# CLASSICAL CORRELATION AND QUANTUM ENTANGLEMENT IN THE MIXED-SPIN ISING-XY
MODEL WITH DZYALOSHINSKII-MORIYA INTERACTION

HAMID ARIAN ZAD  Young Researchers and Elite Club, Mashhad Branch, Islamic
Azad University, Mashhad, Iran<EMAIL_ADDRESS>HOSSEIN
MOVAHHEDIAN  Department of Physics, Shahrood University of Technology,
3619995161, Shahrood, Iran

###### Abstract

In the present work, initially a mixed-three-spin (1/2,1,1/2) cell of a
mixed-$N$-spin chain with Ising-XY model is introduced, for which pair spins
(1,1/2) have Ising-type interaction and pair spins (1/2,1/2) have both XY-type
and Dzyaloshinskii-Moriya(DM) interactions together. An external homogeneous
magnetic field $B$ is considered for the system in thermal equilibrium.
Integer-spins have a single-ion anisotropy property with coefficient $\zeta$.
Then, we investigate the quantum entanglement between half-spins (1/2,1/2), by
means of the concurrence. Classical correlation(CC) for this pair of spins is
investigated as well as the concurrence and some interesting the temperature,
the magnetic field and the DM interaction properties are expressed. Moreover,
single-ion anisotropy effects on the correlation between half-spins is
verified. According to the verifications based on the communication channels
category by D. Rossini, V. Giovannetti and R. Fazio [63], we theoretically
consider such tripartite spin model as an ideal quantum channel, then
calculate its information transmission rate and express some differences in
behaviour between this suggested model and introduced simple models in the
previous works(chains without spin integer and DM interaction) from
information transferring protocol point of view.

###### keywords:

quantum entanglement; classical correlation; channel capacity; Dzyaloshinskii-
Moriya interaction.

##  1 Preliminaries

Heretofore, there are a lot of interests to investigate the various
correlations (whether quantum or classical) for an ideal system [1, 2, 3, 4,
5]. If we would like to verify the quantum correlation between parts of a
system then may be bound us to investigate the entanglement. Quantum
entanglement is a special property which can exist only in the quantum systems
[6, 7, 8, 9]. Thereby, most of researchers confine themselves to verify the
entanglement to understand the behaviour of such systems in the various
situations [9, 10, 11, 12, 13, 14, 15]. In this way, spin models are ideal
candidates for generating and manipulating of entangled states and for
studying the entanglement [3, 16, 17, 18, 19, 20, 21] by verification some
stimulating quantities such as, the concurrence [16, 17, 18], negativity [22,
23, 24], quantum discord [25, 26, 27, 28, 29, 30], quantum disorder [31],
correlation functions [3, 28] and von Neumann entropies [11, 32, 33, 34].
Somewhere, spin models have been studied with the DM interaction [4, 35, 36,
37, 38, 39], that such interaction arises naturally in the perturbation theory
due to the spin-orbit coupling in magnetic systems.

Straightforward researches have been caried out to investigate interaction
between the next-nearest-neighbour sites of a Heisenberg spin model in Refs.
[40, 41, 42, 43]. Such interaction may has an essential role to generate a
Heisenberg model with diamond chain topology by organizing mixture of
particles that have different spins. Motivated by this issue, several studies
have been done on the mixture of different spins with various models and many
interesting results have been reported [44, 45, 46]. Diamond chains as
attractive structures among these spin models were exactly investigated from
quantum entanglement, quantum correlation, phase transitions etc. view points
[47, 48, 49, 50, 51, 52].

The motivation for the study of a diamond chain with the Ising-XXZ model is
that it can describes real materials such as natural mineral azurite
$Cu_{3}(CO_{3})_{2}(OH)_{2}$, where according to experimental results,
theoretical calculations are interestingly reasonable in this case
[53](another polymeric coordination compounds such as $M_{3}(OH)_{2}$ with
spin-1 Heisenberg diamond chain were investigated in the literature [54]).
Another quantum spin models consisting of diamond-shaped cells can be
theoretically suggested and solved. In this regard, we here are interested to
introduce a few body diamond chain with specific model and verify its
bipartite CC and also entanglement in the some physical situations.

In our previous works [24, 30], we analyzed bipartite quantum entanglement in
the mixed-three-spin system (1/2,1,1/2) with two different ‘XXX Heisenberg’and
‘Ising-XY’models in the vicinity of an external homogeneous magnetic field.
This paper has been devoted to verify CC and the quantum entanglement between
half-spins (1/2,1/2) of same as the second model for which an additional DM
interaction is considered between half-spins. Some interesting temperature,the
magnetic field, the DM interaction and the other applied coefficients
properties especially single-ion anisotropy related to the integer spin are
expressed. The main purpose of this work is to provide the exact solution for
the generalized version of the mixed spin-1/2 and spin-1 Ising-XY diamond
chain, which should bring a deep insight into how the thermal and the magnetic
properties depend on the spins-1/2 and spin-1 of the model, in order words, we
are going to understand the spin-1 existence has how much physical effects on
the correlation between spins-1/2.

Forthermore, the suggested model is considered as a memoryless communication
channel between hypothetical sender Alice and the receiver Bob, then quantum
information transmission rate $\mathcal{R}$ [55, 56, 57] is numerically
verified. The ratio $\mathcal{R}$ describes the maximum number of qubits one
can transfer through the channel per unit of time. Before, we proved that
similar mixed-spin model can be considered as a communication channel for
transferring qutrits [58].

The paper is organized as follows. In the next section, we first characterize
the concurrence as a measure of entanglement and CC between the spins
(1/2,1/2), also we have a quick look at the model as a communication
channel(Sec. 3). In Sec. 4 we define our favorite model with an analytical
Hamiltonian and get its eigenvectors and eigenvalues. Then, we extract density
matrix of the bipartite spins (1/2,1/2) from density matrix of the mixed-
three-spin system in representation of the basis states. In Sec. 5, we show
the numerical calculations and simulations of the concurrence and CC between
the spins (1/2,1/2), with respect to the temperature, the magnetic field, the
coupling constant $J$, the single-ion anisotropy $\zeta$, the DM interaction
$D$ and the anisotropy parameter $\gamma$ associated to the XY interaction.
Also, information transmission rate of the mixed-three-spin chain channel is
theoretically investigated. Section 6 is devoted to discussions and a summary
of conclusions.

##  2 Introduction to the Concurrence and the Classical Correlation(CC)

###  2.1 Concurrence

The concurrence that is a measure of entanglement, can be defined for
bipartite spin systems as

$\mathcal{C}_{12}(\rho)=max\\{0,2\lambda-\sum^{4}_{i=1}\lambda_{i}\\},$ (1)

where $\lambda=\max\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$ and
$\lambda_{i}$ are square roots of the eigenvalues of the inner product

$R=\rho\tilde{\rho},$ (2)

with

$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y}){\rho^{\dagger}}(\sigma_{y}\otimes\sigma_{y}),$
(3)

where in the basis states
$\\{\mid{00}\rangle,\mid{01}\rangle,\mid{10}\rangle,\mid{11}\rangle\\}$, the
density matrix of a quantum system with Hamiltonian $H$ in thermal equilibrium
is defined as

$\rho_{eq}=\frac{\exp(-\beta H)}{Tr[\exp(-\beta H)]},$ (4)

where $\beta=1/T$(we set $k_{B}=1$) in which $T$ is the temperature and
$Z=Tr[\exp(-\beta H)]$ is the partition function of the system.
${\rho^{\dagger}}$ denotes the complex conjugation of the density matrix
$\rho$ [16, 17] and

$\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\ i&0\\\ \end{array}\right).$ (5)

Hitherto, the concurrence was explicitly calculated and simulated in terms of
the temperature, the magnetic field, the DM interaction etc. for the various
spin models. In the some of references cited here and references therein, it
has been mentioned that $\mathcal{C}(\rho)$ behaves as sudden death at
striking critical points, which is called “entanglement sudden death”(also
review Refs. [59, 60]). In the following, we investigate the concurrence
changes with respect to the temperature, the magnetic field, the anisotropy
coefficient $\gamma$, the single-ion anisotropy $\zeta$ and the DM interaction
$D$. Moreover, we would like know, that what is the effect of spin-1 existence
in the system on the temperature, the magnetic field and the DM interaction
dependences of the concurrence corresponding to the spins (1/2,1/2),
consequently we extract some interesting outcomes.

