# EDITH : ECG biometrics aided by Deep learning for reliable Individual
auTHentication

Nabil Ibtehaz  Department of Computer Science and Engineering, Bangladesh
University of Engineering and Technology, Dhaka-1205, Bangladesh  Muhammad E.
H. Chowdhury  Department of Electrical Engineering, Qatar University,
Doha-2713, Qatar  Corresponding author<EMAIL_ADDRESS>Amith Khandakar
Department of Electrical Engineering, Qatar University, Doha-2713, Qatar
Serkan Kiranyaz  Department of Electrical Engineering, Qatar University,
Doha-2713, Qatar  M. Sohel Rahman  Department of Computer Science and
Engineering, Bangladesh University of Engineering and Technology, Dhaka-1205,
Bangladesh  Anas Tahir  Department of Electrical Engineering, Qatar
University, Doha-2713, Qatar  Yazan Qiblawey  Department of Electrical
Engineering, Qatar University, Doha-2713, Qatar  Tawsifur Rahman  Department
of Biomedical Physics & Technology, University of Dhaka, Dhaka-1000,
Bangladesh

###### Abstract

In recent years, physiological signal-based authentication has shown great
promises, for its inherent robustness against forgery. Electrocardiogram (ECG)
signal, being the most widely studied biosignal, has also received the highest
level of attention in this regard. It has been proven with numerous studies
that by analyzing ECG signals from different persons, it is possible to
identify them, with acceptable accuracy. In this work, we present, EDITH, a
deep learning-based framework for ECG biometrics authentication system.
Moreover, we hypothesize and demonstrate that Siamese architectures can be
used over typical distance metrics for improved performance. We have evaluated
EDITH using 4 commonly used datasets and outperformed the prior works using a
fewer number of beats. EDITH performs competitively using just a single
heartbeat (96$\sim$99.75% accuracy) and can be further enhanced by fusing
multiple beats (100% accuracy from 3 to 6 beats). Furthermore, the proposed
Siamese architecture manages to reduce the identity verification Equal Error
Rate (EER) to 1.29 %. A limited case study of EDITH with real-world
experimental data also suggests its potential as a practical authentication
system.

Index terms— Authentication, Biometrics, ECG-ID, 1D Convolutional Network, 1D
Siamese Network

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##  1 Introduction

Since the dawn of civilization, the human being has been storing valuable
objects ensuring secrecy and security. Mechanisms such as keys or combination
locks have been used to serve such purposes for centuries. However, with the
development of information technology, our valuable possessions (e.g., bank
accounts) have mostly been shifted to the digital domain. Besides, various
personalized services and amenities have also received similar technological
transformation. The concern of security and secrecy still applies; thus
various measures are adopted to allow only authorized access. Analogous to the
concepts of keys and combination locks, in the digital domain, tokens and
passwords have been being used for decades. But they come with the risk of
being lost or stolen, bringing in the peril of malicious access [1].

Therefore, a multitude of research works have been conducted for decades to
develop resilient authentication systems. There exist a lot works in the
literature that focused on behavioral modalities, such as voice [2], keystroke
[3], gait [4] etc. However, the behavioral authentication systems are often
cumbersome and unreliable. This has made biometric authentication systems more
preferred, which include fingerprint [5], retina [6], facial recognition [7]
etc. Despite being widely adopted, these methods also face a serious issue of
forgery attacks [8].

In recent years, physiological signals have also demonstrated great potential
in authenticating individuals. Physiological signals are in fact more
preferable to other means of authentication as they are hard to counterfeit
and also the individuals need to present themselves to record such signals
[9]. Among various physiological signals electrocardiogram (ECG) [10],
electroencephalography (EEG) [11] and photoplethysmography (PPG) [12] are the
most prominent. Compared to most physiological signals, PPG and ECG signals
can be obtained relatively easily, for example, using simple finger sensors
[13]. Furthermore, ECG signals are more resilient against noises, compared to
PPG or EEG, which has made ECG more preferable as a biometric system [14]. As
a result ECG bands [15], low power circuits [16] and light-weight algorithms
[17] have been developed for ECG based authentication systems. Furthermore,
with the recent introduction of ECG sensors in commercial products like the
Apple Watch and Samsung Galaxy Watch, the possibility of widespread adoption
of ECG biometrics has opened [18].

In this paper, we propose EDITH, which is a deep learning-based ECG biometric
authentication system, EDITH. EDITH comprises a novel convolutional network
architecture based on multiresolution analysis, employing MultiRes blocks and
Spatial Pyramid Pooling in coherent conjunction. To satisfy the reliance on
accurate R-peak detection, we develop a deep learning-based R-peak detector.
Furthermore, we propose a novel Siamese architecture incorporating both
Euclidean and Cosine similarity distance measures. EDITH has been evaluated on
4 benchmark datasets and have consistently outperformed the prior works.

##  2 Related Works

The pioneering work of Biel et al. [10] deserves the most credit for
establishing the potential of ECG signals for human identification. Ever since
this pilot study, ECG, as a biometric has drawn widespread interest, resulting
in a plethora of novel approaches[19]. The different ECG based authentication
systems can broadly be divided into two classes, fiducial and non-fiducial.

The fiducial methods rely on extracting various fiducial landmarks from the
ECG signal. For such approaches, different sub-waves like Q, R, S, P, T are
extracted, and features related to various amplitude, difference, or delay
terms are computed [20, 21, 22]. However, these methods are often criticized
for over-reliance on the fiducial points, slight misdetection of which results
in an erroneous identification [19]. The inability of working with noisy or
variable heart-beats seriously limits these methods. Moreover, for
satisfactory performance, manual intervention is often required, which further
diminishes their utility [23].

To overcome the limitations of the fiducial methods, subsequently, non-
fiducial approaches were investigated. These methods analyze the entire or a
segment of the signal to extract several morphological features [24]. Among
the diverse exploration of non-fiducial methods, the use of Kalman filter
[25], Fourier or Wavelet transformation [26], and generalized S-transformation
[27] are the most notable ones.

Recently, several deep learning-based methods have also been proposed for ECG
biometrics. Such methods have demonstrated much greater accuracy in
identification compared to the traditional approaches. Although such methods
often rely on the detection of R-peaks for proper alignment of the signals,
they still circumvent the flaws of the fiducial approaches. Since R-peak is
the highest and sharpest point of ECG and the QRS complex is less likely to be
affected by physical movement or emotion, they are popularly used for ECG
signal alignment [9]. Convolutional networks, ranging from vanilla
architectures [9], multi-resolutional models [28, 23], residual models [29]
have all demonstrated superior accuracy in this context. Similarly, recurrent
network models such as Long Short-Term Memory (LSTM) [30] or Gated Recurrent
Unit (GRU) [31] have also been successfully applied for precise individual
authentication.

##  3 Motivations and High Level Considerations

Our primary objective is to develop a deep learning model capable of reliable
ECG authentication. Instead of using off-the-shelf networks and transfer
learning, we analyzed the specific challenges that come with this problem and
designed our model accordingly.

###  3.1 Compact Model with Broad Receptive Field

The feasibility of deploying an ECG biometric system depends on the model
being lightweight. This requirement is imposed for several reasons. Firstly,
the end goal would be to run the model on wearable devices, as smartwatches
now carry ECG sensors. Since the typical wearable processors can not be made
powerful due to space and power constraints, making the model simple is
crucial. Secondly, when new users register to the system, they are likely to
provide only a few ECG samples, therefore it would be difficult to obtain a
reasonable amount of samples to properly fine-tune a deep network.

Although it is motivating and suitable to design a shallow network, such
networks are likely to fall behind in performance. Deep networks have a
broader field of vision at the later layers and thus they can make much
complex inferences. This ability to analyze a larger portion of the complex
featuremaps comes at the cost of network complexity. However, in recent times
a few approaches have been proposed to mitigate this. MultiRes blocks [32] can
be used interchangeably with convolutional layers, providing a wider receptive
field through multiresolution analysis, in a compact form. In terms of
pooling, Spatial Pyramid Pooling (SPP) [33] can be used to leverage pooling
operations from different points of view, as if they were computed at much
deeper layers.

Therefore, we use MultiRes blocks and SPP layers to design a compact network
with the essence of a broad field of view found in deeper networks. More
details are presented in Section 4.3.3.

###  3.2 Adapting Siamese Network Distance

In order to solve the problem of ‘Identity Verification’, the standard
pipeline is to extract features or embeddings using a deep learning model and
compute the distance with the templates, based on thresholding which the
verification is performed. The distances are computed using either Euclidean
distance (L2 distance) [23] or cosine similarity [9].

Cosine similarity has been the defacto standard in identity verification. To
investigate the difference in applicability between the two, we extract
embeddings from ECG-ID dataset signals and measure both the cosine distance
$(1-cosine\,\,similarity)$ and L2 or Eucledian distance, and plot histograms
(Fig. 1(a)). It is evident that L2 distance has comparatively more overlaps,
thus the use of cosine similarity is justified. However, when we plot both the
distances in a 2D histogram, separately for genuine (Fig. 1(b)) and imposter
(Fig. 1(c)) comparisons, we see a different view. In both the figures, the
expected region is pointed by a black border and it can be observed that
cosine distance is better at distinguishing the genuine ones whereas L2
distance is more suitable for identifying imposters.

Therefore, we design a novel Siamese network modeling both the two types of
distance. We hypothesize that an adaptive combination of both the distances,
learned by the Siamese network will be capable of verifying identities better.
More details have been presented in Section 4.1.2.

(a)

(b)

(c)

Figure 1: Investigation of Distance Measures.

###  3.3 Multiple Beat Fusion

Although we would like to authenticate the user using a single heartbeat
signal, just like a single fingerprint reading can do, it fails to achieve
satisfactory performance. Not only a single heartbeat signal contains very
little information, but also it succumbs to deviation due to several
variabilities. As a result, in the literature, the common practice is to work
with multiple beats and it greatly boosts the authentication accuracy [30, 29,
23].

