11institutetext: Dept. of Computer Science, Stony Brook University, NY, USA

# Semi-supervised Domain Adaptive Medical Image Segmentation through
Consistency Regularized Disentangled Contrastive Learning

Hritam Basak Dept. of Computer Science, Stony Brook University, NY, USA
Zhaozheng Yin Dept. of Computer Science, Stony Brook University, NY, USA

###### Abstract

Although unsupervised domain adaptation (UDA) is a promising direction to
alleviate domain shift, they fall short of their supervised counterparts. In
this work, we investigate relatively less explored semi-supervised domain
adaptation (SSDA) for medical image segmentation, where access to a few
labeled target samples can improve the adaptation performance substantially.
Specifically, we propose a two-stage training process. First, an encoder is
pre-trained in a self-learning paradigm using a novel domain-content
disentangled contrastive learning (CL) along with a pixel-level feature
consistency constraint. The proposed CL enforces the encoder to learn
discriminative content-specific but domain-invariant semantics on a global
scale from the source and target images, whereas consistency regularization
enforces the mining of local pixel-level information by maintaining spatial
sensitivity. This pre-trained encoder, along with a decoder, is further fine-
tuned for the downstream task, (i.e. pixel-level segmentation) using a semi-
supervised setting. Furthermore, we experimentally validate that our proposed
method can easily be extended for UDA settings, adding to the superiority of
the proposed strategy. Upon evaluation on two domain adaptive image
segmentation tasks, our proposed method outperforms the SoTA methods, both in
SSDA and UDA settings. Code is available at GitHub.

###### Keywords:

Contrastive Learning Style-content disentanglement Consistency Regularization
Domain Adaptation Segmentation.

##  1 Introduction

Despite their remarkable success in numerous tasks, deep learning models
trained on a source domain face the challenges to generalize to a new target
domain, especially for segmentation which requires dense pixel-level
prediction. This is attributed to a large semantic gap between these two
domains. Unsupervised Domain Adaptation (UDA) has lately been investigated to
bridge this semantic gap between labeled source domain, and unlabeled target
domain [29], including adversarial learning for aligning latent
representations [25], image translation networks [26], etc. However, these
methods produce subpar performance because of the lack of supervision from the
target domain and a large semantic gap in style and content information
between the source and target domains. Moreover, when an image’s content-
specific information is entangled with its domain-specific style information,
traditional UDA approaches fail to learn the correct representation of the
domain-agnostic content while being distracted by the domain-specific styles.
So, they cannot be generalized for multi-domain segmentation tasks [6].

Compared to UDA, obtaining annotation for a few target samples is worthwhile
if it can substantially improve the performance by providing crucial target
domain knowledge. Driven by this speculation, and the recent success of semi-
supervised learning (SemiSL), we investigate semi-supervised domain adaptation
(SSDA) as a potential solution. Recently, Liu et al.[16] proposed an
asymmetric co-training strategy between a SemiSL and UDA task, that
complements each other for cross-domain knowledge distillation. Xia et al.[24]
proposed a co-training strategy through pseudo-label refinement. Gu et al.[9]
proposed a new SSDA paradigm using cross-domain contrastive learning (CL) and
self-ensembling mean-teacher. However, these methods force the model to learn
the low-level nuisance variability, which we know is insignificant to the task
at hand. Hence, these methods fail to generalize if similar variational
semantics are absent in the training set. Fourier Domain Adaptation (FDA) [28]
was proposed to address these challenges by a simple yet effective spectral
transfer method. Following [28], we design a new Gaussian FDA to handle this
cross-domain nuisance variability, without explicit feature alignment.

Contrastive learning (CL) is another prospective direction where we enforce
models to learn discriminative information from (dis)similarity learning in a
latent subspace [4, 12]. Liu et al.[17] proposed a margin-preserving
constraint along with a self-paced CL framework, gradually increasing the
training data difficulty. Gomariz et al.[8] proposed a CL framework with an
unconventional channel-wise aggregated projection head for inter-slice
representation learning. However, traditional CL utilized for DA on images
with entangled style and content leads to mixed representation learning,
whereas ideally, it should learn discriminative content features invariant to
style representation. Besides, the instance-level feature alignment of CL is
subpar for segmentation, where dense pixel-wise predictions are indispensable
[1].

To alleviate these three underlined shortcomings, we propose a novel
contrastive learning with pixel-level consistency constraint via disentangling
the style and content information from the joint distribution of source and
target domain. Precisely, our contributions are as follows: (1) We propose to
disentangle the style and content information in their compact embedding space
using a joint-learning framework; (2) We propose encoder pre-training with two
CL strategies: Style CL and Content CL that learns the style and content
information respectively from the embedding space; (3) The proposed CL is
complemented with a pixel-level consistency constraint with dense feature
propagation module, where the former provides better categorization competence
whereas the later enforces effective spatial sensitivity; (4) We
experimentally validate that our SSDA method can be extended in the UDA
setting easily, achieving superior performance as compared to the SoTA methods
on two widely-used domain adaptive segmentation tasks, both in SSDA and UDA
settings.

##  2 Proposed Method

Given the source domain image-label pairs
$\\{(x_{s}^{i},y_{s}^{i})_{i=1}^{\mathbb{N}_{s}}\in\mathcal{S}\\}$, a few
image-label pairs from target domain
$\\{(x_{t1}^{i},y_{t1}^{i})_{i=1}^{\mathbb{N}_{t1}}\in\mathcal{T}1\\}$, and a
large number of unlabeled target images
$\\{(x_{t2}^{i})_{i=1}^{\mathbb{N}_{t2}}\in\mathcal{T}2\\}$, our proposed pre-
training stage learns from images in
$\\{\mathcal{S}\cup\mathcal{T};\mathcal{T}=\mathcal{T}1\cup\mathcal{T}2\\}$ in
a self-supervised way, without requiring any labels. The following fine-tuning
in SSDA considers image-label pairs in $\\{\mathcal{S}\cup\mathcal{T}1\\}$ for
supervised learning alongside unlabeled images $\mathcal{T}2$ in the target
domain for unsupervised prediction consistency. Our workflow is shown in
Figure 1.

Figure 1: Overall workflow of our proposed method. Stage 1: Encoder pre-
training by GFDA and CL on disentangled style and content branches, and pixel-
wise feature consistency module DFPM; Stage 2: Fine-tuning the encoder in a
semi-supervised student-teacher setting.

###  2.1 Gaussian Fourier Domain Adaptation (GFDA)

Manipulating the low-level amplitude spectrum of the frequency domain is the
easiest way for style transfer between domains [28], without notable
alteration in the visuals of high-level semantics. However, as observed in
[28], the generated images consist of incoherent dark patches, caused by
abrupt changes in amplitude around the rectangular mask. Instead, we propose a
Gaussian mask for a smoother transition in frequency. Let,
$\mathcal{F}_{A}(\cdot)$ and $\mathcal{F}_{P}(\cdot)$ be the amplitude and
phase spectrum in frequency space of an RGB image, and $\mathcal{F}^{-1}$
indicates inverse Fourier transform. We define a 2D Gaussian mask $g_{\sigma}$
of the same size as $\mathcal{F}_{A}$, with $\sigma$ being the standard
deviation. Given two randomly sampled images $x_{s}\sim\mathcal{S}$ and
$x_{t}\sim\mathcal{T}$, our proposed GFDA can be formulated as:

$x_{s\to
t}=\mathcal{F}^{-1}[\mathcal{F}_{P}(x_{s}),\mathcal{F}_{A}(x_{t})\odot
g_{\sigma}+\mathcal{F}_{A}(x_{s})\odot(1-g_{\sigma})],$ (1)

where $\odot$ indicates element-wise multiplication. It generates an image
preserving the semantic content from $\mathcal{S}$ but preserving the style
from $\mathcal{T}$. Reciprocal pair $x_{t\to s}$ is also formulated using the
same drill. The source and target images, and the style-transferred versions
$\\{x_{s},x_{s\to t},x_{t},x_{t\to s}\\}$ are then used for contrastive pre-
training below. Visualization of GFDA is shown in the supplementary file.

###  2.2 CL on Disentangled Domain and Content

We aim to learn discriminative content-specific features that are invariant of
the style of the source or target domain, for a better pre-training of the
network for the task at hand. Hence, we propose to disentangle the style and
content information from the images and learn them jointly in a novel
disentangled CL paradigm: Style CL (SCL) and Content CL (CCL). The proposed
SCL imposes learning of domain-specific attributes, whereas CCL enforces the
model to identify the ROI, irrespective of the spatial semantics and
appearance. In joint learning, they complement each other to render the model
to learn domain-agnostic and content-specific information, thereby mitigating
the domain dilemma. The set of images $\\{x_{s},x_{s\to t},x_{t},x_{t\to
s}\\}$, along with their augmented versions are passed through encoder
$\mathcal{E}$, followed by two parallel projection heads, namely style head
$(\mathcal{G}_{\mathcal{S}})$ and content head $(\mathcal{G}_{\mathcal{C}})$
to obtain the corresponding embeddings. Two different losses: style
contrastive loss $\mathcal{L}_{SCL}$ and content contrastive loss
$\mathcal{L}_{CCL}$, are derived below.

Assuming $\\{x_{s},x_{t\to s}\\}$ (along with their augmentations) having
source-style representation (style $A$), and $\\{x_{t},x_{s\to t}\\}$ (and
their augmentations) having target-style representation (style $B$), in style
CL, embeddings from the same domain (style) are grouped together whereas
embeddings from different domains are pushed apart in the latent space.
Considering the $i^{th}$ anchor point $x^{i}_{t}\in\mathcal{T}$ in a minibatch
and its corresponding style embedding
$s^{i}_{t}\leftarrow\mathcal{G}_{\mathcal{S}}(\mathcal{E}(x^{i}_{t}))$ (with
style $B$), we define the positive set consisting of the same target domain
representations as $\Lambda^{+}=\\{s_{t}^{j+},s_{s\to
t}^{j+}\\}\leftarrow\mathcal{G}_{\mathcal{S}}(\mathcal{E}(\\{x_{t}^{j},x_{s\to
t}^{j}\\})),\forall j\in\text{minibatch}$, and negative set having unalike
source domain representation as $\Lambda^{-}=\\{s_{s}^{j-},s_{t\to
s}^{j-}\\}\leftarrow\mathcal{G}_{\mathcal{S}}(\mathcal{E}(\\{x_{s}^{j},x_{t\to
s}^{j}\\})),\forall j\in\text{minibatch}$. Following SimCLR [7] our style
contrastive loss can be formulated as:

$\mathcal{L}_{SCL}=\sum_{i,j}-\log\frac{\exp(sim(s^{i},s^{j+})/\tau)}{\exp(sim(s^{i},s^{j+})/\tau)+\sum_{j\in\Lambda^{-}}\exp(sim(s^{i},s^{j-})/\tau)},$
(2)

where $\\{s^{i},s^{j+}\\}\in\text{style }B;\;s^{j-}\in\text{style }A$,
$sim(\cdot,\cdot)$ defines cosine similarity, $\tau$ is the temperature
parameter [7]. Similarly, we define $\mathcal{L}_{CCL}$ for content head as:

$\mathcal{L}_{CCL}=\sum_{i,j}-\log\frac{\exp(sim(c^{i},c^{j+})/\tau)}{\exp(sim(c^{i},c^{j+})/\tau)+\sum_{j\in\Lambda^{-}}\exp(sim(c^{i},c^{j-})/\tau)},$
(3)

where
$\\{c^{i},c^{j}\\}\leftarrow\mathcal{G}_{C}(\mathcal{E}(\\{x^{i},x^{j}\\}))$.
These contrastive losses, along with the consistency constraint below enforce
the encoder to extract domain-invariant and content-specific feature
embeddings.