###  2.2 Classical correlation

We here recall the concept of CC for the spins (1/2,1/2) briefly. Total
correlation in a bipartite system formed by (sub)systems $\mathscr{A}$ and
$\mathscr{B}$ in a composite Hilbert space
${H}_{bi}={H}^{\mathscr{A}}\otimes{H}^{\mathscr{B}}$ is quantified by the
quantum mutual information [9, 10, 12] as

$\mathcal{I}(\rho_{\mathscr{A}}:\rho_{\mathscr{B}})=S(\rho_{\mathscr{A}})+S(\rho_{\mathscr{B}})-S(\rho_{\mathscr{A}\mathscr{B}}),$
(6)

where, $S(\rho)=-Tr[\rho\log_{2}(\rho)]$ is the von Neumann entropy in which
$\rho_{\mathscr{A(B)}}=Tr_{\mathscr{A(B)}}(\rho)$. The quantum mutual
information includes quantum information and classical one (see Refs. [9,
61]). After a measurement on one of the (sub)systems such as $\mathscr{A}$,
the amount of information obtained about the another (sub)system $\mathscr{B}$
is defined as CC. CC can be defined in terms of POVM measurement [3, 5, 28].
Let us consider a set of projective measurements $\\{\mathscr{B}^{\kappa}\\}$
performed locally only on part $\mathscr{B}$ then, the probability of
measurement outcome $\kappa$ is defined as

$p_{\kappa}=Tr_{\mathscr{A}\mathscr{B}}[(I^{\mathscr{A}}\otimes\mathscr{B}^{\kappa})\rho_{\mathscr{A}\mathscr{B}}(I^{\mathscr{A}}\otimes\mathscr{B}^{\kappa})],$
(7)

where $I^{\mathscr{A}}$ denotes the identity operator for the (sub)system
$\mathscr{A}$. After this measurement, state of the subsystem $\mathscr{A}$ is
described by the conditional density operator

$\rho_{\kappa}=\frac{1}{p_{\kappa}}[(I^{\mathscr{A}}\otimes\mathscr{B}^{\kappa})\rho_{\mathscr{A}\mathscr{B}}(I^{\mathscr{A}}\otimes\mathscr{B}^{\kappa})].$
(8)

The projectors $\mathscr{B}^{\kappa}$ can be characterized as
$\mathscr{B}^{\kappa}=V\Pi^{\kappa}V^{\dagger}$ where,
$\Pi^{\kappa}=|\kappa\rangle\langle\kappa|$ at which $\kappa=\\{0,1\\}$. We
parametrize the matrix $V$ as

$V=\left(\begin{array}[]{cc}cos(\frac{\theta}{2})&e^{-i\phi}sin(\frac{\theta}{2})\\\
e^{i\phi}sin(\frac{\theta}{2})&-cos(\frac{\theta}{2})\\\ \end{array}\right),$
(9)

where $V\in U(2)$, $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$. We define
the suprimum of the difference between the von Neumann entropy
$S(\rho_{\mathscr{A}})$ and the based-on-measurement(POVM) quantum conditional
entropy
$S(\rho_{\mathscr{A}\mathscr{B}}|\\{\mathscr{B}^{\kappa}\\})=\sum_{\kappa}p_{\kappa}S(\rho_{\kappa})$
of the subsystem $\mathscr{A}$ as

$CC(\rho_{\mathscr{A}\mathscr{B}})=sup_{\\{\mathscr{B}^{\kappa}\\}}\\{S(\rho_{\mathscr{A}})-S(\rho_{\mathscr{A}\mathscr{B}}|\\{\mathscr{B}^{\kappa}\\})\\},$
(10)

where
$S_{min}(\rho_{\mathscr{A}\mathscr{B}})=min_{\\{\mathscr{B}^{\kappa}\\}}S(\rho_{\mathscr{A}\mathscr{B}}|\\{\mathscr{B}^{\kappa}\\})$.
CC has been precisely verified in Ref. [26].

##  3 Channel Capacity

Study of the classical information and communication channels was first
characterized by Shannon [62] and its quantum analogous was promoted by von
Neumann [32] and more studies have been devoted to these regimes in the last
decades [33, 56, 63, 64, 65, 66, 67, 68]. The information sent via quantum
communication channels is carried by quantum states(qubits), classical
information(bits) can also be transmitted through quantum channels, namely,
any channel that is able to transmit quantum information can be likewise used
for transmitting classical information. One can find profound detections about
quantum communication channels category in Refs. [68, 69].

Recently, it was proposed for using simple spin chains with specific models as
communication channels [55, 70, 71, 72]. Some interesting suggested models as
communication channel are included Pauli channels [73, 74], depolarizing
channels, dephasing channels [72], spin chains channels [55, 70],
electromagnetic channels [75] and Gaussian channels [64, 76, 77]. Channel
capacity is the maximum rate of a communication channel which information can
be reliably carried. Hence, researchers are interested to study on the
capacity of a channel with memory or memoryless for transmitting or storing
unknown quantum states [69].

When an arbitrary state $\rho$ is propagated through a communication channel
with capacity, it can be wholly characterized by designing mapping protocol as
the bellow form

$\mathcal{M}:\rho_{i}\longrightarrow\rho_{f}=\mathcal{M}[\rho_{i}],$ (11)

where $\rho_{i}$ is the initial state and $\rho_{f}$ is the mapped state
through channel. Such mapping is performed by unitary transformation operator
related to the feature of the channel as

$\mathcal{M}[\rho]=U\rho U^{\dagger}.$ (12)

Caricature of this protocol is shown in Fig. 1.

Figure 1: Caricature of the transferring information through a mixed-spin
chain as a quantum communication channel.