When it comes to working with multiple beats, two approaches are followed in
general. One approach is to merge multiple beats together and train the model
with longer signals [30, 23]. The second approach is to train the model using
single beats but during inference, multiple consecutive beats are taken as
input and a majority voting of their predictions is provided as the final
output [29].

In our work, we followed the second approach, i.e., we train the model with
single heartbeats but during inference, we have the option to ensemble results
from multiple consecutive heartbeats using majority voting. Our rationale is
based on a few considerations. Firstly, training the model on longer signals
would require the model to be more complex but we are less likely to have
sufficient data to train such a model properly. Secondly, not only it is known
that ensemble methods always boost the accuracy, but also this provides us
with the flexibility to work with a single beat if needed (e.g. continuous
authentication [34]). And most importantly, if even one beat deviates from the
standard morphology due to some physiological reasons, it will negatively
affect the entire merged multibeat signal (i.e., approach 1), but ensembling
predictions from multiple beats will be comparatively immune to such
inconsistencies. This is our primary motivation, to make the method more
robust.

##  4 Methodology

###  4.1 Problem Definition

The problem of individual authentication using ECG signals can be divided into
two tasks: Closed Environment Identification and Identity Verification.

####  4.1.1 Closed Environment Identification

For the closed environment identification problem, among a fixed set of $n$
persons, $P=\\{p_{1},p_{2},p_{3},\dots,p_{n}\\}$. The task is to develop a
model $M_{cei}$ that is capable of identifying the person $p_{i}$ when given
an ECG signal of that person $Sig_{i}$. This task, eventually reduces into a
multi-class classification problem.

####  4.1.2 Identity Verification

One major drawback of the former task is that when a new individual is
introduced to the authentication system, the entire pipeline needs to be
retrained with the new data. This becomes cumbersome for practical usage.
Therefore, an alternate task has also been studied in the literature. In the
identity verification task, we are given an independent dataset of persons
($P_{i}={p_{i1},p_{i2},p_{i3},\dots,p_{im}}$) and based on a certain portion
of data, termed as ‘enrollment data’ we generate templates
($T={t_{t1},t_{2},t_{3},\dots,t_{m}}$) for them. Then, the task is to match
the remaining data, termed as ‘evaluation data’ with the templates and verify
the identity. To match a sample $Sig_{a}$ with the template $t_{b}$, i.e.
measuring the similarity or dissimilarity, distances like cosine or Euclidean
distances are used.

###  4.2 Datasets

In order to develop, experiment, and evaluate the EDITH system, we consider 4
different public datasets, which are the main ECG datasets used in ECG based
authentication purposes in the literature [19]. Below we briefly describe
these datasets (please see Table 1 for an overview).

####  4.2.1 ECG-ID

The ECG-ID database [35, 36] is one of the few databases that was developed
for the sole purpose of investigating ECG-based authentication. Therefore,
this dataset is the defacto choice for researchers to work with ECG biometrics
[19]. In this dataset, 20 seconds long ECG signal from lead I, was recorded
from 90 persons. The signals were digitized at 500 Hz with 12-bit resolution
over a nominal $\pm 10$ mV range. For each subject, 2 to 20 session data were
collected. Thus it is the common practice among researchers to consider only 2
sessions for each person to avoid any data imbalance issue [29, 30], which is
also considered in this study.

####  4.2.2 MIT-BIH Arrhythmia

Although the MIT-BIH Arrhythmia dataset [37] is centered around arrhythmia
detection, this dataset has been used to benchmark biometric authentication
accuracy in several works. This dataset contains ECG signals from 47 subjects
suffering from a wide variety of arrhythmias. The recordings were digitized at
360 Hz sampling rate with 11-bit resolution over $10$ mV range of signals. The
ECG signals were recorded in two channels, however, only the first channel was
used for this study.

####  4.2.3 PTB Diagnostic ECG Database

The PTB Diagnostic ECG database [38] contains 549 records from 290 subjects,
out of which 52 individuals are healthy. Thus, for benchmarking authentication
performances only this control group has been used in the literature [23].
This dataset contains ECG records from 15 leads (12 conventional leads and 3
Frank leads). The signals are digitized in 1000 Hz sampling frequency with 16
bits resolution. The ECG signals from the Frank lead (vx) was used in this
study.

####  4.2.4 MIT-BIH NSRDB

The NSRDB database is a part of the MIT-BIH Arrhythmia database [37]. This
database contains signals from 18 subjects, who are free from arrhythmias or
other abnormalities. This database mostly contains Normal Sinus Rhythm, thus
the name NSRDB. 1000 beats from each of the 18 individuals were considered for
this study.

All the datasets were collected from PhysioNet [39] and were resampled to 500
Hz (in case of a different sampling rate). Notebly all these datasets were
already filtered, thus no additional filtering or preprocessing was applied.
However, when applying the proposed method to a different data source, some
preprocessing may be necessary to remove noise and artifacts.

An overview of the datasets has been presented in Table 1

Table 1: Overview of the Datasets Dataset | # Persons |  | Sampling
---
Rate
| Health
---
Condition
Activity |  | Electrode
---
Placement
Reference
ECG-ID | 90 | 500 Hz | Healthy | Sitting | Wrist | [35, 36]
| MIT-BIH
---
Arrhythmia
47 | 360 Hz | Arrhythmia |  | Ambulatory
---
Recording
Chest | [37]
PTB | 290 | 100 Hz | Mixed | At rest | Chest + Limbs | [23]
| MIT-BIH
---
NSRDB
18 | 360 Hz | Healthy |  | Ambulatory
---
Recording
Chest | [37]

In order to perform cross-session evaluation, we used another dataset, CYBHi
[40].

####  4.2.5 CYBHi Dataset

The CYBHi [40] dataset was developed to evaluate ECG biometrics, particularly
in multi-session scenario. The dataset contains two types of data, short-term
and long-term. In the long-term dataset, ECG signals from 63 individuals are
presented which are collected over a period of several days. The ECG signals
were collected from finger using a sampling frequency of 1 kHz.

###  4.3 Proposed Method

The proposed EDITH pipeline takes a continuous ECG signal as input
($f_{s}=500Hz$, resamples otherwise) and detects the R-peaks, based on which
the beats are extracted. Each heartbeats is then passed through a
convolutional neural network (CNN) when dealing with the closed environment
identification task. On the contrary, to perform the identity verification
task they are compared with the stored templates using a siamese network.
Furthermore, the results from multiple beats can be ensembled for improving
the performance. An overview of the pipeline is presented in Fig. 2.

Figure 2: The proposed EDITH Pipeline. From an input ECG signal the R-peaks
are detected using deep learning (MultiResUNet) model, based on which the
heartbeats are segmented. The extracted beats are analyzed using another deep
learning (CNN or Siamese) model for closed environment identification or
identity verification.

####  4.3.1 R-peak Detection

The first step of the proposed method is the R peak detection. It is often
attributed that R peaks being the highest and sharpest point are trivial to
detect [23]. However, for a diverse set of ECG signals, this is not always the
case. As has been demonstrated by recent works [29], popular beat detection
algorithms (e.g. Hamilton [41]) are not consistently sufficient to precisely
locate the R peaks. It was also demonstrated that when working with a single-
beat authentication system, proper detection of R-peaks is the most vital
step, deviating even slightly often diminishes the performance drastically
[29].

Uzair et al. [42] have recently proposed an approach to detect R-peaks using
deep learning technique, which outperforms the traditional methods. In that
work, a 1-D U-Net [43] was used to predict the location of R-peaks, from an
ECG signal. We have deployed our recently proposed variant of U-net
architecture, 1-D MultiResUNet model [32] with deep supervision [44] as done
in [45], to predict the R-peaks from a continuous ECG signal. MultiResUNet
model showed improved performance for 2D applications over the traditional
U-Net architecture which incorporates multiresolution analysis and deep
supervision is proven to improve the performance of deep networks. The
probability map predicted from the model is thresholded and further refined by
some post-processing. It is expected that R-peaks will be considerably distant
from one another. Thus, to minimize false positives, we ignore nearby
predicted beats and replace them with the median prediction. To reduce false
negatives we empirically select a suitable threshold value.

####  4.3.2 Beat Segmentation

After the R-peaks have been detected, the individual heartbeats are segmented
from the ECG signal, following the approach in [29]. The beats are aligned
based on the location of the R-peaks to provide a standardized pattern for the
model. We take a window size of $w$, where $w=256$ samples, and include
$\frac{w}{4}$ and $\frac{3w}{4}$ samples before and after the R-peak
respectively. Furthermore, the extracted heartbeat signals were Z-score
normalized [29].

####  4.3.3 Network Architecture

Developing a deep learning model for a single heartbeat based authentication
system is challenging and difficult for two major reasons: short signal
duration and the requirement of a large database. Short signals will be
affected by pooling layers, for losing probable significant information.
Furthermore, a small(er) dataset prevents us from working with deeper
networks. Consequently, pooling operations have been avoided in the literature
for this purpose [29] and multiresolution analysis has rather been leveraged
instead [28, 23].

Therefore, in our proposed model, we have resorted to using a somewhat shallow
network, incorporating multiresolution analysis with limited pooling
operations. We were motivated to use the MultiRes block as proposed in [32]
and spatial pyramid pooling [33]. The network architecture has been presented
in Fig. 3(a).

The MultiRes block presents a compact form of Inception block [46] and
incorporates a residual connection for aiding in the gradient flow [47]. Both
inception block [23] and residual connections [29] have demonstrated favorable
outcome in ECG based biometrics. We place 3 convolutional layers having
filters of size 32, 64, and 128, respectively and concatenate their
featuremaps, which in essence factorizes larger convolutional operations using
smaller ones. Since we are using a shallow network, we consider convolutional
filters of size 15, to expand the receptive field. We avoid intermediate
pooling operations to prevent loss of information, but after performing the
convolutions for the sake of summarization of the featuremaps, max pooling is
performed. To exploit the multiresolutional nature of our analysis further, we
perform spatial pyramid pooling instead of the vanilla pooling operation.
Three pooling operations, with windows of size of 8, 16, and 32 are performed
in parallel and the outputs are concatenated. Finally, a fully connected layer
with 128 neurons is introduced to generate the embedding for the input
signals. This layer is further regularized with a dropout of 0.25 to avoid
overfitting [48]. All the layers used in this network are batch-normalized
[49] and activated by Rectified Linear Unit (ReLU) activation.

(a)

(b)

Figure 3: The proposed network architecture. 3(a) presents the Closed
Environment Identification model and 3(b) refers to the proposed Siamese model
for Identity Verification.

####  4.3.4 Closed Environment Identification

For the Closed Environment Identification task, the target is to identify a
predefined number of individuals. Therefore, the problem reduces to a Multi-
Class Classification problem. We simply include a softmax layer on top of our
proposed network to perform this (Fig. 3(a)). The number of neurons at the
output layer is kept equal to $n_{persons}$, i.e. the total number of subjects
in the closed environment being considered.

####  4.3.5 Identity Verification

For the identity verification task, on the other hand, we do not have any
fixed number of individuals as our targets; rather a set of templates are
computed to compare with. In the literature, for this task, typically a group
of subjects is used to train a deep learning model, which is then used to
generate embeddings, by removing the top layers. Then, a set of enrollment
signals is considered, containing a few signals from each of the individuals
to evaluate the performance of the method. Templates are computed for the
individuals by taking an average of the embeddings of that individual.

Distance measures like Euclidean distance [23] or cosine similarity [9] are
widely used in the literature, to match test signals with the templates. We
hypothesize that such distance measures may not be capable of weighting the
individual features of the embedding properly, and thus an adaptive method may
be more suitable.

To this end, we propose a Siamese architecture to compute the similarity
between templates and signal embeddings. Siamese networks employ two identical
sub-networks to compare the similarity of two input samples, and they have a
long history in performing verification tasks [50]. The weights of the two
sub-networks are shared together, and hence, they are likely to predict
similar featuremaps for similar input samples. They can be further compared
together, usually by computing the squared differences between the individual
pairs of features. However, drawing motivation to emulate the popular Cosine
similarity distance metric used for this purpose, in addition to computing the
squared differences, we also explore the efficacy of multiplying the
individual pairs of features. We term this as Product Proximity between the
features, as similar valued feature pairs will yield a high multiplication
output, whereas feature pairs with dissimilar values will result in a low
score. Our intuition is that the way squared difference is intended to model
eucledian distance, Product Proximity of the features will represent
correlation or Cosine similarity like metrics. In addition to working with
them separately, we also merge them together and analyze them in fully
connected layers (please refer to Fig. 3(b)). Mathematically,

$\textit{Squared Difference}[i]=(v_{i}-u_{i})^{2},\forall i\in[0\dots d]$ (1)
$\textit{Product Proximity}[i]=(v_{i}\times u_{i}),\forall i\in[0\dots d]$ (2)
$\textit{Combined Metric}[i]=\textbf{concatenate}(\textit{Squared
Difference}[i],\textit{Product Proximity}[i])$ (3)

Here, $v$ and $u$ are $d$-dimensional embeddings generated from a model $\phi$
for inputs $U$ and $V$ respectively, i.e., $v=\phi(V)$, $u=\phi(U)$.

Hence, we take our proposed identification network, pretrained on a disjoint
set of individuals. The top softmax layer and the dropout layers are then
discarded and the top ReLU activation is replaced with a Sigmoid activation
instead so that we can obtain the embeddings in the range of $0\sim 1$. We
take another identical sub-network so that the embeddings for two signals can
be computed in parallel. After the embeddings are computed, we apply two
different types of distance measures between the individual features as
mentioned earlier and we further perform a combination of the two. The
distance measures are further analyzed by the subsequent fully connected
layers and by merging them we predict a similarity score between $0\sim 1$.
The model architecture for the identity verification task has been presented
in Fig. 3(b).

##  5 Experimental Setup

The experiments have been conducted in a server computer with Intel Xeon
@2.2GHz CPU, 24 GB RAM, and NVIDIA TESLA P100 (16 GB) GPU. We implemented the
EDITH pipeline in Python programming language. The deep learning models were
implemented in Keras [51] with Tensorflow backend [52]. The codes are
available in the following github repository.

https://github.com/nibtehaz/EDITH

###  5.1 R-peak Detection Model

In order to train the R-peak Detection Model, we collected the portion of the
ECG signals with ground truth annotation of the R-peaks from the ECG-ID
database. The continuous signals were divided into windows of 1024 samples,
with a step size of 256 samples. Each windowed signal is composed of roughly
three ECG beats. These windowed signals were considered as input to the R-peak
detector model, while the output was a pulse train showing the location of
R-peaks as ‘1’ and ‘0’ elsewhere. In order to apply deep supervision, as
reported in [45], the pulse trains were subsampled accordingly. 80% of the
data was used to train the model and the remaining 20% was used as validation
data. The model was trained for 100 epochs, minimizing binary cross-entropy
loss using the Adam optimizer [53].

###  5.2 Closed Environment Identification Model

For the Closed Environment Identification model, we performed stratified
10-fold cross-validation as well as testing using 60-20-20 train-val-test
split. Thus the data were divided and the CNN models were trained accordingly.
We trained the models for 500 epochs, minimizing categorical crossentropy loss
using Adam optimizer [53]. Notably, except for the PTB dataset, convergence
were reached before 150 epochs for all.

The models were evaluated using the standard accuracy metric as follows:

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$ (4)

Here, TP, TN, FP, FN corresponds to the true positive, true negative, false
positive, false negative predictions respectively.

In addition, we also compute other standard evaluation metrics:

$Precision=\frac{TP}{TP+FP}$ (5) $Recall=\frac{TP}{TP+FN}$ (6)
$F1-Score=\frac{2}{\frac{1}{Precision}+\frac{1}{Recall}}$ (7)

###  5.3 Identity Verification Model

We trained the base model (as mentioned in Section 5.2) using 50% of the
individuals from the ECG-ID dataset.

The training signals were paired with each other to form training samples for
the Siamese model. As we have too many mismatched samoles in comparison to
matched samples, we used Synthetic Minority Oversampling Technique (SMOTE)
[54] to generate synthetic data for matching cases. 21.5 times synthetic
samples were generated for each person as a trade-off between data imbalance
and computation requirements. This enables the model to learn which patterns
are discriminative of matching cases. 80% of the signal pairs were used to
train the model and the rest were used as validation data. Since we predict a
continuous variable as the outcome in this model the Adam optimizer [53] was
used to minimize the Mean squared error (MSE) loss. The model was trained for
75 epochs to predict ‘0’ for mismatches and ‘1’ for matches.

For evaluation, the test data was later split into enrollment and evaluation
data. The enroll signals were used to generate embeddings, and for each
subject, the mean or centroid of the embeddings was considered as the
template. The evaluation signals were matched with the templates for each
individual, and based on a threshold the match-mismatch decision was made.
Correspondingly the standard evaluation metrics, namely False Acceptance Rate
(FAR), False Rejection Rate (FRR), and Equal Error Rate (EER) were computed.

$FAR(th)=\frac{FalseAcceptance(th)}{n_{Imposter}}$ (8)
$FRR(th)=\frac{FalseRejection(th)}{n_{Genuine}}$ (9) $EER=FAR(th_{equal})$
(10) $\exists th_{equal}\in[0\dots 1]\land FAR(th_{equal})=FRR(th_{equal})$

The threshold value $th$ were taken from 0 to 1, and the threshold
$th=th_{equal}$ causing $FAR(th)=FRR(th)$, was used to compute $EER$.

##  6 Results and Discussions

###  6.1 R-peak detection by EDITH

Although detection of R-peaks has not been our primary focus, it turns out
that our proposed R-peak detection model detects R-peaks with reliable
performance. Despite being an auxiliary step of our pipeline, it is on par
with the popular R-peak detection algorithms [41, 55, 56, 57, 58, 59], and
outperforms most both quantitatively and qualitatively.

To present a comparison with the existing algorithms, we detect the R-peaks on
the entire ECG-ID and benchmark the various algorithms based on the ground
truth annotations provided therein. The results are presented in Table 2. We
consider the false positives and false negatives as falsely predicted peaks
and actual peaks that were missed respectively. Besides, to further assess the
time lag of R-peak detection, we propose a metric termed ‘Temporal Error’.
Suppose, a certain R-peak, $r_{i}$ was at timestamp $t_{i}$ and the method
predicts it at timestamp $\hat{t}_{i}$, the Temporal Error is defined as,
$Temporal\;Error=|\hat{t}_{i}-t_{i}|$. The proposed R-peak detector achieves
the least number of false positives, and quite remarkably a negligible
Temporal Error. This ensures a proper segmentation of the heartbeats for
authentication in the next stage of our pipeline.

Table 2: A quantitative evaluation of the proposed R-peak detector. EDITH predicts the least number of false negatives and maintains a negligible Temporal Error. Method | Proposed |  Hamilton [41] |  Christov [55] |  | Engelse-
---
Zeelenberg
[56]
| Pan-
---
Tompkins
[57]
| Stationary
---
Wavelet
Transform [58]
| Two
---
Moving
Average[59]
Detected R-peaks | 2904 | 2873 | 2898 | 2661 | 2883 | 2887 | 2903
False Positives | 132 | 197 | 144 | 109 | 183 | 168 | 173
False Negatives | 196 | 227 | 202 | 439 | 217 | 213 | 197
Temporal Error | $0.69\pm 5.87$ | $37.86\pm 39.41$ | $12.38\pm 24.69$ | $3.82\pm 20.07$ | $25.61\pm 33.57$ | $27.64\pm 32.44$ | $30.28\pm 22.07$

In terms of qualitative assessment of the R-peak detection, we select an ECG
signal from the ECG-ID dataset and detect the R-peaks using the different
methods. The comparative analysis of the R-peak along with the manual ground-
truth annotation (only 10 peaks for each signal is provided in the database)
is presented in Fig. 4(a). Interestingly, the proposed R-peak detector
accurately detects all the peaks, whereas other methods predict false peaks or
miss R-peaks, and usually yield a high time lag between the detected and the
actual peak. The superior performance of the proposed method is further
illustrated in Fig. 4(b), where similar observations can be drawn over the
segmented beats based on the detected R-peaks, which is the primary concern in
this context. It is evident that the beats detected by the proposed approach
are properly aligned, almost identical to the ground-truth. On the other hand,
the traditional methods sometimes not only missed beats but also failed in
aligning them properly. Notably, missing beats as well as misaligned ones
adversely affects the identification accuracy as pointed out in [29].