###  2.3 Consistency Constraint

The disentangled CL aims to learn global image-level representation, which is
useful for instance discrimination tasks. However, segmentation is attributed
to learning dense pixel-level representations. Hence, we propose an additional
Dense Feature Propagation Module (DFPM) along with a momentum encoder
$\mathcal{E}^{\prime}$ with exponential moving average (EMA) of parameters
from $\mathcal{E}$. Given any pixel $m$ of an image $x$, we transform its
feature $f^{m}_{\mathcal{E}^{\prime}}$ obtained from $\mathcal{E}^{\prime}$ by
propagating other pixel features from the same image:

$\tilde{f}^{m}_{\mathcal{E}^{\prime}}=\sum_{\forall n\in
x}\mathcal{K}(f_{\mathcal{E}^{\prime}}^{m})\otimes\cos(f_{\mathcal{E}^{\prime}}^{m},f_{\mathcal{E}^{\prime}}^{n})$
(4)

where $\mathcal{K}$ is a linear transformation layer, $\otimes$ denotes matmul
operation. This spatial smoothing of learned representation is useful for
structural sensitivity, which is fundamental for dense segmentation tasks. We
enforce consistency between this smoothed feature
$\tilde{f}_{\mathcal{E}^{\prime}}$ from $\mathcal{E}^{\prime}$ and the regular
feature $f_{\mathcal{E}}$ from $\mathcal{E}$ as:

$\mathcal{L}_{Con}=\sum_{[{d}(m,n)<Th]}-\big{[}\cos(\tilde{f}^{m}_{\mathcal{E}^{\prime}},f^{n}_{\mathcal{E}})+\cos(f^{m}_{\mathcal{E}},\tilde{f}^{n}_{\mathcal{E}^{\prime}})\big{]}$
(5)

where ${d}(\cdot,\cdot)$ indicates the spatial distance, $Th$ is a threshold.
The overall pre-training objective can be summarized as:

$\mathcal{L}_{Pre}=\lambda_{1}\mathcal{L}_{SCL}+\lambda_{2}\mathcal{L}_{CCL}+\mathcal{L}_{Con}$
(6)

###  2.4 Semi-supervised Fine-tuning

The pre-training stage is followed by semi-supervised fine-tuning using a
student-teacher framework [20]. The pre-trained encoder $\mathcal{E}$, along
with a decoder $\mathcal{D}$ are used as a student branch, whereas an
identical encoder-decoder network (but differently initialized) is used as a
teacher network. We compute a supervised loss on the labeled set
$\\{\mathcal{S}\cup\mathcal{T}1\\}$ along with a regularization loss between
the prediction of the student and teacher branches on the unlabeled set
$\\{\mathcal{T}2\\}$ as:

$\mathcal{L}_{Sup}=\frac{1}{\mathbb{N}_{s}+\mathbb{N}_{t1}}\sum_{x^{i}\in\\{\mathcal{S}\cup\mathcal{T}1\\}}CE\left[\mathcal{D}_{S}\left(\mathcal{E}_{S}(x^{i})\right),y^{i}\right]$
(7)
$\mathcal{L}_{Reg}=\frac{1}{\mathbb{N}_{t2}}\sum_{x^{i}\in\\{\mathcal{T}2\\}}CE\left[\mathcal{D}_{S}\left(\mathcal{E}_{S}(x^{i})\right),\mathcal{D}_{T}\left(\mathcal{E}_{T}(x^{i})\right)\right]$
(8)

where $CE$ indicates cross-entropy loss, $\mathcal{E}_{S}$, $\mathcal{D}_{S}$,
$\mathcal{E}_{T}$, $\mathcal{D}_{T}$ indicate the student and teacher encoder
and decoder networks. The student branch is updated using a consolidated loss
$\mathcal{L}=\mathcal{L}_{Sup}+\lambda_{3}\mathcal{L}_{Reg}$, whereas the
teacher parameters $(\theta_{T})$ are updated using EMA from the student
parameters $(\theta_{S})$:

$\theta_{T}(t)=\alpha\theta_{T}(t-1)+(1-\alpha)\theta_{S}(t)$ (9)

where $t$ tracks the step number, and $\alpha$ is the momentum coefficient
[11].

In summary, the overall SSDA training process contains pre-training
(subsection 2.1-subsection 2.3) and fine-tuning (subsection 2.4), whereas, we
only use the student branch $(\mathcal{E}_{S},\mathcal{D}_{S})$ for inference.

##  3 Experiments and Results

Datasets: We evaluate our work on two different DA tasks to evaluate its
generalizability: (1) Polyp segmentation from colonoscopy images in Kvasir-SEG
[13] and CVC-EndoScene Still [22], and (2) Brain tumor segmentation in MRI
images from BraTS2018 [18]. Kvasir and CVC contain 1000 and 912 images
respectively and were split into $4:1$ training-testing sets following [12].
BraTS consists of brain MRIs from 285 patients with T1, T2, T1CE, and FLAIR
scans. The data was split into $4:1$ train-test ratio, following [16].
Source$\to$Target: We perform experiments on $CVC\to Kvasir$ and $Kvasir\to
CVC$ for polyp segmentation, and $T2\to\\{T1,T1CE,FLAIR\\}$ for tumor
segmentation. The SSDA accesses $10-50\%$ and $1-5$ labels from the target
domain for the two tasks, respectively. For UDA, only $\mathcal{S}$ is used
for $\mathcal{L}_{Sup}$, whereas $\mathcal{T}1\cup\mathcal{T}2$ is used for
$\mathcal{L}_{Reg}$. Implementation details: Implementation is done in a
PyTorch environment using a Tesla V100 GPU with 32GB RAM. We use U-Net [19]
backbone for the encoder-decoder structure, and the projection heads
$\mathcal{G}_{\mathcal{S}}$ and $\mathcal{G}_{\mathcal{C}}$ are shallow FC
layers. The model is trained for $300$ epochs for pre-training and $500$
epochs for fine-tuning using an ADAM optimizer with a batch size of $4$ and a
learning rate of $1e-4$. $\lambda 1,\lambda 2,\lambda 3$, and $Th$ are set to
$0.75,0.75,0.5,0.6$, respectively by validation, $\tau,\alpha$ are set to
$0.07,0.999$ following [11]. Augmentations include random rotation and
translation. Metrics: Segmentation performance is evaluated using Dice
Similarity Score (DSC) and Hausdorff Distance (HD).

|  |  | CVC $\rightarrow$ Kvasir |  | Kvasir $\rightarrow$ CVC
---|---|---|---|---|---
Task | Method | Target label | DSC$\uparrow$ | HD$\downarrow$ |  | DSC$\uparrow$ | HD$\downarrow$
No DA | Source only | $0\%$L | 62.2 | 5.6 |  | 53.9 | 6.2
UDA | PCEDA [27] | $0\%$L | 73.6 | 4.4 |  | 70.1 | 4.7
ASN [21] | $0\%$L | 80.1 | 3.6 |  | 83.7 | 3.7
BDL [14] | $0\%$L | 77.8 | 4.0 |  | 81.7 | 4.1
CoFo [12] | $0\%$L | 82.8 | 3.6 |  | 81.1 | 3.5
FDA [28] | $0\%$L | 80.4 | 3.9 |  | 75.1 | 4.2
Ours | 0%L | 83.8 | 3.4 |  | 84.5 | 3.1
SSDA | DLD [23] | 10%L | 84.2 | 3.2 |  | 85.1 | 3.1
ACT [16] | 10%L | 86.9 | 3.0 |  | 87.3 | 2.9
SLA [6] | 10%L | 85.5 | 3.1 |  | 86.2 | 3.3
FSM [26] | 10%L | 85.8 | 3.4 |  | 86.2 | 3.1
Ours | 10%L | 87.7 | 2.9 |  | 86.9 | 2.7
DLD [23] | 50%L | 87.6 | 2.8 |  | 87.9 | 2.6
ACT [16] | 50%L | 89.4 | 2.6 |  | 90.3 | 2.4
SLA [6] | 50%L | 88.6 | 2.7 |  | 89.3 | 2.8
FSM [26] | 50%L | 89.1 | 2.6 |  | 89.8 | 2.5
Ours | 50%L | 90.6 | 2.4 |  | 90.8 | 2.2
Supervised | Source+Target | 100%L | 92.1 | 2.1 |  | 93.8 | 2.0

Table 1: Comparison with state-of-the-art UDA and SSDA methods for polyp
segmentation on KVASIR and CVC. SSDA results are shown for 10%-labeled (10%L)
and 50%-labeled (50%L) data in the target domain. The results of cited methods
are directly reported from the corresponding papers. No DA: the encoder-
decoder model trained only using labeled data from the source domain is
applied to the target domain without adaptation. Supervised: model is trained
using all labeled data from source and target domains. The best and second-
best results are highlighted in RED and BLUE, respectively.

|  |  | DSC$\uparrow$ | HD$\downarrow$
---|---|---|---|---
Task | Method | Target Label | T1 | T1CE | FLAIR | T1 | T1CE | FLAIR
No DA | Source only |  $0$L | 3.9 | 6.0 | 64.4 | 56.9 | 50.8 | 30.4
UDA | SSCA [15] |  $0$L | 59.3 | 63.5 | 82.9 | 12.5 | 11.2 | 7.9
SIFA [5] |  $0$L | 51.7 | 58.2 | 68.0 | 19.6 | 15.0 | 16.9
DSA [10] |  $0$L | 57.7 | 62.0 | 81.8 | 14.2 | 13.7 | 8.6
DSFN [30] |  $0$L | 57.3 | 62.2 | 78.9 | 17.5 | 15.5 | 13.8
Ours | 0L | 60.7 | 64.4 | 83.3 | 11.1 | 10.9 | 7.3
SSDA | DLD [23] | 1L | 65.8 | 66.5 | 81.5 | 12.0 | 10.3 | 7.1
ACT [16] | 1L | 69.7 | 69.7 | 84.5 | 10.5 | 10.0 | 5.8
ACT-EMD [16] | 1L | 67.4 | 69.0 | 83.9 | 10.9 | 10.3 | 6.4
SLA [6] | 1L | 64.7 | 66.1 | 82.3 | 12.2 | 10.5 | 7.1
Ours | 1L | 72.2 | 71.9 | 85.8 | 10.0 | 9.5 | 5.2
DLD [23] | 5L | 67.8 | 68.3 | 83.3 | 11.2 | 9.9 | 6.6
ACT [16] | 5L | 71.3 | 70.8 | 85.0 | 10.0 | 9.8 | 5.2
ACT-EMD [16] | 5L | 70.3 | 69.8 | 84.4 | 10.4 | 10.2 | 5.7
SLA [6] | 5L | 67.2 | 71.2 | 83.1 | 11.7 | 10.1 | 6.8
Ours | 5L | 73.1 | 72.4 | 86.1 | 9.7 | 9.3 | 4.8
Supervised | Source+Target | all labeled | 73.6 | 72.9 | 86.6 | 9.5 | 9.1 | 4.6

Table 2: Comparison with state-of-the-art UDA and SSDA methods for whole tumor
segmentation on BraTS2018, where source domain is T2. SSDA results are
demonstrated for 1-labeled (1L) and 5-labeled (5L) data in the target domain.

###  3.1 Performance on SSDA

Quantitative comparison of our proposed method with different SSDA methods
[23, 16, 6, 26] for both tasks are shown in Table 1 and Table 2. ACT [16]
simply ignores the domain gap and only learns content semantics, resulting in
substandard performance on the BraTS dataset that has a significant domain
gap. FSM [26], on the other hand, is adaptable to learning explicit domain
information, but lacks strong pixel-level regularization on its prediction,
resulting in subpar performance. We address both of these shortcomings in our
work, resulting in superior performance on both tasks. Other methods like [23,
6], which are originally designed for natural images, lack critical refining
abilities even after fine-tuning for medical image segmentation and hence are
far behind our performance in both tasks. The margins are even higher for less
labeled data (1L) on the BraTS dataset, which is promising considering the
difficulty of the task. Moreover, our method produces performance close to its
fully-supervised counterpart (last row in Table 1 and Table 2), using only a
few target labels.

###  3.2 Performance on UDA

Unlike SSDA methods, UDA fully relies on unlabeled data for domain-invariant
representation learning. To analyze the effectiveness of DA, we extend our
model to the UDA setting (explained in section 3[Source$\to$Target]) and
compare it with SoTA methods [15, 5, 12, 28, 30, 10, 27, 14] in Table 1 and
Table 2. Methods like [12, 21] rely on adversarial learning for aligning
multi-level feature space, which is not effective for small-sized medical
data. Other methods [27, 14] rely on an image-translation network but fail in
effective style adaptation, resulting in source domain-biased subpar
performance. Our method, although relies on FDA [28], outperforms it with a
large margin of upto $12.5\%$ DSC for polyp segmentation, owing to its
superior learning ability of disentangled style and content semantics. Similar
results are observed for the BraTS dataset in Table 2, where our work achieved
a margin of upto $2.4\%$ DSC than its closest performer.

###  3.3 Ablation Experiments

We perform a detailed ablation experiment, as shown in Table 3. The
effectiveness of disentangling and joint-learning of style and content
information is evident from the experiment (b)&(c) as compared to (a), where
the introduction of SCL and CCL boosts overall performance significantly.
Moreover, when combined together (experiment (d)), they provide a massive
$9.54\%$ and $8.52\%$ DSC gain over traditional CL (experiment (a)) for
$CVC\to Kvasir$ and $Kvasir\to CVC$, respectively. This also points out a
potential shortfall of traditional CL: its inability to adapt to a complex
domain in DA. The proposed DFPM (experiment (e)) provides local pixel-level
regularization, complementary to the global disentangled CL, resulting in a
further boost in performance ($\sim 1.5\%$). We have similar ablation study
observations on the BraTS2018 dataset, which is provided in the supplementary
file, along with some qualitative examples along with available ground truth.

Experiment# | Stage 1 | Stage 2 |  | CVC $\rightarrow$ Kvasir |  | Kvasir $\rightarrow$ CVC
---|---|---|---|---|---|---
TCL | SCL | CCL | DFPM | SemiSL |  | DSC$\uparrow$ | HD$\downarrow$ |  | DSC$\uparrow$ | HD$\downarrow$
(a) | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$ |  | 81.7 | 4.4 |  | 82.1 | 4.2
(b) | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$ |  | 83.2 | 3.9 |  | 84.7 | 3.5
(c) | $\times$ | $\times$ | $\checkmark$ | $\times$ | $\checkmark$ |  | 84.5 | 3.8 |  | 85.4 | 3.1
(d) | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ |  | 89.5 | 2.8 |  | 89.1 | 2.4
(e) | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | 90.6 | 2.4 |  | 90.8 | 2.2

Table 3: Ablation experiment for polyp segmentation in SSDA(50%L) setting to
identify the contribution of individual components. TCL: traditional CL [4],
SCL: proposed style CL, CCL: proposed content CL. The last row, highlighted in
RED, indicates our results.

##  4 Conclusion

We propose a novel style-content disentangled contrastive learning, guided by
a pixel-level feature consistency constraint for semi-supervised domain
adaptive medical image segmentation. To the best of our knowledge, this is the
first attempt for SSDA in medical image segmentation using CL, which is
further extended to the UDA setting. Our proposed work, upon evaluation on two
different domain adaptive segmentation tasks in SSDA and UDA settings,
outperforms the existing SoTA methods, justifying its effectiveness and
generalizability.

Supplementary File: Semi-supervised Domain Adaptive Medical Image Segmentation
through Consistency Regularized Disentangled Contrastive Learning Hritam
Basak, Zhaozheng Yin

Figure 2: Visualization of our proposed GFDA module, FT: Fourier Transform,
IFT: Inverse Fourier Transform; (A) The Gaussian spectral transfer method of
changing image style without altering semantic content information; (B)
Qualitative comparison of our proposed method along with traditional FDA
method with a fixed rectangular kernel. Clearly, GFDA results in smoother and
noise-free intensity transitions in the reconstructed images. Figure 3:
Qualitative analysis of our segmentation performance for polyp segmentation on
target-domain Kvasir-SEG and CVC datasets. Figure 4: Qualitative analysis of
our segmentation performance for tumor segmentation from BraTS2018 dataset on
target-domain T1, T1CE, and FLAIR modalities, where T2 is used as source-
domain.

| Stage 1 | Stage 2 | T2$\to$T1 |  | T2$\to$T1CE |  | T2$\to$FLAIR
---|---|---|---|---|---|---|---
Experiment # | TCL | SCL | CCL | DFPM | SemiSL | DSC$\uparrow$ | HD$\downarrow$ |  | DSC$\uparrow$ | HD$\downarrow$ |  | DSC$\uparrow$ | HD$\downarrow$
(a) | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | 63.6 | 11.6 |  | 62.8 | 11.8 |  | 77.6 | 8.8
(b) | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$ | 67.4 | 10.7 |  | 66.4 | 10.4 |  | 80.3 | 7.9
(c) | $\times$ | $\times$ | $\checkmark$ | $\times$ | $\checkmark$ | 67.8 | 10.6 |  | 67.7 | 10.3 |  | 80.9 | 7.7
(d) | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | 72.2 | 10.0 |  | 71.7 | 9.7 |  | 85.3 | 5.1
(e) | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 73.1 | 9.7 |  | 72.4 | 9.3 |  | 86.1 | 4.8

Table 4: Ablation experiment for tumor segmentation on BraTS2018 dataset in
SSDA(5L) setting to identify the contribution of individual components. TCL:
traditional CL [2], SCL: proposed style CL, CCL: proposed content CL. The last
row, highlighted in RED, indicates our results.

## References

  * [1] Basak, H., Chattopadhyay, S., Kundu, R., Nag, S., Mallipeddi, R.: Ideal: Improved dense local contrastive learning for semi-supervised medical image segmentation. In: ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 1–5. IEEE (2023)
  * [2] Basak, H., Ghosal, S., Sarkar, R.: Addressing class imbalance in semi-supervised image segmentation: A study on cardiac mri. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 224–233. Springer (2022)
  * [3] Basak, H., Yin, Z.: Pseudo-label guided contrastive learning for semi-supervised medical image segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 19786–19797 (2023)
  * [4] Chaitanya, K., Erdil, E., Karani, N., Konukoglu, E.: Contrastive learning of global and local features for medical image segmentation with limited annotations. Advances in Neural Information Processing Systems 33, 12546–12558 (2020)
  * [5] Chen, C., Dou, Q., Chen, H., Qin, J., Heng, P.A.: Synergistic image and feature adaptation: Towards cross-modality domain adaptation for medical image segmentation. In: Proceedings of the AAAI conference on artificial intelligence. vol. 33, pp. 865–872 (2019)
  * [6] Chen, S., Jia, X., He, J., Shi, Y., Liu, J.: Semi-supervised domain adaptation based on dual-level domain mixing for semantic segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 11018–11027 (2021)
  * [7] Chen, T., Kornblith, S., Norouzi, M., Hinton, G.: A simple framework for contrastive learning of visual representations. In: International conference on machine learning. pp. 1597–1607. PMLR (2020)
  * [8] Gomariz, A., Lu, H., Li, Y.Y., Albrecht, T., Maunz, A., Benmansour, F., Valcarcel, A.M., Luu, J., Ferrara, D., Goksel, O.: Unsupervised domain adaptation with contrastive learning for oct segmentation. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Part VIII. pp. 351–361. Springer (2022)
  * [9] Gu, R., Zhang, J., Wang, G., Lei, W., Song, T., Zhang, X., Li, K., Zhang, S.: Contrastive semi-supervised learning for domain adaptive segmentation across similar anatomical structures. IEEE Transactions on Medical Imaging 42(1), 245–256 (2022)
  * [10] Han, X., Qi, L., Yu, Q., Zhou, Z., Zheng, Y., Shi, Y., Gao, Y.: Deep symmetric adaptation network for cross-modality medical image segmentation. IEEE transactions on medical imaging 41(1), 121–132 (2022)
  * [11] He, K., Fan, H., Wu, Y., Xie, S., Girshick, R.: Momentum contrast for unsupervised visual representation learning. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 9729–9738 (2020)
  * [12] Huy, T.D., Huyen, H.C., Nguyen, C.D., Duong, S.T., Bui, T., Truong, S.Q.: Adversarial contrastive fourier domain adaptation for polyp segmentation. In: 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI). pp. 1–5. IEEE (2022)
  * [13] Jha, D., Smedsrud, P.H., Riegler, M.A., Halvorsen, P., de Lange, T., Johansen, D., Johansen, H.D.: Kvasir-seg: A segmented polyp dataset. In: MultiMedia Modeling: 26th International Conference, MMM 2020, Daejeon, South Korea, January 5–8, 2020, Proceedings, Part II 26. pp. 451–462. Springer (2020)
  * [14] Li, Y., Yuan, L., Vasconcelos, N.: Bidirectional learning for domain adaptation of semantic segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 6936–6945 (2020)
  * [15] Liu, X., Xing, F., El Fakhri, G., Woo, J.: Self-semantic contour adaptation for cross modality brain tumor segmentation. In: 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI). pp. 1–5. IEEE (2022)
  * [16] Liu, X., Xing, F., Shusharina, N., Lim, R., Jay Kuo, C.C., El Fakhri, G., Woo, J.: Act: Semi-supervised domain-adaptive medical image segmentation with asymmetric co-training. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Proceedings, Part V. pp. 66–76. Springer (2022)
  * [17] Liu, Z., Zhu, Z., Zheng, S., Liu, Y., Zhou, J., Zhao, Y.: Margin preserving self-paced contrastive learning towards domain adaptation for medical image segmentation. IEEE Journal of Biomedical and Health Informatics 26(2), 638–647 (2022)
  * [18] Menze, B.H., Jakab, A., Bauer, S., Kalpathy-Cramer, J., Farahani, K., et al.: The multimodal brain tumor image segmentation benchmark (brats). IEEE transactions on medical imaging 34(10), 1993–2024 (2014)
  * [19] Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation. In: International Conference on Medical image computing and computer-assisted intervention. pp. 234–241. Springer (2015)
  * [20] Tarvainen, A., Valpola, H.: Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. Advances in neural information processing systems 30 (2017)
  * [21] Tsai, Y.H., Hung, W.C., Schulter, S., Sohn, K., Yang, M.H., Chandraker, M.: Learning to adapt structured output space for semantic segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 7472–7481 (2019)
  * [22] Vázquez, D., Bernal, J., Sánchez, F.J., Fernández-Esparrach, G., López, A.M., Romero, A., Drozdzal, M., Courville, A.: A benchmark for endoluminal scene segmentation of colonoscopy images. Journal of healthcare engineering 2017 (2017)
  * [23] Wang, Z., Wei, Y., Feris, R., Xiong, J., Hwu, W.M., Huang, T.S., Shi, H.: Alleviating semantic-level shift: A semi-supervised domain adaptation method for semantic segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops. pp. 936–937 (2020)
  * [24] Xia, Y., Yang, D., Yu, Z., Liu, F., Cai, J., Yu, L., Zhu, Z., Xu, D., Yuille, A., Roth, H.: Uncertainty-aware multi-view co-training for semi-supervised medical image segmentation and domain adaptation. Medical image analysis 65, 101766 (2020)
  * [25] Xing, F., Cornish, T.C.: Low-resource adversarial domain adaptation for cross-modality nucleus detection. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Proceedings, Part VII. pp. 639–649. Springer (2022)
  * [26] Yang, C., Guo, X., Chen, Z., Yuan, Y.: Source free domain adaptation for medical image segmentation with fourier style mining. Medical Image Analysis 79, 102457 (2022)
  * [27] Yang, Y., Lao, D., Sundaramoorthi, G., Soatto, S.: Phase consistent ecological domain adaptation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 9011–9020 (2020)
  * [28] Yang, Y., Soatto, S.: Fda: Fourier domain adaptation for semantic segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 4085–4095 (2020)
  * [29] Yao, K., Su, Z., Huang, K., Yang, X., Sun, J., Hussain, A., Coenen, F.: A novel 3d unsupervised domain adaptation framework for cross-modality medical image segmentation. IEEE Journal of Biomedical and Health Informatics 26(10), 4976–4986 (2022)
  * [30] Zou, D., Zhu, Q., Yan, P.: Unsupervised domain adaptation with dual-scheme fusion network for medical image segmentation. In: IJCAI. pp. 3291–3298 (2022)


# Layers of classicality in the compatibility of measurements

Arindam Mitra1,2 1Optics and Quantum Information Group, The Institute of
Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India.
2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India.

###### Abstract

The term “Layers of classicality” in the context of quantum measurements, was
introduced in [T. Heinosaari, Phys. Rev. A 93, 042118 (2016)]. The strongest
layer among these consists of the sets of observables that can be broadcast
and the weakest layer consists of the sets of compatible observables. There
are several other layers in between those two layers. In this work, we study
their physical and geometric properties and show the differences and
similarities among the layers in these properties. In particular we show that:
(i) none of the layers of classicality respect transitivity property, (ii) the
concept like degree of broadcasting similar to degree of compatibility does
not exist, (iii) there exist informationally incomplete POVMs that are not
individually broadcastable, (iv) a set of broadcasting channels can be
obtained through concatenation of broadcasting and non-disturbing channels,
(v) unlike compatibility, other layers of classicality are not convex, in
general. Finally, we discuss the relations among these layers. More
specifically, we show that specific type of concatenation relations among
broadcasting channels decide the layer in which a pair of observables resides.

##  I Introduction

The incompatibility of observables is one of the origins of nonclassicality in
quantum theory. A set of POVMs is compatible if those can be measured
simultaneously. Otherwise, they are incompatible [1]. Today we all know the
connections between incompatibility, nonlocality and steering [2, 3].
Nonclassical features like Bell inequality violation as well as the
demonstration of steering is only possible using incompatible measurements [4,
5]. In quantum information theory, incompatibility of measurements provide an
advantage over compatible measurements in several information-theoretic tasks
[6, 7].
The impossibility of universal cloning of quantum state is known as the No-
cloning theorem[8]. Broadcasting is the weaker version of cloning. Today we
know that only commuting states can be simultaneously broadcast [9].
Similarly, there are no-cloning theorems for a single POVM as well as for a
set of POVMs [10, 11, 12, 13].
Compatible measurement does not show any nonclassical feature. But the
compatibility of observables is the weakest layer of classicality. The other
layers from stronger to weaker can be written as (1) broadcastable sets of
observables, (2) one-side broadcastable sets of observables, (3) mutually
nondisturbing observables, and (4) nondisturbing observables [14].

In the Ref. [14], the existence of hierarchies of these layers have been
proved and complete characterisation for qubit observables is presented and
also the the work presented in Ref. [14] is the first work in this research
direction. In the present article, we discuss the properties of individual
layers and the present work may be considered as the immediate development in
this research direction subsequent to the work prsentated in the Ref. [14].
More specifically in this paper, we mainly tackle the following questions- (i)
what are the differences in properties between compatibility and any other
layer of classicality? (ii) are there any similarities in the properties
between compatibility and any other layer of classicality? (iii) what are the
connections among different layers of classicality? Our results open up
several research avenues which has been summarised in the conclusion.

The rest of this paper is organized as follows. In section II, we briefly
review the incompatibility of quantum observables, quantum channels and the
layers of classicality. In section III, we discuss our main results i.e.,
physical as well as geometric properties of different layers of classicality
and relations among different layers of classicality. More specifically, in
the subsection III.1, we show that none of the layers of classicality respect
transitivity property. Then we show that the concept like degree of
broadcasting similar to degree of compatibility does not exist. We also show
the existence of informationally incomplete POVMs that are not individually
broadcastable. Next we show that a set of broadcasting channels can be
obtained through concatenation of broadcasting and non-disturbing channels. In
the subsection III.2, we show that unlike compatibility, other layers of
classicality are not convex, in general. Finally in the subsection III.3, We
show that specific type of concatenation relations among the broadcasting
channels decide the layer in which a pair of observables resides. Finally, in
section IV we summarize the work and discuss the future outlook.

##  II Preliminaries

In this section, we briefly discuss on compatibility, quantum channels,
quantum instruments and the layers of classicality.

###  II.1 Observables and Compatibility

Let, $A$ be an observable acting on $d$ dimensional Hilbert space
$\mathcal{H}$ with number of outcomes $n_{A}$. Then we denote the outcome set
of $A$ as $\Omega_{A}=\\{1,....,n_{A}\\}$. We denote the set of all
observables acting on $d$ dimensional Hilbert space $\mathcal{H}$ with outcome
set $X$ as $\mathcal{O}^{d}_{X}$ and with outcome set $Y$ as
$\mathcal{O}^{d}_{Y}$. Then we denote the cartesian product of these two sets
as $\mathcal{O}^{d}_{XY}=\mathcal{O}^{d}_{X}\times\mathcal{O}^{d}_{Y}$. Then
the set $\mathcal{O}^{d}_{XY}$ is the set of all the pairs $(A,B)$ $\forall
A\in\mathcal{O}^{d}_{X},\forall B\in\mathcal{O}^{d}_{Y}$. From now on, through
out the paper, we will consider all observables are acting on $d$ dimensional
Hilbert space $\mathcal{H}$ unless specifically mentioned. A pair of
observables $(A,B)\in\mathcal{O}^{d}_{XY}$ is compatible iff there exist an
observable $G\in\mathcal{O}^{d}_{Z}$ where $Z=X\times Y$, such that

$\displaystyle A(x)=\sum_{y}G(x,y)~{}\forall x\in X$ (1) $\displaystyle
B(y)=\sum_{x}G(x,y)~{}\forall y\in Y.$ (2)

We denote the set of all compatible pairs acting on $d$ dimensional Hilbert
space $\mathcal{H}$ as $\mathcal{O}^{d}_{comp,XY}$[15]. Clearly,
$\mathcal{O}^{d}_{comp,XY}\subseteq\mathcal{O}^{d}_{XY}$. Also
$\mathcal{O}^{d}_{comp,XY}$ is convex, where for $p\in[0,1]$, the convex
combination of two observables $A\in\mathcal{O}^{d}_{X}$ and
$B\in\mathcal{O}^{d}_{X}$ is defined as $pA+(1-p)B=\\{pA(x)+(1-p)B(x)\\}$
[16].
An observable $A=\\{A(x)\\}$ is called commutative if
$A(x)A(y)=A(y)A(x)~{}\forall x,y\in\Omega_{A}$. Throughout the paper, we
restrict ourselves to the pairs of observables. A pair of observables
$(A,B)\in\mathcal{O}^{d}_{XY}$ is mutually commuting if $A(x)B(y)=B(y)A(x)$
for all $x\in X$ and $y\in Y$.

###  II.2 Quantum Channels

We denote the set of density matrices on a Hilbert space $\mathcal{H}$ as
$\mathcal{S}(\mathcal{H})$. We also denote the set of linear operators on
Hilbert space $\mathcal{H}$ as $\mathcal{L}(\mathcal{H})$. A CPTP map
$\Theta:\mathcal{S}(\mathcal{H}_{in})\rightarrow\mathcal{S}(\mathcal{H}_{out})$
is called a quantum channel, where $\mathcal{H}_{in}$ and $\mathcal{H}_{out}$
are two Hilbert spaces [15]. The dual channel of $\Lambda$ is defined as a map
$\Lambda^{*}:\mathcal{L}(\mathcal{H}_{out})\rightarrow\mathcal{L}(\mathcal{H}_{in})$
which satisfies the equation
$\text{Tr}[\Lambda(\rho)A(x)]=\text{Tr}[\rho\Lambda^{*}(A(x))]$ for all
$x\in\Omega_{A}$, for all states $\rho\in\mathcal{S}(\mathcal{H}_{in})$ and
all observable $A=\\{A(x)\\}$ acting on $\mathcal{H}_{out}$. We denote
composition of two channels
$\Lambda_{1}:\mathcal{S}(\mathcal{H}_{in})\rightarrow\mathcal{S}(\mathcal{H}^{\prime})$
and
$\Lambda_{2}:\mathcal{S}(\mathcal{H}^{\prime})\rightarrow\mathcal{S}(\mathcal{H}_{out})$
as $\Lambda_{2}\circ\Lambda_{1}$, where $\mathcal{H}^{\prime}$ is another
Hilbert space. From the definition of dual channel we have

$(\Lambda_{2}\circ\Lambda_{1})^{*}=\Lambda_{1}^{*}\circ\Lambda_{2}^{*}.$ (3)

We denote the set of all channels for which
$\mathcal{H}_{in}=\mathcal{H}_{out}=\mathcal{H}$ as $\mathcal{C}^{d}$.

Our definition of broadcasting channel is same as the definition given in
[14].

######  Definition 1.

A channel
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$
with $\mathcal{H}_{A}=\mathcal{H}_{B}=\mathcal{H}$ is called a broadcasting
channel.

We denote the set of all broadcasting channels acting on
$\mathcal{S}(\mathcal{H})$ as $\mathcal{C}^{d}_{broad}$. Let us now define a
property of broadcasting channels which we named as local changeability.

######  Definition 2.

A broadcasting channel $\Lambda_{1}\in\mathcal{C}^{d}_{broad}$ is called
locally changeable to another broadcasting channel
$\Lambda_{2}\in\mathcal{C}^{d}_{broad}$ if there exist two channels
$\Sigma_{1}\in\mathcal{C}^{d}$ and $\Sigma_{2}\in\mathcal{C}^{d}$ such that

$\Lambda_{2}=(\Sigma_{1}\otimes\Sigma_{2})\circ\Lambda_{1}$ (4)

and we denote it as $\Lambda_{2}\preceq_{local}\Lambda_{1}$. If both
$\Lambda_{2}\preceq_{local}\Lambda_{1}$ and
$\Lambda_{1}\preceq_{local}\Lambda_{2}$ hold then they are called locally
interchangeable and we denote it as $\Lambda_{1}\simeq_{local}\Lambda_{2}$.

We will use this property to construct relations between different layers of
classicality.

###  II.3 Quantum Instruments

An instrument $\mathcal{I}$ with outcome set $X$ is the collection of CP maps
$\mathcal{I}=\\{\mathcal{I}_{x}:\mathcal{L}(\mathcal{H})\rightarrow\mathcal{L}(\mathcal{K})\\}$
such that $\mathcal{I}^{\mathcal{C}}=\sum_{x}\mathcal{I}_{x}$ is a quantum
channel, where $\mathcal{H}$ and $\mathcal{K}$ are two Hilbert spaces. An
observable $A\in\mathcal{O}^{d}_{X}$ can be implemented using $\mathcal{I}$ if
for all $\rho\in\mathcal{S}(\mathcal{H})$ and $x\in X$, $\text{Tr}[\rho
A(x)]=\text{Tr}[\mathcal{I}_{x}(\rho)]$. Such an instrument is known as
$A$-compatible instrument and the channel $\mathcal{I}^{\mathcal{C}}$ is
called compatible with $A$[15].

###  II.4 Layers of Classicality

In this subsection, we discuss briefly on the layers of classicality [14].

Figure 1: (Colour online) Different layers of classicality.

####  II.4.1 Broadcastable pairs of observables

######  Definition 3.

An observable $A\in\mathcal{O}^{d}_{X}$ is broadcast by a broadcasting channel
$\Lambda\in\mathcal{C}^{d}_{broad}$ if for all
$\rho\in\mathcal{S}(\mathcal{H})$, $x\in X$

$\displaystyle\text{Tr}[\rho A(x)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)A(x)\otimes\mathbb{I}]=\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes
A(x)].$ (5)

A pair of observables $(A,B)\in\mathcal{O}^{d}_{XY}$ is broadcastable if there
exists a broadcasting channel $\Lambda$ which can broadcast both observables
$A$ and $B$. We denote the set of all broadcastable pairs of observables in
$\mathcal{O}^{d}_{XY}$ as $\mathcal{O}^{d}_{broad,XY}$.
It is proved in [14] that a mutually commuting pair of commutative observables
acting on finite dimensional Hilbert space is broadcastable. It is also proved
in [14] that a pair of qubit observables are broadcastable iff those
observables are commutative and mutually commuting.

####  II.4.2 One-side broadcastable pairs of observables

######  Definition 4.

A pair of observables $(A,B)\in\mathcal{O}_{XY}$ is one-side broadcastable if
there exists a broadcasting channel $\Lambda\in\mathcal{C}^{d}_{broad}$ such
that for all $\rho\in\mathcal{S}(\mathcal{H})$, $x\in X$ and $y\in Y$

$\displaystyle\text{Tr}[\rho
A(x)]=\text{Tr}[\Lambda(\rho)A(x)\otimes\mathbb{I}],$ (6)
$\displaystyle\text{Tr}[\rho B(y)]=\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes
B(y)]$ (7)

hold. We denote the set of all one side-broadcastable pairs of observables in
$\mathcal{O}^{d}_{XY}$ as $\mathcal{O}^{d}_{1-side-broad,XY}$.

####  II.4.3 Nondisturbing pairs of observables

######  Definition 5.

An observable $A\in\mathcal{O}^{d}_{X}$ can be measured without disturbing
$B\in\mathcal{O}^{d}_{Y}$ if there exists an instrument
$\mathcal{I}=\\{\mathcal{I}(x)\\}$ such that for all
$\rho\in\mathcal{S}(\mathcal{H})$, $x\in X$ and $y\in Y$

$\displaystyle\text{Tr}[\rho A(x)]=\text{Tr}[\mathcal{I}_{x}(\rho)],$ (8)
$\displaystyle\text{Tr}[\rho
B(y)]=\text{Tr}[\mathcal{I}^{\mathcal{C}}(\rho)B(y)].$ (9)

A pair of observables $(A,B)$ is called nondisturbing if either $A$ can be
measured without disturbing $B$ or $B$ can be measured without disturbing $A$
[14]. We denote the set of all nondisturbing pairs of observables in
$\mathcal{O}^{d}_{XY}$ as $\mathcal{O}^{d}_{nondist,XY}$. Some of the
properties of nondisturbing pairs of observables has been discussed in the
article [17]. In the dimension more than two, the complete characterization of
nondisturbing pairs is difficult and requires separate investigation.

####  II.4.4 Mutually nondisturbing pairs of observables

######  Definition 6.

A pair of observables $(A,B)\in\mathcal{O}_{XY}$ is mutually nondisturbing if
$A$ can be measured without disturbing $B$ and $B$ can be measured without
disturbing $A$.

We denote the set of all mutually nondisturbing pairs of observables in
$\mathcal{O}^{d}_{XY}$ as $\mathcal{O}^{d}_{mut-nondist,XY}$.
It is proved in [14] that
$\mathcal{O}^{d}_{broad,XY}\subseteq\mathcal{O}^{d}_{1-side-
broad,XY}\subseteq\mathcal{O}^{d}_{mut-
nondist,XY}\subseteq\mathcal{O}^{d}_{nondist,XY}\subseteq\mathcal{O}^{d}_{comp,XY}$.
For any observable, if all of its POVM elements are multiple of identity, then
the observable is called a trivial observable. Otherwise, it is called non-
trivial observable. Any mutually commuting pair of non-trivial qubit
observables is necessarily commutative [14]. We now write down the following
proposition which is originally proved in [14].

######  Proposition 1.

For a pair of non-trivial qubit observables $A$ and $B$, the following
statements are equivalent.

  1. 1.

The pair $(A,B)$ is one-side broadcastable.

  2. 2.

The pair $(A,B)$ is mutually nondisturbing.

  3. 3.

The pair $(A,B)$ is nondisturbing.

  4. 4.

The pair $(A,B)$ is mutually commuting.

  5. 5.

The pair $(A,B)$ is broadcastable.

##  III Main Results

In this section, we discuss the main results of our work. At first, we discuss
the physical properties of some layers of classicality. Then, we discuss
geometric properties of different layers of classicality. Finally, we discuss
the connection among different layers.

###  III.1 Physical properties of some of the layers of classicality

In this section, we discuss mainly the physical properties of broadcasting,
one-side broadcasting and compatibility. Let us start with a simple property.
We know that there exist three non-trivial observables $A,B$, and $C$ such
that $(A,B)$ and $(B,C)$ are compatible pairs but the pair $(A,C)$ is not
compatible i.e., compatibility does not respect transitivity property. Our
next proposition shows that none of the layers of classicality respect this
property.

######  Proposition 2.

There exist non-trivial observables $A$, $B$, and $C$ such that the pairs
$(A,B)$ and $(B,C)$ reside in a particular layer of classicality, while
$(A,C)$ does not reside in that layer of classicality.

###### Proof.

Consider three non-trivial sharp observables acting on
$\mathcal{H}_{2}\otimes\mathcal{H}_{2}$

$\displaystyle A$
$\displaystyle=\\{\ket{+0}\bra{+0},\ket{-0}\bra{-0},\ket{01}\bra{01},\ket{11}\bra{11}\\}$
(10) $\displaystyle B$
$\displaystyle=\\{\mathbb{I}\otimes\ket{0}\bra{0},\ket{01}\bra{01},\ket{11}\bra{11}\\}$
(11) $\displaystyle C$
$\displaystyle=\\{\ket{00}\bra{00},\ket{10}\bra{10},\ket{01}\bra{01},\ket{11}\bra{11}\\}$
(12)

where $\ket{0},\ket{1}$ are the eigenvectors of $\sigma_{z}$ and
$\ket{+},\ket{-}$ are the eigenvectors of $\sigma_{x}$ and $\mathcal{H}_{2}$
is Hilbert space with dimension $d=2$.
Clearly, these observables are commutative and also the pairs $(A,B)$ and
$(B,C)$ are mutually commuting. Therefore, they are
broadcastable $\implies$ one side broadcastable $\implies$ mutually
nondisturbing $\implies$ nondisturbing $\implies$ compatible.
But the pair $(A,C)$ is mutually noncommuting. Therefore, this pair is
incompatible $\implies$ disturbing $\implies$ mutually disturbing $\implies$
not one-side broadcastable $\implies$ not broadcastable.

So, the pairs $(A,B)$ and $(B,C)$ reside in all of these layers of
classicality. But the pair $(A,C)$ does not reside in any of these layers of
classicality. ∎

######  Definition 7.

An observable $A_{p}$ is called an unsharp version of the observable $A$ with
unsharp parameter $p$ iff
$A_{p}=\\{A_{p}(x)=pA(x)+(1-p)\frac{\mathbb{I}}{n_{A}}\\}$ for all
$x\in\Omega_{A}$, where $n_{A}$ is the number of outcomes of $A$ and
$p\in(0,1]$.

Now we show a basic difference in a property between broadcasting and
compatibility. Consider a pair of observables $(A,B)$ with number of outcomes
$n_{A}$ and $n_{B}$. It is well known that for the particular value of
$p,q\in(0,1]$, the compatibility of the pair $(A_{p},B_{q})$ does not imply
compatibility of the pair $(A,B)$. But for broadcasting, this is not the case.
The following theorem will clarify that.

######  Theorem 1.

If the pair $(A_{p},B_{q})$ is broadcastable for some $p,q\in(0,1]$, then the
pair $(A,B)$ is also broadcastable.

###### Proof.

Suppose the pair $(A_{p},B_{q})$ is broadcastable by the broadcasting channel
$\Lambda$ and $p,q\in(0,1]$. Then for all $x\in\Omega_{A_{p}}$ and
$\rho\in\mathcal{S}(\mathcal{H})$ we have

$\displaystyle\text{Tr}[\rho
A_{p}(x)]=\text{Tr}[\Lambda(\rho)(A_{p}\otimes\mathbb{I})]$ $\displaystyle
or,~{}$
$\displaystyle\text{Tr}[\rho(pA(x)+(1-p)\frac{\mathbb{I}}{n_{A}})]=\text{Tr}[\Lambda(\rho)((pA(x)$
$\displaystyle\hskip
108.12054pt+(1-p)\frac{\mathbb{I}}{n_{A}})\otimes\mathbb{I})]$ $\displaystyle
or,~{}$ $\displaystyle
p\text{Tr}[\rho(A(x)]+\frac{(1-p)}{n_{A}}=p\text{Tr}[\Lambda(\rho(A(x)\otimes\mathbb{I})]$
$\displaystyle\hskip 139.41832pt+\frac{(1-p)}{n_{A}}$ $\displaystyle or,~{}$
$\displaystyle\text{Tr}[\rho(A(x)]=\text{Tr}[\Lambda(\rho(A(x)\otimes\mathbb{I})].$
(13)

Similarly, one can prove for all $x\in\Omega_{A_{p}}$ $y\in\Omega_{B_{q}}$ and
$\rho\in\mathcal{S}(\mathcal{H})$

$\displaystyle\text{Tr}[\rho A(x)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)(\mathbb{I}\otimes A(x))],$ (14)
$\displaystyle\text{Tr}[\rho B(y)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)(\mathbb{I}\otimes B(y)],$ (15)
$\displaystyle\text{Tr}[\rho B(y)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)(B(y)\otimes\mathbb{I})].$ (16)

Therefore, the pair $(A,B)$ is broadcastable by the same channel $\Lambda$. ∎

The highest value of a unsharp parameter for which the the unsharp version of
a pair is compatible, is known as degree of compatibility[1]. The Theorem 1
suggests that there is no concept like degree of broadcasting similar to
degree of compatibility. In other words broadcastability of observables can
not be quantified in this way!

One immediate corollary of Theorem 1 is

######  Corollary 2.

If a compatible pair $(A_{p},B_{q})$ is the unsharp version of an incompatible
pair $(A,B)$ for some $p,q\in(0,1]$ then the pair $(A_{p},B_{q})$ is not
broadcastable.

As we already know that there exist incompatible pairs of observables which
has non-zero compatibility region, we conclude from the Corollary 2 that
$\mathcal{O}^{d}_{broad,XY}\subset\mathcal{O}^{d}_{comp,XY}$. Similarly, it is
easy to prove that

######  Corollary 3.

If the pair $(A,B)$ is broadcastable, then the pair $(A_{p},B_{q})$ is
broadcastable for all $p,q\in[1,0]$.

Our, next proposition is very important and describes the relation between
broadcasting of a compatible pair and broadcasting of it’s joint observable.

######  Proposition 3.

For any compatible pair $(A,B)\in\mathcal{O}^{d}_{comp,XY}$, if the joint
observable is broadcastable by a channel $\Lambda\in\mathcal{C}^{d}_{broad}$,
then the pair is also broadcastable by the same channel $\Lambda$.

###### Proof.

Let the pair $(A,B)\in\mathcal{O}^{d}_{XY}$ is compatible and
$G\in{\mathcal{O}_{X\times Y}}$ be the joint observable of it. Therefore for
all $x\in X$ and $y\in Y$,

$\displaystyle A(x)=\sum_{y}G(x,y);B(y)=\sum_{x}G(x,y).$ (17)

Suppose that $G$ is broadcastable by $\Lambda\in\mathcal{C}^{d}_{broad}$.

Therefore,

$\displaystyle\text{Tr}[\rho A(x)]$
$\displaystyle=\text{Tr}[\rho\sum_{y}G(x,y)]$
$\displaystyle=\sum_{y}\text{Tr}[\rho G(x,y)]$
$\displaystyle=\sum_{y}\text{Tr}[\Lambda(\rho)G(x,y)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[\Lambda(\rho)\sum_{y}G(x,y)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[\Lambda(\rho)A(x)\otimes\mathbb{I}]$ (18)

Similarly, one can prove that

$\displaystyle\text{Tr}[\rho A(x)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes A(x)]$ (19)
$\displaystyle\text{Tr}[\rho B(y)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)B(y)\otimes\mathbb{I}]$ (20)
$\displaystyle\text{Tr}[\rho B(y)]$
$\displaystyle=\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes B(y)]$ (21)

Hence, the proposition is proved. ∎

The immediate corollary of above proposition is

######  Corollary 4.

If a compatible pair is not broadcastable, then their joint observable is also
not broadcastable.

We clarify this through our next example.

######  Example 1.

Consider unsharp versions of a pair of spin-$\frac{1}{2}$ observables along
$x$ and $y$ direction respectively with unsharpness parameter
$\lambda=\frac{1}{\sqrt{2}}$. This pair is compatible[18]. But from the
Corollary 2, we get that this pair is not broadcastable. Now, the joint
observable of this pair is
$G=\\{\frac{1}{2}\ket{+\vec{n_{1}}}\bra{+\vec{n_{1}}},\frac{1}{2}\ket{-\vec{n_{1}}}\bra{-\vec{n_{1}}},\frac{1}{2}\ket{+\vec{n_{2}}}\bra{+\vec{n_{2}}},$
$\frac{1}{2}\ket{-\vec{n_{2}}}\bra{-\vec{n_{2}}}\\}$, where,
$\vec{n_{1}}=(1,1,0)$ and $\vec{n_{2}}=(1,-1,0)$. Therefore, from Corollary 4
we get that this joint observable is not broadcastable. It should be noted
that $G$ is not informationally complete.

Therefore, Corollary 2 and Corollary 4 together with example 1 enable us to
write down the following theorem.

######  Theorem 2 (No-broadcasting theorem for a single informationally
incomplete POVM).

There exist some informationally incomplete POVMs which are not individually
broadcastable.

This theorem immediately raises a question-What is the minimum amount of
extracted information which prohibits quantum broadcasting of observables? We
keep this as open question. Broadcasting is the weaker version of cloning, and
therefore the Theorem 2 is a generalization of the no-cloning theorem for a
single POVM given in [13].
Now, we define nondisturbing quantum channels with respect to a observable.

######  Definition 8.

A channel $\Theta:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$
is nondisturbing with respect to an observable $A$ if for all
$\rho\in\mathcal{S}(\mathcal{H})$,

$\displaystyle\text{Tr}[\rho
A(x)]=\text{Tr}[\Theta(\rho)A(x)]~{}\forall~{}x\in\Omega_{A}$ (22)

or, equivalently

$\Theta^{*}(A(x))=A(x)~{}\forall~{}x\in\Omega_{A}.$ (23)

A channel is nondisturbing for a set of observables iff it is nondisturbing
for all observables in that set. Now, we prove our next proposition.

######  Proposition 4.

If the observable $A$ is broadcastable by the channel $\Lambda$ and if
$\Theta:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$ is a
nondisturbing channel with respect to $A$ then $A$ is also broadcastable by
$\Lambda\circ\Theta$.

###### Proof.

The observable $A$ is broadcastable by $\Lambda$. Then for all
$\rho\in\mathcal{S}(\mathcal{H})$ and for all $x\in\Omega_{A}$

$\displaystyle\text{Tr}[(\Lambda\circ\Theta)(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[\Lambda(\Theta(\rho))A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[\Theta(\rho)A(x)]$ $\displaystyle=\text{Tr}[\rho
A(x)]$ (24)

Similarly one can prove that

$\text{Tr}[(\Lambda\circ\Theta)(\rho)\mathbb{I}\otimes A(x)]=\text{Tr}[\rho
A(x)].$ (25)

Therefore, $A$ is broadcastable by $\Lambda\circ\Theta$.

∎

For an observable, in this way, one can get the set of different broadcasting
channels that broadcast the observable. It is an open question whether this
set has the highest element i.e., a broadcasting channel in this set from
which all other broadcasting channels that can broadcast the observable, can
be constructed through concatenation with nondisturbing channels. It is also
an open question whether this set has the lowest element i.e., a broadcasting
channel in this set which can be constructed from all other broadcasting
channels that broadcast the observable through concatenation with
nondisturbing channels.
The results similar to the Theorem 1, Corollaries 2, 3, 4 and Propositions 3,
4 can be similarly shown hold for one-side broadcastable pairs of observables.

###  III.2 Geometric properties of different layers of classicality

In this section, we discuss the geometric properties of the layers of
classicality. Suppose $\chi_{XY}\subseteq\mathcal{O}^{d}_{XY}$ is a set of
pairs of observables with outcome set $X$ and $Y$ respectively. Then we denote
the set of broadcasting channels which can broadcast all pairs of observables
in the set $\chi$ as $\Gamma^{d}_{broad}(\chi_{XY})$. Similarly, for a set of
broadcasting channels $\xi\subseteq\mathcal{C}^{d}_{broad}$, we denote the set
of all pairs of observables with outcome set $X$ and $Y$ respectively, which
can be broadcast by all broadcasting channels in the set $\xi$ as
$\mathcal{O}^{d}_{broad,XY}(\xi)$. So, clearly
$\cup_{\Lambda\in\mathcal{C}^{d}_{broad}}\mathcal{O}^{d}_{broad,XY}(\Lambda)=\mathcal{O}^{d}_{broad,XY}$.

######  Proposition 5.

For any pair of observables $(A,B)\in\mathcal{O}^{d}_{XY}$,
$\Gamma^{d}_{broad}(\\{(A,B)\\})$ is convex.

###### Proof.

Suppose both broadcasting channels $\Lambda_{1}$ and $\Lambda_{2}$ can
broadcast the pair $(A,B)$. Then, for all $p\in[0,1]$, $x\in X$ and
$\rho\in(\mathcal{S}(\mathcal{H}))$

$\displaystyle\text{Tr}[(p\Lambda_{1}+(1-p)\Lambda_{2})(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=~{}$ $\displaystyle
p\text{Tr}[\Lambda_{1}(\rho)A(x)\otimes\mathbb{I}]+(1-p)\text{Tr}[\Lambda_{2}(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=~{}$ $\displaystyle p\text{Tr}[(\rho A(x)]+(1-p)\text{Tr}[\rho
A(x)]$ $\displaystyle=~{}$ $\displaystyle\text{Tr}[\rho A(x)]$ (26)

In a similar way, one can easily prove that for all $p\in[0,1]$, $x\in X$,
$y\in Y$ and $\rho\in(\mathcal{S}(\mathcal{H}))$

$\displaystyle\text{Tr}[(p\Lambda_{1}+(1-p)\Lambda_{2})(\rho)\mathbb{I}\otimes
A(x)]=\text{Tr}[\rho A(x)]$ (27)

and

$\displaystyle\text{Tr}[(p\Lambda_{1}+(1-p)\Lambda_{2})(\rho)B(y)\otimes\mathbb{I})]$
$\displaystyle=~{}$
$\displaystyle\text{Tr}[(p\Lambda_{1}+(1-p)\Lambda_{2})(\rho)\mathbb{I}\otimes
B(y)]$ $\displaystyle=~{}$ $\displaystyle\text{Tr}[(\rho B(y)]$ (28)

Therefore, the broadcasting channel
$\Lambda^{\prime}=p\Lambda_{1}+(1-p)\Lambda_{2}$ can the pair $(A,B)$ for all
$p\in[0,1]$.

∎

Similarly, we prove our next proposition.

######  Proposition 6.

For any channel $\Lambda\in\mathcal{C}^{d}_{broad}$, the set
$\mathcal{O}^{d}_{XY,broad}(\Lambda)$ is convex.

###### Proof.

Suppose both the pairs $(A_{1},B_{1})\in\mathcal{O}^{d}_{broad,XY}$ and
$(A_{2},B_{2})\in\mathcal{O}^{d}_{broad,XY}$ are broadcast by same channel
$\Lambda$. Then for all $p\in[0,1]$, $x\in X$ and
$\rho\in(\mathcal{S}(\mathcal{H}))$

$\displaystyle\text{Tr}[\Lambda(\rho)(pA_{1}(x)+(1-p)A_{2}(x))\otimes\mathbb{I}]$
$\displaystyle=~{}$ $\displaystyle
p\text{Tr}[\Lambda(\rho)A_{1}(x)\otimes\mathbb{I}]+(1-p)\text{Tr}[\Lambda(\rho)A_{2}(x)\otimes\mathbb{I}]$
$\displaystyle=~{}$ $\displaystyle p\text{Tr}[\rho
A_{1}(x)]+(1-p)\text{Tr}[(\rho)A_{2}(x)]$ $\displaystyle=~{}$
$\displaystyle\text{Tr}[\rho(pA_{1}(x)+(1-p)A_{2}(x))].$ (29)

In a similar way, one can easily prove that for all $p\in[0,1]$, $x\in X$,
$y\in Y$ and $\rho\in(\mathcal{S}(\mathcal{H}))$

$\displaystyle\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes(pA_{1}(x)+(1-p)A_{2}(x))]$
$\displaystyle=~{}$ $\displaystyle\text{Tr}[\rho(pA_{1}(x)+(1-p)A_{2}(x))].$
(30)

and

$\displaystyle\text{Tr}[\rho(pB_{1}(y)+(1-p)B_{2}(y))]$ $\displaystyle=~{}$
$\displaystyle\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes(pB_{1}(y)+(1-p)B_{2}(y))]$
(31) $\displaystyle=~{}$
$\displaystyle\text{Tr}[\Lambda(\rho)(pB_{1}(y)+(1-p)B_{2}(y))\otimes\mathbb{I}].$
(32)

So, $\Lambda$ can broadcast the pair $(pA_{1}+(1-p)A_{2},pB_{1}+(1-p)B_{2})$
for all $p\in[0,1]$. ∎

Therefore, $\mathcal{O}^{d}_{broad,XY}$ is union of these convex set. Now,
suppose $(A_{1},B_{1})\in\mathcal{O}_{broad,XY}$ and
$(A_{2},B_{2})\in\mathcal{O}_{broad,XY}$ are two pairs of observables. Let,
$\lambda$ be an arbitrary number and $\lambda\in[0,1]$. Then consider another
pair of observables $(A^{\lambda},B^{\lambda})$ such that $A^{\lambda}=\lambda
A_{1}+(1-\lambda)A_{2}$ and $B^{\lambda}=\lambda B_{1}+(1-\lambda)B_{2}$. Then
the following theorem holds.

######  Theorem 3.

If two pairs $(A_{1},B_{1})\in\mathcal{O}_{broad,XY}$ and
$(A_{2},B_{2})\in\mathcal{O}_{broad,XY}$ are not broadcastable by the same
channel i.e., if
$\Gamma^{d}_{broad}(\\{(A_{1},B_{1}),(A_{2},B_{2})\\})=\emptyset$ where
$\emptyset$ is a null set, then for all $p,q\in[0,1]$ and $p\neq q$,

$\Gamma^{d}_{broad}(\\{(A^{p},B^{p}),(A^{q},B^{q})\\})=\emptyset.$

###### Proof.

Suppose $(A^{\lambda},B^{\lambda})$ is broadcastable for all $\lambda\in[0,1]$
by the same broadcasting channel $\Lambda$. Then for all $\lambda\in[0,1]$,
$x\in X$ and $\rho\in(\mathcal{S}(\mathcal{H}))$,

$\displaystyle\text{Tr}[\rho(\lambda A_{1}(x)+(1-\lambda)A_{2}(x))]$
$\displaystyle=~{}$ $\displaystyle\text{Tr}[\Lambda(\rho)(\lambda
A_{1}(x)+(1-\lambda)A_{2}(x))\otimes\mathbb{I}]$ (33) $\displaystyle=~{}$
$\displaystyle\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes(\lambda
A_{1}(x)+(1-\lambda)A_{2}(x))]$ (34)

and

$\displaystyle\text{Tr}[\rho(\lambda B_{1}(x)+(1-\lambda)B_{2}(x))]$
$\displaystyle=~{}$ $\displaystyle\text{Tr}[\Lambda(\rho)(\lambda
B_{1}(x)+(1-\lambda)B_{2}(x))\otimes\mathbb{I}]$ (35) $\displaystyle=~{}$
$\displaystyle\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes(\lambda
B_{1}(x)+(1-\lambda)B_{2}(x))]$ (36)

Now, let

$\displaystyle\text{Tr}(\rho
A_{1}(x))-\text{Tr}[\Lambda(\rho)A_{1}(x)\otimes\mathbb{I}]=p_{1}(x)$ (37)
$\displaystyle\text{Tr}(\rho
A_{2}(x))-\text{Tr}[\Lambda(\rho)A_{2}(x)\otimes\mathbb{I}]=q_{1}(x)$ (38)
$\displaystyle\text{Tr}(\rho
B_{1}(y))-\text{Tr}[\Lambda(\rho)B_{1}(y)\otimes\mathbb{I}]=r_{1}(y)$ (39)
$\displaystyle\text{Tr}(\rho
B_{2}(y))-\text{Tr}[\Lambda(\rho)B_{2}(y)\otimes\mathbb{I}]=s_{1}(y)$ (40)

and

$\displaystyle\text{Tr}(\rho
A_{1}(x))-\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes A_{1}(x)]=p_{2}(x)$ (41)
$\displaystyle\text{Tr}(\rho
A_{2}(x))-\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes A_{2}(x)]=q_{2}(x)$ (42)
$\displaystyle\text{Tr}(\rho
B_{2}(y))-\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes B_{1}(y)]=r_{2}(y)$ (43)
$\displaystyle\text{Tr}(\rho
B_{2}(y))-\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes B_{2}(y)]=s_{2}(y)$ (44)

Now it is to be noted that for all $i\in\\{1,2\\}$ and $x\in X$ and $y\in Y$,
the numbers $p_{1}(x),q_{i}(x),r_{i}(y),s_{i}(y)$ are the differences of
probabilities and all of these are independent of $\lambda$ and depend on
$\Lambda$ and $\rho$. Therefore, moduli of them are less than or equals to 1.
It is also to be noted that as $\Lambda$ can not broadcast individual pairs
$(A_{1},B_{1})$ and $(A_{2},B_{2})$ simultaneously and therefore, for all
$i\in\\{1,2\\}$ and $x\in X$ and $y\in Y$, the numbers
$p_{1}(x),q_{i}(x),r_{i}(y),s_{i}(y)$ simultaneously can not be zero. Now
suppose one of those numbers, say for example $p_{1}(x)\neq 0$ for some $x\in
X$. Then from equations (33),(37) and (38) we have for all $\lambda\in[0,1]$

$\displaystyle\lambda p_{1}(x)=(1-\lambda)(-q_{1})$ $\displaystyle
p_{1}(x)=\frac{(\lambda-1)}{\lambda}(q_{1})$ (45) $\displaystyle\mid
p_{1}(x)\mid=\mid\frac{(1-\lambda)}{\lambda}\mid\mid q_{1}\mid.$ (46)

So, if $q_{1}(x)=0$ above equation is satisfied only for $\lambda=0$. Now
suppose $q_{1}(x)\neq 0$. Then, R.H.S of the equation(46) will vary with
$\lambda$. But L.H.S. of the equation(46) is independent of $\lambda$ and will
not change. So, the equation(46) can be satisfied only for a particular value
of $\lambda$ and can not be satisfied for all $\lambda\in[0,1]$.

Furthermore, note that for a particular channel of $\Lambda$, for
$\lambda<\frac{\mid q_{1}(x)\mid}{1+\mid q_{1}(x)\mid}$, R.H.S of equation(46)
is greater than 1. Therefore, since $\mid p_{1}(x)\mid\leq 1$, the equation
(33) can not be satisfied for all $\lambda\leq\frac{\mid q_{1}(x)\mid}{1+\mid
q_{1}(x)\mid}$. Therefore, for any one or more than one of the numbers
$p_{1}(x),q_{i}(x),r_{i}(y),s_{i}(y)$ for all $i\in\\{1,2\\}$ and $x\in X$ and
$y\in Y$ to be greater than $0$, one can similarly prove that
equations(33),(34)(III.2) and (36) can be satisfied simultaneously at most for
a particular value of $\lambda\in[0,1]$. Therefore, a broadcasting channel
$\Lambda$ can broadcast the pair $(A^{\lambda},B^{\lambda})$ at most for a
particular value of $\lambda$. Hence, the theorem is proved. ∎

Our next proposition will clarify the geometric property of
$\mathcal{O}^{d}_{broad,XY}$ and the other layers.

######  Proposition 7.

$\mathcal{O}^{d}_{broad,XY}$, $\mathcal{O}^{d}_{1-side-broad,XY}$,
$\mathcal{O}^{d}_{mut-nondist,X,Y}$ and $\mathcal{O}^{d}_{nondist,XY}$ are not
convex, in general.

###### Proof.

We prove this by showing a counter example. We know that for two bloch vectors
$\vec{n_{1}}$ and $\vec{n_{2}}$

$[\vec{n_{1}}.\vec{\sigma},\vec{n_{2}}.\vec{\sigma}]=2i(\vec{n_{1}}\times\vec{n_{2}}).\vec{\sigma}.$
(47)

$\\{\sigma_{i}|i=1,2,3\\}$ are pauli matrices. We denote the eigen basis of
$\sigma_{z}$ and $\sigma_{x}$ as $\\{\ket{0},\ket{1}\\}$ and
$\\{\ket{+},\ket{-}\\}$ respectively. Now consider two observables $A_{1}$ and
$A_{2}$ acting on two dimensional Hilbert space such that

$\displaystyle A_{1}(1)$ $\displaystyle=\frac{1}{2}\ket{0}\bra{0},$ (48)
$\displaystyle A_{1}(2)$ $\displaystyle=\frac{1}{2}\ket{1}\bra{1},$ (49)
$\displaystyle A_{1}(3)$
$\displaystyle=\frac{3}{8}\ket{0}\bra{0}+\frac{1}{8}\ket{1}\bra{1},$ (50)
$\displaystyle A_{1}(4)$
$\displaystyle=\frac{1}{8}\ket{0}\bra{0}+\frac{3}{8}\ket{1}\bra{1}$ (51)

and

$\displaystyle A_{2}(1)$ $\displaystyle=\frac{1}{2}\ket{+}\bra{+},$ (52)
$\displaystyle A_{2}(2)$ $\displaystyle=\frac{1}{2}\ket{-}\bra{-},$ (53)
$\displaystyle A_{2}(3)$
$\displaystyle=\frac{1}{8}\ket{+}\bra{+}+\frac{3}{8}\ket{-}\bra{-},$ (54)
$\displaystyle A_{2}(4)$
$\displaystyle=\frac{3}{8}\ket{+}\bra{+}+\frac{1}{8}\ket{-}\bra{-}.$ (55)

Again, consider two sharp spin $\frac{1}{2}$ observables $B_{1}$ and $B_{2}$
acting on two dimensional Hilbert space such that

$\displaystyle B_{1}(1)=\ket{0}\bra{0};\hskip
8.5359ptB_{1}(2)=\ket{1}\bra{1},$ (56) $\displaystyle
B_{2}(1)=\ket{+}\bra{+};\hskip 2.84544ptB_{2}(2)=\ket{-}\bra{-}.$ (57)

Clearly, the both of the pairs $(A_{1},B_{1})$ and $(A_{2},B_{2})$ are
commutative and mutually commuting and therefore, both of the pairs are
individually broadcastable.

Now, let $A^{\prime}=\frac{1}{2}A_{1}+\frac{1}{2}A_{2}$ and
$B^{\prime}=\frac{1}{2}B_{1}+\frac{1}{2}B_{2}$. We know that a pair of qubit
observables are broadcastable iff those observables are commutative and
mutually commuting[14]. If we show that $A^{\prime}$ is not commutative then
the pair $(A^{\prime},B^{\prime})$ is not broadcastable .

Now, it is to be noted that

$\displaystyle A^{\prime}(1)$
$\displaystyle=\frac{1}{2}\left[\frac{1}{2}\ket{0}\bra{0}+\frac{1}{2}\ket{+}\bra{+}\right]$
$\displaystyle=\frac{1}{8}\left[2.\mathbb{I}+(\sigma_{z}+\sigma_{x})\right].$
(58)

Similarly,

$\displaystyle A^{\prime}(3)$
$\displaystyle=\frac{1}{2}\left[\frac{3}{8}\ket{0}\bra{0}+\frac{1}{8}\ket{1}\bra{1}+\frac{1}{8}\ket{+}\bra{+}+\frac{3}{8}\ket{-}\bra{-}\right]$
$\displaystyle=\frac{1}{16}\left[4.\mathbb{I}+(\sigma_{z}-\sigma_{x})\right].$
(59)

Therefore, using equation (47) we get $[A^{\prime}(1),A^{\prime}(3)]\neq 0$.
Hence, we have proved that $\mathcal{O}^{2}_{broad,X_{1}Y_{1}}$ is not convex,
where $X_{1}=\\{1,2,3,4\\}$ and $Y_{1}=\\{1,2\\}$. Several other counter
examples exist to prove this proposition. Now as the pairs $(A_{1},B_{1})$,
$(A_{2},B_{2})$ and $(A^{\prime},B^{\prime})$ are the pairs of non-trivial
qubit observables, from the Proposition 1, we get that
$\mathcal{O}^{2}_{1-side-broad,X_{1}Y_{1}}$, $\mathcal{O}^{2}_{mut-
nondist,X_{1},Y_{1}}$ and $\mathcal{O}^{2}_{nondist,X_{1}Y_{1}}$ are also not
convex. This is one of the essential differences between compatibility and any
other layer of classicality in geometric properties. ∎

There may be some other differences in geometric properties. But it needs
further investigation to find out that.
From the proof of propositions 5,6 and theorem3 it is clear that similar
results hold for one-side broadcasting.
Below in the Table 1, the comparison of several properties of the different
layers of classicality has been presented.

Name of the layers |  Transitivity property |  Convexity property |  Degree of the layer
---|---|---|---
Broadcastable pairs |  $\times$ (Prop. 2) |  $\times$ (Prop. 7) |  $\times$ (Th.1)
One-side Broadcastable pairs |  $\times$ (Prop. 2) |  $\times$ (Prop. 7) |  $\times$ (similar to Th.1)
Mutually non-disturbing pairs |  $\times$ (Prop. 2) |  $\times$ (Prop. 7) |  ?
Non-disturbing pairs |  $\times$ (Prop. 2) |  $\times$ (Prop. 7) |  ?
Compatible pairs |  $\times$ (Prop. 2) |  ✓(Ref.[16]) |  ✓(Ref.[1])
Table 1:  Comparison of transitivity property, convexity property and
existence of the concept of degree of the layers (i.e., the highest value of
unsharp parameter for which the unsharp version of a pair reside in a
particular layer) is presented in this table. (✓) indicates that a particular
property is satisfied by a layer or a concept exist for that layer, ($\times$)
indicates the opposite to that of (✓) and (?) indicates that it is not known
whether a particular property is satisfied by a layer or a concept exist for
that layer.

###  III.3 Relations among the layers of classicality

Before investigating relations among different layers of classicality let us
define some strict non-overlapping layers of classicality.

######  Definition 9.

Let $(A,B)\in\mathcal{O}_{XY}$ be a pair of observables. Then

  1. 1.

The pair is called weakly compatible if the pair is compatible, but not
nondisturbing.

  2. 2.

The pair is called weakly nondisturbing if the pair is nondisturbing, but not
mutually nondisturbing.

  3. 3.

The pair is called weakly mutually nondisturbing if the pair is mutually
nondisturbing, but not one-side broadcastable.

  4. 4.

The pair is called weakly one-side broadcastable if the pair is one-side
broadcastable, but not broadcastable.

This layers in Definition 9 are the strips between two successive layers of
classicality and can be easily visualised from the figure 1. Let us now
investigate the relation between broadcasting and one-side broadcasting. In
the following theorems and proposition we will see that specific concatination
relations between the broadcasting channels decide the resident of the pair in
a particular layer. We start with our next theorem.

######  Theorem 4 (Broadcasting and one-side broadcasting).

Consider a pair $(A,B)\in\mathcal{O}^{d}_{XY}$ and also suppose that $A$ is
broadcastable by $\Lambda_{1}$ and $B$ is broadcastable by $\Lambda_{2}$. Then
the pair is atleast one-side broadcastable if there exist two channels
$\Sigma,\Theta\in\mathcal{C}^{d}$ such that
$\Lambda_{2}=(\Theta\otimes\Sigma)\circ\Lambda_{1}$, where
$\Theta\in\mathcal{C}^{d}$ is a nondisturbing channel with respect to $A$ and
$\Sigma\in\mathcal{C}^{d}$ is an arbitrary channel. Along with this, if the
pair is also not broadcastable, then it is weakly one-side broadcastable.

###### Proof.

$A$ is broadcast by $\Lambda_{1}$. We know that dual channels are unital. Then
for all $x\in X$, and $\rho\in(\mathcal{S}(\mathcal{H}))$ we have

$\displaystyle\text{Tr}[\Lambda_{2}(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[(\Theta\otimes\Sigma)\Lambda_{1}(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[\Lambda_{1}(\rho)(\Theta\otimes\Sigma)^{*}(A(x)\otimes\mathbb{I})]$
$\displaystyle=\text{Tr}[\Lambda_{1}(\rho)(\Theta^{*}(A(x))\otimes\Sigma^{*}(\mathbb{I}))]$
$\displaystyle=\text{Tr}[\Lambda_{1}(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}(\rho A(x))$ (60)

Now $B$ is broadcast by $\Lambda_{2}$. Then for $y\in Y$ and
$\rho\in\mathcal{S}(\mathcal{H})$,

$\displaystyle\text{Tr}(\rho
B(y))=\text{Tr}[\Lambda_{2}(\rho)\mathbb{I}\otimes B(y)]$ (61)

From equations (60) and equation (61), we get that the pair $(A,B)$ is one-
side broadcastable. ∎

Before, we investigate the relation between one-side broadcasting and
nondisturbance, let us write down the proposition which is originally proved
in [14].

######  Proposition 8.

An observable $A\in\mathcal{O}^{d}_{X}$ can be measured without disturbing an
observable $B\in{\mathcal{O}^{d}_{Y}}$ iff there exists a $d^{\prime}$
dimensional ancilla Hilbert space $\mathcal{K}$, a channel
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}\otimes\mathcal{H})$
and an observable $A^{\prime}\in\mathcal{O}^{d^{\prime}}_{X}$ acting on the
ancilla Hilbert space $\mathcal{K}$ such that for any state
$\rho\in\mathcal{S}(\mathcal{H})$, $x\in X$ and $y\in Y$, we have

$\displaystyle\text{Tr}[\rho(A(x))]=\text{Tr}[\Lambda(\rho)A^{\prime}(x)\otimes\mathbb{I}]$
(62)
$\displaystyle\text{Tr}[\rho(B(y))]=\text{Tr}[\Lambda(\rho)\mathbb{I}\otimes
B(y)].$ (63)

Now let us investigate the connection between broadcasting and nondisturbance.
Our next theorem states one of the possible relations.

######  Theorem 5 (Broadcasting and nondisturbance).

If $A\in\mathcal{O}^{d}_{X}$ is broadcastable by the broadcasting channel
$\Lambda_{1}\in\mathcal{C}^{d}_{broad}$ and $B\in\mathcal{O}^{d}_{Y}$ is
broadcastable by the broadcasting channel
$\Lambda_{2}\in\mathcal{C}^{d}_{broad}$, then $A$ can be measured without
disturbing $B$ if $\Lambda_{2}$ is locally changeable to $\Lambda_{1}$. Along
with this, if the pair is also not mutually nondisturbing, then it is weakly
nondisturbing.

###### Proof.

If $\Lambda_{2}$ is locally changeable to $\Lambda_{1}$ then there exist two
channels $\Sigma_{1},\Sigma_{2}\in\mathcal{C}^{d}$ such that

$\Lambda_{1}=(\Sigma_{1}\otimes\Sigma_{2})\circ\Lambda_{2}.$ (64)

Since, $A$ is broadcastable by $\Lambda_{1}$, for all
$\rho\in\mathcal{S}(\mathcal{H})$ and for all $x\in X$

$\displaystyle\text{Tr}[\rho A(x)]$
$\displaystyle=\text{Tr}[\Lambda_{1}(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[(\Sigma_{1}\otimes\Sigma_{2})\circ\Lambda_{2}(\rho)A(x)\otimes\mathbb{I}]$
$\displaystyle=\text{Tr}[\Lambda_{2}(\rho)\Sigma_{1}^{*}(A(x))\otimes\Sigma_{2}^{*}(\mathbb{I})]$
$\displaystyle=\text{Tr}[\Lambda_{2}(\rho)\Sigma_{1}^{*}(A(x))\otimes\mathbb{I})]$
(65)

Since $B$ is broadcast by $\Lambda_{2}$, all $\rho\in\mathcal{S}(\mathcal{H})$
and for all $y\in Y$

$\displaystyle\text{Tr}[\rho
B(y)]=\text{Tr}[\Lambda_{2}(\rho)\mathbb{I}\otimes B(y)].$ (66)

Choosing $A^{\prime}=\\{A^{\prime}(x)\\}=\\{\Sigma_{1}^{*}(A(x))\\}$ and from
equation (65), equation (66), and Proposition 8 we get that the pair $(A,B)$
is a nondisturbing pair. Hence, the theorem is proved. ∎

From Theorem 5 we immediately get our next proposition.

######  Proposition 9 (Broadcasting and mutual nondisturbance).

If $A\in\mathcal{O}^{d}_{X}$ is broadcastable by the broadcasting channel
$\Lambda_{1}\in\mathcal{C}^{d}_{broad}$ and $B\in\mathcal{O}^{d}_{Y}$ is
broadcastable by the broadcasting channel
$\Lambda_{2}\in\mathcal{C}^{d}_{broad}$, then $A$ and $B$ are at least
mutually nondisturbing if $\Lambda_{2}$ and $\Lambda_{1}$ are locally
interchangeable. Along with this, if the pair is also not one-side
broadcastable, then it is weakly mutually nondisturbing.

The results similar to Theorem 5 and Proposition 9 hold if $A$ and $B$ are
one-side broadcastable by $\Lambda_{1}$ and $\Lambda_{2}$ respectively to two
different sides.
We obtained connections between different layers of classicality. To find out
more connections similar to the connections derived in this subsection, it
needs further investigation.

##  IV Conclusion

We have studied the physical and geometric properties of different layers of
classicality. We have shown that such properties of different layers of
classicality have differences as well as similarities. In particular, We have
shown that: (i) none of the layers of classicality respect transitivity
property, (ii) the concept like degree of broadcasting similar to degree of
compatibility does not exist, (iii) there exist informationally incomplete
POVMs that are not individually broadcastable, (iv) a set of broadcasting
channels can be obtained through concatenation of broadcasting and non-
disturbing channels, (v) unlike compatibility, other layers of classicality
are not convex, in general. Finally, we have obtained connections among
different layers of classicality. All of our results are valid for all finite
dimensions except Propostion 7. It is not known whether Proposition 7 is valid
for the dimensions more than two.
This work opens up several new avenues of research on the compatibility of
POVMs and the other layers of classicality. Firstly, it is not known whether
the set of all broadcasting channels that broadcast a particular observable
has the greatest or the lowest element. Secondly, we do not know which layers
are open or closed among the layers of classicality, except for compatibility.
Thirdly, we do not know whether the set of all nondisturbing pairs of
observables and the set of all mutually nondisturbing pairs of observables are
non-convex in higher dimensions. It will be also interesting to construct
wittnesses and resource theory for the different layers. Fourthly, till now we
do not have a full set of connections among different layers of classicality
and also do not have the full set of mathematical properties. Fifthly, it is
possible to generalise atleast some of our results for a set of $n$
observables with $n>2$.
An important avenue for future research is to find out how some of these
layers of classicality provide advantages over other layers of classicality in
several information-theoretic tasks.

##  V Acknowledgements

I would like to thank my advisor Prof. Sibasish Ghosh and my co-advisor Prof.
Prabha Mandayam for their support and valuable comments on this work.

## References

  * Heinosaari  _et al._ [2016] T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, Journal of Physics A: Mathematical and Theoretical 49, 123001 (2016).
  * Quintino  _et al._ [2014] M. T. Quintino, T. Vértesi, and N. Brunner, Joint measurability, einstein-podolsky-rosen steering, and bell nonlocality, Physical review letters 113, 160402 (2014).
  * Stevens and Busch [2014] N. Stevens and P. Busch, Steering, incompatibility, and bell-inequality violations in a class of probabilistic theories, Physical Review A 89, 022123 (2014).
  * Son  _et al._ [2005] W. Son, E. Andersson, S. M. Barnett, and M. Kim, Joint measurements and bell inequalities, Physical Review A 72, 052116 (2005).
  * Uola  _et al._ [2015] R. Uola, C. Budroni, O. Gühne, and J.-P. Pellonpää, One-to-one mapping between steering and joint measurability problems, Physical review letters 115, 230402 (2015).
  * Wootters and Fields [1989] W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased measurements, Annals of Physics 191, 363 (1989).
  * Cerf  _et al._ [2002] N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Security of quantum key distribution using d-level systems, Physical review letters 88, 127902 (2002).
  * Wootters and Zurek [1982] W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299, 802 (1982).
  * Barnum  _et al._ [1996] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Physical Review Letters 76, 2818 (1996).
  * Ferraro  _et al._ [2006] A. Ferraro, M. Galbiati, and M. G. Paris, Cloning of observables, Journal of Physics A: Mathematical and General 39, L219 (2006).
  * Ferraro and Paris [2007] A. Ferraro and M. G. Paris, Joint measurements on qubits and cloning of observables, Open Systems & Information Dynamics 14, 149 (2007).
  * Alvarez-Rodriguez  _et al._ [2014] U. Alvarez-Rodriguez, M. Sanz, L. Lamata, and E. Solano, Biomimetic cloning of quantum observables., Scientific Reports 4, 4910 (2014).
  * Rastegin [2010] A. E. Rastegin, No-cloning theorem for a single povm, Quantum Information & Computation 10, 971 (2010).
  * Heinosaari [2016] T. Heinosaari, Simultaneous measurement of two quantum observables: Compatibility, broadcasting, and in-between, Physical Review A 93, 042118 (2016).
  * Heinosaari and Ziman [2011] T. Heinosaari and M. Ziman,  _The mathematical language of quantum theory: from uncertainty to entanglement_ (Cambridge University Press, 2011).
  * Carmeli  _et al._ [2019] C. Carmeli, T. Heinosaari, and A. Toigo, Quantum incompatibility witnesses, Physical review letters 122, 130402 (2019).
  * Heinosaari and Wolf [2010] T. Heinosaari and M. M. Wolf, Nondisturbing quantum measurements, Journal of mathematical physics 51, 092201 (2010).
  * Busch [1986] P. Busch, Unsharp reality and joint measurements for spin observables, Physical Review D 33, 2253 (1986).