This paper puts substantial limits on the amount of information that can be
transmitted reliably along a mixed-three-spin chain memoryless channel. The
action of a generic quantum channel denoted on a single system as
$\mathcal{E}_{1}$ can be defined as $\mathcal{E}_{N}=\mathcal{E}_{1}^{\otimes
N}$, where $N$ represents the number of channel uses. The quantum capacity
$\mathcal{Q}$ measured in qubits per channel use, is defined as

$\mathcal{Q}=\max\limits_{N\rightarrow\infty}\frac{\mathcal{Q}_{N}}{N},$ (13)

where
$\mathcal{Q}_{N}=\max_{\rho}\big{[}\mathcal{S}(\mathcal{E}_{N}(\rho))-\mathcal{S}_{e}(\rho,\mathcal{E}_{N})\big{]}$,
which denotes the maximum coherent information. $\mathcal{S}$ represents the
von Neumann entropy, and $\mathcal{S}_{e}$ is the entropy exchange, namely
$\mathcal{S}_{e}(\rho,\mathcal{E})=\mathcal{S}\big{(}(\mathds{1}_{\mathcal{H}}\otimes\mathcal{E})(\mid\Psi_{\rho}\rangle\langle\Psi_{\rho}\mid)\big{)}$,
and state $\mid\Psi_{\rho}\rangle$ is any purification of $\rho$ by means of a
reference quantum system $\mathcal{H}$, i.e.
$\rho=Tr_{\mathcal{H}}\big{[}\mid\Psi_{\rho}\rangle\langle\Psi_{\rho}\mid)\big{]}$.

The transfer protocol can be as the follows: (I) sender (Alice) applies a SWAP
operation $S_{A}$ for the set of unknown states
$|\psi_{n}\rangle_{A}=\mid{\psi_{n},\cdots\psi_{3},\psi_{2},\psi_{1}}\rangle$,
and first part of the spin chain $C_{A}$ (see Fig. 2). Here, it is assumed
that the spin chain is initially in a separable state
$\mid{\psi}\rangle_{C}=\mid{\downarrow,\circlearrowleft,\downarrow}\rangle$
which, here states of spins-half and spin-integer of the tripartite system are
set up in the $S_{z}$ and $J_{z}$ down states
$\mid{0}\rangle=\mid{-1/2}\rangle=\mid{\downarrow}\rangle$ and
$\mid{-1}\rangle=\mid{\circlearrowleft}\rangle$ respectively, (II) after time
$t=\tau$, the receiver (Bob) tries to recover the information sent from Alice
by means of some decoding protocols, namely, Bob will extract information sent
by applying a SWAP operator $S_{B}$. This operator couples Bob,s memory with
$C_{B}$ which corresponds to the reduced density operator for the $N$-th
qubit. In fact, after time $t=\tau$, this protocol can be described as a
completely positive trace preserving (CPTP) mapping from input state density
matrices to output state density matrices, in the form

$\rho_{A}\longrightarrow\mathcal{M}[\rho_{A}]=Tr^{T_{B}}[U(\rho_{A}\otimes\rho_{C})U^{\dagger}],$
(14)

Figure 2: Schematic representation of the mixed-three-spin chain as a quantum
communication channel.

where $\mathcal{M}$ represents the mapping scenario of a real memoryless
channel and $U$ is the unitary transformation which describes joint evolution
of the composite Hamiltonian $H_{AC}=H^{A}\otimes H^{C}$, $Tr^{T_{B}}$ denotes
to trace over all spins except receiver B.

In the present work, it is considered a scenario where the sender and the
receiver use their spins belong to the spin chain for encoding and decoding
information(classical or quantum). From communication theory point of view,
this consideration is not enough persuasive, but on the one hand, the
consequences can be treated analytically.

For doing transmission protocol, suppose that Alice has a memory with state
$|\Psi\rangle_{A}=\cdots|\psi_{3}\rangle\otimes|\psi_{2}\rangle\otimes|\psi_{1}\rangle$,
where
$|\psi_{i}\rangle=\alpha^{\prime}_{i}|\downarrow\rangle+\beta^{\prime}_{i}|\uparrow\rangle$($i=\\{1,2,3,\cdots\\}$
and $|\alpha^{\prime}_{i}|^{2}+|\beta^{\prime}_{i}|^{2}=1$). By starting the
transmission protocol($t=0$) by Alice via coupling first memory element
$|\psi_{1}\rangle$ with state of the first chain spin $C_{A}$ through SWAP
gate $S_{A}(1)$, memory element $|\psi_{1}\rangle$ will replace with the state
of $C_{A}$. We here assumed that the mixed-three-spin chain be initially in
state $|\downarrow\circlearrowleft\downarrow\rangle$. This procedure can be
characterized as

$\begin{array}[]{lcl}\big{(}\cdots|\psi_{3}\rangle\otimes|\psi_{2}\rangle\otimes|\psi_{1}\rangle\big{)}_{A}\otimes|\downarrow\circlearrowleft\downarrow\rangle_{C}\otimes\big{(}|\downarrow\rangle\otimes|\downarrow\rangle\otimes|\downarrow\rangle\cdots\big{)}_{B}\xrightarrow{S_{A}(1)}\\\
\big{(}\cdots|\psi_{3}\psi_{2}\downarrow\rangle\big{)}_{A}\big{(}\alpha^{\prime}_{1}|\downarrow\circlearrowleft\downarrow\rangle+\beta^{\prime}_{1}|\uparrow\circlearrowleft\downarrow\rangle\big{)}_{C}\big{(}|\downarrow\rangle\otimes|\downarrow\rangle\otimes|\downarrow\rangle\cdots\big{)}_{B}.\end{array}$
(15)

After time evolution $\tau$, the first memory element $|\psi_{1}\rangle$
embedded in $C_{A}$ after first SWAP $S_{A}(1)$, spreads along the mixed-
three-spin chain, thereby, the total state (15) becomes

$\begin{array}[]{lcl}\big{(}\cdots|\psi_{3}\psi_{2}\downarrow\rangle\big{)}_{A}\otimes\big{(}\big{(}\alpha^{\prime}_{1}|\downarrow\circlearrowleft\downarrow\rangle+\beta^{\prime}_{1}\Upsilon_{13}(\tau)|\downarrow\circlearrowleft\uparrow\rangle\big{)}_{C}\otimes\big{(}|\downarrow\downarrow\downarrow\cdots\rangle\big{)}_{B},\end{array}$
(16)

where

$\begin{array}[]{lcl}\Upsilon_{13}(\tau)=\leftidx_{C}\langle\downarrow\circlearrowleft\downarrow|e^{\frac{-iH\tau}{\hbar}}|\downarrow\circlearrowleft\uparrow\rangle_{C}\end{array}$
(17)

which is the probability amplitude of finding the spin up ($|\uparrow\rangle$)
in the 3-th part of the mixed-three-spin chain($C_{B}$ presented in Fig. 2).
In Sec. 5, we will use these statements to verify the transmission rate of the
channel numerically.

##  4 Suggested Spin Model and Theoretical Background

We introduce Hamiltonian of the mixed-$N$-spin system with Ising-XY model
(mixed-spin chain shown in Fig. 1) which is in an external homogeneous
magnetic field $B$, as the follows

$\begin{array}[]{lcl}H=\sum\limits_{i=1}^{M}\big{(}(1+\gamma){S}^{x}_{i,1}{S}^{x}_{i,3}+(1-\gamma){S}^{y}_{i,1}{S}^{y}_{i,3}\big{)}\\\
+\sum\limits_{i=1}^{M}\vec{B}\cdot(\vec{S}_{i,1}+\vec{S}_{i,3})+\sum\limits_{i=1}^{M}\vec{D}\cdot(\vec{S}_{i,1}\times\vec{S}_{i,3})\\\
+\sum\limits_{i=1}^{M}J({S}^{z}_{i,1}{J}^{z}_{i,2}+{J}^{z}_{i,2}{S}^{z}_{i,3})+\sum\limits_{i=1}^{M}\zeta({J}^{z}_{i,2})^{2}+\sum\limits_{i=1}^{M}\vec{B}\cdot\vec{J}_{i,2},\end{array}$
(18)

where $i$ denotes the number of mixed-three-spin cell in the chain, $\gamma$
is anisotropy parameter, $J$ is the Ising coupling between the spins (1,1/2),
$D$ is the DM interaction between spins-half of the cell and $\zeta$ is the
single-ion anisotropy parameter considered for spins-integer.
$\vec{S}=\\{{S}^{x},{S}^{y},{S}^{z}\\}$ and
$\vec{J}=\\{{J}^{x},{J}^{y},{J}^{z}\\}$ are spin operators(with $\hbar=1$)
which, are introduced as the following matrices

