of Malta, Msida MSD 2080, Malta  Mirko
Consiglio  Department of Physics, University of Malta, Msida MSD 2080, Malta
Karol Życzkowski  Institute of Theoretical Physics, Jagiellonian University,
Łojasiewicza 11, 30-348 Kraków, Poland  Center for Theoretical Physics, Polish
Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland

(April 29, 2024)

###### Abstract

Quantum state transfer protocols are a major toolkit in many quantum
information processing tasks, from quantum key distribution to quantum
computation. To assess performance of a such a protocol, one often relies on
the average fidelity between the input and the output states. Going beyond
this scheme, we analyze the entire probability distribution of fidelity,
providing a general framework to derive it for the transfer of single- and
two-qubit states. Starting from the delta-like shape of the fidelity
distribution, characteristic to perfect transfer, we analyze its broadening
and deformation due to realistic features of the process, including non-
perfect read-out timing. Different models of quantum transfer, sharing the
same value of the average fidelity, display different distributions of
fidelity, providing thus additional information on the protocol, including the
minimum fidelity.

##  I Introduction

Faithful transfer of quantum information between different locations or
different parts of the same quantum hardware is a basic building block of
several quantum information processing protocols and it is crucial for a
variety of applications [1, 2]. Since the seminal paper by Bose [3], spin
chains emerged as suitable models to describe quantum data buses allowing one
for high fidelity quantum state transfer (QST). One usually considers short
distance communication, or more generally, the transfer of a state from site
${\cal A}$ to ${\cal B}$, typically located at two ends of the chain.

A great variety of different protocols which employ spin systems as
transmission channels have been devised and experimentally realized [4, 5, 6,
7, 8], achieving either ‘perfect’ or ‘almost perfect’ QST [9]. Broadly
speaking, they can be divided into different classes, according to the
physical mechanism on which they rely in order to achieve the transmission –
see e.g. Ref. [10]. Examples consist of protocols based on time-dependent
couplings [11], including pulsed and general non-adiabatic driving [12, 13,
14, 15, 16], fully engineered interactions [17, 18, 19, 20, 21], ballistic
transfer [22, 23, 24], Rabi-like oscillations between eigenstates having
support on the sending and receiving sites [25, 26, 27, 28, 29, 30]. All of
these methods have been shown to achieve transmission perfectly or with a high
fidelity. These approaches have been generalized in various ways to describe
setup with more than one receiver, to study routing of quantum states [31,
32], distribution of entanglement [33, 34, 35, 14, 36], shared between two or
among several qubits [37, 38, 39, 40]. State transfer in the presence of noise
and disorder due to fluctuating local fields along the spin chain analyzed in
[41, 33, 42], has been simulated with a quantum computing device [43].

To assess the quality of the state transmission through the spin chain
quantitatively, it is customary to employ the average fidelity. Indeed, the
fidelity, $F\in[0,1]$, provides a measure of the overlap between the input
state of the sending part, $\ket{\psi_{\cal A}}$, with the output state $\rho$
of the receiving part. Its expectation value, $\langle
F\rangle=\langle\psi_{\cal A}|\rho|\psi_{\cal A}\rangle$, averaged over the
set of all pure states of the sender, is an indicator of the average quality
of the transmission protocol.

However, in several cases, the average fidelity does not provide a full
characterization of the transfer process, which can, instead, be obtained by
analysing the full statistics of the values taken by the fidelity over the
Hilbert space of the sending part. This information is contained in the
probability distribution function (PDF) of fidelity.

The only case in which the distribution of the fidelity does not provide any
new information with respect to the average is that of perfect state transfer,
where the distribution becomes a delta function, peaked on the value $F=1$.
However, a perfect QST typically poses severe requirements, such as an
extremely high level of control of the parameters of spin network, or a very
precise extraction time at the receiving site. If these requirements are not
met, the ideal PDF of the fidelity can acquire a width, and its functional
form may provide us with a way to characterize the physical process underlying
the state transfer.

Distribution of fidelity between random quantum states was analyzed in [44],
while ref. [45] studies the distribution between random input pure states and
the output an a given quantum operation. The aim of this work is to
investigate the distribution of fidelity for state transfer in the case of a
non-precise read-out timing on the receiving side. This assumption leads to
two specific forms of the distribution depending on the transmission
mechanism. Thus, we will demonstrate explicitly that the equivalence at the
level of the average fidelity is not enough to conclude that different
protocols are fully equivalent, as they can give rise to different
distributions of fidelity in presence of imperfections.

This paper is organized as follows. In Section II we highlight techniques to
obtain the distribution of fidelity for state transfer protocols. After
recalling in Section III a general approach to evaluate the fidelity for spin-
chain-based transfer using the Kraus representation of the channel connecting
the sender to the receiver, we apply the general method in Section IV, in
which three different protocols are analyzed. We consider the one-qubit and
two-qubits state transfer protocol. In the latter case the distribution
fidelity is derived for states with a fixed value of entanglement. The paper
is concluded in Section V with some final remarks.

##  II Distribution of fidelity

For a given protocol, we can evaluate the fidelity for any sender state; its
expression will vary with the details of the dynamics, but, generally
speaking, it will depend on the parameters with which we identify the state to
be sent, and on the time elapsed since the beginning of the protocol, i.e.
$F\equiv F(t;\alpha,\beta,\dots)$. To obtain the probability distribution of
fidelity at a given time $t$ we need to exploit its functional dependence on
the state parameters.

Denote the mean fidelity averaged over the others parameters $F(\alpha)=\int
F(\alpha,\beta,\cdots)d\beta\cdots$. With a minor abuse of notation, we can
write the inverse functions, $\\{\alpha(F),\beta(F),\cdots\\}$. Then we arrive
at the following form of the probability distribution function,

$\text{pdf}(F)=\text{pdf}\left(\alpha(F)\right)\left|\frac{\partial\alpha(F)}{\partial
F}\right|,\;\;\;\;\;\text{with}\;\;0\leq F\leq 1.$ (1)

The primary concern at this point is to derive a clear expression for the
fidelity over time, and to establish a parametrization that allows us to
utilize Eq. 1. Generally, this task is not straightforward. Therefore, in the
subsequent discussion, we aim to outline a road-map to achieve the desired
outcome.

##  III General approach to the transfer map

We consider a general network system in which each node hosts a $d$-level
system, with Hilbert space given by $\mathcal{H}=\mathbb{C}^{d}$. As pointed
out above, we are interested in evaluating the fidelity between a pure input
state $\ket{\psi_{\cal A}}$ written at $t=0$ on a set of sending sites
$\mathcal{A}$, and the state $\rho_{\mathcal{B}}(t)$ retrieved, at the end-
time of the protocol, $t$, from a different set of receiving sites
$\mathcal{B}$; namely,

$F(t)=\bra{\psi_{\mathcal{A}}}\rho_{\mathcal{B}}(t)\ket{\psi_{\mathcal{A}}}.$
(2)

Assuming that the rest of the system, which we call
${\mathcal{A}^{\mathsf{c}}}$, is initialized in _fixed_ state
$\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$, the total initial state is
$\ket{\psi_{\mathcal{A}}}\otimes\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$ and the
reduced density matrix of the receiving sites $\mathcal{B}$ is obtained after
tracing out the complementary degrees of freedom,
${\mathcal{B}^{\mathsf{c}}}$:

$\displaystyle\rho_{\mathcal{B}}(t){=}\text{Tr}_{{\mathcal{B}^{\mathsf{c}}}}{\left\\{U(t)\Big{(}\\!\ket{\psi_{\mathcal{A}}}\otimes\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}\\!\Big{)}\Big{(}\\!\bra{\psi_{\mathcal{A}}}\otimes\bra{\psi_{\mathcal{A}^{\mathsf{c}}}}\\!\Big{)}U^{\dagger}(t)\right\\}}$
$\displaystyle{=}\sum_{\phi_{\mathcal{B}^{\mathsf{c}}}}\bra{\phi_{\mathcal{B}^{\mathsf{c}}}}U(t)\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}\\!\Big{(}\ket{\psi_{\mathcal{A}}}\bra{\psi_{\mathcal{A}}}\Big{)}\\!\bra{\psi_{\mathcal{A}^{\mathsf{c}}}}U^{\dagger}(t)\ket{\phi_{\mathcal{B}^{\mathsf{c}}}}\leavevmode\nobreak\
,$ (3)

Figure 1: Sketch of the transmission and illustration of our notation: the
state $\ket{\psi}$ is prepared at the sending sites, ${\cal A}$, and its
information content is spread all over the system, ${\cal A}\bigcup{\cal
A}^{c}$, due to the unitary evolution $U$. The state is later retrieved at the
receiving sites ${\cal B}$ .

where $U(t)$ is the unitary time evolution operator describing the dynamics of
the full system, and $\\{\ket{\phi_{{\mathcal{B}^{\mathsf{c}}}}}\\}$ denotes a
complete orthonormal basis of ${\mathcal{B}^{\mathsf{c}}}$. The second line of
Section III provides the Kraus decomposition of the completely positive linear
operator
$\Lambda(t):\mathcal{H}_{\mathcal{A}}{\rightarrow}\mathcal{H}_{\mathcal{B}}$,
which maps the initial state of $\mathcal{A}$ onto the state of $\mathcal{B}$
at time $t$. Explicitly,

$\rho_{\mathcal{B}}(t){=}\Lambda(t)\Big{[}\ket{\psi_{\mathcal{A}}}\bra{\psi_{\mathcal{A}}}\Big{]}{=}\sum_{\phi_{\mathcal{B}^{\mathsf{c}}}}E_{\phi_{\mathcal{B}^{\mathsf{c}}}}\ket{\psi_{\mathcal{A}}}\bra{\psi_{\mathcal{A}}}E_{\phi_{\mathcal{B}^{\mathsf{c}}}}^{\dagger},$
(4)

with
$E_{\phi_{\mathcal{B}^{\mathsf{c}}}}(t)=\bra{\phi_{\mathcal{B}^{\mathsf{c}}}}U(t)\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$.
Substituting this expression for $\rho_{\mathcal{B}}$ into Eq. 2, we obtain
the fidelity in the form

$F(t)=\sum_{\phi_{\mathcal{B}^{\mathsf{c}}}}|\bra{\psi_{\mathcal{A}}}E_{\phi_{\mathcal{B}^{\mathsf{c}}}}(t)\ket{\psi_{\mathcal{A}}}|^{2}.$
(5)

This general expression for $F$ will be utilised in the following for
different state transfer channels.

###  III.1 Dynamics with U(1) symmetry

The Kraus operators $E_{\phi_{\mathcal{B}^{\mathsf{c}}}}(t)$ entering Eq. 5
can be cumbersome to evaluate for arbitrary channels. However, in the presence
of symmetries in the system’s dynamics, some general assumptions can be made
about their form. Let us assume that a conserved quantity $\mathbf{Q}$ exists
whose spectral decomposition (taking into account possible degeneracies) is
given by

$\mathbf{Q}=\sum_{q}q\sum_{d=1}^{d_{q}}\ket{q_{d}}\bra{q_{d}}\leavevmode\nobreak\
.$ (6)

Then the evolution operator $U(t)$ reads

$U(t)=\sum_{q}\sum_{d=1}^{d_{q}}\sum_{d^{\prime}=1}^{d_{q}}U_{q}^{dd^{\prime}}(t)\ket{q_{d}}\bra{q_{d^{\prime}}}=U_{1}\oplus
U_{2}\oplus\cdots,$ (7)

where $U_{q}^{dd^{\prime}}=\bra{q_{d}}U(t)\ket{q_{d^{\prime}}}$. The above
equation entails that $U(t)$ does not couple different subspaces of the
conserved quantity $\mathbf{Q}$. In order to take advantage of the presence of
invariant subspaces, each Kraus operator appearing in Eq. 4 can be expressed
as

$\displaystyle E_{\phi_{\mathcal{B}^{\mathsf{c}}}}{=}$
$\displaystyle\bra{\phi_{\mathcal{B}^{\mathsf{c}}}}U(t)\ket{\psi_{{\mathcal{A}^{\mathsf{c}}}}}=$
$\displaystyle=$
$\displaystyle\sum_{\phi_{\mathcal{B}}}\sum_{\phi_{\mathcal{A}}}\bra{\phi_{\mathcal{B}}}\bra{\phi_{\mathcal{B}^{\mathsf{c}}}}U(t)\ket{\psi_{{\mathcal{A}^{\mathsf{c}}}}}\ket{\phi_{\mathcal{A}}}\;\ket{\phi_{\mathcal{B}}}\bra{\phi_{\mathcal{A}}}.$
(8)

Inserting Eq. 7 into the latter expression, we realize that if neither
$\ket{\phi_{\mathcal{B}},\phi_{\mathcal{B}^{\mathsf{c}}}}$ nor
$\ket{\psi_{{\mathcal{A}^{\mathsf{c}}}},\phi_{\mathcal{A}}}$ belong to the
$q$-subspace, the corresponding element of
$E_{\phi_{\mathcal{B}^{\mathsf{c}}}}$ is null.

If we now assume that $\mathbf{Q}$ is a sum of local observables, namely

$\mathbf{Q}{=}\sum_{i=1}^{N}\mathbf{q}_{i}{=}\\!\left(\sum_{i\in\mathcal{A}}\mathbf{q}_{i}{+}\\!\sum_{j\in{\mathcal{A}^{\mathsf{c}}}}\mathbf{q}_{j}\right)\\!{=}\\!\left(\sum_{i\in\mathcal{B}}\mathbf{q}_{i}{+}\\!\sum_{j\in{\mathcal{B}^{\mathsf{c}}}}\mathbf{q}_{j}\right)\leavevmode\nobreak\
,$ (9)

where $\mathbf{q}_{i}$ is a _local_ operator acting on each local
$\mathcal{H}_{i}$, then
$q=q_{\mathcal{A}^{\mathsf{c}}}+q_{\mathcal{A}}=q_{\mathcal{B}^{\mathsf{c}}}+q_{\mathcal{B}}$,
which implies that only the Kraus matrix elements fulfilling the above
conditions are non-zero. In particular, these considerations apply to the case
of a spin system with, e.g., conserved magnetization along the quantization
axis, holding for Hamiltonians with global $\mathrm{U}(1)$ symmetry.

##  IV Spin chain-based state transfer

In this section we provide some explicit cases in order to apply the general
framework outlined above. In particular, we will consider the transfer of one
and two spin-$1/2$ states over a network of coupled spins. In this case, we
take the sender to reside at the first sites of the chain, while the receiver
is located at the opposite edge, e.g., at site $N$ for the single qubit state
transfer, or $\\{N{-}1,N\\}$ in the $2$ qubit case.

We consider a network $\mathbf{G}$ of $N$ sites, each hosting a qubit, whose
dynamics is generated by an $\mathrm{U}(1)$-symmetric Hamiltonian of the form:

$\mathbf{H}_{\mathbf{G}}=\sum_{i\neq
j}^{N}J_{ij}\left(\sigma^{i}_{x}\sigma^{j}_{x}{+}\sigma^{i}_{y}\sigma^{j}_{y}{+}\Delta_{ij}\sigma^{i}_{z}\sigma^{j}_{z}\right)+\sum_{i=1}^{N}B_{i}\sigma_{z}^{i}\leavevmode\nobreak\
,$ (10)

where $\sigma_{x/y/z}^{i}$ are the Pauli matrices for the qubit residing at
the $i$-th site, $B_{i}$ are static magnetic fields and $J_{ij},\Delta_{ij}$
are coupling strengths.

This Hamiltonian is symmetric under rotation around the $z$-axis; thus, in
this context, the quantity $\mathbf{Q}=\sum_{i=1}^{N}\sigma_{i}^{z}$ is,
indeed, conserved.

###  IV.1 One-qubit transfer

The first case under investigation is the transfer of the state of one qubit.
This entails that the only allowed values of $q$ are
$0+q_{\mathcal{A}^{\mathsf{c}}}$ and $1+q_{\mathcal{A}^{\mathsf{c}}}$, where
$q_{\mathcal{A}^{\mathsf{c}}}=\expectationvalue{\sum_{j\in{\mathcal{A}^{\mathsf{c}}}}\sigma^{j}_{z}}{\psi_{\mathcal{A}^{\mathsf{c}}}}$.
Hence, there are only two classes of Kraus operators corresponding to
$q_{\mathcal{B}^{\mathsf{c}}}=0,1$ and their explicit expressions depend on
the state of the channel at the time the transfer protocol begins, which we
specify in the following subsections.

#### Initial vacuum state for the channel —

If we take $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}=\ket{0\dots 0}$, then
$q_{\mathcal{A}^{\mathsf{c}}}=0$ and the Kraus operators in Section III.1 are
given by

$E_{0}{=}\begin{pmatrix}1&0\\\
0&a_{1}^{N}(t)\end{pmatrix},\;E_{1}^{j}{=}\begin{pmatrix}0&a_{j}^{N}(t)\\\
0&0\end{pmatrix}\;\text{for}\;j{=}2,\ldots,N,$ (11)

where we have renamed $U_{1}^{ij}(t)=a_{i}^{j}(t)$. The class
$q_{\mathcal{B}^{\mathsf{c}}}=1$ can be cast as a single Kraus operator,

$\displaystyle
E_{1}{=}\begin{pmatrix}0&\sqrt{1-\left|a_{1}^{N}(t)\right|^{2}}\\\
0&0\end{pmatrix}\leavevmode\nobreak\ .$ (12)

By parametrizing the sender state as

$\displaystyle\ket{\psi_{\mathcal{A}}}=\cos\left(\frac{\theta}{2}\right)\ket{0}+e^{i\phi}\sin\left(\frac{\theta}{2}\right)\ket{1},$
$\displaystyle\text{with}\;\;0\leq\theta\leq\pi\;\;\text{and}\;\;0\leq\phi\leq
2\pi,$ (13)

the fidelity (5) turns into a function of $\theta$, without $\phi$ dependence,
due to the fact that the Hamiltonian in Eq. 10 possesses $\mathrm{U}(1)$
symmetry and the initial state of the channel
$\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$ is an eigenstate of the total
magnetization $\mathbf{Q}$. It turns out that

$F(t,\theta)=\bigl{(}1-|a_{1}^{N}(t)|^{2}\bigr{)}s^{2}c^{2}+\left|a_{1}^{N}(t)s^{2}+c^{2}\right|^{2},$
(14)

where we used the completeness relation $\sum_{j=1}^{N}|a_{j}^{N}|^{2}=1$ and
defined $s\equiv\sin(\theta/2),c\equiv\cos(\theta/2)$.

The PDF for the angle $\theta$ of uniformly distributed pure states on the
Bloch sphere is given by

$\text{pdf}(\theta)=\dfrac{1}{2}\sin\theta,\qquad\text{with}\;\;0\leq\theta\leq\pi,$
(15)

so that, by straightforward application of Eq. 1, we obtain a closed
expression for the distribution of fidelity reported in Appendix A.

We are now in position to compare the distribution of fidelity obtained from
different transfer protocols. We will consider the weak-coupling approach (W)
[25], the barrier protocol (B) [28], and the perfect transfer protocol (P)
[17] – see Fig. 2 and Appendix B for details. A sketch of these three
protocols provided in Fig. 2 emphasizes their main differences: the coupling
constants $J_{n}$ are depicted by red segments, whose number indicates the
strength of coupling. For the W protocol, the wavy segment stands for a very
weak $J$. Shadows around the dots are used to represent local magnetic fields:
the larger the shadow, the stronger the field. All of these approaches admit
an optimal time transfer $t_{\text{opt}}$ at which the average fidelity over
all possible input states reaches the maximum value. It is exactly $\langle
F\rangle=1$ for the P-protocol, whereas for the W- and B-protocols the unit
value is reached asymptotically, although at the expense of a longer transfer
time. This means that, for all practical purposes, at the optimal time the
protocol is perfect, so that $\text{pdf}(F)=\delta(F-1)$.

Figure 2: Sketch of the three protocols considered in this work. The coupling
constants $J_{n}$ are depicted by red segments, whose number represent the
strength of coupling. In the W protocol, the wavy segment stands for a very
weak $J$. The dots symbolize the sites, while their shadows represent the
local magnetic fields, where a larger shadow indicates a stronger magnetic
field. The barrier (${\mathbf{B}}$) protocol assumes uniform couplings and a
strong magnetic field applied to the nearest-neighbor of the senders and the
receivers. The weak coupling (${\mathbf{W}}$) protocol is defined by weaker
couplings between the chain and the senders (receivers). In the perfect
(${\mathbf{P}}$) protocol one assumes to be able to engineer all of the
couplings between the sites. Details of each protocol are discussed in
Appendix B.

On the other hand, if the read-out timing is not precise, or if, for
experimental reasons, one is satisfied with an $\langle F\rangle$ slightly
smaller than $1$, by choosing a “compromise setting”, then, different
protocols display very different behaviours of the pdf, as reported in Fig. 3.

In particular, as one can notice from the insets in Fig. 3, and discussion in
Appendix B, these three protocols show different sensitivities to the optimal
time. The average fidelity for the B-protocol, oscillates in time and shows
very narrow peaks, quite close to each other. On the other hand, the
P-protocol and the W-protocol display a single, but much broader peak, with a
reduced dependence on the choice of the time $t_{\text{opt}}$.

As a result of the different dynamics, the W- and B-protocols display quite a
similar distribution of fidelities, with a lower bounded support. The
$\text{pdf}(F)$ for the P-protocol, instead, shows tails extending to much
smaller values of $F$, as seen from the top panel of Fig. 3. In the lower
panel, we show the fidelity distributions under the condition that the average
fidelity is set to a smaller value, $\langle F\rangle<1$, common to all of the
three protocols. In this case, one can see that the W- and P-protocols,
display the same $\text{pdf}(F)$, peaked at $F=1$ and with a tail extending
down to a minimum fidelity value, $F_{min}$. The $\text{pdf}(F)$ for the
B-protocol, instead, has a peak at a smaller $F$, with a tail extending on its
right side, up to $F=1$.

The features displayed by the PDFs can be understood by analyzing the dynamics
of the spin system in the three cases. In particular, setting the average
fidelity to a given value amounts to choosing both the absolute value of the
transition amplitude $a_{1}^{N}$ and its real part (see the expression for
$\langle F\rangle$ reported in Appendix B). It turns out that for both the P-
and W-protocols, this amplitude can be taken to be either real or purely
imaginary, so that it is fully determined by the choice of $\langle F\rangle$.
Thus, these two approaches to state transfer have the same amplitude for a
given $\langle F\rangle$. Now, by virtue of Eq. 26, they enjoy the same
distribution with $\theta$.

Interestingly, from Eq. 14, it is also possible to derive the states that are
transmitted with minimum fidelity at fixed transition amplitude $a_{1}^{N}$.
This is an important quantity in QST protocols as it allows the definition of
a lower bound on the reliability of the transferred state. For the P- and
W-protocols, where the transition amplitude $a_{1}^{N}$ is real (for $N$ odd)
or imaginary (for $N$ even), we can set $a_{1}^{N}$ to be real (possibly by
application of a constant magnetic field in the even $N$ case). Hence the
state with $\theta=\pi$, i.e., $\ket{\psi_{\mathcal{A}}}=\ket{1}$, yields the
minimum fidelity
$F(t,\pi)=\left(a_{1}^{N}\right)^{2}=\left(\sqrt{2\left(3\langle
F\rangle-1\right)}-1\right)^{2}$ for all values of $a_{1}^{N}$. On the other
hand, for the B-protocol, where no constant magnetic field is applied for
maximising the average fidelity, the state that minimizes Eq. 14 is dependent
on the complex amplitude $a_{1}^{N}$. Expressing the transition amplitude in
polar form $a_{1}^{N}=re^{i\phi}$, we obtain that the minimum fidelity
obtained in Eq. 14 for

$\displaystyle\begin{cases}\theta=\cos^{-1}\left(\frac{r^{2}-1}{2r\left(r-\cos\phi\right)}\right),&r>\frac{1}{3}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \phi\in\left[\phi^{*},2\pi-\phi^{*}\right]\\\
\theta=\pi,&r>\frac{1}{3}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\phi\notin\left[\phi^{*},2\pi-\phi^{*}\right]\\\
\theta=\pi,&r\leq\frac{1}{3}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ 0\leq\phi<2\pi\end{cases}\leavevmode\nobreak\ $
(16)

where $\phi^{*}=\cos^{-1}\left(\frac{3r^{2}-1}{2r}\right)$. Eq. 16 illustrates
the fact that, for the cases of interest where $r$ approaches unity, the state
that mimimizes the fidelity in Eq. 14 can be either
$\ket{\psi_{\mathcal{A}}}=\ket{1}$ or a state lying in the neighborhood of the
equatorial plane as $\theta\rightarrow\pi/2$, depending on the phase $\phi$ of
the transition amplitude $a_{1}^{N}$. Equipped with the knowledge of the state
that achieves the maximum fidelity, i.e., $\ket{0}$ for all protocols, and the
state that achieves the minimum fidelity we can now explain the positions of
the peaks in the $\text{pdf}(F)$ in Fig. 3. For the W- and P-protocol, the
rate of change of the fidelity with $\theta$ is less at $\theta=0$ than at
$\theta=\pi$, implying that there are more states in the neighborhood of
$\ket{0}$ than in that of $\ket{1}$ in a fixed interval of the maximum and
minimum fidelity, respectively. On the other hand, for the B-protocol, when
the minimum fidelity is achieved for a state lying close to the equatorial
plane, the opposite holds.

Figure 3:  Probability distribution of fidelity for three protocols analyzed:
(top) Reading time is set to up the $2\%$ of the optimal one, which depends on
the choice of the protocol. The average fidelity, in all of the three cases,
is greater than $0.99$ – see the insets; (bottom) Average fidelity is fixed to
$0.99$ and the minimal fidelity of the protocol B is higher when compared to
the W- and P-protocols. Results are obtained for $N=22$ sites, with
$h_{0}=200$ for the B-protocol, and $J_{0}=1/200$ for the W-protocol – see
Appendix B for details.

####  $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}\neq\text{vacuum}$ —

In order to show the role played by the Kraus operators, now we assume that
the initial state $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$ belongs to the
1-excitation subspace and that it is uniformly spread over the entire system:

$\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}=\frac{1}{\sqrt{N{-}2}}\sum_{j=2}^{N-1}\ket{j}.$
(17)

Figure 4:  PDF of the fidelity for $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$
when the channel is in the single-excitation state given by Eq. 17. The three
protocols have very different behaviour despite their average fidelity being
fixed to $\langle F\rangle=0.99$. The $B$-protocol shows a higher minimum
fidelity, compared to $W$ and $P$ protocols. The latter protocols, however,
transfer more states with higher fidelity. The fugure reports the case for
$N=15$, $h_{0}=100$ for the B protocol and $J_{0}=1/100$ for the W protocol.

With these assumptions we have three different classes of Kraus operators
corresponding to $\phi_{\mathcal{B}^{\mathsf{c}}}{\in}\,q=\\{0,1,2\\}$ (i.e.
we can have $0,1$ or $2$ excitations in ${\mathcal{B}^{\mathsf{c}}}$):

$\displaystyle
E_{0}=\frac{1}{\sqrt{N{-}2}}\sum_{j=2}^{N{-}1}\begin{pmatrix}0&a_{j}^{N}\\\
0&0\end{pmatrix},$ (18) $\displaystyle
E_{1}=\frac{1}{\sqrt{N{-}2}}\sum_{j=2}^{N{-}1}\begin{pmatrix}b_{1j}^{kN}&0\\\
0&a_{j}^{N}\end{pmatrix},$ $\displaystyle\text{for}\;k$
$\displaystyle=\\{1,\ldots,N{-}1\\},$ $\displaystyle
E_{2}=\frac{1}{\sqrt{N{-}2}}\sum_{j=2}^{N{-}1}\begin{pmatrix}0&0\\\
b_{1j}^{kl}&0\end{pmatrix},$ $\displaystyle\text{for}\;k<l$
$\displaystyle=\\{1,\ldots,N{-}1\\}.$

where we have introduced the two site transition amplitudes
$U_{2}^{ij}(t)=b_{i_{1}i_{2}}^{j_{1}j_{2}}(t)$ for all the pairs $\\{i,j\\}$.

As in the previous example, the transfer fidelity can be written as a
quadratic function of $\cos\theta$, and, again, inverting this relation and
using Eq. 1, it is possible to obtain an analytic expression of
$\text{pdf}(F)$, see Appendix A for details. In this case, also the average
fidelity $\langle F\rangle$ depends on the $2$-excitations transition
amplitudes:

$\langle
F\rangle=\frac{1}{3}+\frac{1}{6(N-2)}\sum_{k=1}^{N-1}\left|\sum_{j=2}^{N-1}a_{j}^{k}+b^{kN}_{1j}\right|^{2}.$
(19)

Fig. 4 shows the probability distribution corresponding to three protocols at
fixed average fidelity $\langle F\rangle=0.99$. The shapes of the
distributions are different: the $B$-protocol displays a minimum fidelity
greater than the $W$ and $P$ protocols, but, for most of the states, achieves
a lower fidelity than the $W$\- and $P$\- ones, which, on the other hand,
display a maximum achievable fidelity.

###  IV.2 Two-qubit transfer

Consider now transfer of a two-qubit quantum state: In this case the dynamics
takes place in three $q$ subspaces and the Kraus operators act on a two-qubit
subspace. Assuming $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}$ to be initially in
the vacuum state ($q_{\mathcal{A}^{\mathsf{c}}}=0$), the three classes of
Kraus operators are now given by:

$\displaystyle E_{0}=\begin{pmatrix}1&0&0&0\\\\[2.15277pt]
0&a_{N}^{2}&a_{N}^{1}&0\\\\[2.15277pt]
0&a_{N{-}1}^{2}&a_{N{-}1}^{1}&0\\\\[2.15277pt] 0&0&0&b_{N{-}1N}^{12}\\\
\end{pmatrix},$ (20a) $\displaystyle
E_{1}^{j}=\begin{pmatrix}0&a_{j}^{2}&a_{j}^{1}&0\\\\[2.15277pt]
0&0&0&b_{jN}^{12}\\\\[2.15277pt] 0&0&0&b_{jN{-}1}^{12}\\\\[2.15277pt]
0&0&0&0\\\ \end{pmatrix},\leavevmode\nobreak\ j=\\{1,\ldots,N{-}2\\},$ (20b)
$\displaystyle E_{2}^{kj}=\begin{pmatrix}0&0&0&b_{kj}^{12}\\\\[2.15277pt]
0&0&0&0\\\\[2.15277pt] 0&0&0&0\\\\[2.15277pt] 0&0&0&0\\\
\end{pmatrix},\leavevmode\nobreak\ k<j=\\{1,\ldots,N{-}2\\}.$ (20c)

Writing a family of two-qubit states in the Schmidt form,

$\ket{\psi_{\mathcal{A}}}=\sqrt{\frac{1-s}{2}}\ket{00}+\sqrt{\frac{1+s}{2}}\ket{11},\quad-1\leq
s\leq 1$ (21)

we assure that any other pure state can be obtained form it by local unitary
transformations. Notice that the parameter $|s|$ quantifies the separability
of the state, as the concurrence of $\ket{\psi_{\mathcal{A}}}$ reads
$C=\sqrt{1-s^{2}}$. Due to Eq. 5 the fidelity now has the form

$F(t,U_{1},U_{2},C)=\sum_{\phi_{\mathcal{B}^{\mathsf{c}}}}|\bra{\psi_{\mathcal{A}}}U_{1}^{\dagger}U_{2}^{\dagger}\,E_{\phi_{\mathcal{B}^{\mathsf{c}}}}(t)\,U_{1}U_{2}\ket{\psi_{\mathcal{A}}}|^{2}.$
(22)

Averaging over all possible local unitaries, the fidelity turns out to be a
function of concurrence alone:

$F(t,C)=\int\\!\\!F(t,u_{1},u_{2},C)du_{1}du_{2}=A(t)-B(t)C^{2}$ (23)

where $A(t)$ and $B(t)$ are functions involving the transition amplitudes $a$s
and $b$s entering the Kraus operators in Eq. 20a. The lenghty expressions of
these functions are reported in Section A.3. From Eq. 23 it is straightforward
to see that the minimum fidelity (averaged over entanglement) achievable
through the specific protocol is given by the minimum between $A(t)$ and
$A(t)+B(t)$.

Making use of the distribution of concurrence for two-qubit random pure states
[46, 47]:

$\text{pdf}(C)=3C\sqrt{1-C^{2}}$ (24)

and inserting into Eq. 1, we derive the distribution of fidelity for the
transfer of a two-qubit state with any degree of entanglement,

$\text{pdf}(F)=\frac{3}{2|B(t)|}\sqrt{\frac{F-A(t)}{B(t)}}$ (25)

with $(F{-}A(t))/B(t)>0$.
Two-qubit fidelity distribution derived in Eq. 25 is used to obtain Fig. 5
relative to the cases of the three different protocols ($W,B,P$) at fixed
average fidelity $\langle F\rangle=0.99$. The three distributions have the
same shape, but that of the $B$ protocol is narrower, with a minimum fidelity
greater than that obtained by the $W$ and $P$ protocols.

Figure 5:  Probability distribution of the fidelity for the 2-qubits transfer
averaged over local unitary operations at fixed average fidelity $\langle
F\rangle=0.99$. The minimum fidelity of the protocol $B$ is greater than this
achieved in protocols $W$ and $P$. Data are obtained for $N=9$ and $h_{0}=200$
for protocol $B$ and $J_{0}=1/200$ for protocol $W$.

##  V Discussion and Conclusions

To characterize the quality of QST, previous works have considered the average
transfer fidelity between the initial state and its image with respect to a
state transfer protocol, or, in a few cases, also the variance of fidelity
[48, 40]. In this contribution we go beyond these results and focus on the
entire probability distribution of fidelity for various state transfer
protocols. It is demonstrated that transmission protocols, which are
equivalent with respect to the average transfer fidelity, display different
behaviors in terms of their reliability, and worst case performances.

We exploited a global symmetry of the Hamiltonian describing the state
transfer in order to evaluate analytically the probability distribution
function of the fidelity for one and two qubits states. We derived the
corresponding distributions of fidelity using explicit expressions for the
Kraus operators describing the dynamical map that sends the input into the
output state. Allowing for non-optimal scenarios, like errors in the reading
times, we showed how different protocols, although resulting in the same
average fidelity, can display very different distributions. Furthermore, for
the one-qubit QST, we found an exact relation between the sender-receiver
transition amplitude and the minimum fidelity of the protocol.

For the two-qubit transfer, we used the entanglement probability distribution
of the input state to characterize the PDF of the entanglement averaged
fidelity. We have left the examination of the mixed state transfer and the
potential presence in the channel of noise and/or dissipation that may impact
the dynamics for future analysis, as well as the extension of the transfer
protocols for a larger number of subsystems, $n>2$.

Finally, considering the general framework presented for obtaining one- and
two-qubit distribution of fidelity, our results can be straightforwardly
applied to state transfer protocols different that those considered here.

###### Acknowledgements.

TJGA ackowledges funding by the European flagship on quantum technologies
(‘ASPECTS’ consortium 101080167). MC acknowledges funding by the Tertiary
Education Scholarships Scheme and the MCST Research Excellence Programme 2022
on the QVAQT project at the University of Malta. SL acknowledge support by MUR
under PRIN Project No. 2017 SRN-BRK QUSHIP. KZ was supported by the QuantERA
II Programme financed by the National Science Centre, Poland, under the
contract number 2021/03/Y/ST2/00193. The views and opinions expressed are
however those of the authors only and do not necessarily reflect those of the
European Union. Neither the European Union nor the granting authority can be
held responsible for them.

##  Appendix A Calculation of probability distribution of fidelity

In this section we provide the details for the evaluation of the PDF of the
fidelity for the different cases analyzed in the main text.

###  A.1 $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}=\text{vacuum}$

Firstly note that Eq. 14 can be recast as a quadratic function of $\cos\theta$

$F=\frac{|a_{1}^{N}|^{2}{-}\text{Re}[a_{1}^{N}]}{2}\cos^{2}(\theta){+}\frac{1{-}|a_{1}^{N}|^{2}}{2}\cos(\theta)+\frac{1{+}\text{Re}[a_{1}^{N}]}{2},$
(26)

which is easily solved for $\theta$.
Defining $a{=}(|a_{1}^{N}|^{2}-\text{Re}[a_{1}^{N}])/2$,
$b{=}(1-|a_{1}^{N}|^{2})/2$ and $c{=}(1+\text{Re}[a_{1}^{N}])/2$ we find the
expression of $\theta(F)\equiv F^{-1}$ to be

$\theta^{(\pm)}(F)=-\frac{b}{2a}\pm\frac{\sqrt{\Delta(F)}}{2a}$ (27)

with $\Delta(F)=b^{2}-4a(c-F)$.
Applying Eq. 1 we obtain

$\text{pdf}(F)=\text{pdf}^{\,(-)}(F)+\text{pdf}^{\,(+)}(F),$ (28)

where each term is defined for $\absolutevalue{\theta(F)^{(\pm)}}\leq 1$ and
is given by

$\text{pdf}^{\,(\pm)}(F)=\frac{1}{2}\sqrt{1{-}\theta^{(\pm)}(F)^{2}}\,\absolutevalue{\left(\Delta(F)\sqrt{1{-}\theta^{(\pm)}(F)^{2}}\right)^{-1}}.$

###  A.2 $\ket{\psi_{\mathcal{A}^{\mathsf{c}}}}\neq\text{vacuum}$

As mentioned in the main body of the paper, in this case the fidelity is also
a quadratic function of $\cos\theta$,

$\displaystyle F(\theta)$
$\displaystyle=\left(\frac{1}{4}-\frac{3\alpha+\beta-3\gamma}{4(N-2)}\right)\cos^{2}\theta$
$\displaystyle+\left(\frac{1}{2}-\frac{\alpha+\gamma}{2(N-2)}\right)\cos\theta$
$\displaystyle+\left(\frac{1}{4}+\frac{\alpha+\beta-\gamma}{4(N-2)}\right),$
(29)

with $\alpha=\absolutevalue{\sum_{j=2}^{N-1}a_{j}^{N}}^{2}$,
$\beta=\sum_{k=1}^{N-1}\absolutevalue{\sum_{j=2}^{N-1}a_{j}^{N}+b_{1j}^{kN}}^{2}$
and $\gamma=\sum_{k=1}^{N-1}\absolutevalue{\sum_{j=2}^{N-1}b_{1j}^{kN}}^{2}$.

Redefining the coefficients $a,b,c$ it is possible to proceed as in the
previous section in the evaluation of the $\text{pdf}(F)$.

###  A.3 Two-qubit transfer

As discussed in the main text, averaging over local unitaries acting on the
two qubits input state, the fidelity can be written as a function of the
initial concurrence $C$ (see Eq. 23) as:

$F(t,C)=A(t)-B(t)C^{2}$ (30)

where the time dependent coefficients $A$ and $B$ are given by

$\displaystyle A(t){=}$
$\displaystyle\frac{1}{72}\big{(}14+2\sum_{j=1}^{N-2}|a^{1}_{j}+b^{12}_{jN}|^{2}+|a^{2}_{j}+b^{12}_{jN-1}|^{2}+$
(31) $\displaystyle
2\,\text{Re}\big{[}2a^{1}_{N-1}a^{*2}_{N}{+}2b^{12}_{N-1N}{+}4(a^{1}_{N-1}{+}a^{2}_{N})(1{+}b^{*12}_{N-1N})\big{]}+$
$\displaystyle
6\big{(}|a^{1}_{N-1}|^{2}+|a^{2}_{N}|^{2}+|b^{12}_{N-1N}|^{2}\big{)}\big{)}$
$\displaystyle B(t){=}$
$\displaystyle\frac{1}{72}\big{(}{-}10+5\sum_{j=1}^{N-2}|a^{1}_{j}+b^{12}_{jN}|^{2}+|a^{2}_{j}+b^{12}_{jN-1}|^{2}-$
(32) $\displaystyle
2\,\text{Re}\big{[}4a^{1}_{N-1}a^{*2}_{N}{+}4b^{12}_{N-1N}{-}(a^{1}_{N-1}{+}a^{2}_{N})(1{+}b^{*12}_{N-1N})\big{]}+$
$\displaystyle
6(|a^{1}_{N-1}|^{2}+|a^{2}_{N}|^{2}+|b^{12}_{N-1N}|^{2})\big{)}$

For example, in the three instances reported in Eq. 25, respectively for the
three protocols $W,B,P$, at fixed average fidelity $\langle F\rangle=0.99$,
the values of these functions are:

$\begin{cases}B)\qquad&$A=0.9904$\,,\,$B=-0.0006$\\\
W)\qquad&$A=0.9912$\,,\,$B=-0.0021$\\\
P)\qquad&$A=0.9910$\,,\,$B=-0.0017$.\end{cases}$ (33)

##  Appendix B Details of the B-, W- and P-protocols

Recalling equation Eq. 10 of the main text, the three protocols $(W,B,P)$ are
defined by different choices of the couplings and the magnetic fields (see
Fig. 2) as reported in Table 1. For all the three cases one has
$\Delta_{ij}=0$. The weak-coupling protocol $W$ is defined by having the first
and last couplings much smaller than the rest ($J_{0}\ll J$) and no local
magnetic fields.

protocol | couplings | magnetic field
---|---|---
W | $J\delta_{i,i{+}1}$ | $B_{i}{=}0$
$J_{1,2}{=}J_{N{-}1,N}=J_{0}$
B | $J\delta_{i,i{+}1}$ | $B_{i}{=}0$
$B_{2}{=}B_{N{-}1}=h_{0}$
P | $J\sqrt{i(N-i)}\delta_{i,i{+}1}$ | $B_{i}{=}0$

Table 1: Values of coupling constants and local magnetic fields that define
the three protocols considered in this work. Figure 6:  Average fidelity
relative to three different protocols plotted against a rescaled time with
respect to the relative optimal time $t_{\text{opt}}$. No auxiliary magnetic
field is applied on the chain (left panel) compared to a constant magnetic
field assuring that at the optimal time $t_{\text{opt}}$ the average fidelity
is as close as possible to unity (right). No significant differences occur for
protocol $B$, whereas for protocols $W$ and $P$ the auxiliary field is
necessary. Both panels refer to a chain of $N=22$ sites with $h_{0}=200$ for
protocol $B$ and $J_{0}=1/200$ for protocol $W$.

The barrier protocol $B$ has uniform couplings in all the chain and strong
magnetic fields only on the second and the last but one sites ($h_{0}\gg J$).
The perfect protocol $P$ is characterized by engineered couplings and no
magnetic fields.

For the one qubit QST with the chain initially in the vacuum, as in Section
IV.1, expressing the transition amplitude in polar form
$a_{1}^{N}=re^{i\phi}$, the fidelity averaged over all possible pure state is
given by [3]:

$\langle F\rangle=\frac{1}{2}+\frac{r\cos{\phi}}{3}+\frac{r^{2}}{6}$ (34)

The phase $\phi$ of the transition amplitude $a_{1}^{N}$ can be adjusted
applying a uniform magnetic field such that at the optimal time
$t_{\text{opt}}$ one retrieves $\cos{\phi}=1$. This is shown in Fig. 6: on the
left we report the behavior of $\langle F\rangle$ for the three protocols
without the auxiliary magnetic field, and on the right an uniform magnetic
field $B_{\text{aux}}=\text{Arg}(a_{1}^{N}(t_{\text{opt}}))/t_{\text{opt}}$ is
applied.

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