(a)

(b)

Figure 4: A qualitative evaluation of the proposed R-peak detector. 4(a)
presents the R-peaks detected by various methods from a sample ECG signal of
the ECG-ID database. 4(b) demonstrates the segmented beats based on the
detected R-peaks using those methods.

Furthermore, we used our R-peak detection model to detect R-peaks from the PTB
and NSRDB datasets, without any modification or fine-tuning. In both cases,
our model was capable of detecting the R-peaks reliably. However, for the lack
of manual ground-truth annotation, it was not possible to evaluate the
performance directly. Nevertheless, the high level of accuracy in identifying
the individuals implicitly points out to the overall robustness of the R-peak
detector.

###  6.2 Comparative Evaluations for the Closed Environment Identification

####  6.2.1 Closed Environment Identification

In order to assess the efficacy of EDITH for the closed environment
identification task, we follow the two traditional evaluation schemes,
stratified 10-fold cross-validation, and 60-20-20 data split. EDITH manages to
accurately identify the individuals consistently in all datasets. The accuracy
plots of EDITH across the four datasets are shown in Fig. 5(a). It can be
noticed that for 10-fold cross-validation, the accuracy is slightly higher,
which could be due to the availability of additional training data.
Nevertheless, in all the experiments, EDITH achieves a remarkable performance
in the closed environment identification. In addition, we also present the
standard classification metrics in Table 3. It can be observed that for MIT-
BIH, PTB, and NSRDB datasets, the proposed method achieves balanced precision
and recall scores which is also reflected at the high F1-Score. Although in
the ECG-ID dataset recall is slightly worse, the precision is comparatively
higher. This signifies that the method reduces false positives at the cost of
false negatives. This proves beneficial as for an authentication system
restricting unauthorized access is more concerning. Nevertheless, the recall,
i.e. authorized access can be improved by ensembling the result of multiple
beats.

(a)

(b)

Figure 5: Closed Environment Identification performance of EDITH. In 5(a) the bar plots represent the accuracy obtained using a single heartbeat in each dataset. On the other hand, 5(b) illustrates the gain in performance when fusing predictions from multiple beats together. Table 3: Closed Environment Identification train-val-test split single-beat performance Dataset | Precision | Recall | F1-Score | Accuracy
---|---|---|---|---
ECG-ID | 98.053 % | 94.481 % | 96.234 % | 96.247 %
MIT-BIH | 98.429 % | 98.072 % | 98.243 % | 98.170 %
PTB | 99.711 % | 99.702 % | 99.699 % | 99.702 %
NSRDB | 99.586 % | 99.417 % | 99.499 % | 99.500 %

####  6.2.2 Fusing Multiple Beats for Improved Performance

In the literature, it has been demonstrated that combining multiple beats
greatly increases authentication accuracy [30, 29]. This is expected due to
the information gain from the other beats. Therefore, we have experimented
with performing a majority voting over multiple consecutive beats to further
improve the performance of EDITH. From the results presented in Fig. 5(b), it
can be observed that increasing the number of beats significantly improves the
performance. We have limited our analysis up to 8 beats, as prior studies have
presented this as an optimal trade-off between performance and practicality
[60, 61, 9]. In this regard, EDITH shows great promise by achieving the
perfect $100\%$ accuracy using only 3 (ECG-ID, PTB) and 6 (MIT-BIH, NSRDB)
beats.

In Table 4, we have presented a comparative analysis of closed environment
identification performance of EDITH using the 60-20-20 split on the four
different datasets, along with the state-of-the-art results. Here, we only
list the recent, top-performing deep learning-based approaches and it can be
observed that EDITH outperforms them either by scoring higher or by achieving
the same performance with less number of beats.

Table 4: Comparison with existing methods (Closed Environment Identification) ECG-ID | MIT-BIH
---|---
Method |  | Number
---
of beats
| Evaluation
---
Scheme
Accuracy (%) | Method |  | Number
---
of beats
| Evaluation
---
Scheme
Accuracy (%)
[30] | 9 | Validation score | 100 | [30] | 9 | Validation score | 100
[29] | 8 | Validation score | 100 | [29] | 6 | Validation score | 100
[23] | 2 | 10 Fold CV | 98.24 | [23] | 2 | 10 Fold CV | 95.99
[31] | 9 | Test data | 98.60 | [31] | 9 | Test data | 98.40
EDITH | 3 | Test data | 100 | EDITH | 6 | Test data | 100
PTB | NSRDB
Method |  | Number
---
of beats
| Evaluation
---
Scheme
Accuracy (%) | Method |  | Number
---
of beats
| Evaluation
---
Scheme
Accuracy (%)
[23] | 2 | 10 Fold CV | 100 | [23] | 2 | 10 Fold CV | 97.17
[9] | 8 | Test data | 100 | [28] | 2-3 | Test data | 95.1
EDITH | 3 | Test data | 100 | EDITH | 6 | Test data | 100

####  6.2.3 Cross-Session Evaluation

For biometric authentication protocols, it is more meaningful to evaluate them
using multi-session data, i.e., training the models with the data from a
particular session and test them with data from a different session. Very few
ECG databases provide us with multi-session data, ECG-ID and CYBHi are two of
them. Therefore, we analyze the cross-session accuracy using these two
datasets. We follow the similar evaluation criteria as followed by other
works. We train the model using session 1 data and use the data from session 2
for evaluation, and vice versa. For this experiment, we just consider a single
beat, in order to compare with other methods.

The result for cross-session analysis is presented in Table 5. As it is
intuitive and also reported by [62], identification accuracy sharply falls
when tested on data from a different session, due to several variabilities.
Despite that our model handles this quite well, as the accuracy on the ECG-ID
dataset is still above 90%. However BYCHi dataset being a more complex one,
our performance falls below 80%, still, it is over 10% better than the
previous best-performing work [62]. For the ECG-ID dataset, although [63]
performs better in such scenarios, their performance is not balanced, e.g. for
some individuals, their model fails whereas our individual level accuracies
are always over 70%.

Table 5: Cross-Session Evaluation Dataset | Method |  | Session 1
---
Accuracy (%)
| Session 2
---
Accuracy (%)
ECG-ID | [29] | 83.33 | -
[63] | 88.47-96.78 | 77.12-97.78
EDITH | 91.67 | 92.671
CYBHi | [62] | 61 | 60.3
EDITH | 71.597 | 73.929

###  6.3 Identity Verification Experiments

####  6.3.1 Siamese Architecture vs Cosine Similarity

As mentioned in the earlier sections, following the standard protocols to
benchmark the identity verification task, we first train a model using a
subset of individuals from the ECG-ID dataset and use that model to predict
embeddings for the test signals. The test signals are further divided into two
parts, enrollment and evaluation. It is common practice to keep 50% data for
enrollment and use the rest for evaluation [23]. However, we experimented with
different sizes of enrollment data and different numbers of beats. The EER
values obtained for the different configurations are presented in Fig. 6. In
addition, we include the results using cosine similarity and the resultant
improvements achieved by using the Siamese network.

Figure 6:  Equal Error Rate (EER) obtained in different configurations for
identity verification task. In addition to presenting the results for the
proposed Siamese network, corresponding results using cosine similarity and
the improvement in using Siamese network are also presented.

From Fig. 6, we can observe that increasing the size of enrollment data at
first gradually improves the performance, but then it starts to degrade. This
can be explained as follows: if an enrollment set is limited, diversity in
patterns may be insufficient (underfitting). If it is too large, this will
enforce a bias on the matching task (overfitting). In our experiments, we have
found that 40% of the entire data used as the enrollment data yields the best
performance. This is a reasonable setting, considering the prior works [30,
23], where 50% data is used. On the other hand, increasing the number of beats
almost consistently improves the performance. Again, it is evident that using
the proposed Siamese network invariably yields better performance compared to
the standard cosine similarity metric.

In Table 6, the results obtained by the comparative evaluation against the
prior approaches are presented, where EDITH achieves the top performance,
registering a better EER value than others.

Table 6: Comparison with existing methods for Identity Verification using the ECG-ID dataset. Method | Zaghouani et al.[64] | Chun et al.[65] | Chu et al.[23] | Salloum et al.[30] | EDITH
---|---|---|---|---|---
EER | 15 % | 5.2 % | 2 % | 1.7 % |  1.29 %

####  6.3.2 Case Study (Failure Scenarios)

We further analyze some cases where EDITH failed in verifying the identities.
Fig. 7 presents some relevant examples. For the true acceptance and true
rejection cases, the reasons for the correct decision are clearly visible
(negligible S region for true accept case vs. prominent S region for true
reject case) through visual comparison of the input with the template.
However, for some cases, signals from other individuals closely mimicked the
pattern of the templates (false accept). On the contrary, in some situations,
the ECG signal from the same person may deviate significantly, and hence, it
hardly matches the template (salient S region in false reject). The level of
variability and inconsistency in the ECG signals are responsible for such
erroneous decisions, this is the primary motivation for analyzing multiple
beats instead of a single one.

Figure 7: Example cases of identity verification. Here, we can observe that
signals from the same (blue) /different (red) individuals, match/mismatches
with the templates. However, due to the variability in ECG signal, sometimes
signal from a different person may match with the templates (false accept),
similarly signal from the same person may deviate too much from the templates
(false reject).

####  6.3.3 Result on Other Datasets

We also evaluated our identity verification performance on the other datasets.
Since very few works were evaluated on datasets other than ECG-ID, we could
not present any meaningful comparison. We present the EER and ROC curves in
Fig. 8 and Fig. 9 respectively. The results here presented are for single beat
only with 50% enrollment rate. The scores on other datasets are less than ECG-
ID dataset because the model was adapted for ECG-ID dataset and just fine-
tuned on the other datasets.

(a) ECG-ID (EER = 4.27%)

(b) MIT-BIH (EER = 6.36%)

(c) PTB (EER = 5.66%)

(d) NSRDB (EER = 5.17%)

Figure 8: EER Curves of Identity Verification on Different Datasets.

(a) ECG-ID
(AUC = 94.41%)

(b) MIT-BIH
(AUC = 87.76%)

(c) PTB
(AUC = 84.88%)

(d) NSRDB
(AUC = 86.29%)

Figure 9: ROC Curves of Identity Verification on Different Datasets.

###  6.4 Investigation on model interpretability

The improved performance of machine learning often comes at the cost of
interpretability. However, for security and biometrics applications,
interpretation of algorithms is imperative. Not only this will enable us to
assert that the model is learning actual significant information, but also it
will act as a defense mechanism against spoofing or counterfeiting attacks.
Recently, considerable focus has been put on interpreting the deep learning
models, instead of considering them merely as black-boxes [66].

With the above backdrop, we investigate the interpretability of our proposed
method. First of all, to analyze the discriminative features extracted by the
model for a random signal from the ECG-ID database (Fig. 10(a)), we visualize
the feature maps computed by the first convolutional layer (Fig. 10(b)).
Although the earlier feature maps alone do not sufficiently signify, still we
can obtain some rough ideas about the features. For example, it can be
observed that some filter seeks for the R (19), S (7), T (24) regions, while
some filter just merely denoise (12) or invert (26) the signal.

(a)

(b)

Figure 10: Demonstration of the features learned by the filters. For a random
signal from the ECG-ID database (10(a)), we visualize the output featuremaps
of the filters from the first convolutional layer (10(b)). Some filters have
identified the R (19), S (7), T (24) regions, whereas some just denoise (12)
or invert (26) the signal.

Saliency maps [67] provide further insights into the interpretability of the
models, which highlight the parts of the signal that influence the most in
making the predictions. Saliency maps can be generated by computing the
gradient of the loss function with respect to the individual timestamps of the
input signal [68]. For example, for an $n$ sample long input signal,
$X=[x_{1},x_{2},x_{3},\dots,x_{n}]$, we can compute the saliency map,
$S=[s_{1},s_{2},s_{3},\dots,s_{n}]$ as $s_{i}=|\frac{\delta L}{\delta
x_{i}}|,\forall i\epsilon[1\dots n]$, where $L$ is the value of the loss
function.

Fig. 11 presents the saliency maps of 3 different signals from 3 different
persons from the ECG-ID database, where the regions with high saliency scores
are colored in red. For Person 1, we have a clear S region and vague Q region,
but for Person 12 we observe the opposite. Thus, in order to discriminate
them, one would look for the Q and S regions, which are the ones sought by the
model as demonstrated in Fig. 11. Since signals from both the person have a
conspicuous R peak, the model emphasizes that as well. However, for Person 13,
the R peak is less dominant, thus the model has put little emphasis there.
This analysis indicates the model’s capability of identifying the intelligible
fiducial landmarks and making the prediction accordingly.

Figure 11: Saliency maps for randomly selected 3 different persons from the
ECG-ID database. It is interesting to observe that the model has learned the
fiducial landmarks like Q,R,S sub-waves, without any explicit supervision.

###  6.5 Evaluation with Real-World Data

In order to assess the practical utility of the proposed method, we also
perform a limited trial. We collect ECG signals from 9 persons using HealthyPi
v4111https://healthypi.protocentral.com/. This kit contains an ECG front end
of Texas Instruments (TI) ADS1292R with 24-bit resolution, having a signal-to-
noise ratio (SNR) of 107 dB. The ECG signals are recorded at 125 Hz, which are
upsampled to 500 Hz using bicubic interpolation for our analysis.

We take the R-peak detector model and the closed environment identification
model trained on the ECG-ID database and investigate their applicability with
this completely different source of data. The model successfully detects the
R-peaks, leading to a proper segmentation of the beats. In Fig. 12(a) we have
presented an overlapping plot of the ECG beats for different persons and it
can be observed that all the beats have been properly segmented except one.

(a)

(b)

Figure 12: Evaluation with real-world data. (12(a)) illustration of reliable
R-peak detection and (12(b)) the embeddings for different individuals obtained
using the closed environment identification model.

Similarly, we pass the detected beats to the identification model and compute
the embeddings. We present a 2-dimensional t-distributed stochastic neighbor
embedding (TSNE) [69] plot of the embeddings in Fig. 12(b). It can be observed
that the embeddings from individual persons are clustered together, which
indicates that a basic fine-tuning or template matching would be able to
identify the persons.

Therefore, this limited study demonstrates the potential of EDITH in real-
world authentication applications. It is indeed promising that the model was
able to generalize to this extent, despite the data being captured using a
different device through a different protocol, and that too with a different
sampling rate and quantization.

To sum up, owing to the continual adversarial attacks and forgery attempts,
security researchers are in constant pursuit of more reliable and robust
biometric authentication systems. Leveraging physiological signals seems to be
the next pivotal step in this regard, due to the inherent defense mechanism
thereof against forgery. The proposed EDITH system is a step towards attaining
this objective. We have demonstrated superior performance of EDITH over the
existing methods in different tasks using various datasets. Furthermore, EDITH
circumvents certain limitations of the contemporary methods, for instance,
inconsistent R-peak detection and the need for longer ECG signals.

##  7 Conclusion

The domains of security and biometrics have always been an arms race. As
efforts were put in developing more advanced authentication systems, a similar
level of striving (or perhaps, even more) was in place to forge them. For
example, 3D facial biometrics was introduced to overcome the limitations of 2D
face scanning biometrics, as mere images were able to spoof them. But still,
those systems were still vulnerable to 3D face model based counterfeiting
attacks [70]. As a result, researchers from this domain have not only been
attempting to improve the existing systems but also are constantly looking for
more reliable alternatives. In this regard, physiological signal based
biometrics, bring in greater promises as they are comparatively difficult to
fake.

In this work, we have presented EDITH, a biometric authentication system based
on ECG signals and deep learning techniques. Similar to [29] EDITH was
developed to work with a single heartbeat, but its performance can be enhanced
by merging multiple beats. We propose a novel architecture leveraging
multiresolution analysis. As a result, EDITH outperforms existing methods by
achieving better accuracy with less number of beats. Furthermore, we present a
competitive alternative to the widely adopted Cosine similarity-based identity
verification. The proposed novel Siamese architecture, incorporating both
Euclidean and Cosine-like distances consistently surpasses Cosine similarity-
based template matching. Another remarkable achievement of EDITH lies in its
ability to learn the fiducial points of ECG signals without any explicit
supervision. Furthermore, a limited trial with real-world ECG data
corroborates the potential of EDITH as a practical authentication system.

Perhaps, physiological signal based biometrics are yet to be sufficient for
real-world deployment. Nevertheless, for the favorable characteristics they
bring in, such means of authentication is worth investigating further. The
future directions of this research can be manifold. It is indeed true that
owing to the variability and diversity using a single heartbeat is unlikely to
offer reliable authentication. Despite this, it deserves further investigation
to attain the holy grail of single beat authentication. Analysis of variation
in ECG signals with time along with different states of the individual is
another avenue that deserves more attention to make authentication more robust
and steadfast.

## Acknowledgement

This work was supported by Grant NPRP12S-0227-190164 from the Qatar National
Research Fund, a member of Qatar Foundation, Doha, Qatar and the claims made
herein are solely the responsibility of the authors. Open Access funding
provided by the Qatar National Library.

## Competing interests

The authors declare no competing interests.

## Author Contributions

Nabil Ibtehaz: Conceptualization, Methodology, Software, Validation, Formal
analysis, Writing - Original Draft, Writing - Review & Editing. Muhammad E. H.
Chowdhury: Conceptualization, Writing - Original Draft, Writing - Review &
Editing, Supervision, Project administration. Amith Khandakar: Data Curation,
Methodology, Visualization, Resources, Writing - Review & Editing. Serkan
Kiranyaz: Conceptualization, Supervision, Writing - Review & Editing. M. Sohel
Rahman: Conceptualization, Supervision, Writing - Review & Editing. Anas
Tahir: Data Curation, Investigation, Resources, Writing - Review & Editing.
Yazan Qiblawey: Data Curation, Investigation, Resources, Writing - Review &
Editing. Tawsifur Rahman: Methodology, Formal analysis, Writing - Review &
Editing. All the authors reviewed, and revised the draft and approved the
final version of the article before submission.

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##  Appendix A Definitions and Properties of Smooth Entropies

In this appendix, we summarize the properties of the smooth conditional
entropies that are used in the main text. For the properties of the purified
distance used in some of the proofs, see Appendix B.

###  A.1 Basic Properties

######  Lemma 12

(duality: see e.g. [29]) For any subnormalized pure state $|\psi\rangle$ on
system $ABC$, and for any $\epsilon>0$, $H_{\rm
max}^{\epsilon}(A|B)_{\psi}=-H_{\rm min}^{\epsilon}(A|C)_{\psi}$.

######  Lemma 13

(monotonicity: Theorem 18 in [29] and Theorem 6.2 in [26]) For any
$\rho^{AB}\in{\mathcal{S}}_{\leq}({\mathcal{H}}^{AB})$,
$0\leq\epsilon\leq\sqrt{{\rm Tr}[\rho]}$, any unital CPTP map
${\mathcal{E}}:A\rightarrow C$ and any CPTP map ${\mathcal{F}}:B\rightarrow
D$, it holds that $H_{\rm min}^{\epsilon}(A|B)_{\rho}\leq H_{\rm
min}^{\epsilon}(C|D)_{{\mathcal{E}}\otimes{\mathcal{F}}(\rho)}$.

######  Lemma 14 (isometric invariance: Lemma 13 in [29])

For any $\epsilon\geq 0$,
$\rho^{AB}\in{\mathcal{S}}_{\leq}({\mathcal{H}}^{AB})$ and any linear
isometries $U:A\rightarrow C$ and $V:B\rightarrow D$, $H_{\rm
min}^{\epsilon}(A|B)_{\rho}=H_{\rm
min}^{\epsilon}(C|D)_{{\mathcal{U}}\otimes{\mathcal{V}}(\rho)}$.

######  Lemma 15 (additivity: see Section I C in [19])

For any $\rho\in{\mathcal{S}}({\mathcal{H}}^{AB})$ and
$\sigma\in{\mathcal{S}}({\mathcal{H}}^{CD})$, it holds that

$\displaystyle\\!\\!H_{\rm max}(AC|BD)_{\rho\otimes\sigma}=H_{\rm
max}(A|B)_{\rho}+H_{\rm max}(C|D)_{\sigma}.\\!\\!$ (357)

######  Lemma 16

(superadditivity: Lemma A.2 in [15]) For any states $\rho^{AB}$, $\sigma^{CD}$
and any $\epsilon,\epsilon^{\prime}\geq 0$, it holds that

$\displaystyle\\!\\!H_{\rm
min}^{\epsilon+\epsilon^{\prime}}(AC|BD)_{\rho\otimes\sigma}\geq H_{\rm
min}^{\epsilon}(A|B)_{\rho}\\!+\\!H_{\rm
min}^{\epsilon^{\prime}}(C|D)_{\sigma}.\\!\\!\\!$ (358)

######  Lemma 17 (chain rule: see [31])

For any $\epsilon>0$, $\epsilon^{\prime},\epsilon^{\prime\prime}\geq 0$ and
$\rho\in{\mathcal{S}}_{\leq}({\mathcal{H}}^{ABC})$, it holds that

$\displaystyle H_{\rm
min}^{\epsilon+\epsilon^{\prime}+2\epsilon^{\prime\prime}}(AB|C)_{\rho}$
$\displaystyle\geq H_{\rm min}^{\epsilon^{\prime}}(B|C)_{\rho}+H_{\rm
min}^{\epsilon^{\prime\prime}}(A|BC)_{\rho}-f(\epsilon),$ (359) $\displaystyle
H_{\rm min}^{\epsilon^{\prime}}(AB|C)_{\rho}$ $\displaystyle\leq H_{\rm
max}^{\epsilon^{\prime\prime}}(B|C)_{\rho}+H_{\rm
min}^{\epsilon+\epsilon^{\prime}+2\epsilon^{\prime\prime}}(A|BC)_{\rho}+2f(\epsilon),$
(360)

where

$\displaystyle f(\epsilon):=-\log{(1-\sqrt{1-\delta^{2}})}.$ (361)

######  Lemma 18

(dimension bounds: Corollary of Lemma 20 in [29]) For any state $\rho^{AB}$
and $\epsilon\geq 0$, it holds that

$\displaystyle H_{\rm min}^{\epsilon}(A|B)_{\rho}$
$\displaystyle\geq-\log{d_{A}},$ (362) $\displaystyle H_{\rm
max}^{\epsilon}(A|B)_{\rho}$ $\displaystyle\leq\log{d_{A}}.$ (363)

######  Lemma 19 (dimension bound: Lemma 21 in [9])

For any state $\rho^{ABC}$ and $\epsilon>0$, it holds that

$\displaystyle H_{\rm min}^{\epsilon}(AB|C)_{\rho}\leq H_{\rm
min}^{\epsilon}(A|C)_{\rho}+\log{d_{B}}.$ (364)

######  Lemma 20

(continuity) For any $\epsilon,\delta\geq 0$, any $\rho^{AB}$ and
$\sigma^{AB}\in{\mathcal{B}}^{\delta}(\rho)$, it holds that

$\displaystyle H_{\rm min}^{\epsilon+\delta}(A|B)_{\rho}\geq H_{\rm
min}^{\epsilon}(A|B)_{\sigma}.$ (365)

######  Proof:

Let $\hat{\sigma}^{AB}\in{\mathcal{B}}^{\epsilon}(\sigma)$ be such that
$H_{\rm min}^{\epsilon}(A|B)_{\sigma}=H_{\rm min}(A|B)_{\hat{\sigma}}$. Due to
the triangle inequality for the purified distance, it holds that

$\displaystyle P(\rho,\hat{\sigma})\leq
P(\rho,\sigma)+P(\sigma,\hat{\sigma})\leq\epsilon+\delta,$ (366)

which implies $\hat{\sigma}\in{\mathcal{B}}^{\epsilon+\delta}(\rho)$. Thus, we
obtain Inequality (365) as

$\displaystyle H_{\rm min}^{\epsilon}(A|B)_{\sigma}=H_{\rm
min}(A|B)_{\hat{\sigma}}\leq\sup_{\hat{\rho}\in{\mathcal{B}}^{\epsilon+\delta}(\rho)}H_{\rm
min}(A|B)_{\hat{\rho}}=H_{\rm min}^{\epsilon+\delta}(A|B)_{\rho}.$ (367)

$\blacksquare$

######  Lemma 21

(one-dimensional system.) Suppose that $d_{A}=1$. For any $\epsilon\geq 0$ and
$\rho\in{\mathcal{S}}({\mathcal{H}}^{AB})$, it holds that

$\displaystyle 0\leq H_{\rm
min}^{\epsilon}(A|B)_{\rho}\leq-\log{(1-2\epsilon)},$ (368) $\displaystyle
0\geq H_{\rm max}^{\epsilon}(A|B)_{\rho}\geq\log{(1-2\epsilon)}.$ (369)

######  Proof:

Since $d_{A}=1$, there exists a fixed vector
$\mbox{$|e\rangle$}\in{\mathcal{H}}^{A}$ such that
$I^{A}=\mbox{$\mbox{$|e\rangle$}\\!\mbox{$\langle e|$}$}$ and that any
$\tilde{\rho}\in{\mathcal{S}}_{\leq}({\mathcal{H}}^{AB})$ is represented as
$\mbox{$\mbox{$|e\rangle$}\\!\mbox{$\langle
e|$}$}^{A}\otimes\tilde{\rho}^{B}$. Due to the definition of the smooth
conditional min entropy, we have

$\displaystyle H_{\rm min}^{\epsilon}(A|B)_{\rho}$ $\displaystyle\geq H_{\rm
min}(A|B)_{\rho}$ (370)
$\displaystyle=\sup_{\sigma^{B}\in{\mathcal{S}}_{=}({\mathcal{H}}^{B})}H_{\rm
min}(A|B)_{\rho|\sigma}$ (371) $\displaystyle\geq H_{\rm
min}(A|B)_{\rho^{AB}|\rho^{B}}$ (372)
$\displaystyle=\sup\\{\lambda\in\mathbb{R}|2^{-\lambda}I^{A}\otimes\rho^{B}\geq\rho^{AB}\\}$
(373)
$\displaystyle=\sup\\{\lambda\in\mathbb{R}|2^{-\lambda}\mbox{$\mbox{$|e\rangle$}\\!\mbox{$\langle
e|$}$}^{A}\otimes\rho^{B}\geq\mbox{$\mbox{$|e\rangle$}\\!\mbox{$\langle
e|$}$}^{A}\otimes\rho^{B}\\}$ (374) $\displaystyle=0.$ (375)

This implies the first inequality in (368). To prove the second inequality in
(368), let $\hat{\rho}\in{\mathcal{B}}^{\epsilon}(\rho)$ and
$\sigma^{B}\in{\mathcal{S}}_{=}({\mathcal{H}}^{B})$ be such that

$\displaystyle H_{\rm min}^{\epsilon}(A|B)_{\rho}=H_{\rm
min}(A|B)_{\hat{\rho}}=H_{\rm min}(A|B)_{\hat{\rho}|\sigma}.$ (376)

By definition, it holds that

$\displaystyle 2^{-H_{\rm
min}^{\epsilon}(A|B)_{\rho}}I^{A}\otimes\sigma^{B}\geq\hat{\rho}^{AB},$ (377)

which is equivalent to

$\displaystyle 2^{-H_{\rm
min}^{\epsilon}(A|B)_{\rho}}\mbox{$\mbox{$|e\rangle$}\\!\mbox{$\langle
e|$}$}^{A}\otimes\sigma^{B}\geq\mbox{$\mbox{$|e\rangle$}\\!\mbox{$\langle
e|$}$}^{A}\otimes\hat{\rho}^{B}.$ (378)

By taking the trace in both sides, we obtain

$\displaystyle 2^{-H_{\rm min}^{\epsilon}(A|B)_{\rho}}\geq{\rm
Tr}[\hat{\rho}].$ (379)

The R.H.S. of the above inequality is evaluated as

$\displaystyle{\rm
Tr}[\hat{\rho}]=\|\hat{\rho}\|_{1}\geq\|\rho\|_{1}-\|\rho-\hat{\rho}\|_{1}\geq
1-2\epsilon,$ (380)

where the last line follows from (6) and the condition
$\hat{\rho}\in{\mathcal{B}}^{\epsilon}(\rho)$. This implies the second
inequality in (368). Inequality (369) follows due to the duality relation
(Lemma 12). $\blacksquare$

###  A.2 Classical-Quantum States

######  Lemma 22 (Lemma A.5 in [15])

For any state $\rho^{ABK}\in{\mathcal{S}}_{=}({\mathcal{H}}^{ABK})$ in the
form of

$\displaystyle\rho^{ABK}=\sum_{k}p_{k}\rho_{k}^{AB}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K},$ (381)

where $\rho_{k}\in{\mathcal{S}}_{=}({\mathcal{H}}^{AB})$, $\mbox{$\left\langle
k|k^{\prime}\right\rangle$}=\delta_{k,k^{\prime}}$ and $\\{p_{k}\\}_{k}$ is a
normalized probability distribution, it holds that

$\displaystyle H_{\rm min}(A|BK)_{\rho}=-\log\left(\sum_{k}p_{k}\cdot
2^{-H_{\rm min}(A|B)_{\rho_{k}}}\right).$ (382)

######  Lemma 23 (Lemma A.7 in [15])

For any state
$\rho^{ABK_{1}K_{2}}\in{\mathcal{S}}_{\leq}({\mathcal{H}}^{ABK_{1}K_{2}})$ in
the form of

$\displaystyle\rho^{ABK_{1}K_{2}}=\sum_{k}p_{k}\rho_{k}^{AB}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K_{1}}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K_{2}},$ (383)

where $\mbox{$\left\langle k|k^{\prime}\right\rangle$}=\delta_{k,k^{\prime}}$,
and for any $\epsilon\geq 0$, it holds that

$\displaystyle H_{\rm min}^{\epsilon}(AK_{1}|BK_{2})_{\rho}=H_{\rm
min}^{\epsilon}(A|BK_{2})_{\rho}=H_{\rm min}^{\epsilon}(A|BK_{1})_{\rho}.$
(384)

######  Lemma 24 (Lemma 29 in [33])

In the same setting as in Lemma 23, it holds that

$\displaystyle H_{\rm max}^{\epsilon}(AK_{1}|BK_{2})_{\rho}=H_{\rm
max}^{\epsilon}(A|BK_{2})_{\rho}=H_{\rm max}^{\epsilon}(A|BK_{1})_{\rho}.$
(385)

######  Lemma 25

Consider a state in the form of

$\displaystyle\rho^{ACK}=\sum_{k}p_{k}\rho_{k}^{A}\otimes\sigma_{k}^{C}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K}.$ (386)

For any $\epsilon>0$, it holds that

$\displaystyle H_{\rm min}^{\epsilon}(A|CK)_{\rho}=H_{\rm
min}^{\epsilon}(A|K)_{\rho}\geq 0.$ (387)

######  Proof:

It is straightforward to verify that there exists a quantum operation
${\mathcal{E}}:K\rightarrow CK$ such that
$\rho^{ACK}={\mathcal{E}}(\rho^{AK})$. Due to the monotonicity of the smooth
conditional min entropy under operations on the conditioning system, we have

$\displaystyle H_{\rm min}^{\epsilon}(A|K)_{\rho^{AK}}\leq H_{\rm
min}^{\epsilon}(A|CK)_{\rho^{ACK}}=H_{\rm
min}^{\epsilon}(A|CK)_{{\mathcal{E}}(\rho^{AK})}\leq H_{\rm
min}^{\epsilon}(A|K)_{\rho^{AK}},$ (388)

which implies $H_{\rm min}^{\epsilon}(A|CK)_{\rho}=H_{\rm
min}^{\epsilon}(A|K)_{\rho}$. The non-negativity follows due to Lemma 22 as

$\displaystyle H_{\rm min}^{\epsilon}(A|K)_{\rho}\geq H_{\rm
min}(A|K)_{\rho}=-\log\left(\sum_{k}p_{k}\cdot 2^{-H_{\rm
min}(A)_{\rho_{k}}}\right)\geq-\log\left(\sum_{k}p_{k}\right)=0,$ (389)

which completes the proof. $\blacksquare$

###  A.3 Classically-labelled Pure States

######  Lemma 26

Consider a state in the form of

$\displaystyle\rho^{ABCK}=\sum_{k}p_{k}\mbox{$\mbox{$|\psi_{k}\rangle$}\\!\mbox{$\langle\psi_{k}|$}$}^{ABC}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K}.$ (390)

For any $\epsilon>0$, it holds that

$\displaystyle H_{\rm max}^{\epsilon}(A|BK)_{\rho}=-H_{\rm
min}^{\epsilon}(A|CK)_{\rho}.$ (391)

######  Proof:

It is straightforward to verify that a purification of the state $\rho$,
defined by (390), is given by

$\displaystyle\mbox{$|\psi_{\rho}\rangle$}^{ABCKK^{\prime}}=\sum_{k}\sqrt{p_{k}}\mbox{$|\psi_{k}\rangle$}^{ABC}\mbox{$|k\rangle$}^{K}\mbox{$|k\rangle$}^{K^{\prime}}.$
(392)

Due to the duality of the smooth conditional entropies (Lemma 12), we have

$\displaystyle H_{\rm max}^{\epsilon}(A|BK)_{\rho}=H_{\rm
max}^{\epsilon}(A|BK)_{\psi_{\rho}}=-H_{\rm
min}^{\epsilon}(A|CK^{\prime})_{\psi_{\rho}}=-H_{\rm
min}^{\epsilon}(A|CK)_{\rho},$ (393)

which completes the proof. $\blacksquare$

######  Lemma 27

Consider the same setting as in Lemma 26. For any $\epsilon>0$, it holds that

$\displaystyle H_{\rm min}^{\epsilon}(A|K)_{\rho}=H_{\rm
min}^{\epsilon}(B|K)_{\rho}.$ (394)

######  Proof:

To prove (394), let $\hat{\rho}^{AK}\in{\mathcal{B}}^{\epsilon}(\rho)$ and
$\varsigma\in{\mathcal{S}}_{=}({\mathcal{H}}^{K})$ be such that

$\displaystyle H_{\rm min}^{\epsilon}(A|K)_{\rho}=H_{\rm
min}(A|K)_{\hat{\rho}}=H_{\rm min}(A|K)_{\hat{\rho}|\varsigma}.$ (395)

With ${\mathcal{C}}$ being the completely dephasing operation on $K$ with
respect to the basis $\\{\mbox{$|k\rangle$}\\}_{k}$, it holds that

$\displaystyle P({\mathcal{C}}(\hat{\rho}),{\mathcal{C}}(\rho))\leq
P(\hat{\rho}),\rho)\leq\epsilon.$ (396)

In addition, if

$\displaystyle 2^{-\lambda}I^{A}\otimes\varsigma^{K}\geq\hat{\rho}^{AK},$
(397)

then

$\displaystyle 2^{-\lambda}I^{A}\otimes{\mathcal{C}}(\varsigma)^{K}\geq{\rm
id}^{A}\otimes{\mathcal{C}}^{K}(\hat{\rho}^{AK}).$ (398)

Thus, without loss of generality, we may assume that both $\hat{\rho}^{AK}$
and $\varsigma$ are diagonal in $\\{\mbox{$|k\rangle$}\\}_{k}$. That is, we
may assume that $\hat{\rho}^{AK}$ and $\varsigma$ are in the form of

$\displaystyle\hat{\rho}^{AK}=\sum_{k}\hat{p}_{k}\hat{\rho}_{k}^{A}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K},\quad\varsigma=\sum_{k}q_{k}\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}.$ (399)

Suppose that the Schmidt decomposition of $|\psi_{k}\rangle$ is given by

$\displaystyle\mbox{$|\psi_{k}\rangle$}=\sum_{j}\sqrt{\mu_{j|k}}\mbox{$|e_{j|k}\rangle$}^{A}\mbox{$|f_{j|k}\rangle$}^{B},$
(400)

Define linear operators $v_{k}:{\mathcal{H}}^{A}\rightarrow{\mathcal{H}}^{B}$
and
$V:{\mathcal{H}}^{A}\otimes{\mathcal{H}}^{K}\rightarrow{\mathcal{H}}^{B}\otimes{\mathcal{H}}^{K}$
by

$\displaystyle v_{k}:=\sum_{j}\mbox{$|f_{j|k}\rangle$}^{B}\mbox{$\langle
e_{j|k}|$}^{A}\quad(\forall k)$ (401)

and $V:=\sum_{k}v_{k}\otimes\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K}$. It is straightforward to verify that
$\rho^{BK}=V\rho^{AK}V^{\dagger}$. Thus, due to the monotonicity of the
purified distance under trace non-increasing CP maps (Lemma 7 in [29]), it
holds that

$\displaystyle P(\rho^{BK},V\hat{\rho}^{AK}V^{\dagger})\leq
P(\rho^{AK},\hat{\rho}^{AK})\leq\epsilon.$ (402)

Applying $V$ to the both sides in condition (397), it follows that

$\displaystyle 2^{-\lambda}V(I^{A}\otimes\varsigma^{K})V^{\dagger}\geq
V\hat{\rho}^{AK}V^{\dagger}.$ (403)

Noting that $I^{B}\geq(v_{k}^{\dagger}v_{k})^{B}$, this implies that

$\displaystyle 2^{-\lambda}I^{B}\otimes\varsigma^{K}\geq
V\hat{\rho}^{AK}V^{\dagger}.$ (404)

Thus, we arrive at

$\displaystyle H_{\rm min}^{\epsilon}(A|K)_{\rho}\leq H_{\rm
min}^{\epsilon}(B|K)_{\rho}.$ (405)

By exchanging the roles of $A$ and $B$, we also obtain the converse
inequality. This completes the proof of Equality (394).

##  Appendix B Properties of The Purified Distance

We summarize the properties of the purified distance, used in Appendix A to
prove the properties of the smooth conditional entropies.

######  Lemma 28

(monotonicity: Lemma 7 in [29]) For any subnormalized states
$\rho,\sigma\in{\mathcal{S}}_{\leq}({\mathcal{H}})$ and for any completely
positive trace non-increasing map ${\mathcal{E}}$, it holds that
$P(\rho,\sigma)\geq P({\mathcal{E}}(\rho),{\mathcal{E}}(\sigma))$.
Consequently, for any linear isometry ${\mathcal{U}}$, it holds that
$P(\rho,\sigma)=P({\mathcal{U}}(\rho),{\mathcal{U}}(\sigma))$

######  Lemma 29

For any normalized state $\rho$ on system $A$ and any normalized pure state
$|\phi\rangle$ on system $AB$, the purified distance satisfies

$\displaystyle
P(\rho^{A},\phi^{A})=\min_{|\psi\rangle^{AB}}P(\mbox{$\mbox{$|\psi\rangle$}\\!\mbox{$\langle\psi|$}$},\mbox{$\mbox{$|\phi\rangle$}\\!\mbox{$\langle\phi|$}$})=\sqrt{1-\max_{|\psi\rangle^{AB}}|\langle\psi|\phi\rangle|^{2}},$
(406)

where the minimum and the maximum are taken over all purifications
$|\psi\rangle$ of $\rho$.

######  Proof:

Follows from Definition 4 and Lemma 8 in [29]. $\blacksquare$

######  Lemma 30

Consider a state $\Gamma$ on $KAB$ and a pure state $|\Psi\rangle$ on $KABC$
in the form of

$\displaystyle\mbox{$|\Psi\rangle$}=\sum_{k}\sqrt{p_{k}}\mbox{$|k\rangle$}^{K}\mbox{$|\psi_{k}\rangle$}^{ABCD},\quad\quad\Gamma=\sum_{k}p_{k}\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}^{K}\otimes\mbox{$\mbox{$|\gamma_{k}\rangle$}\\!\mbox{$\langle\gamma_{k}|$}$}^{AB}.$
(407)

There exists a set of pure states $\\{\mbox{$|\phi_{k}\rangle$}\\}_{k}$ on
$CD$ such that, for the state

$|\tilde{\Gamma}\rangle$
$\displaystyle=\sum_{k}\sqrt{p_{k}}\mbox{$|k\rangle$}^{K}\mbox{$|\gamma_{k}\rangle$}^{AB}\mbox{$|\phi_{k}\rangle$}^{CD},$
(408)

it holds that

$\displaystyle\\!\\!\\!P\left(\mbox{$\mbox{$|\tilde{\Gamma}\rangle$}\\!\mbox{$\langle\tilde{\Gamma}|$}$}\\!,\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$}\right)=P\left(\Gamma^{KAB},\>{\mathcal{C}}^{K}\\!\circ\\!{\rm
Tr}_{CD}(\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$})\right),\\!\\!$
(409)

where ${\mathcal{C}}$ is the completely dephasing operation on $K$ with
respect to the basis $\\{\mbox{$|k\rangle$}\\}_{k}$.

######  Proof:

It is straightforward to verify that a purification of the state
${\mathcal{C}}^{K}\\!\circ\\!{\rm
Tr}_{CD}(\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$})$ is given by

$\displaystyle\mbox{$|\Psi_{p}\rangle$}=\sum_{k}\sqrt{p_{k}}\mbox{$|k\rangle$}^{K}\mbox{$|\psi_{k}\rangle$}^{ABCD}\mbox{$|k\rangle$}^{K^{\prime}},$
(410)

and that any purification of the state $\Gamma^{KAB}$ to the system
$KABCDK^{\prime}$ is in the form of

$\displaystyle\mbox{$|\Gamma_{p}\rangle$}=\sum_{k}\sqrt{p_{k}}\mbox{$|k\rangle$}^{K}\mbox{$|\gamma_{k}\rangle$}^{AB}\mbox{$|\xi_{k}\rangle$}^{CDK^{\prime}},$
(411)

with $\\{\mbox{$|\xi_{k}\rangle$}\\}_{k}$ being a set of orthogonal states. A
simple calculation yields

$\displaystyle|\langle\Psi_{p}|\Gamma_{p}\rangle|=\sum_{k}p_{k}|(\langle\psi_{k}|^{ABCD}\langle
k|^{K^{\prime}})(|\gamma_{k}\rangle^{AB}|\xi_{k}\rangle^{CDK^{\prime}})|.$
(412)

The maximum of the above quantity over all orthogonal
$\\{\mbox{$|\xi_{k}\rangle$}\\}_{k}$ is achieved by
$\\{\mbox{$|\xi_{k}\rangle$}\\}_{k}$ that is decomposed into
$|\xi_{k}\rangle^{CDK^{\prime}}=|\phi_{k}\rangle^{CD}|k\rangle^{K^{\prime}}$.
Using this $\\{|\phi_{k}\rangle\\}_{k}$, we define a state
$|\tilde{\Gamma}\rangle$ by

$\displaystyle\mbox{$|\tilde{\Gamma}\rangle$}:=\sum_{k}\sqrt{p_{k}}\mbox{$|k\rangle$}^{K}\mbox{$|\gamma_{k}\rangle$}^{AB}\mbox{$|\phi_{k}\rangle$}^{CD}$
(413)

and a purification of $\Gamma^{KAB}$ by

$\displaystyle\mbox{$|\Gamma_{p}^{*}\rangle$}:=\sum_{k}\sqrt{p_{k}}\mbox{$|k\rangle$}^{K}\mbox{$|\gamma_{k}\rangle$}^{AB}\mbox{$|\phi_{k}\rangle$}^{CD}\mbox{$|k\rangle$}^{K^{\prime}}.$
(414)

It follows that

$\displaystyle\max_{\\{\xi_{k}\\}_{k}}|\langle\Psi_{p}|\Gamma_{p}\rangle|=|\langle\Psi_{p}|\Gamma_{p}^{*}\rangle|$
(415)
$\displaystyle=\sum_{k}p_{k}|\langle\psi_{k}|^{ABCD}|\gamma_{k}\rangle^{AB}|\phi_{k}\rangle^{CD}|$
(416) $\displaystyle=|\langle\Psi|\tilde{\Gamma}\rangle|.$ (417)

In addition, the states $|\Psi_{p}\rangle$ and $|\Gamma_{p}^{*}\rangle$ are
obtained by a linear isometry $P^{K\rightarrow
KK^{\prime}}:=\sum_{k}\mbox{$|k\rangle$}^{K}\mbox{$|k\rangle$}^{K^{\prime}}\mbox{$\langle
k|$}$ from $|\Psi\rangle$ and $|\tilde{\Gamma}\rangle$as

$\displaystyle\mbox{$|\Psi_{p}\rangle$}=P^{K\rightarrow
KK^{\prime}}\mbox{$|\Psi\rangle$},\quad\mbox{$|\Gamma_{p}^{*}\rangle$}=P^{K\rightarrow
KK^{\prime}}\mbox{$|\tilde{\Gamma}\rangle$}$ (418)

Thus, due to the property of the purified distance (Lemma 29 and Lemma 28), it
follows that

$\displaystyle
P\left(\mbox{$\mbox{$|\tilde{\Gamma}\rangle$}\\!\mbox{$\langle\tilde{\Gamma}|$}$},\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$}\right)=P\left(\mbox{$\mbox{$|\Gamma_{p}^{*}\rangle$}\\!\mbox{$\langle\Gamma_{p}^{*}|$}$},\mbox{$\mbox{$|\Psi_{p}\rangle$}\\!\mbox{$\langle\Psi_{p}|$}$},\right)=P\left(\Gamma^{KAB},\>{\mathcal{C}}^{K}\\!\circ\\!{\rm
Tr}_{CD}(\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$})\right),$
(419)

which completes the proof. $\blacksquare$

######  Lemma 31

Consider the same setting as in Lemma 30, and assume that $C$ and $D$ are
composite systems $C_{0}M_{C}$ and $D_{0}M_{D}$, respectively, where $M_{C}$
and $M_{D}$ are isomorphic quantum systems with an orthonormal basis
$\\{\mbox{$|m\rangle$}\\}_{m}$. In addition, suppose that the state $\Psi$ is
classically coherent in $M_{C}M_{D}$, i.e., that

$\displaystyle\|\langle m|^{M_{C}}\langle
m^{\prime}|^{M_{D}}\mbox{$|\Psi\rangle$}\|\propto\delta_{m,m^{\prime}}.$ (420)

Then, without loss of generality, we may assume that the states
$|\phi_{k}\rangle$ are classically coherent in $M_{C}M_{D}$.

######  Proof:

It is straightforward to verify that the state $\Psi$ is classically coherent
in $M_{C}M_{D}$ if and only if all $\psi_{k}$ are classically coherent in
$M_{C}M_{D}$. Consequently, the maximum of each term in (416) is achieved by
$\phi_{k}$ that is classically coherent in $M_{C}M_{D}$, which completes the
proof. $\blacksquare$

######  Lemma 32

(gentle measurement: Lemma 5 in [20] and Corollary of Lemma 7 in [5]) Let
$\epsilon\in(0,1]$, $\rho\in{\mathcal{S}}({\mathcal{H}})$ and
$\Lambda\in{\mathcal{L}}({\mathcal{H}})$ be such that $0\leq\Lambda\leq I$ and
${\rm Tr}[\Lambda\rho]\geq 1-\epsilon$. It holds that

$\displaystyle\|\rho-\sqrt{\Lambda}\rho\sqrt{\Lambda}\|_{1}\leq
2\sqrt{\epsilon},\quad
P(\rho,\sqrt{\Lambda}\rho\sqrt{\Lambda})\leq\sqrt{2\epsilon}.$ (421)

Figure 9:  The construction of encoding and decoding operations in the proof
of the direct part is depicted. (i) is obtained by cancelling out
$G_{\sigma}U$ and $(G_{\sigma}U)^{\dagger}$, corresponding to Inequality (171)
obtained from (153) and (154). (ii) follows from the fact that the state
$|\Psi_{\sigma}\rangle$ is obtained from $|\Psi\rangle$ by applying $P$ and
$G_{\sigma}$, due to (158). In (iii), we trace out $Z\equiv Z_{L}Z_{R}$ and
apply the completely dephasing operation ${\mathcal{C}}$ to
$X^{\prime}Y^{\prime}Z^{\prime}$. See Inequalities (173) and (176) that are
obtained from (171). Note that the source state $\Psi_{s}$ is obtained from
$|\Psi\rangle$ and $|\Psi_{\sigma}\rangle$ as (160).