$\begin{array}[]{lcl}S^{x}=\frac{1}{2}\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right),{S^{y}}=\frac{1}{2}\left(\begin{array}[]{cc}0&-i\\\ i&0\\\
\end{array}\right),{S^{z}}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right),\end{array}$ (19)
$\begin{array}[]{lcl}{J^{x}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 0&1&0\\\
\end{array}\right),{J^{y}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\
i&0&-i\\\ 0&i&0\\\
\end{array}\right),{J^{z}}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\\\ \end{array}\right).\end{array}$ (20)

We here consider $M=1$ and $\vec{B}=B_{z}$ and $\vec{D}=D_{z}$, which are
homogeneous magnetic field and DM interaction in the $z$-direction. Note that
here all of introduced parameters are considered dimensionless parameters.
Eigenvectors of the Hamiltonian of the mixed-three-spin (1/2,1,1/2) chain are
given by

$\begin{array}[]{lcl}\mid{\phi_{1}}\rangle=a\mid{\uparrow,\bigcirc,\downarrow}\rangle+\mid{\downarrow,\bigcirc,\uparrow}\rangle,\quad\mid{\phi_{2}}\rangle=b\mid{\uparrow,\bigcirc,\downarrow}\rangle+\mid{\downarrow,\bigcirc,\uparrow}\rangle,\\\
\mid{\phi_{3}}\rangle=e\mid{\uparrow,\circlearrowright,\uparrow}\rangle+\mid{\downarrow,\circlearrowright,\downarrow}\rangle,\quad\mid{\phi_{4}}\rangle=f\mid{\uparrow,\circlearrowright,\uparrow}\rangle+\mid{\downarrow,\circlearrowright,\downarrow}\rangle,\\\
\mid{\phi_{5}}\rangle=g\mid{\uparrow,\bigcirc,\uparrow}\rangle+\mid{\downarrow,\bigcirc,\downarrow}\rangle,\quad\mid{\phi_{6}}\rangle=h\mid{\uparrow,\bigcirc,\uparrow}\rangle+\mid{\downarrow,\bigcirc,\downarrow}\rangle,\\\
\mid{\phi_{7}}\rangle=j\mid{\uparrow,\circlearrowleft,\downarrow}\rangle+\mid{\downarrow,\circlearrowleft,\uparrow}\rangle,\quad\mid{\phi_{8}}\rangle=k\mid{\uparrow,\circlearrowleft,\downarrow}\rangle+\mid{\downarrow,\circlearrowleft,\uparrow}\rangle,\\\
\mid{\phi_{9}}\rangle=l\mid{\uparrow,\circlearrowleft,\uparrow}\rangle+\mid{\downarrow,\circlearrowleft,\downarrow}\rangle,\quad\mid{\phi_{10}}\rangle=m\mid{\uparrow,\circlearrowleft,\uparrow}\rangle+\mid{\downarrow,\circlearrowleft,\downarrow}\rangle,\\\
\mid{\phi_{11}}\rangle=n\mid{\uparrow,\circlearrowright,\downarrow}\rangle+\mid{\downarrow,\circlearrowright,\uparrow}\rangle,\quad\mid{\phi_{12}}\rangle=o\mid{\uparrow,\circlearrowright,\downarrow}\rangle+\mid{\downarrow,\circlearrowright,\uparrow}\rangle,\end{array}$
(21)

where

$\begin{array}[]{lcl}a=-\frac{i(i-D)}{\sqrt{1+D^{2}}},\quad
b=\frac{i(i-D)}{\sqrt{1+D^{2}}},\\\
e=-\frac{1}{2}\frac{\gamma}{B+\frac{{\zeta}}{2}-\frac{1}{2}\sqrt{({\zeta}+2B)^{2}+4J({\zeta}+J+2B)+\gamma^{2}}+J},\\\
f=-\frac{1}{2}\frac{\gamma}{B+\frac{{\zeta}}{2}+\frac{1}{2}\sqrt{({\zeta}+2B)^{2}+4J({\zeta}+J+2B)+\gamma^{2}}+J},\\\
g=-\frac{1}{2}\frac{\gamma}{-\frac{1}{2}\sqrt{\gamma^{2}+4B^{2}}+B},h=-\frac{1}{2}\frac{\gamma}{\frac{1}{2}\sqrt{\gamma^{2}+4B^{2}}+B},\\\
j=-\frac{i(i-D)}{-\zeta+\sqrt{1+\zeta^{2}+D^{2}}},\quad
k=-\frac{i(i-D)}{-\zeta-\sqrt{1+\zeta^{2}+D^{2}}},\\\
l=\frac{1}{2}\frac{\gamma}{\frac{\zeta}{2}-B+\frac{1}{2}\sqrt{(\zeta-2B)^{2}+4J(\zeta+J-2B)+\gamma^{2}}+J},\\\
m=\frac{1}{2}\frac{\gamma}{\frac{\zeta}{2}-B-\frac{1}{2}\sqrt{(\zeta-2B)^{2}+4J(\zeta+J-2B)+\gamma^{2}}+J},\\\
n=\frac{i(i-D)}{-\zeta-\sqrt{1+\zeta^{2}+D^{2}}},\quad
o=\frac{i(i-D)}{-\zeta+\sqrt{1+\zeta^{2}+D^{2}}},\\\ \end{array}$ (22)

and the corresponding eigenvalues are

$\begin{array}[]{lcl}E_{1}={\zeta}+\frac{1}{2}\sqrt{1+D^{2}},\quad
E_{2}={\zeta}-\frac{1}{2}\sqrt{1+D^{2}}\\\
E_{3}=B+\frac{\zeta}{2}+\frac{1}{2}\sqrt{(\zeta+2B)^{2}+4J(\zeta+J+2B)+\gamma^{2}},\\\
E_{4}=B+\frac{\zeta}{2}-\frac{1}{2}\sqrt{(\zeta+2B)^{2}+4J(\zeta+J+2B)+\gamma^{2}},\\\
E_{5}={\zeta}+\frac{1}{2}\sqrt{\gamma^{2}+4B^{2}},\quad
E_{6}={\zeta}-\frac{1}{2}\sqrt{\gamma^{2}+4B^{2}}\\\
E_{7}=\frac{\zeta}{2}-B+\frac{1}{2}\sqrt{\zeta^{2}+D^{2}+1},\quad
E_{8}=\frac{\zeta}{2}-B-\frac{1}{2}\sqrt{\zeta^{2}+D^{2}+1}\\\
E_{9}=\frac{\zeta}{2}-B+\frac{1}{2}\sqrt{(\zeta-2B)^{2}+4J(\zeta+J-2B)+\gamma^{2}},\\\
E_{10}=\frac{\zeta}{2}-B-\frac{1}{2}\sqrt{(\zeta-2B)^{2}+4J(\zeta+J-2B)+\gamma^{2}},\\\
E_{11}=\frac{\zeta}{2}+B+\frac{1}{2}\sqrt{\zeta^{2}+D^{2}+1},\quad
E_{12}=\frac{\zeta}{2}+B-\frac{1}{2}\sqrt{\zeta^{2}+D^{2}+1}\\\ \end{array}$
(23)

In the basis states representation, the total density matrix of the considered
tripartite system which is a thermal equilibrium state can be characterized by
using latest equations. Hence, the density matrix of the pair spins (1/2,1/2)
can be expressed as

$\mathbf{\rho^{T_{2}}_{13}}=\frac{1}{Z}\left(\begin{array}[]{cccc}\delta&0&0&\varsigma\\\
0&P&\xi&0\\\ 0&\xi^{*}&Q&0\\\ \varsigma^{*}&0&0&\chi\end{array}\right),$ (24)

where $T_{2}$ is partial trace over second spin(note that the matrix is
symmetric). $\\{\delta,\varsigma,P,Q,\xi,\chi\\}$ are functions of $T$, $B$,
$D$, $\gamma$, $J$ and $\zeta$, and also mixture of components of the total
density matrix.

##  5 Exact Numerical Solution

In order to provide a detailed analytical and numerical simulation, here, we
use the concurrence as a measure of entanglement for the bipartite
(sub)system, also CC is investigated and simulated numerically as well as the
concurrence.

###  5.1 Correlation functions

With regard to the geometric of correlation functions, we will calculate the
concurrence and CC for the particular case bipartite spins (1/2,1/2) whose the
density operator is presented in the form (24). Arrays of this matrix can be
characterized as the following equations

$\begin{array}[]{lcl}\delta=\frac{1}{4}(1+\mathcal{G}^{i}_{z}+\mathcal{G}^{j}_{z}+\mathcal{G}^{ij}_{zz}),\quad
P=\frac{1}{4}(1+\mathcal{G}^{i}_{z}-\mathcal{G}^{j}_{z}-\mathcal{G}^{ij}_{zz}),\\\
Q=\frac{1}{4}(1-\mathcal{G}^{i}_{z}+\mathcal{G}^{j}_{z}-\mathcal{G}^{ij}_{zz}),\quad\chi=\frac{1}{4}(1-\mathcal{G}^{i}_{z}-\mathcal{G}^{j}_{z}+\mathcal{G}^{ij}_{zz}),\\\
\varsigma=\frac{1}{4}(\mathcal{G}^{ij}_{xx}-\mathcal{G}^{ij}_{yy}),\quad\xi=\frac{1}{4}(\mathcal{G}^{ij}_{xx}+\mathcal{G}^{ij}_{yy}),\end{array}$
(25)

where $\mathcal{G}^{k}_{z}=\langle\sigma_{z}^{k}\rangle$ with $k=\\{i,j\\}$,
is the magnetization density at site $k$ and
$\mathcal{G}^{ij}_{\mu\nu}=\langle\sigma_{\mu}^{i}\sigma_{\nu}^{j}\rangle$
with $\mu,\nu=\\{x,y,z\\}$ denote spin-spin correlation functions at sites $i$
and $j$. Note that the expectation value can be defined as
$Tr[\rho_{13}\mathcal{G}]$. If we introduce the elements

$\begin{array}[]{lcl}\mathcal{E}_{1}=\xi+\xi^{*}+\varsigma+\varsigma^{*},\quad\mathcal{E}_{2}=\xi+\xi^{*}-\varsigma+\varsigma^{*},\\\
\mathcal{E}_{3}=\delta+\chi-
P-Q,\quad\mathcal{E}_{4}=\delta-\chi-P+Q,\quad\mathcal{E}_{5}=\delta-\chi+P-Q,\end{array}$
(26)

in accordance with the reconstructed density matrix (24) as

$\begin{array}[]{lcl}\rho^{T2}_{13}=\frac{1}{4Z}\big{(}I\otimes
I+\sum\limits_{i=1}^{3}\mathcal{E}_{i}(\sigma^{i}\otimes\sigma^{i})+\mathcal{E}_{4}(I\otimes\sigma^{3})+\mathcal{E}_{5}(\sigma^{3}\otimes
I)\big{)},\end{array}$ (27)

then, we obtain a simple equation for CC of the bipartite system represented
in Eq. (10) as

$\begin{array}[]{lcl}CC(\rho_{\mathscr{A}\mathscr{B}})=\frac{(1-\mathcal{E})}{2}\log_{2}(1-\mathcal{E})+\frac{(1+\mathcal{E})}{2}\log_{2}(1+\mathcal{E}),\end{array}$
(28)

in which
$\mathcal{E}=\max\\{|\mathcal{E}_{1}|,|\mathcal{E}_{2}|,|\mathcal{E}_{3}|\\}$.

####  5.1.1 Concurrence

The reduced density matrix presented in Eq. (24) has whatever is needed about
the bipartite spins (1/2,1/2), hence the concurrence can be readily obtained
as [78]

$\begin{array}[]{lcl}\mathcal{C}(\rho_{\mathscr{A}\mathscr{B}})=2\max\\{\max\big{(}0,|\frac{\varsigma}{Z}|-P\big{)},|\frac{\xi}{Z}|-(\frac{\delta\chi}{Z^{2}})^{\frac{1}{2}}\\}.\end{array}$
(29)

The concurrence (29) as function of the temperature $T$ and the magnetic field
$B$ at fixed values of the anisotropy, the single-ion anisotropy and DM
interaction parameters is shown in Fig. 3. As illustrated in this figure, the
concurrence at low temperature and weak magnetic field is maximum
($\mathcal{C}(\rho)=1$), on the other hand, this quantity is minimum
($\mathcal{C}(\rho)=0$) at high temperature and strong magnetic field. This
essential property of the concurrence has been studied for various spin models
in the previous works [16, 17, 38] and here it is true for our favorite
bipartite (sub)system and is compatible with the previous works.

In the some of used references, authors gained a critical temperature at which
the concurrence vanishes, but here we see that with changes of the magnetic
field, the concurrence vanishes at different critical temperatures. Indeed, by
inspecting Fig. 4, one can observe that for the various magnetic fields, the
concurrence diagrams vanish in the various critical temperatures and that is
because, the concurrence of the bipartite (sub)system strongly depends on the
some extra parameters except the temperature and the magnetic field, such as
the single-ion anisotropy and DM interaction parameters.

Figure 3: Concurrence of the spins (1/2,1/2), with respect to the temperature
and the magnetic field at fixed values of $\zeta=J$ and $D=5J$, for: (a)
$\gamma=0.2J$; (b) $\gamma=0.8J$.

Figure 4: Concurrence of the spins (1/2,1/2) as function of the temperature at
fixed values of $\zeta=J$ and $D=5J$ in the various magnetic fields for: (a)
$\gamma=0.2J$; (b) $\gamma=0.8J$.

Also, it is explicitly seen by increasing the anisotropy $\gamma$, the
concurrence vanishes at higher critical temperatures for the fixed values of
the magnetic field(blue dash line and green dot-dashed line). Meanwhile, at
weak magnetic field, the concurrence not entirely be zero in the higher
temperatures. In the stronger anisotropy $\gamma$, this phenomenon becomes
more clear(Figs. 3(b) and 4(b)). As a result, because of verifying the
entanglement for such quantum system at very low temperatures (near $T=0$) is
practically difficult, it can be easier by increasing of the anisotropy.
Indeed, one can use the anisotropy as an entanglement controller in higher
temperatures for such model.

Figure 5: Concurrence of the spins (1/2,1/2), with respect to the single-ion
anisotropy and DM interaction parameters at low temperature $T=0.15J$ and
fixed $B=J$ for: (a) $\gamma=0.2J$; (b) $\gamma=0.8J$.

Figure 5 represents the concurrence of the spins (1/2,1/2) with respect to the
single-ion anisotropy and the DM interaction parameters at low
temperature($T=0.15J$) and fixed homogeneous magnetic field $B=J$. As
illustrated in this figure, for the fixed values of the single-ion anisotropy,
with decrease of the DM interaction from its high values, the concurrence
decreases until reaches a minimum in which a sudden change occurs in a special
critical DM interaction. Here, state of the bipartite (sub)system will change,
indeed a phase transition occurs in this critical point. with increase of
$\zeta$ this critical point tends to the stronger DM interaction.

If we consider a line that connects these critical points, we see that the
concurrence increases with increase of the DM interaction for the region upper
than this line, namely, the concurrence is proportional to the DM interaction
in this region. On the other hand, for the region lower than the connection
line, the concurrence decreases with increase of the DM interaction.

This function decreases with increase of the single-ion anisotropy $\zeta$ at
fixed values of $D$ until reaches that minimum in which phase transition
occurs as we mentioned before. Hence, the concurrence reaches its maximum
value in the strong DM interaction and small single-ion anisotropy. Shape of
the concurrence digram is changed by increasing the anisotropy $\gamma$. These
properties are obviously presented in Figs. 6 and 7.

Figure 6:  Concurrence of the spins(1/2,1/2) as function of the DM interaction
parameter $D$ at low temperature $T=0.15J$ at fixed values of the single-ion
anisotropy and $B=J$ for: (a) $\gamma=0.2J$; (b) $\gamma=0.8J$.

As shown in Fig. 6(a), maximum amount of the concurrence decreases with
increase of the single-ion anisotropy $\zeta$. Moreover, in this figure it is
obviously visible, that those critical points in which the concurrence reaches
its minimum at low temperature tend to the stronger DM interaction, where
$\zeta>0$ and $\gamma$ are considered fixed. For the strong single-ion
anisotropy property $\zeta>4J$, we found that the minimum amount of the
concurrence becomes zero, namely, in this condition state of the bipartite
(sub)system is a separable state. With increase of the anisotropy parameter
$\gamma$(from 0.2 to 0.8), the critical points shift to the weaker DM
interaction(Fig. 6(b)).

In the present paper, we obliged ourselves to investigate the pairwise
entanglement of spins (1/2,1/2) from the single-ion anisotropy point of view,
which is merely considered for integer-spins in the total spin chain. Fig. 7
depicts the concurrence as function of the single-ion anisotropy $\zeta$ at
fixed values of $D$ at low temperature. As illustrated in this figure, one can
see that the pairwise entanglement sorely depends on the single-ion anisotropy
$\zeta$. With increase of the single-ion anisotropy in the tripartite system,
the concurrence of the bipartite (sub)system decreases for fixed values of
$D$.

Figure 7: Concurrence of the spins(1/2,1/2) as function of the single-ion
anisotropy parameter $\zeta$ at low temperature $T=0.15J$ at fixed values of
the DM interaction and $B=J$ for: (a) $\gamma=0.2J$; (b) $\gamma=0.8J$.

In the almost weak DM interaction(red solid line and green dot-dashed line),
the concurrence decreases by increasing the single-ion anisotropy from zero
and reaches a minimum in which phase transition occurs. With further increase
of the single-ion anisotropy, a sudden change happens in the concurrence
behaviour at a special critical single-ion anisotropy $\zeta^{c}$ and for
$\zeta>\zeta^{c}$ this quantity decreases smoothly. By increasing the
anisotropy $\gamma$, this sudden change(the concurrence minimum point) occurs
at almost weaker the single-ion anisotropy for $D=0$(red solid line in Fig.
7), while for $D>0$ happens at higher single-ion anisotropy(green dot-dashed
line). This behaviour is slightly complex but comprehensible and a reason that
make our favorite system appealing to study. In the strong DM interaction, the
concurrence decreases almost independent of the anisotropy $\gamma$ with
increase of the single-ion anisotropy parameter $\zeta$(black dot line).

Figure 8: Contour plots of the concurrence as function of the anisotropy
$\gamma$ and the DM interaction $D$ at fixed values of $\zeta=J$ at low
temperature $T=0.15J$ for: (a) $B=0$; (b) $B=J$; (c) $B=2J$; (d) $B=3J$; (e)
$B=4J$; (f) $B=5J$, for spins (1/2,1/2). Colour bars represent changes of the
concurrence from $C=0$(black regions) to $C=1$(red regions).

For better understanding the behaviour of the concurrence with respect to the
anisotropy $\gamma$ and DM interaction, we knew interest that depict the
anisotropy and the DM interaction dependences of the concurrence at low
temperatures at fixed values of the magnetic field, in the form of contour
plots as illustrated in Fig. 8. This figure shows that in this circumstances,
the concurrence is symmetric versus the anisotropy and the DM interaction
parameters for various magnetic fields. Also, it can be readily seen that in
the strong DM interaction i.e., $D>|2J|$ for $0\leq B\leq 2J$(Figs.
8(a)$-$8(c)) and $D>|BJ|$ for $2J<B\leq 4J$(Figs. 8(d)$-$8(e)), the
concurrence is maximum. In the presence of the strong magnetic field $B>4J$
the concurrence behaviour dramatically alters. We explain this exotic
behaviour in the different intervals of the magnetic field in the following.

If one follows Fig. 8 step by step then realizes that the thermal concurrence
as a measure of pairwise entanglement has an exotic behaviour versus
increasing the magnetic field. This quantity for interval $0\leq B\leq 2J$
behaves different from interval $2J<B\leq 4J$, also its behaviour for $B>4J$
is generally different from former intervals. Namely, for the first interval,
with increase of the magnetic field from zero to $2J$ the concurrence arises
at the weaker DM interaction(the blue region gradually decreases) as far as
the concurrence does not vanish even at $D=0$ and $\gamma\approx 0$. While, at
the second interval, with increase of the magnetic field, the concurrence
arises at stronger the DM interaction($D\approx|BJ|$), which width changes of
the blue region can present this exotic behaviour. Consequently, for the
strong magnetic field, the field influence on the concurrence behaviour will
overcome the DM interaction. Finally, with regard to Fig. 8(f) for $B>4J$ the
maximum amount of the concurrence is entirely limited to the DM interaction
interval $D<|BJ|$ and strong anisotropy $\gamma\approx|J|$.

Figure 9:  Surfaces of constant geometric concurrence of state $\rho(AB)$
defined in Eq. (24) at low temperature $T=0.15$ for: (a) $\zeta=J$ and
$C(\rho)=0.03$, (b) $\zeta=J$ and $C(\rho)=0.3$, (c) $\zeta=J$ and
$C(\rho)=0.8$. It is clear that the geometry of the concurrence is symmetric
versus absolute values of the DM interaction and the anisotropy, while there
is not any symmetry versus the magnetic field axes.

In Fig. 9, we plot level surfaces of geometric concurrence of the state
$\rho(AB)$ defined in Eq. (24) at low temperature and fixed value of
$\zeta=J$. Using this figure, one can recognize that in what regions seeks
existence of the entangled states for the bipartite (sub)system. Obviously,
surfaces of the constant geometric concurrence are symmetric versus DM
interaction and anisotropy $\gamma$, while we can not find any symmetry versus
the magnetic field axes. Selecting of this special model may be reson of this
symmetry breaking and one can choose a model for which symmetry always be
established.

####  5.1.2 Classical correlation

With regard to Sec. 2 and references therein, we start to explain the
configuration of the CC between the spins (1/2,1/2) as well as its
concurrence, for realizing some extra physical behaviours of the (sub)system
which are rare in the other investigated spin models in the previous works. We
here verify this quantity which can be exist in the both classical and quantum
systems as function of the various parameters then, compare it with the
concurrence. Finally, we get some interesting outcomes.

Figure 10:  CC of the spins (1/2,1/2) as function of the temperature and the
magnetic field at fixed values of $D=5J$ and $\zeta=J$ for: (a) $\gamma=0.2J$;
(b) $\gamma=0.8J$.

CC as function of the temperature and the magnetic field at fixed values of
the single-ion anisotropy and the DM interaction parameters is shown in Fig.
10. Let us divide this figure to four segments: (i) at low temperatures($T\ll
J$) and interval $0.5J\lesssim B\lesssim 3.5J$, CC is maximum and in the
outside of this interval this quantity vanishes. Also here, by increasing the
temperature from zero this quantity gradually vanishes; (ii) at interval
$0\leq B\lesssim 1$, by increasing the temperature from zero CC arises, and
with further increase of the temperature until $T\approx 3J$, this quantity
reaches a maximum value then vanishes; (iii) at $B\gtrsim 2J$, by increasing
the temperature from $T\approx 0.5J$, CC arises as far as at high temperatures
this quantity reaches another maximum then vanishes again; (iv) for $T\gtrsim
3J$ at interval $0.5J\lesssim B\lesssim 1.5J$, with increase of the
temperature, this quantity arises again and reaches a maximum smaller that
other. It is clear that, with increase of the anisotropy the maximum value of
CC decreases in the latest three region (ii), (iii) and (iv) but it increases
in the first region (i).

If we look smartly at Fig. 3 from the top perspective, we realize that for low
temperatures by increasing the anisotropy, the CC behaviour will almost
becomes similar to the quantum entanglement for the spins (1/2,1/2). The
subject matter is the behaviour of both functions at low temperature which
almost become the same with increasing the anisotropy. Indeed, the anisotropy
has the unification ability on the concurrence and CC functions at low
temperature.

CC as function of the single-ion anisotropy and the DM interaction at low
temperatures and fixed value of the magnetic field is shown in Fig. 11. As
shown in this figure, in the strong DM interaction($D\gtrsim 5J$), CC is
maximum independent of the single-ion anisotropy changes. But for the weaker
DM interaction, with increase of the single-ion anisotropy, this quantity
vanishes at a special region, then arises again and reaches another maximum.
Such region that presents a critical DM interaction domain versus the single-
ion anisotropy parameter is depicted in Fig. 12.

Figure 11:  CC contour plots for the spins (1/2,1/2) as function of the
single-ion anisotropy and the DM interaction at low temperature $T=0.15J$ and
fixed $B=J$ for: (a) $\gamma=0.2J$; (b) $\gamma=0.8J$. Figure 12:  The
critical DM interaction $D^{c}$ versus single-ion anisotropy $\zeta$ at low
temperatures $T=0.15J$ and fixed $B=J$ for: (a) $\gamma=0.2J$(boxes); (b)
$\gamma=0.8J$(circles). Note that the critical DM interaction points for case
$\gamma=0.2J$ is above of the dashed line which represents linear equation
$D=0.75\zeta$, while they are below of it for $\gamma=0.8J$.

This figure shows that by increasing the single-ion anisotropy at low
temperatures and fixed magnetic field $B=J$, CC vanishes at higher critical DM
interaction for both $\gamma=0.2J$ and $\gamma=0.8J$ where, for the case of
$\gamma=0.2J$ these critical points are above line $D=0.75\zeta$(dashed line),
but for the case of $\gamma=0.8J$, they are below this line. This is means
that, for Fig. 11(a) we have $0.75\zeta<D^{c}<\zeta$ while, for Fig. 11(b)
inequality $0.5\zeta<D^{c}<0.75\zeta$ is established(also here, there are some
points at which CC is maximum at weak DM interaction and small the single-ion
anisotropy). By comparing Fig. 11(a) with Fig. 6(a) one can gain some
likenesses and differences in behaviour of the concurrence and CC. For
example, at fixed values of $\zeta>0$ with increase of the DM interaction from
zero, CC decreases from a maximum and reaches a minimum value at a critical
point. With further increase of the DM interaction, this function increases
and reaches another maximum, just like the concurrence. By increasing $\zeta$
the critical point tend to the stronger the DM interaction. Unlike the
concurrence, both CC maxima before and after the critical point are equivalent
for $\zeta\geq 2J$. Another achievements can be obtained by comparing all
represented figures corresponding to the concurrence and CC.

###  5.2 Transmission rate

Quantum transmission rate for the introduced protocol in Fig. 2 can be
obtained with regard to the quantum channel capacity of the memoryless channel
map $\mathcal{M}[\rho]$. This capacity that was defined as the maximum amount
of the quantum information reliably transmitted per use of the channel
$\mathcal{M}$ (see Ref. [65]), was used for a spin chain channel in Ref. [63]
and is given by

$\mathcal{Q}(\eta)=\max\limits_{p\in[0,1]}\\{H_{2}(\eta
p)-H_{2}((1-\eta)p)\\},$ (30)

where, $H_{2}(X)=-X\log_{2}(X)-(1-X)\log_{2}(1-X)$ is the dyadic Shannon
entropy.

The rate of this channel for $\varepsilon$ uses in the time interval
$T=\varepsilon\tau$ can be defined as

$\mathcal{R}\equiv\lim\limits_{\varepsilon\rightarrow\infty}\frac{\varepsilon\mathcal{Q}(\eta)}{\varepsilon\tau}=\frac{\mathcal{Q}(\eta)}{\tau}.$
(31)

Assume a simple spin-1/2 Heisenberg model with general Hamiltonian $H_{G}$.
The operator of the total $z$-component of the spin, given by
$\sigma_{total}^{z}=\sum_{i}\sigma_{i}^{z}$ is conserved, namely
$\big{[}\sigma_{total}^{z},H_{G}\big{]}=0$. Hence, the Hilbert space
$\mathcal{H}_{G}$ decomposes into invariant subspaces, each of which is a
distinct orthogonal eigenvector of the operator $\sigma_{total}^{z}$. Here,
the transfer amplitude $\eta$ in Eq. (30) is obtained using confined
Hamiltonian $\mathcal{H}_{G}$. So, perfect and efficient quantum state
transfer is happened for the such quantum chain with $\mathcal{R}=1$. But, for
our suggested model with Hamiltonian (18) as a communication channel, we
obtain the transfer amplitude $\eta$ using general Hamiltonian $H_{G}$ of the
system, and prove that the general Hamiltonian of the such system with few
body is a capable operator to investigate the transmission protocol. Hence, we
focus on the maximum and minimum amount of the rate $\mathcal{R}$.

For our model, the transfer amplitude $\eta$ is a sinusoidal function of
$\tau$ just the same one in Ref. [63], but with changeable period
$\pi/2(1-iD)$, where $i=\sqrt{-1}$, namely

$\eta=|\Upsilon_{13}(\tau)|^{2}=|\sin(2(1-iD)\tau|^{2}.$ (32)

By setting Eq. (32) in Eq. (30), we can get the quantum transmission rate
(31). Figure 13 depicts this rate versus time evolution $\tau$ numerically for
various fixed values of the DM interaction. With regard to this figure,
information transferring rate roughly depends on the DM interaction, i.e., for
$D=0.05$(Fig. 13(a)) with the pass of time, maximum transmission rate
increases between time interval $0<\tau<10$ and reaches a biggest peak which
represents maximum quantum transmission rate. By increasing the DM interaction
$D$ from 0.05 to 0.5 the biggest peak of the rate occurs with the pass of less
time and its intensity increases almost ten times(from $\mathcal{R}=0.25$ to
$\mathcal{R}=2.5$).

Figure 13: Quantum transmission rate versus time $\tau$ for: (a) $D=0.05$; (b)
$D=0.1$; (c) $D=0.3$; (d) $D=0.5$.

With regard to this protocol, we can realize that an unknown qubit can be
reliably transferred through a mixed-three-spin (1/2,1,1/2) channel which
spins (1/2,1/2) have both XY and DM interactions together, without considering
spin-integer. Namely, the transmission rate is independent of the Ising
coupling between spins (1,1/2) and single-ion anisotropy parameter related to
the spin-integer. But as noted, it is roughly dependent on the DM interaction.
If one looks carefully at the Fig. 13 and follows limited time interval
$0<\tau<1$ then, can realize that with increase of the DM interaction, the
transmission rate of the channel arises gradually and then reaches its maximum
value in this interval.

##  6 Summary and conclusions

We have introduced a mixed-$N$-spin Ising-XY model, then focused on a mixed-
three-spin (1/2,1,1/2) cell of it to investigate CC and quantum entanglement
between spins (1/2,1/2) in the vicinity of a homogeneous magnetic field. Here,
both XY and DM interactions between spins (1/2,1/2) also Ising interaction
between spins (1,1/2) and a single-ion anisotropy for spin-integer have been
considered. In this paper, we restricted ourselves to a finite chain with
mixed-three-spin (1/2,1,1/2) of a larger mixed-$N$-spin chain, and because it
is generally difficult to study a spin chain with large length, small-size
systems can be good options for obtaining some information about large-size
systems. Fortunately, small-size cells of a large-size spin chain can also
depict well properties of the large-size chain such as the temperature, the
magnetic(here the DM interaction and the single-ion anisotropy for the
suggested triangular cell) and the quantum correlation properties.

In conclusion, we found that in the fixed values of $\zeta=J$ and $D=5J$, the
concurrence is maximum at low temperature and weak magnetic field, but with
increase of the temperature this quantity decreases until vanishes at a
critical temperature. This critical temperature is changed with the magnetic
field changes, namely, for various magnetic fields we have different critical
temperatures. With increase of the anisotropy, these critical points shift to
the higher temperatures.

Also, at low temperature and fixed value of $B=J$, the concurrence is maximum
at strong DM interaction and zero single-ion anisotropy, and by increasing the
single-ion anisotropy also decreasing the DM interaction this quantity
decreases until reaches a minimum at which phase transition occurs. This is
means that the concurrence as a measure of entanglement associated to the
half-spins, is roughly dependent on the single-ion anisotropy related to the
spin-1. Some critical pints of the DM interaction and the single-ion
anisotropy have been numerically presented in which the concurrence suddenly
changes.

Moreover, at low temperatures, we investigated the concurrence as function of
the DM interaction and anisotropy at various fixed values of the magnetic
field, and realized that this quantity is symmetric in the DM interaction and
the anisotropy framework and will extremely change with increase of the
magnetic field. As a result, the concurrence has a different behaviour for
field intervals $0\leq B\leq 2J$, $2J<B\leq 4J$ and $B>4J$.

Surfaces of constant geometric concurrence of the (sub)system (1/2,1/2) with
state $\rho(AB)$ at low temperature and fixed $\zeta=J$ have been presented in
this work. We concluded that these surfaces depict the position of regions
contained the entangled states properly. By using these surfaces we can easier
detect circumstances in which the (sub)system state is entangled.

In that follows, we verified CC for the bipartite spins (1/2,1/2), and
understood that at fixed values of $\zeta=J$ and $D=5J$ this term has a exotic
behaviour in the temperature and the magnetic field framework. As an
interesting outcome, we showed that at low temperatures, by increasing the
anisotropy, the behaviour of CC will almost be same as the concurrence. Also,
this term is investigated at low temperature and fixed magnetic field $B=J$
with respect to the DM interaction and the single-ion anisotropy. Hence, we
gained a set of the critical DM interaction in which CC is vanished, and by
increasing the anisotropy $\gamma$ range of this set of the critical points is
changed in the DM interaction and the single-ion anisotropy framework(Fig.
12).

Finally, we theoretically considered the tripartite mixed-spins (1/2,1,1/2) as
a communication channel with capacity, then analyzed quantum information
transmission rate of it. To perform the transmission protocol, we assumed that
an unknown qubit can be reliably transferred through this channel. Here, among
the introduced parameters for the favorite system, the transmission rate is
just dependent on the DM interaction and that is because of the special Ising-
XY considered model for the spin chain which is not mentioned in the previous
works. As another interesting result, we showed that by increasing the DM
interaction between spins (1/2,1/2), maximum transmission rate occurs at the
less time interval and this channel without considering the spin-1, can
reliably transfer quantum information.

One can compute another properties of the channel such as amplitude damping
channel, entanglement-assisted classical capacity of the channel etc. for our
suggested model and obtains some interesting outcomes.

## References

## References

  * [1] T. Werlang, C. Trippe, G. A. P. Ribeiro and G. Rigolin, Phys. Rev. Lett. 105 (2010) 095702.
  * [2] G. F. Zhang, Z. T. Jiang and A. Abliz, Annals of Physics 326 (2011) 867-875.
  * [3] T. R. d. Oliveira, A. Saguia and M. S. Sarandy, Eur. Phys. Lett. 100 (2012) 60004.
  * [4] G. F. Zhang, H. Fan, A. L. Ji, Z. T. Jiang, A. Abliz and W. M. Liu, Annals of Physics 326 (2011) 2694-2701.
  * [5] P. R. Wells Jr., C. M. Chaves, J. A. e. Castro and B. Koiller, arXiv:1305.5818v1.
  * [6] M. Horodecki, J. Oppenheim and A. Winter, arXiv:0505062v1.
  * [7] M. Horodecki, J. Oppenheim and A. Winter, Communications in Mathematical Physics 269 (2007) 107.
  * [8] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. 81 (2009) 865-942.
  * [9] N. J. Cerf and C. Adami, Physica D 120 (1998) 62-81.
  * [10] N. J. Cerf and C. Adami, Fundamental Theories of Physics 81 (1997) 77-84.
  * [11] N. J. Cerf and C. Adami, Phys. Rev. Lett. 79 (1997) 5194.
  * [12] K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein, Phys. Rev. A 58 (1998) 883.
  * [13] G. Evenbly, R. N. C. Pfeifer, V. Pico, S. Iblisdir, L. Tagliacozzo, I. P. McCulloch and G. Vidal, