# Multi-objective Feature Selection with Missing Data in Classification

Yu Xue, Yihang Tang, Xin Xu, Jiayu Liang, Ferrante Neri Yu Xue, Yihang Tang
and Xin Xu are with the School of Computer and Software, Nanjing University of
Information Science and Technology, Nanjing 210044, China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>xuxin1975715@hotmail.com)Jiayu Liang;
Tianjin Key Laboratory of Autonomous Intelligent Technology and System,
Tiangong University, Tianjin 300387, China<EMAIL_ADDRESS>Ferrante Neri; School of Computer Science, University of Nottingham, UK
(email<EMAIL_ADDRESS>
###### Abstract

Feature selection (FS) is an important research topic in machine learning.
Usually, FS is modelled as a+ bi-objective optimization problem whose
objectives are: 1) classification accuracy; 2) number of features. One of the
main issues in real-world applications is missing data. Databases with missing
data are likely to be unreliable. Thus, FS performed on a data set missing
some data is also unreliable. In order to directly control this issue plaguing
the field, we propose in this study a novel modelling of FS: we include
reliability as the third objective of the problem. In order to address the
modified problem, we propose the application of the non-dominated sorting
genetic algorithm-III (NSGA-III). We selected six incomplete data sets from
the University of California Irvine (UCI) machine learning repository. We used
the mean imputation method to deal with the missing data. In the experiments,
k-nearest neighbors (K-NN) is used as the classifier to evaluate the feature
subsets. Experimental results show that the proposed three-objective model
coupled with NSGA-III efficiently addresses the FS problem for the six data
sets included in this study.

###### Index Terms:

Feature selection, Multi-objective, Optimization, NSGA-III, Missing data

##  I Introduction

Alarge number of data sets contains a lot of irrelevant or redundant features
(useless features). Useless features not only waste computing cost, but also
decrease the performance of classification [1]. Without prior information
about the data, useless features are often difficult to be identified. Useless
features are detrimental during classification tasks since they often lead to
a low accuracy and high computational cost. Feature selection (FS) is the
process of identification and elimination of these useless features.

In the past few decades, researchers have proposed many heuristic FS methods.
With respect to the logic used to assess the quality of the selected feature
(feature subset), they are categorised as 1) filter [2] and 2) wrapper [3]
methods. Filter methods use some specific functions to evaluate the usage of
the features. According to the different evaluation functions, the filter
methods can be divided into distance-based, consistency-based, dependency-
based and information-based. Some well-known filter methods belonging to this
category are Koller’s [4], Relief[5] and Set Cover[6]. Unlike filter methods,
wrapper methods [7] make use of a specific classifier as its evaluation
function, and use classification accuracy to evaluate the candidate selected
features. Wrapper methods are usually more accurate than filter methods, since
they directly take the classifier as the evaluation function for feature
subsets. On the other hand, the employment of the classifier is time-
consuming.

In order to be effectively applied, wrapper methods are usually coupled with
meta-heuristics that search the space by trying to perform the least function
call the possible. Meta-heuristics used to address FS problems can be divided
into two categories 1) those methods that encode FS as a single-objective
problem by identifying a specific feature or through the linear combination of
multiple objectives; 2) those methods that attempt to simultaneously address
multiple criteria of FS and make use of multi-objective optimization
algorithms to solve it.

Some examples of meta-heuristics that treat FS as a single-objective problem
are brain storm optimization [8], differential evolution [9], artificial bee
colony (ABC) [10] and particle swarm optimization (PSO) [11]. Some other
studies, while still modelling FS as a single-objective problem, embed
accuracy and solution size within the algorithmic logic. For example, Zhang
$et$ $al$. [12] presented a variable length PSO to make the particles have
different shorter lengths. Xue $et$ $al$. [13] devised a PSO algorithm with
adaptive parameters and strategies for FS with multiple classifiers. Besides,
Xue $et$ $al$. [14] also proposed a self-adaptive PSO for FS with large-scale
data sets, in order to strengthen the ability of PSO in solving FS problem. In
addition, to prevent the loss of excellent offspring, Zhang $et$ $al$. [15]
designed a new memory strategy and applied it to bare bones PSO so as to
balance the exploration ability of the algorithm.

In fact, FS can be seemed as a multi-objective optimization problem [16, 17].
For example, sometimes the classification accuracy is high but the number of
features is also large, so multiple objectives need to be considered at the
same time. For FS problems, the multi-objective FS methods can provide sets of
relatively optimal solutions instead of a single solution. Some popular multi-
objective algorithms [18, 19, 20, 21, 22] such as non-dominated sorting
genetic algorithm II (NSGA-II) [23], multi-objective evolutionary algorithm
with domain decomposition (MOEA/D) [24] and multi-objective PSO [25, 26] are
often used for the multi-objective FS problems.

Recently, the problem of FS has been addressed by coding it as a multi-
objective problem. For example, Xue $et$ $al$. [27] studied two ideas of
integrating non-dominated sorting strategy and crowding strategy into PSO
respectively. Besides, for the multi-objective FS problem, Nguyen $et$ $al$.
[28] proposed a hybrid/memetic PSO algorithm whose search potential has been
augmented by a local search. Zhang $et$ $al$. [29] observed that FS
intrinsically contains multiple conflicting objectives and thus proposed an
improved version of MOEA/D. Hancer $et$ $al$. [10] proposed a multi-objective
version of ABC for FS. To solve the problem of multi-objective FS efficiently,
Zhang $et$ $al$. [30] improved ABC with a parameterless search mechanism and
designed a new multi-objective FS algorithm. Moreover, Zhang $et$ $al$. [31]
proposed a multi-objective PSO based on the cost of features, which takes the
time cost and classification accuracy as two objectives. All the above studies
only considered two objectives. In most cases, the FS objectives considered
simultaneously are classification accuracy and solution size.

Many studies about FS considered complete data sets. However, in the real-
world applications, missing data is a common phenomenon due to various
unexpected reasons. For example, in the investigation and study, the data may
be missing due to negligence of the researcher, the cost of obtaining data is
too high, personal privacy involved in data collection, and so on. Missing
data have an impact on the formulation of FS problems as it may select
unreliable features. To solve the FS problems with missing data, we need to
firstly deal with the incomplete data sets. A popular approach to deal with
missing data is data imputation, i.e. an interpolation approach that
reconstructs missing data on the basis present in the data set. The common
imputation methods include mean imputation method [32], regression imputation
method [33], hot deck imputation method [34], and k-means clustering [35]. In
the present paper, to process incomplete data, we make use of the _mean
imputation approach_ : for each feature, we interpolate the missing values
using the average of the data available.

After the application of the mean imputation approach, this paper proposes the
modelling of the reliability of the data through a third objective of the
multi-objective optimization problem. More specifically, unlike the studies in
the literature, this paper not only considers the classification accuracy and
solution size, but also introduces the missing rate for FS in order to enhance
upon the reliability of FS. Thus, the problem is modelled as a three-objective
optimization problem. Since the proposed model causes an increase in the
complexity of the problem, we propose the use of non-dominated sorted genetic
algorithm III (NSGA-III) [36].

The remainder of the paper is organized as follows. Section II describes the
imputation method to pre-process the data. In Section III, we briefly outline
the NSGA-III algorithm in the context of FS. Section IV describes the
experimental design while Section V gives the experimental results. Section
VI, provides the conclusion of this study.

##  II Mean Imputation Method

Missing data is a frequent problem in machine learning. The FS methods should
be correspondingly changed if the data sets have missing data. When the
missing rate of the data set is less than 1%, the influence on experimental
results can be ignored. The missing rate of 1% $\sim$ 5% will slightly affect
the experimental results, but it can be controlled. However, if the missing
rate is greater than 5%, the results of the experiment would be affected.
Therefore, for obtaining the reliable results, we need to use the effective
values to estimate the missing values. Some methods have been proposed to deal
with the missing data. Armina $et$ $al$. [37] summarized some imputation
methods for missing values. For example, Krause $et$ $al$. [38] designed
amultiple imputation based on sophisticated imputation models. Amiri $et$
$al$. [39] introduced a fuzzy-rough methods to handle missing data. Donder
$et$ $al$. [40] introduced some imputation methods such as single imputation
and multiple imputation to get complete data sets. In this study we employ the
mean imputation method in single imputation to interpolate the missing data.
We chose this method since it is well-suited to handle large data sets thanks
to its low computational complexity and hence modest execution time, see [40].
The mean imputation method is divided into fixed distance imputation method
and non-distance imputation method [41]. This paper uses non-distance mean
imputation method which is described as follows.

Let $\left[v_{i,j}\right]$ be the incomplete data set which is here
interpreted as a matrix where some of the entries are empty,
$v_{i,j}=\emptyset$ for some $i$ and $j$. Those entries that are not empty are
normalised between $0$ and $1$, i.e. $v_{i,j}\in\left[0,1\right]$. The row
index $i$ indicate the instance whilst the column index $j$ indicates the
$j^{th}$ feature. The mean imputation method used in this study estimates the
missing entries alongside the column $j$ by replacing the empty entries with
$Ave_{j}$ calculated in the following way:

$Ave_{j}=\frac{\sum\limits_{i=1}^{N}{v_{i,j}}}{N-lm_{j}}$ (1)

where $lm_{j}$ is the number of missing entries associated with the feature
$j$, $N$ is the total number of all instances.

##  III Three-Objective Feature Selection Problems and NSGA-III Algorithm

The FS problem is encoded as a multi-objective optimisation problem where its
candidate solution is represented by a vector of real numbers. Let us consider
a data set with $n$ features

$\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$ (2)

where $x_{i}\in[0,1]$. It must be remarked that the candidate solution
$\mathbf{x}$ has the same structure of the row vector of the data set
$\mathbf{v_{i}}=\left(v_{i,1},v_{i,2},\ldots,v_{i,n}\right)$.

In order to evaluate the candidate solution $\mathbf{x}$, the objective
functions are calculated in the following way. At first, the binary vector

$\mathbf{z}=\left(z_{1},z_{2},\ldots,z_{n}\right)$ (3)

is generated by means of the equation

$z_{i}=\left\\{{\begin{array}[]{*{20}{c}}{1,\begin{array}[]{*{20}{c}}{}\hfil&{\begin{array}[]{*{20}{c}}{x_{i}\geq\theta}\end{array}}\end{array}}\\\
{0,\begin{array}[]{*{20}{c}}{}\hfil&{x_{i}<\theta}\end{array}}\end{array}}\right.$

(4)

where $\theta$ is a threshold value that determines whether or not a feature
is selected.

When the vector element (design variable) $x_{i}$ is greater than $\theta$
then $z_{i}$ is set equal to 1. The assignment $z_{i}=1$ denotes that the
$i^{th}$ feature is selected. Conversely, if the vector $x_{i}$ is smaller
than $\theta$ then $z_{i}$ is set equal to 0. The assignment $z_{i}=0$
represents that the $i^{th}$ feature is not selected. In other words, the
candidate solution $\mathbf{x}$ can be interpreted as a vector whose elements
represent the probability of a feature to be selected (or discarded).

Then, with the generated $\mathbf{z}$ that represents the data set after some
features (columns) have been removed, three objectives are calculated. These
three objectives aim to assess: 1) classification accuracy; 2) solution size;
3) missing rate.

For the first objective, i.e. errors of classification accuracy, we used the
K-NN classifier, with $k$ set to $5$, and we implemented $l$-fold cross-
validation method ($l$=10). The formula to calculate the classification
accuracy is given as follows:

${\color[rgb]{0,0,0}A_{cor}=\left(\frac{{1}}{l}\sum\limits_{i=1}^{l}\frac{{N_{Cor}}}{{N_{All}}}\right)\times
100\%}$ (5)

where $N_{Cor}$ denotes the amount of test samples that are correctly
predicted, $N_{All}$ denotes the number of all test samples. However, in this
paper, as we use the non-dominant relationship for comparison, we introduced
the classification error rate to evaluate the performance. The formula to
calculate the classification error rate and thus the first objective $f_{1}$
is given by:

$f_{1}(\mathbf{x})=1-A_{cor}$ (6)

The solution size is another objective that can be formulated as follows:

$f_{2}(\mathbf{x})=\sum\limits_{i=1}^{n}{z_{i}}$ (7)

In other words, one of the criteria is to remove as many features as possible
that is to have as many zeros as possible within the vector $\mathbf{z}$.

Classification error rate and solution size are two objectives commonly used
in traditional multi-objective FS problems. Besides these two objectives, we
introduce in this study the missing rate as the third objective. Thus, the FS
problem is extended into a three-objective FS problem. The purpose of adding
the third objective is to consider the reliability of the selected features.
The missing rate refers to the percentage of missing data in the selected
feature set with respect to the missing data in the original data set. At
first. we store the serial number of the selected feature into the vector $y$.
The number of missing data in the selected feature set is indicated with $lm$,
and it is calculated as:

$lm=\sum\limits_{j=1}^{f_{2}(\mathbf{x})}{lm_{y_{j}}}$ (8)

where $lm_{y_{j}}$ shows how many missing values in the ${y_{j}}^{th}$
feature. Next, the following formula shows how we calculate the number of
missing values in the original data set $la$:

$la=\sum\limits_{j=1}^{n}{lm_{j}}$ (9)

After having obtained $lm$ and $la$, the missing rate can be calculated as
follows:

$f_{3}(\mathbf{x})=\frac{{l_{m}}}{l_{a}}\times 100\%$ (10)

###  III-A NSGA-III algorithm

NSGA-III [36] is a popular algorithm for multi-objective optimization. It’s
main feature is the so-called reference point-based selection method.

Input: reference points $R$, parent population $P_{t}$

Output: $P_{t+1}$

1 Initialize $S_{t}$ = 0, $i$ = 1

2$Q_{t}$=Recombination+Mutation($P_{t}$);

3$R_{t}$=$P_{t}\cup Q_{t}$

4$(F_{1},F_{2},...)$=Fast-nondominated-sort($R_{t}$)

5repeat

6 $S_{t}=S_{t}\cup F_{i}$ and $i=i+1$

7until  _$|S_{t}|\geq N$_ ;

8Last front to be included: $F_{l}=F_{i}$

9if  _$|S_{t}|=N$_ then

10 $P_{t+1}=S_{t}$, break

11 end if

12else

13 $P_{t+1}=S_{t}=\cup_{j=1}^{l-1}F_{j}$

14 Select the point from $F_{l}:K=N-|P_{t+1}|$

15 Based on the selection of reference points:
$P_{t+1}:Selection(K,S_{t},\rho_{j},R,F_{l})$

16 end if

17

18

Algorithm 1 NSGA-III algorithm

Briefly, the basic idea of the NSGA-III is described as follows: firstly, it
constructs a set of reference points, and randomly generates an initialization
population $P_{t}$ of $N$ individuals, then uses binary crossover and
polynomial mutations to generate new populations $Q_{t}$, and combines the
$P_{t}$ and the $Q_{t}$ for fast non-dominated sorting. After that, $N$
individuals are chosen to enter the offspring population through the non-
dominated rank, and the reference point mechanism is used for selection in the
case when the selection cannot be made through the non-dominated rank [36].
The structure of NSGA-III is outlined in Algorithm 1.

###  III-B Fast non-donminated sorting

Input: population $P$

Output: $F_{i}$

1

2Initialize $S_{p}=\phi$, $np$ = 0

3for  _each $p\in P$_ do

4

5 for  _each $q\in P$_ do

6 Compare p with q: if  _$p$ dominates $q$_ then

7 $S_{p}=S_{p}\cup\\{q\\}$

8 end if

9 else

10 $np=np+1$

11 end if

12

13 end for

14 if  _$np=0$_ then

15

16 $F_{1}=F_{1}\cup\\{p\\}$

17 $p_{rank}=1$

18 end if

19

20 end for

21$i=1$

22while  _$F_{i}\neq\phi$_ do

23 $Q=\phi$

24 foreach  _$p\in F_{I}$_ do

25 foreach  _$q\in S_{p}$_ do

26 $n_{q}$=$n_{q}-1$

27 if  _$n_{q}=0$_ then

28 $q_{rank}$=$i$+1

29 $Q=Q\cup\\{q\\}$

30 end if

31

32 end foreach

33

34 end foreach

35 $i$=$i$+1

36 $F_{i}$=$Q$

37 end while

38

39

Algorithm 2 Fast-nondominated-sort($R_{t}$)

The most important part of the fast non-dominated sorting is the non-dominated
relationship [23]. When comparing non-dominated relationships, two parameters
need to be calculated. The first one is $n_{p}$ to count the amount of
individuals which dominate $p$, and the second is $S_{p}$ to store the
individuals which dominated by $p$. The process of fast non-donminated sorting
is described as follows:

First, we initialize $S_{p}$ and $n_{p}$ by setting $S_{p}$ to null set and
$n_{p}$ to 0. Then, the algorithm traverses each individual in the population
$P$, and compare this individual to all the remaining individuals in $P$. For
example, when it traverses to $p$, it compares $p$ with $q$. If $p$ dominates
$q$, $q$ is added to $S_{p}$. If $q$ dominates $p$, $n_{p}$ is increased by 1.
After traversing all the individuals, if $n_{p}$ = 0, $p$ is put into $F_{1}$.
Next, it initializes the rank number $i$ to 1. Finally, for each individual
$p$ in $F_{1}$, all individuals $q$ in $S_{p}$ are traversed. Whenever
traversing to $q$, $n_{q}$ is reduced by 1. When $n_{q}=0$, $q$ will be put
into $F_{2}$ and rank of $q$ is set to 2. Next, the rank is increased by 1 and
the algorithm enters the loop until $F_{i}=\phi$.

###  III-C Selection method based on reference points

Input: $K$, $S_{t}$, $\rho_{j}$ (the number of individuals associated with
reference point $j$ in $F_{1}$ to $F_{l-1}$), $R$, $F_{l}$

Output: $P_{t+1}$

1

2for  _each $s\in S_{t}$_ do

3 for  _each $r\in R$_ do

4 Compute $V(s,r)=s-r^{\mathrm{T}}s/||r||$

5 end for

6

7 Obtain the closest reference point $\pi(s)$ with individual $s$ :
$\pi(s)=r:argmin_{r\in R}V(s,r)$

8 Obtain the distance between $s$ and $\pi(s)$: $d(s)=V(s,\pi(s))$

9 end for

10

11$\rho_{j}=\sum_{s\in F_{c}}(\pi(s)=j)$ where $c=1,...,l-1$

12$k$=1

13while  _$k <K$_ do

14 $J_{min}$=$\\{j:argmin_{j\in R}\rho_{j}\\}$

15 $j_{r}$=random($J_{min}$)

16 $N_{j}$ = $\\{s:\pi(s)=j_{r},s\in F_{l}\\}$

17 if  _$N_{j}\neq 0$_ then

18

19 if  _$\rho_{j}=0$_ then

20

21 $P_{t+1}$ = $P_{t+1}$ $\cup\\{s\\}$ ($s:argmin_{s\in N_{j}}d(s)$)

22 end if

23 else

24 $P_{t+1}$ = $P_{t+1}$ $\cup random(N_{j})$

25 end if

26

27 $\rho_{j}=\rho_{j}+1$

28 $F_{l}=F_{l}/s$

29 $k=k+1$

30 end if

31 else

32 $R=R/j_{r}$

33 end if

34

35 end while

36

37

Algorithm 3 Selection($K$, $S_{t}$, $\rho_{j}$, $R$, $F_{l}$)

After the fast non-donminated ranking, individuals are put into the next
offspring population according to the non-donminated rank. When the offspring
individuals cannot be selected with the non-donminated rank, we use the
reference point selection method. First, we construct the reference points by
using the methods devised by Das and Denniss [42]. This method generates
$C_{P+M-1}^{P}$ isometric reference points on an equilateral triangle whose
apexes are (1,0,0), (0,1,0) and (0,0,1), and this coordinate axis is based on
the ideal point $p^{ideal}_{j}$ as the origin. $M$ is the objective dimension.
Each objective is divided into $P$ parts. Fig.1 depicts an example of a
reference point set with three objectives and each objective is separated into
Four parts.

Figure 1:  The coordinates of reference points with three objectives and each
objective divided into four parts.

The distance between each reference point is $\frac{{1}}{P}$, and the
coordinate $r_{j}$ calculation formula of each reference point is given as
follows.

$r_{j}=(r_{1},r_{2},...,r_{j}),j=1,2,...,M$ (11)

$r_{j}\in\left\\{0,1/P,...,P/P\right\\},\sum\limits_{j=1}^{M}{r_{j}=1}$ (12)

Then each individual needs to link to a reference point. Firstly, we compute
the reference line that is between the reference point and the origin
$p^{ideal}_{j}$. Then, we calculate the vertical distance $V$ from each
individual in the population $S_{t}$ to each reference line, and the formula
of vertical distance $V$ is given as follows:

$V(s,r)=s-r^{\mathrm{T}}s/||r||$ (13)

Finally, we associate the individual with the reference point corresponding to
the closest reference line.

When we add the individuals to the next generation $P_{t+1}$ from $F_{l}$.
Firstly, we randomly select a reference point with the least individual
association in $F_{1}$ to $F_{l-1}$, and then obtain the $N_{j}$, where
$N_{j}$ represents the amount of individuals associating with the $j_{r}$ in
the current frontier $F_{l}$. Next, if $N_{j}=0$ (it means that no individual
in $F_{l}$ associates with the $j_{r}$), and then a $j_{r}$ is replaced. But
if $N_{j}\neq 0$ (it means that there are individuals in $F_{l}$ associating
with the $j_{r}$), and next if $\rho_{j}=0$ (it means that no individual
associates with $j$ between $F_{1}$ and $F_{l-1}$), the individual with the
shortest distance is selected. Otherwise, an individual is randomly selected.

Algorithm 3 shows the pseudo-code of NSGA-III selection.

##  IV Experimental Setup

Table I displays the six incomplete data sets from the University of
California Irvine (UCI) Machine Learning Repositor.

With reference to Table I, DS$i$ represents the $i^{th}$ data set, DN
represents the names of data sets. NoI indicates the number of individuals in
the data sets, Dim denotes the data set dimension, and AoC denotes the amount
of classes, MR denotes the percentage of missing values in the data sets. It
can be observed that the highest missing rate is 24.9%.

TABLE I: Information of Datasets

NO. DN NoI Dim AoC MR DS1 processed.va.data 200 14 5 24.9% DS2 Heart-h 294 14
2 19.0% DS3 Hepatitis 155 20 2 5.4% DS4 Tumor.data 339 18 21 3.7% DS5
processed. switzerland.data 123 14 4 15.8% DS6 arrhythmia.data 452 279 16 6.0%

###  IV-A Preparation work

First, we use the mean imputation method to fill in the missing values in the
data sets. Then, we separate each data set into two parts, i.e., the training
set and test set. 70% of the examples of the initial data set are chosen as
the training set at random, and the remaining examples are used as test set.
K-NN method is utilized for evaluating the fitness value of the feature subset
and the 10-fold cross-validation is utilized for measuring classification
accuracy.

###  IV-B Benchmark algorithms and parameter settings

To verify the effectiveness of NSGA-III on three objectives FS problems, four
algorithms of NSGA-II [23], SPEA-II [43], IBEA [44] and KnEA [45] are used for
comparison. These comparison algorithms are run within the PLATEMO [46]
platform. The specification of the platform is fundamental in accordance with
the study reported in [47]. Each algorithm has run 30 times on the six data
sets. Each run has been stopped when the computational budget on the number of
fitness evaluation ($NFE$) was reached.

With reference to [36], the parameters of NSGA-III are set in the following
way: $NFE$ = 100000, $\theta$ = 0.6 see eq. (4), number of objectives $M$ = 3
, population size $PS$ = 100, the upper bound of individuals $up$ = 1, and the
lower bound $ub$ = 0.

###  IV-C Performance metrics

In order to evaluate the performance of the NSGA-III algorithm on the multi-
objective FS problems, we introduce two indicators. The first is the inverted
generational distance (IGD) [48]. The descriptions of the IGD is given as
follows:

$IGD(D,Z)=\frac{1}{|Z|}\sum\limits_{i=1}^{|z|}{min_{j=1to|D|}ed(z_{i},d_{i})}$
(14)

where $D$ denotes the non-dominated solution set, and $Z$ is the objective
solution set. $ed(z_{i},d_{i})$ represents the minimum Euclidean distance from
the individual in $Z$ to population $D$. The smaller the value of IGD, the
better the distribution and convergence quality of the solutions.

The second is hyper-volume metric (HV) [49]. HV is a kind of quality judgment
of the test algorithm by comprehensively evaluating the convergence,
extensiveness and distribution of the solution set of the multi-objective
optimization algorithm, see [50].

##  V Experimental Results

TABLE II: MEAN VALUES AND STANDARD DEVIATIONS OF IGD VALUES OBTAINED BY THE
FIVE ALGORITHMS ON THE TRAINING SETS

Datasets NSGA-II SPEA-II IBEA KnEA NSGA-III MV SD MV SD MV SD MV SD MV SD DS1
IGD 1.87E-02 8.02E-03 3.60E-02 7.86E-02 2.87E-01 2.64E-01 3.13E-02 0.0497
9.04E-02 8.97E-02 T-sig - - + - DS2 IGD 0.105 9.50E-02 5.04E-02 5.32E-02
4.51E-01 2.57E-01 4.90E-02 4.87E-02 4.97E-02 4.75E-02 T-sig + = + = DS3 IGD
2.12E-01 9.50E-02 7.90E-02 4.33E-02 1.55E-01 5.91E-02 5.28E-02 2.31E-02
1.60E-01 6.80E-02 T-sig + - = - DS4 IGD 8.22E-02 4.96E-02 7.07E-02 1.74E-02
5.49E-02 2.82E-02 1.66E-02 9.85E-03 4.71E-02 3.69E-02 T-sig + + = - DS5 IGD
6.41E-04 3.56E-04 7.58E-02 1.49E-01 4.31E-03 1.20E-03 5.51E-02 1.18E-01
3.20E-04 2.83E-04 T-sig + + + + DS6 IGD 4.87E+00 3.83E+00 4.11E+00 2.34E+00
2.60E+00 1.62E+00 2.33E+00 2.10E+00 1.57E+00 2.28E+00 T-sig + + + =

TABLE III: MEAN VALUES AND STANDARD DEVIATION OF IGD VALUES OBTAINED BY THE
FIVE ALGORITHMS ON THE TEST SETS

Datasets NSGA-II SPEA-II IBEA KnEA NSGA-III MV SD MV SD MV SD MV SD MV SD DS1
IGD 1.85E-01 1.53E-01 3.05E-01 1.94E-01 1.21E-01 1.35E-01 4.44E-01 3.31E-01
1.20E-01 1.09E-01 T-sig + + = + DS2 IGD 3.61E-01 1.96E-01 7.09E-01 1.18E-01
1.10E+00 3.13E-01 3.38E-01 1.58E-01 3.83E-01 2.19E-01 T-sig = + + = DS3 IGD
2.91E-01 2.17E-01 4.49E-01 4.49E-01 3.47E-01 1.66E-01 4.67E-01 3.00E-01
5.77E-01 2.02E-01 T-sig - = + = DS4 IGD 5.24E-01 2.65E-01 3.09E-01 2.12E-01
4.74E-01 2.55E-01 3.73E-01 2.58E-01 3.65E-01 2.00E-01 T-sig + = = = DS5 IGD
1.31E+00 5.19E-01 6.51E-01 5.71E-01 2.90E-01 2.50E-01 3.91E-01 3.35E-01
7.14E-01 4.60E-01 T-sig + - - - DS6 IGD 8.86E+00 2.29E+00 6.63E+00 3.96E+00
4.96E+00 1.17E+00 3.67E+00 2.03E+00 3.47E+00 1.72E+00 T-sig + + + =

The experimental results on the six data sets are listed in Table II, III, IV,
V. In these Tables, MV indicates the mean value while SD indicates the
standard deviation. The best mean values are highlighted in bold. T-sig
indicates the statistical significance of the results according to the T-test
with confidence level $95\%$. The “+” denotes that NSGA-III is significantly
better than the comparison approach, the “-” denotes that the comparison
approach is significantly better than NSGA-III, and “=” denotes that NSGA-III
and comparison approach have similar results.

Numerical results on the training sets in Table II show that the mean values
obtained by NSGA-III on the six data sets are smaller, and the corresponding
standard deviations are smaller too. Through comparing NSGA-III with NSGA-II,
SPEA-II, IBEA and KnEA, it is found that NSGA-III is significantly better than
NSGA-II and it outperforms NSGA-II on five data sets with significant
difference. NSGA-III performs similar to NSGA-II on one training set. NSGA-III
is superior to SPEA-II on three training sets with significant difference. The
results of NSGA-III are similar to SPEA-II on two training sets. When
comparing NSGA-III with IBEA, it is found that the IGD values obtained by
NSGA-III on the four training sets are smaller than IBEA with significant
difference. The performance of NSGA-III on two training sets is similar to
that of IBEA. The IGD value of NSGA-III is smaller than KnEA on one data set
with significant difference, and is similar to KnEA on two training sets.
Through the above analysis, we can get that NSGA-III is better than NSGA-II,
SPEA-II and IBEA when using IGD index, and it has similar performance to that
of KnEA. By analyzing Table III, we can reach the same conclusion on the test
sets: NSGA-III is still superior to NSGA-II, SPEA-II and IBEA, and it has a
similar performance to that of KnEA.

TABLE IV: MEAN VALUES AND STANDARD DEVIATIONS OF HV VALUES OBTAINED BY THE
FIVE ALGORITHMS ON THE TRAINING SETS

Datasets NSGA-II SPEA-II IBEA KnEA NSGA-III MV SD MV SD MV SD MV SD MV SD DS1
HV 5.38E-02 2.77E-03 4.87E-02 3.25E-03 4.36E-02 5.12E-02 5.35E-02 2.68E-03
1.78E-01 8.52E-03 T-sig + + + + DS2 HV 6.29E-02 4.71E-03 1.38E-01 6.64E-03
1.18E-01 1.27E-01 1.15E-01 6.13E-03 1.51E-01 7.72E-03 T-sig + + + + DS3 HV
2.41E-01 8.25E-03 1.91E-01 5.02E-03 1.92E-01 2.13E-01 1.75E-01 3.47E-03
2.64E-01 6.74E-03 T-sig + + + + DS4 HV 1.59E+00 1.12E-02 1.92E+00 1.54E-02
5.01E-01 5.16E-01 5.18E-01 4.82E-03 1.35E+00 8.39E-03 T-sig - - + + DS5 HV
1.04E-01 2.97E-03 8.32E-02 2.12E-03 5.32E-04 5.57E-04 1.10E-01 4.26E-03
1.03E-01 2.00E-03 T-sig = + + - DS6 HV 3.92E-02 1.92E-02 1.16E-01 1.60E-02
3.22E-03 5.49E-03 2.73E-03 8.94E-04 3.27E-02 1.11E-02 T-sig = - = +

TABLE V: MEAN VALUES AND STANDARD DEVIATIONS OF HV VALUES OBTAINED BY THE FIVE
ALGORITHMS ON THE TEST SETS

Datasets NSGA-II SPEA-II IBEA KnEA NSGA-III MV SD MV SD MV SD MV SD MV SD DS1
HV 5.09E-04 3.10E-04 3.30E-04 3.20E-04 0.00E+00 0.00E+00 5.06E-04 2.97E-04
2.24E-02 4.91E-02 T-sig + + + + DS2 HV 4.06E-02 4.80E-03 1.67E-02 1.78E-03
2.80E-01 2.54E-02 3.80E-02 4.54E-03 1.09E-01 1.06E-02 T-sig + + - + DS3 HV
1.32E-03 1.20E-03 5.04E-03 1.95E-03 3.85E-03 2.07E-03 5.23E-03 2.04E-03
1.23E-02 3.33E-03 T-sig + + + + DS4 HV 4.42E-01 2.91E-02 3.26E-01 2.01E-02
3.84E-01 2.59E-02 3.83E-01 2.56E-02 3.30E-01 1.73E-02 T-sig - = - - DS5 HV
6.19E-02 2.51E-03 8.78E-02 3.41E-03 5.28E-04 2.22E-05 6.16E-02 3.53E-03
6.46E-02 1.90E-03 T-sig + - + = DS6 HV 7.17E-02 1.74E-02 7.19E-05 2.63E-04
3.13E-02 1.08E-02 1.17E-02 6.81E-03 1.46E-01 1.52E-02 T-sig + + + +

Table IV and Table V show the results in terms of HV for the training and test
sets respectively. Table IV shows that for three data sets, NSGA-III displays
a significantly better performance than NSGA-II and in two cases NSGA-II and
NSGA-III have a similar performance. NSGA-III performs better than SPEA-II on
four training sets. NSGA-III is superior to IBEA and KnEA on five training
sets. The results in Table V clearly show that NSGA-III is superior to other
algorithms in the majority of cases in terms of HV. According to our
interpretation, the reason for the good performance of NSGA-III is that it
constructs a uniformly distributed reference point system, so that the
selected offspring are evenly distributed in the objective space, reducing the
situation of the offspring gathering together, which improves the distribution
of offspring, and also increase the diversity of offspring.

On all data sets, each algorithm has obtained a set of solutions with a non-
dominated rank of 1. In order to depict the advantages of NSGA-III with
respect to its competitors, Fig.2 and Fig.3 show the Pareto fronts of NSGA-III
and the other four algorithms considered in this study on the six data sets
used in the experiments. Fig.2 shows the results on the training sets while
Fig.3 presents the results on test sets. In each subfigure, the x-axis
represents the classification error rate $f_{1}$, the y-axis represents the
solution sizes $f_{2}$, and the z-axis represents the missing rate $f_{3}$.

DS 1 DS 2

DS 3 DS 4

DS 5 DS 6

Figure 2: Classification accuracy,solution size and missing rate of different
algorithms on training sets (DS 1- DS 6).

DS 1 DS 2

DS 3 DS 4

DS 5 DS 6

Figure 3: Classification accuracy,solution size and missing rate of different
algorithms on test sets (DS 1- DS 6).

Results in Fig.2 and Fig.3 show that the distribution of the solutions in the
non-dominated sets detected by NSGA-III are more uniform than those detected
the other algorithms. Furthermore, NSGA-III is better than other algorithms in
terms of classification accuracy, solution size and missing rate on both
training and test sets. On the basis of the results we obtained, we conclude
that NSGA-III has better performance for FS problems with three objectives.
According to our interpretation, the reason why NSGA-III is superior to other
algorithms is that it uses the method of reference point selection, which
associates individuals with each reference point, and effectively selects
individuals with less correlation with reference points. This mechanism
enables distribution uniform of the populations, which then promotes the
offspring population to produce more diverse individuals in the following
generations.

##  VI Conclusion

This paper proposes a novel interpretation of FS problem in data science with
a specific reference to data sets with missing data. Unlike classical studies
in the literature that use accuracy and size of the solutions as quality
metrics, we propose the simultaneous inclusion of a third metric, that is the
missing rate. This modelling poses a three-objective optimization problem that
is addressed by means of an ad-hoc implementation of NSGA-III.

In order to demonstrate the effectiveness of the proposed approach, we tested
NSGA-III on six incomplete data sets from the the UCI machine learning
repository and compared them against four popular algorithms for multi-
objective optimization. Numerical results show the overall superiority NSGA-
III to the other methods considered in this study in terms of IGD and HV.

Although the performance of our NSGA-III implementation is promising, we feel
that there is some margin for improvement. Future research will investigate
the integration of knowledge-based features associated to the FS to the
selection mechanism of NSGA-III.

## Acknowledgements

This work was partially supported by the National Natural Science Foundation
of China (61876089, 61876185, 61902281), the opening Project of Jiangsu Key
Laboratory of Data Science and Smart Software(No.2019DS301), the Science and
Technology Program of Ministry of Housing and Urban-Rural Development
(2019-K-141), the Engineering Research Center of Digital Forensics, Ministry
of Education, the Entrepreneurial team of sponge City (2017R02002), and the
PAPD.

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# SVD Entanglement Entropy of Chiral Dirac Oscillators

Yuvraj Singh<EMAIL_ADDRESS>S.N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata 700106, India  Department of Physics,
Shiv Nadar Institution of Eminence, Gautam Buddha Nagar, Uttar Pradesh 201314,
India  Rabin Banerjee<EMAIL_ADDRESS>S.N. Bose National Centre for Basic
Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India

(July 2024)

###### Abstract

We discuss the SVD entanglement entropy, which has recently come up as a
successor to the pseudo entropy. This paper is a first-of-its-kind application
of SVD entanglement entropy to a system of chiral Dirac oscillators which
prove to be natural to study the SVD formalism because the two chiral
oscillator ground states can be taken as the pre-selected and post-selected
states. We argue how this alternative for entanglement entropy is better and
more intuitive than the von Neumann one to study quantum phase transition. We
also provide as an illustrative example, a new generalized proof of the SVD
entanglement entropy being $\log 2$ for a pair of Bell states that differ from
each other by relative phases.

INTRODUCTION
Entanglement entropy is a measure of the quantum correlation or entanglement
between two subsystems in a quantum system. It has become an important
parameter in studying the quantum properties of any emergent quantum system.
Entanglement entropy is usually computed by the definition of entropy given by
von Neumann [1] and is called the von Neumann entanglement entropy. Here we
consider a single state and compute the entanglement entropy which is a
measure of quantum correlations among the degrees of freedom for the given
state. Recently however a new way has been devised to calculate entanglement
entropy using the SVD (Singular Value Decomposition) entanglement entropy
proposed in [2]. In this we take two states and form a density matrix from
these states rather than taking the same state, these states are called pre-
selection and post-selection states, and then the entanglement entropy gives
the measure of quantum correlations among the degrees of freedom for the post-
selection and pre-selection states. It is observed that this result of SVD
entanglement entropy comes out close to the usual von Neumann entanglement
entropy. After the original work [2] some papers citing SVD entropy have also
appeared [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] where it has been considered as
a further generalization of the pseudo entropy [14].

Here we concern ourselves with the entanglement entropy related to the states
of the Dirac oscillator which is a relativistic fermionic harmonic oscillator
in the presence of a constant magnetic field. These oscillators were first
studied in [15, 16] and prove to be a very interesting system to study in
quantum optics and condensed matter physics. In recent decades a lot of work
has been put into the theoretical [17, 18] as well as experimental advancement
[19] of this in terms of its thermal properties [20], further generalizations
[21, 22] and its applications to cosmology and gravitational systems [23, 24,
25] have also come up. In work done by Bermudez et al. in [26], it was shown
that these Dirac oscillators in the presence of a constant magnetic field
exhibit a quantum phase transition. This study calculated the analytical
expression for order parameter, and canonical quantum fluctuations for the
quantum phase transition, furthermore, entanglement properties between the
degrees of freedom in the relativistic ground state were also studied. The
entanglement entropy turns out to be divergent at the point of phase
transition which is seen as a common feature of quantum phase transition [27,
28, 29, 30].

In this work, we have shown how SVD entanglement entropy is used to calculate
the entanglement entropy for a system exhibiting chiral behavior. Our results
show that the behavior of SVD entanglement entropy is similar to [26] but the
physical interpretation is more intuitive than the von Neumann entanglement
entropy. The SVD entanglement entropy is the entanglement across the quantum
phase transition of the degrees of freedom of the ground states. In both
cases, the value of entanglement entropy is divergent at the critical point
indicative of a quantum phase transition [2, 30].

As a prelude to the study of chiral Dirac oscillators, we provide as an
illustrative example, a new generalized proof of the SVD entanglement entropy
being $\log 2$ for a pair of Bell states that differ from each other by
relative phases. This is a generalization of a result given in [2] where they
have considered a specific example of a Bell state and the same Bell state but
with a relative phase as pre-selection and post-selection states respectively.
This work is also a first-of-its-kind example of the application of SVD
entanglement entropy in a quantum optical system. All the other studies are
based on holographic CFTs as cited above.

The organization of this paper is as follows. In the first section, we
introduce the SVD entanglement entropy and its basic properties, brief the
work done in [2] along with the generalized example for Bell states. In the
second section, we introduce the formalism related to the chiral Dirac
oscillator. Non-relativistic oscillators and chirality have been studied in a
fundamental and field-theoretic manner by [31, 32, 33] its connection to
material science and topological Chern-Simons gauge theory has been pointed
out in [34, 35]. In the third section, we introduce our main result and
compare it with the old result for entanglement entropy. Conclusions are given
in the last section.

##  1 SVD ENTANGLEMENT ENTROPY

There have been numerous ways that try to incorporate a post-selection and
pre-selection state into the calculation of entanglement entropy, as it is a
useful formalism for treating problems in quantum gravity, black hole
information paradox, condensed matter physics [36, 37, 14, 38, 39, 40, 41, 42,
43] and quantum information theory [44, 45, 46, 47]. The aim of this section
is to introduce this new definition of entropy as elaborated in [2]. This
definition is the general definition for entanglement entropy which tries to
solve the problem of considering pre-selection and post-selection states.

We are already familiar with the von Neumann entropy for a bipartite system.
If $\rho$ represents the density operator of the system then, the von Neumann
entropy is given by,

$S=-Tr(\rho\log\rho)$

But this definition of entropy uses only a single state and that state may be
entangled or not.

The SVD (single value decomposition) entanglement entropy uses two states a
pre-selected state and a post-selected state to calculate the entanglement
entropy. It is proposed to be a natural generalization of the von Neumann
entanglement entropy.

Consider a Hilbert space that is decomposed into two Hilbert spaces
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ if we take
$|\psi_{1}\rangle$ as the pre-selected state and $|\psi_{2}\rangle$ as the
post-selected state, then we can define the reduced transition matrix[2],

$\tau^{1|2}:=\frac{|\psi_{1}\rangle\langle\psi_{2}|}{\langle\psi_{1}|\psi_{2}\rangle}$
(1)

whereas the unnormalized version is,

$\tau^{1|2}=|\psi_{1}\rangle\langle\psi_{2}|$

Now if we take the partial trace,

$\tau^{1|2}_{A}=Tr_{B}[\tau^{1|2}]$

Using this we define,

$\rho^{1|2}_{A}:=\frac{\sqrt{(\tau^{1|2}_{A})^{\dagger}(\tau^{1|2}_{A})}}{Tr\left[\sqrt{(\tau^{1|2}_{A})^{\dagger}(\tau^{1|2}_{A})}\right]}$
(2)

This operator is a density operator. Using this we define the SVD entropy as,

$S_{SVD}(\tau^{1|2}_{A}):=-Tr[\rho^{1|2}_{A}\log\rho^{1|2}_{A}]$ (3)

Hence, the functional form of SVD and von Neumann entanglement entropy is
similar.[2]
Now, we try to summarise the general definition and physical interpretation
(in the context of information theory) as given in [2].
General Definition:  Given a space $\mathbb{M}_{m}$ of $m\times m$ matrices.
Then $S_{SVD}$ is defined by,

$S_{SVD}:\mathbb{M}_{m}\longrightarrow\left[0,\infty\right)$
$S_{SVD}(A)=S_{vN}\left(\frac{|A|}{||A||_{1}}\right)=-\sum^{m}_{i=1}\frac{\sigma_{i}}{||A||_{1}}\log\left(\frac{\sigma_{i}}{||A||_{1}}\right)$

where $|A|:=\sqrt{A^{\dagger}A}$ and,
$||A||_{1}:=Tr|A|=\sum^{m}_{i=1}\sigma_{i}$ ($\sigma_{i}$ are eigenvalues of
$|A|$). Also note if $A=0$ then $S_{SVD}=0$.[2]
Physical Interpretation:  von Neumann entropy coincides with the average
number of Bell pairs that can be distilled from a given bipartite state via
local operations and classical communication[48, 1]. This result can also be
proved for SVD entropy.[2]

We can also diagonalize the reduced transition matrix,

$\tau^{1|2}=U\cdot\Lambda\cdot V^{\dagger}$

where $U$ and $V$ are unitary matrices, and $\Lambda$ is the diagonal matrix
such that $\Lambda=diag(\lambda_{1},\ldots,\lambda_{d_{A}})$ and
$\lambda_{i}\geq 0,\forall i\in\\{1,\ldots,d_{A}\\}$. Then we can define the
normalized eigenvalues as:

$\hat{\lambda}_{i}=\frac{\lambda_{i}}{\sum\lambda_{i}}$

Clearly, $\hat{\lambda}_{i}$ are eigenvalues of $\rho^{1|2}_{A}$, then we
have:

$S_{SVD}=-\sum^{d_{A}}_{i}\hat{\lambda}_{i}\log\hat{\lambda}_{i}$

These are all parts of standard results and definitions done in [2].

###  1.1 General Result for the Bell States

Now, we make a general statement about the SVD entropy of the Bell states. In
[2] two states belonging to a bipartite system of bell states
($\mathcal{H}_{A}\otimes\mathcal{H}_{B}$) have been considered as:

$\begin{split}|\psi_{1}\rangle&=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)\\\
|\psi_{2}\rangle&=\frac{1}{\sqrt{2}}(|00\rangle+\phi|11\rangle)\end{split}$
(4)

where $\phi\in\mathbf{C}$, $|\phi|=1$, and $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$ are the pre-selected and post-selected states respectively.
Now we consider the following generalized states belonging to a bipartite
system of bell states ($\mathcal{H}_{A}\otimes\mathcal{H}_{B}$).

$\begin{split}|\psi_{1}\rangle&=a|\alpha\beta\rangle+b|\gamma\delta\rangle,\\\
|\psi_{2}\rangle&=c|\tilde{\alpha}\tilde{\beta}\rangle+d|\tilde{\gamma}\tilde{\delta}\rangle\end{split}$
(5)

where
$\alpha,\beta,\gamma,\delta,\tilde{\alpha},\tilde{\beta},\tilde{\gamma}$, and
$\tilde{\delta}\in\\{0,1\\}$ and $a,b,c,d$ $\in\mathbf{C}$,
$|a|,|b|,|c|,|d|=1$. The transition matrix is given by,

$\tau^{1|2}=\frac{|\psi_{1}\rangle\langle\psi_{2}|}{\langle\psi_{1}|\psi_{2}\rangle}$
$\tau^{1|2}=\frac{ac^{*}|\alpha\beta\rangle\langle\tilde{\alpha}\tilde{\beta}|+ad^{*}|\alpha\beta\rangle\langle\tilde{\gamma}\tilde{\delta}|+ba^{*}|\gamma\delta\rangle\langle\tilde{\alpha}\tilde{\beta}|+bd^{*}|\gamma\delta\rangle\langle\tilde{\gamma}\tilde{\delta}|}{a^{*}c\langle\alpha\beta|\tilde{\alpha}\tilde{\beta}\rangle+a^{*}d\langle\alpha\beta|\tilde{\gamma}\tilde{\delta}\rangle+b^{*}c\langle\gamma\delta|\tilde{\alpha}\tilde{\beta}\rangle+b^{*}d\langle\gamma\delta|\tilde{\gamma}\tilde{\delta}\rangle}$

The reduced transition matrix is,

$\tau^{1|2}_{A}=Tr_{B}(\tau^{1|2})$
$\tau^{1|2}_{A}=\frac{ac^{*}|\alpha\rangle\langle\tilde{\alpha}|(\langle\beta|\tilde{\beta}\rangle)+ad^{*}|\alpha\rangle\langle\tilde{\gamma}|(\langle\beta|\tilde{\delta}\rangle)+bc^{*}|\gamma\rangle\langle\tilde{\alpha}|(\langle\delta|\tilde{\beta}\rangle)+bd^{*}|\gamma\rangle\langle\tilde{\gamma}|(\langle\delta|\tilde{\delta}\rangle)}{a^{*}c\langle\alpha\beta|\tilde{\alpha}\tilde{\beta}\rangle+a^{*}d\langle\alpha\beta|\tilde{\gamma}\tilde{\delta}\rangle+b^{*}c\langle\gamma\delta|\tilde{\alpha}\tilde{\beta}\rangle+b^{*}d\langle\gamma\delta|\tilde{\gamma}\tilde{\delta}\rangle}$

Since these are Bell states:

  * •

Case I: If without loss of generality we assume that $\alpha=\beta=0$
$\implies$ $\gamma$, $\delta=1$ (Bell states).

$\tau^{1|2}_{A}=\frac{ac^{*}|0\rangle\langle\tilde{\alpha}|(\langle
0|\tilde{\beta}\rangle)+ad^{*}|0\rangle\langle\tilde{\gamma}|(\langle
0|\tilde{\delta}\rangle)+bc^{*}|1\rangle\langle\tilde{\alpha}|(\langle
1|\tilde{\beta}\rangle)+bd^{*}|1\rangle\langle\tilde{\gamma}|(\langle
1|\tilde{\delta}\rangle)}{a^{*}c\langle
00|\tilde{\alpha}\tilde{\beta}\rangle+a^{*}d\langle
00|\tilde{\gamma}\tilde{\delta}\rangle+b^{*}c\langle
11|\tilde{\alpha}\tilde{\beta}\rangle+b^{*}d\langle
11|\tilde{\gamma}\tilde{\delta}\rangle}$

Also for the denominator to be non-zero $\tilde{\alpha}=\tilde{\beta}$ and
$\tilde{\gamma}=\tilde{\delta}$. Since we are working with Bell states
$\implies$ $\tilde{\alpha}\neq\tilde{\gamma}$.
Again, without loss of generality assume that, $\tilde{\gamma}=0$ and
$\tilde{\alpha}=1$, then,

$\tau^{1|2}_{A}=\frac{\left[ad^{*}|0\rangle\langle 0|+bc^{*}|1\rangle\langle
1|\right]}{(a^{*}d+b^{*}c)}$ (6)

If we had used the complimentary values i.e. $\tilde{\gamma}=1$ and
$\tilde{\alpha}=0$, then,

$\tau^{1|2}_{A}=\frac{\left[ac^{*}|0\rangle\langle 0|+bd^{*}|1\rangle\langle
1|\right]}{(a^{*}c+b^{*}d)}$ (7)

  * •

Case II: If $\alpha=\delta$ then $\beta=\gamma$, $\alpha\neq\beta$ and
$\gamma\neq\delta$, so without loss of generality if we assume that $\alpha=0$
then $\beta=1$, $\gamma=1$, and $\delta=0$.

$\tau^{1|2}_{A}=\frac{ac^{*}|0\rangle\langle\tilde{\alpha}|(\langle
1|\tilde{\beta}\rangle)+ad^{*}|0\rangle\langle\tilde{\gamma}|(\langle
1|\tilde{\delta}\rangle)+bc^{*}|1\rangle\langle\tilde{\alpha}|(\langle
0|\tilde{\beta}\rangle)+bd^{*}|1\rangle\langle\tilde{\gamma}|(\langle
0|\tilde{\delta}\rangle)}{a^{*}c\langle
01|\tilde{\alpha}\tilde{\beta}\rangle+a^{*}d\langle
01|\tilde{\gamma}\tilde{\delta}\rangle+b^{*}c\langle
10|\tilde{\alpha}\tilde{\beta}\rangle+b^{*}d\langle
10|\tilde{\gamma}\tilde{\delta}\rangle}$

Also for the denominator to be non-zero $\tilde{\alpha}\neq\tilde{\beta}$ and
$\tilde{\gamma}\neq\tilde{\delta}$. Since we are working with Bell states
$\implies$ $\tilde{\alpha}=\tilde{\delta}$ and $\tilde{\beta}=\tilde{\gamma}$.
Again, without loss of generality assume that, $\tilde{\alpha}=1$ and
$\tilde{\beta}=0$, then,

$\tau^{1|2}_{A}=\frac{\left[ad^{*}|0\rangle\langle 0|+bc^{*}|1\rangle\langle
1|\right]}{(a^{*}d+b^{*}c)}$ (8)

If we had assumed the complimentary values i.e. $\tilde{\alpha}=0$ and
$\tilde{\beta}=0$, then,

$\tau^{1|2}_{A}=\frac{\left[ac^{*}|0\rangle\langle 0|+bd^{*}|1\rangle\langle
1|\right]}{(a^{*}c+b^{*}d)}$ (9)

Now, we evaluate the density matrix from the reduced transition matrix.

$\rho^{1|2}_{A}=\frac{\sqrt{(\tau^{1|2}_{A})^{\dagger}\tau^{1|2}_{A}}}{Tr\left(\sqrt{(\tau^{1|2}_{A})^{\dagger}\tau^{1|2}_{A}}\right)}$
$\rho^{1|2}_{A}=\frac{\sqrt{\frac{|ad|^{2}|0\rangle\langle
0|+|bc|^{2}|1\rangle\langle
1|}{(a^{*}d+bc^{*})(ad^{*}+b^{*}c)}}}{Tr\left(\sqrt{\frac{|ad|^{2}|0\rangle\langle
0|+|bc|^{2}|1\rangle\langle 1|}{(a^{*}d+bc^{*})(ad^{*}+b^{*}c)}}\right)}$
$\rho^{1|2}_{A}=\frac{|0\rangle\langle 0|+|1\rangle\langle 1|}{2}$
$S_{SVD}=-Tr\left(\rho^{1|2}_{A}\log\rho^{1|2}_{A}\right)=\log 2$ (10)

Thus the entropy for Bell states is $\log 2$ which is equal to the value of
von Neumann entropy for Bell states. This result confirms the validity of SVD
entanglement entropy as a generalization of the usual von Neumann entanglement
entropy for two different states. In [2] the $\log 2$ result was obtained for
the particular choice (4).

##  2 CHIRAL DIRAC OSCILLATORS

In the previous example, introducing by hand a phase among the Bell states,
the pre-selected and post-selected states were defined. Chiral Dirac
oscillators, on the other hand, provide a natural setting for obtaining these
states by considering the opposite chiralities. The Dirac equation gives the
Dirac oscillator in its usual framework [15, 16]:

$i\hbar\frac{\partial|\psi\rangle}{\partial
t}=[c\boldsymbol{\alpha}(\textbf{p}-im\beta\omega\textbf{r}+e\textbf{A})+\beta
mc^{2}]|\psi\rangle$ (11)

where m is the relativistic mass of the fermion, $-e$ is the electric charge,
$|\psi\rangle$ stands for the Dirac four-component spinor, r and p represent
position and momentum operators, $\omega$ is the Dirac oscillator frequency,
$c$ is the speed of light and $\beta:=diag(\mathbf{I},-\mathbf{I})$,
$\alpha_{j}:=off-diag(\sigma_{j},\sigma_{j})$ are the Dirac matrices related
to the usual Pauli matrices. This relativistic oscillator involves two phonon
modes. Now, if we introduce the magnetic field by minimal coupling
$\textbf{p}\longrightarrow\textbf{p}+e\textbf{A}$ where A is the vector
potential related to the magnetic field. In the two-dimensional setup, the
Dirac matrices become the Pauli matrices [26, 49], then (11) becomes:

$i\hbar\frac{\partial|\psi\rangle}{\partial
t}=\left[c\sum^{2}_{j=1}\sigma_{j}(p_{j}-im\beta\sigma_{z}x_{j}+eA_{j})+\sigma_{z}mc^{2}\right]|\psi\rangle$
(12)

where $|\psi\rangle$ is a two-component spinor that mixes spin-up and spin-
down with positive and negative energies. The non-minimal coupling of Dirac
oscillator ($\textbf{p}-im\beta\omega\textbf{r}$) and the minimal coupling of
the magnetic field endows the particle with intrinsic left-handed and right-
handed chirality respectively [26, 50, 51].

###  2.1 Mapping onto a Simultaneous JC-AJC Hamiltonian

The above Hamiltonian in (11) can be mapped onto a simultaneous Jaynes
Cummings and Anti-Jaynes-Cummings Hamiltonian[52]. Here we enlist the final
results of this mapping and the associated energy spectrum and ground states
(next section) obtained in [26, 51] to proceed with the discussion of entropy
in the third section.

$H=\delta\sigma_{z}-H^{\circlearrowright}_{JC}(g_{r})+H^{\circlearrowleft}_{AJC}(g_{l})$
(13)

where $\delta=mc^{2}$ stands for the detuning parameter proportional to the
rest mass energy, $H^{\circlearrowright}_{JC}(g_{r})$ represents a right-
handed Jaynes-Cummings Hamiltonian

$H^{\circlearrowright}_{JC}(g_{r})=g_{r}\sigma^{+}\tilde{a}_{r}+g_{r}^{*}\sigma^{-}\tilde{a}_{r}^{\dagger}$

with $g_{r}:=imc^{2}\sqrt{2\tilde{\xi}}$ as the interaction coupling strength.
Analogously, the term $H^{\circlearrowleft}_{AJC}(g_{l})$ stands for a left-
handed Anti-Jaynes-Cummings interaction

$H^{\circlearrowleft}_{AJC}(g_{l})=g_{l}\sigma^{+}a_{l}^{\dagger}+g_{l}^{*}\sigma^{-}a_{l}$

with a similar coupling strength $g_{l}:=imc^{2}\sqrt{2{\xi}}$
Now we introduce the circular annihilation-creation operators
($\tilde{a}_{r}\>\text{and}\>a_{l}$) used in the above equations as defined in
[26, 51]. Using the axial gauge ($A_{0}=0$) the magnetic field in the Dirac
oscillator is defined as $\mathbf{B}=B\mathbf{e}_{z}$ with the vector
potential $\mathbf{A}:=\frac{B}{2}\left[-y,x,0\right]$. We describe the
dynamics by two frequencies, the Dirac oscillator frequency $\omega$ and the
cyclotron frequency $\omega_{c}=eB/m$. Hence, the annihilation-creation
operators are:

$a_{i}=\frac{1}{\sqrt{2}}\left(\frac{1}{\triangle}r_{i}-i\frac{\triangle}{\hbar}p_{i}\right),\tilde{a}_{i}=\frac{1}{\sqrt{2}}\left(\frac{1}{\tilde{\triangle}}r_{i}-i\frac{\tilde{\triangle}}{\hbar}p_{i}\right)$

where $\triangle:=\sqrt{\hbar/m\omega}$,
$\tilde{\triangle}:=\sqrt{\hbar/m\tilde{\omega}}$ and
$\tilde{\omega}=w_{c}/2$. The basic commutation relations are,

$\begin{split}[a_{i},\tilde{a}_{j}]=\frac{1}{2}\left[\frac{\triangle}{\tilde{\triangle}}-\frac{\tilde{\triangle}}{\triangle}\right]\delta_{ij},\\\
[a_{i},\tilde{a}_{j}^{\dagger}]=\frac{1}{2}\left[\frac{\triangle}{\tilde{\triangle}}+\frac{\tilde{\triangle}}{\triangle}\right]\delta_{ij}\end{split}$
(14)

We have introduced $i=x,y$ to account for two possible directions for
equations of motion. Also note that $\tilde{\xi}=\hbar\tilde{\omega}/mc^{2}$
and $\xi=\hbar\omega/mc^{2}$.
So the circular annihilation-creation operators for each frequency are:

$\tilde{a}_{r}=\frac{1}{\sqrt{2}}(\tilde{a}_{x}-i\tilde{a}_{y}),\>\>\>\>\tilde{a}_{l}=\frac{1}{\sqrt{2}}(\tilde{a}_{x}+i\tilde{a}_{y}),\>\>\>\>a_{r}=\frac{1}{\sqrt{2}}(a_{x}-ia_{y}),\>\>\>\>a_{l}=\frac{1}{\sqrt{2}}(a_{x}+ia_{y})$
(15)

###  2.2 Exact Solution: Energy Spectrum and Eigenstates

In this section, we summarise the results of a unitary transformation that
converts the bi-chromatic Hamiltonian in (13) into a monochromatic JC(AJC)
term that includes a bosonic degree of freedom with a certain chirality that
depends upon external parameters $\xi>\tilde{\xi}(\xi<\tilde{\xi})$ [26, 51].

####  2.2.1 Left-Handed Regime $\tilde{\xi}<\xi$

Under the above conditions, the Hamiltonian given in (13) can be mapped onto a
single-mode anti-Jaynes-Cummings Hamiltonian by using the unitary
transformation:

$U_{\alpha}:=e^{\alpha(a_{l}\tilde{a}_{r}-\tilde{a}_{r}^{\dagger}a_{l}^{\dagger})}$
(16)

where the real parameter $\alpha$ depends on the relative strength of the
oscillator and magnetic couplings:

$\alpha:=\frac{1}{\lambda}\text{arctanh}\left(\frac{\lambda\sqrt{\tilde{\omega}}}{\sqrt{\omega}-\mu\sqrt{\tilde{\omega}}}\right)$
(17)

with $\mu:=(\triangle/\tilde{\triangle}-\tilde{\triangle}/\triangle)/2$ and
$\lambda:=\sqrt{\mu^{2}+1}$. Two Hermitian operators related through a unitary
operator share a common spectrum, hence in the weak magnetic field regime:

$E=\pm E_{n_{l}}=\pm mc^{2}\sqrt{1+2*\zeta_{l}(\xi,\tilde{\xi})(n_{l}+1)}$
(18)

where $\zeta_{l}(\xi,\tilde{\xi}):=\xi-\tilde{\xi}-2\mu(\xi\tilde{\xi})^{1/2}$
is related to the initial relevant parameters and $n_{l}=0,1,...$ represents
the number of left-handed quanta. This transformation can be immediately
related to a two-mode squeezing operator in the context of quantum optics with
squeezing parameter $z:=-\alpha\tilde{\mu}/2\in\mathbf{R}$ [53]. The action of
such a squeezing operator (16) over left-handed chiral Fock states gives rise
to SU(1,1) coherent states
$|z,n_{l}\rangle:=U^{\dagger}_{\alpha}|n_{l}\rangle|vac\rangle_{r}$ [26, 54].

$|z,n_{l}\rangle=\cosh^{-(n_{l}+1)}|z|\sum^{\infty}_{m=0}\sqrt{\frac{(m+n_{l})!}{n_{l}!m!}}(-1)^{m}\tanh^{m}|z||m+n_{l},m\rangle$
(19)

For the fermionic ground state

$E_{g}=mc^{2}$ (20)

$|g\rangle=|z,0\rangle|\chi_{\uparrow}\rangle=\frac{1}{\cosh|z|}\sum^{\infty}_{m=0}(-1)^{m}\tanh^{m}|z||m,m\rangle|\chi_{\uparrow}\rangle$
(21)

and is interpreted as a spin-up squeezed vacuum state, where the squeezing
parameter $z:=-\alpha\tilde{\mu}/2$ depends on the relative coupling strengths
$\xi,\tilde{\xi}$.

####  2.2.2 Right-Handed Regime $\tilde{\xi}>\xi$

In this condition, we transform the Hamiltonian into a Jaynes-Cummings
Hamiltonian by the unitary transformation.

$U_{\tilde{\alpha}}:=e^{\tilde{\alpha}(a_{l}\tilde{a}_{r}-\tilde{a}_{l}^{\dagger}a_{l}^{\dagger})}$
(22)

where the parameter $\tilde{\alpha}$ is,

$\tilde{\alpha}=\frac{1}{\lambda}\text{arctanh}\left(\frac{\lambda\sqrt{\omega}}{\sqrt{\tilde{\omega}}+\mu\sqrt{\omega}}\right)$

The energy spectrum becomes,

$\tilde{E}=\pm\tilde{E}_{n_{r}}=\pm
mc^{2}\sqrt{1+2\zeta_{r}(\xi,\tilde{\xi})(n_{r}+1)}$ (23)

where $\zeta_{r}(\xi,\tilde{\xi}):=\tilde{\xi}-\xi+2\mu(\tilde{\xi}\xi)^{1/2}$
and $n_{r}=0,1,...,$ represents the number of right-handed quanta. In this
case, the squeezing parameter becomes
$\tilde{z}:=-\tilde{\alpha}\tilde{\mu}/2\in\mathcal{R}$ and the SU(1,1)
coherent states
$|\tilde{z},n_{r}\rangle:=U^{\dagger}_{\tilde{\alpha}}|vac\rangle_{l}|n_{r}\rangle$[26].

$|\tilde{z},n_{r}\rangle=\cosh^{-(n_{r}+1)}|\tilde{z}|\sum^{\infty}_{\tilde{m}=0}\sqrt{\frac{(\tilde{m}+n_{r})!}{n_{r}!\tilde{m}!}}(-1)^{\tilde{m}}\tanh^{\tilde{m}}|\tilde{z}||\tilde{m}+n_{r},\tilde{m}\rangle$
(24)

We also see how the fermionic ground state in this case differs from the left-
handed regime one (21) as:

$E_{\tilde{g}}=mc^{2}\sqrt{1+2\zeta_{r}}$ (25)
$|\tilde{g}\rangle=\tilde{C}_{0+}|\tilde{z},0\rangle|\chi_{\uparrow}\rangle-i\tilde{C}_{0-}|\tilde{z},1\rangle|\chi_{\downarrow}\rangle$
(26)

where $\tilde{C}_{n_{r}\pm}:=[(\tilde{E}_{n_{r}}\pm
mc^{2}).2\tilde{E}_{n_{r}}]^{1/2}$, and is interpreted as a spin-up squeezed
vacuum state, where the squeezing parameter
$\tilde{z}:=-\tilde{\alpha}\tilde{\mu}/2$ depends on the relative coupling
strengths $\xi,\tilde{\xi}$.

####  2.2.3 Critical Regime $\tilde{\xi}=\xi$

In this regime, the magnetic field coupling cancels the effect of the Dirac
string couple, hence we get a free relativistic fermionic particle [26]. The
energy spectrum is given by:

$E^{c}=\pm E^{c}_{\textbf{p}}=\pm\sqrt{m^{2}c^{4}+\textbf{p}^{2}c^{2}}$ (27)

where $\textbf{p}=(p_{x},p_{y})$ stands for two dimensional fermion momentum.
We can see that the energy is the usual free particle energy. The eigenstates
are described as:

$|\pm E^{c}_{\textbf{p}}\rangle=\sqrt{\frac{mc^{2}\pm E^{c}_{\textbf{p}}}{\pm
2E^{c}_{\textbf{p}}}}\left(|\chi_{\uparrow}\rangle+\frac{c(p_{x}+ip_{y})}{mc^{2}\pm
E^{c}_{\textbf{p}}}|\chi_{\downarrow}\rangle\right)|\textbf{p}\rangle$ (28)

where $\textbf{p}:=|p_{x},p_{y}\rangle$ are two-dimensional plane wave
solutions.

A schematic showing a simplified version of the set-up is shown in Figure 1.
Using these results Bermudez et al. in [26] showed how the quantum phase
transition occurs by plotting the energy spectrum as a function of relative
coupling strength, studied the order parameter, the divergence of quantum
fluctuations, and the entanglement entropy. The dependence of the magnetic
field provides a control parameter to test the properties of the system such
as chirality, squeezing, phonon statistics, and entanglement.

Figure 1: This cartoon represents the classical trajectory of a particle in a
cyclotron. The various arcs represent the energy levels: $n_{l}=0,1,...$ and
$n_{r}=0,1,...$ for left-handed regime and right-handed regime. Note that it
is obvious that the energy levels do not exist in classical scenario, so the
figure is just a concept demonstration not an accurate representation of the
set-up.

##  3 ENTANGLEMENT AT THE CRITICAL POINT

Entanglement is one of the key features of a quantum mechanical system. It is
a property that manifests a lot of the non-local phenomena in quantum
mechanics. In the current system, [26] showed that there exists a quantum
phase transition, along with angular momentum as the order parameter. Another
feature is that in the given left-hand and right-hand regimes, each regime has
different ground states. This section is concerned with providing a measure of
the entanglement of these states via the entanglement entropy. In the previous
work [26] the entanglement entropy of these ground states had been calculated
using the von Neumann entropy.
Here, we have tried to use the above-defined SVD entropy to calculate the
entanglement entropy. The presence of a left-handed regime ground state and
right-handed regime ground state serves as a preselection and postselection
state [2] respectively for the phenomena of quantum phase transition. We argue
that SVD entropy here can provide a better understanding of the entanglement
between the ground states near the critical limit of the quantum phase
transition. The SVD entropy can be interpreted as the entanglement entropy
across the ground states near the critical point of the quantum phase
transition. Since von Neumann entropy is the entropy of the individual states,
it does not take into consideration both the left-handed and right-handed
regime ground states simultaneously. It quantifies the entanglement between
different relativistic degrees of freedom. von Neumann entropy only shows us
the behavior of the individual states near the critical point of phase
transition meanwhile, SVD entropy shows how the states are getting entangled
with the states post-phase transition as the system nears a phase transition.
Hence, SVD entropy quantifies the entanglement between different relative
degrees of freedom of the different ground states. We also try to provide some
physical insight into the calculated result that has been missing in [26]
where they focused on the von Neumann definition.
We begin by defining our Hilbert space. The Hilbert space is a tripartite
system, consisting of continuous variables associated with chiral degrees of
freedom, and the discrete variables are associated with spin degrees of
freedom. To calculate the SVD entropy we would require the transition matrix
for the ground states. We define the transition matrix from the given ground
states of the left-handed regime $(|g\rangle)$ and the right-handed regime
$(|\tilde{g}\rangle)$ [2].

$\tau^{g|\tilde{g}}=\frac{|g\rangle\langle\tilde{g}|}{\langle
g|\tilde{g}\rangle}$ (29)

where $|g\rangle$ is the pre-selected state and $|\tilde{g}\rangle$ is the
post-selected state. In order to calculate this matrix. We are considering the
ground states.
For the left-handed regime $(\tilde{\xi}<\xi)$ (10,12):

$|g\rangle=|z,0\rangle|\chi_{\uparrow}\rangle$

and,

$|z,n_{l}\rangle=U^{\dagger}_{\alpha}|n_{l}\rangle|vac_{r}\rangle.$

So,

$|z,0\rangle|\chi_{\downarrow}\rangle=\frac{1}{\cosh|z|}\sum^{\infty}_{m=0}(-1)^{m}\tanh^{m}|z||m,m\rangle|\chi_{\uparrow}\rangle$

For the right-handed regime $(\tilde{\xi}>\xi)$ (15,17):

$|\tilde{g}\rangle=\tilde{C}_{0+}|\tilde{z},0\rangle|\chi_{\uparrow}\rangle-i\tilde{C}_{0-}|\tilde{z},1\rangle|\chi_{\downarrow}\rangle$

and,

$|\tilde{z},n_{r}\rangle=U^{\dagger}_{\tilde{\alpha}}|vac_{l}\rangle|n_{r}\rangle$
$|\tilde{z},0\rangle=\frac{1}{\cosh|\tilde{z}|}\sum^{\infty}_{\tilde{m}=0}(-1)^{\tilde{m}}\tanh^{m}|\tilde{z}||\tilde{m},\tilde{m}\rangle$

Now,

$|g\rangle\langle\tilde{g}|=\tilde{C}^{*}_{0+}|z,0\rangle\langle\tilde{z},0||\chi_{\uparrow}\rangle\langle\chi_{\uparrow}|+\iota\tilde{C}^{*}_{0-}|z,0\rangle\langle\tilde{z},1||\chi_{\uparrow}\rangle\langle\chi_{\downarrow}|$
(30)

and,

$\langle
g|\tilde{g}\rangle=\frac{\tilde{C}_{0+}}{\cosh|z|\cosh|\tilde{z}|}\sum^{\infty}_{m=0}\tanh^{m}|z|\tanh^{m}|\tilde{z}|.$
(31)

Taking the partial trace of $\tau^{g|\tilde{g}}$ with respect to two out of
three indices at a time $l,r,s$ to get three reduced transition matrices.

$\begin{split}\tau^{g|\tilde{g}}_{l}&=Tr_{s}(Tr_{r}\tau^{g|\tilde{g}})\\\
\tau^{g|\tilde{g}}_{l}&=\frac{\frac{\tilde{C}_{0+}}{\cosh|z|\cosh|\tilde{z}|}\sum^{\infty}_{n_{l}=0}\tanh^{n_{l}}|z|\tanh^{n_{l}}|\tilde{z}||n_{l}\rangle\langle
n_{l}|}{\langle g|\tilde{g}\rangle}\end{split}$ (32)

$\begin{split}\tau^{g|\tilde{g}}_{r}&=Tr_{s}(Tr_{r}\tau^{g|\tilde{g}})\\\
\tau^{g|\tilde{g}}_{r}&=\frac{\frac{\tilde{C}_{0+}}{\cosh|\tilde{z}|\cosh|\tilde{z}|}\sum^{\infty}_{n_{r}=0}\tanh^{n_{r}}|z|\tanh^{n_{r}}|\tilde{z}||n_{r}\rangle\langle
n_{r}|}{\langle g|\tilde{g}\rangle}\end{split}$ (33)

$\begin{split}\tau^{g|\tilde{g}}_{s}&=Tr_{l}(Tr_{r}\tau^{g|\tilde{g}})\\\
\tau^{g|\tilde{g}}_{s}&=|\chi_{\uparrow}\rangle\langle\chi_{\uparrow}|\end{split}$
(34)

We note that the reduced transition matrix for right and left degrees of
freedom is effectively the same. Also, the reduced transition matrix for
spinorial degrees of freedom is a pure state hence, its entanglement entropy
would be zero ($S_{SVD}^{s}=0$), this means that there is no spin-orbit
entanglement between the discrete and continuous degrees of freedom of both
ground states. This value of zero coincides with the left-handed regime von
Neumann entanglement entropy for $s$ degree of freedom given in [26].
The density matrices for the reduced transition matrices according to the
scheme given in [2] will be:

$\begin{split}\rho^{g|\tilde{g}}_{r}&=(1-\tanh|z|\tanh|\tilde{z}|)\sum^{\infty}_{n_{r}=0}(\tanh^{n_{r}}|z|\tanh^{n_{r}}|\tilde{z}||n_{r}\rangle\langle
n_{r}|)\\\
\rho^{g|\tilde{g}}_{l}&=(1-\tanh|z|\tanh|\tilde{z}|)\sum^{\infty}_{n_{l}=0}(\tanh^{n_{l}}|z|\tanh^{n_{l}}|\tilde{z}||n_{l}\rangle\langle
n_{l}|)\end{split}$ (35)

Now we calculate the SVD entanglement entropy [2]

$\begin{split}S_{SVD}^{l\>or\>r}&=-Tr(\rho^{g|\tilde{g}}_{l\>or\>r}\>\log\rho^{g|\tilde{g}}_{l\>or\>r})\\\
S_{SVD}^{l\>or\>r}&=\frac{\tanh|z|\tanh|\tilde{z}|}{\tanh|z|\tanh|\tilde{z}|-1}\log(\tanh|z|\tanh|\tilde{z}|)-\log(1-\tanh|z|\tanh|\tilde{z}|)\end{split}$
(36)

when $|z|\approx|\tilde{z}|$,

$S_{SVD}^{l\>or\>r}=\sinh^{2}|z|\log(1+\mathop{\mathrm{cosech^{2}}}|z|)+\log(\cosh^{2}|z|)$
(37)

which is exactly equal to the von Neumann entropy of the left-handed regime
given in [26].

Let us focus on the domain of SVD entropy that is $|z|$ and $|\tilde{z}|$. The
values for $z$ only exist in one half of the domain (the domain is the
coupling strength ratio $\tilde{\xi}/\xi$) and the values for $\tilde{z}$ only
exist in the other half of the domain (Figure 2). This shows that the SVD
entropy as a function of both $z$ and $\tilde{z}$ is only valid for points
close to the critical point (where $|z|\approx|\tilde{z}|$ and
$\tilde{\xi}/\xi\approx 1$) when it is plotted against the coupling strength
ratio.

Figure 2: $z$ and $\tilde{z}$ are plotted as a function of the coupling
strength ratio. Note that their value diverges at the critical region and the
graphs are not mirror images of each other, there is an asymmetry. Figure 3:
SVD entropy for $l$ and $r$ chiral degrees of freedom is plotted as a function
of the coupling strength ratio. The entanglement entropy of the chiral degrees
of freedom with the rest of the system diverges at the critical region. Figure
4: This figure shows more closely the graph for SVD entropy. Near the critical
point, the values for SVD entropy become more valid. The point where the graph
cuts off is where the value is diverging. Note that the asymmetry present in
Figure 1 is also manifest here Figure 5: This is a rough sketch of Figure 4.
The pie charts under the graph represent the degrees of freedom of the ground
states ($l,r,s$). The dotted arrow line indicates entanglement for those
degrees of freedom across the critical point. Since the SVD entanglement
entropy for spinorial d.o.f. is 0, there is no arrow connecting those across
the critical point. Figure 6: Comparison between SVD entropy and von Neumann
entropy for different chiral degrees of freedom. For the left-handed regime,
the von Neumann entropy for $l$ and $r$ degrees of freedom is equal [26]. Note
how the values of left-regime von Neumann and the SVD entropy have the same
behavior near the critical point. Figure 7: This figure shows more closely the
comparison of behaviors between both the definitions of entropies.

The SVD entanglement entropy is plotted (Figure 3) by considering the values
of $z$ and $\tilde{z}$ outside their domain as we have assumed
$|z|\approx|\tilde{z}|$. The most accurate representation of SVD entanglement
entropy is near the critical point (Figure 4). The schematic in Figure 5
explains the plot in Figure 4, it shows how the entanglement is across the
states at the critical point in the quantum phase transition. Since at the
critical point, we can assume that for some value of the coupling strength
ratio near the critical point we have approximately equal values of $|z|$ and
$|\tilde{z}|$. Numerically, we have taken the values of $\tilde{z}$ starting
from the critical point and used those together with the values of
$z$(starting from near the origin) to plot the graph for the left-hand side of
Figure 3 and similarly, we have taken values from near the origin for $z$
along with values for $\tilde{z}$ (starting from the critical point) to plot
the right-hand side of Figure 3.
As shown above for the $|z|\approx|\tilde{z}|$ case we see that the value for
the SVD entanglement entropy matches the von Neumann entropy given in [26].
This becomes apparent near the critical point of the plot Figure 7. Figure 6
and Figure 7 is the plot for comparison between von Neumann and SVD
entanglement entropy. Note that the value for SVD entanglement entropy is
different in the right-handed regime, as compared to the results for von
Neumann one. This is an important point of distinction between both the
entropies. We also see that SVD entanglement entropy gives us one value for
entropy for both $l$ and $r$ degrees of freedom and also for the left-handed
and right-handed regime, which arguably makes it a better parameter to study
quantum phase transition.

##  4 CONCLUSION AND DISCUSSION

We examined a relativistic spin-1/2 Dirac oscillator in a constant magnetic
field. The relativistic Hamiltonian can be mapped to Jaynes-Cummings and
anti–Jaynes-Cummings terms [26], describing interactions between the spinor
and chirally intrinsic bosons. These models are crucial in several areas of
theoretical and experimental research, and reveal the interplay between
opposite chiralities. A. Bermudez et al. in [26] showed us two distinct phases
under weak and strong magnetic fields, each with opposite chirality. In the
intermediate regime, a complex interaction between chiralities leads to a
quantum phase transition. In this work we discuss the entanglement entropy of
the hybrid ground states obtained from this system across the phase transition
in more detail using the SVD entanglement entropy.

The SVD entanglement entropy is a generalization of the von Neumann entropy.
It uses a preselection and a post-selection state to calculate the
entanglement entropy of the system. Arthur J. Parzygnat et al. in [2] extended
the von Neumann entropy concept from density matrices to general square
matrices, defining it as SVD entropy. They showed that SVD entropy possesses
similar properties to von Neumann entropy, including additivity, invariance
under two-sided independent unitary transformations, and a modified form of
concavity. In addition to the examples provided in [2] we provided a general
example (or proof) for SVD entanglement entropy of Bell states which turns out
to be $\log 2$ similar to the von Neumann case.

The entanglement properties that were brought forth from the SVD definition,
turned out to show reasonably similar behavior as the von Neumann case, the
values diverge at the critical point which is indicative of the signature of a
quantum phase transition. Hence, both definitions are qualitatively similar at
times but quantitative differences occur. Further applications of SVD
entanglement entropy can be seen in quantum information and quantum
computation theory. A relation between entropy and computational complexity
can be sought out through the SVD entanglement entropy between a quantum
algorithm’s initial and final states. Further applications of SVD entanglement
entropy as an indicator of phase transition can be done in a geometrical
spacetime context, wherein the phase transition of a particular quantum field
is triggered by geometric changes in the background spacetime.

Entanglement is seen as a measure of complexity from an information theory
point of view [1], for the entanglement entropy to diverge at the critical
point, it means that the complexity and the quantum correlations of the
degrees of freedom are divergent due to entanglement between the degrees of
freedom of left-handed and right-handed regime ground state. This sort of
behavior intuitively makes sense in a classical system exhibiting phase
transition, as at the critical point the system undergoes a transition hence
the information at the critical point of the system is in-deterministic
because of its transitory nature [27, 28, 29, 30]. For a quantum system, it is
explicitly shown by our work and [26] how the phase transition happens using
entropic calculations, through first principle rather than statistical
arguments.

##  5 ACKNOWLEDGEMENT

One of the authors (YS) has been supported by the Summer Research Program-2024
by S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt
Lake, Kolkata 7000106, India.

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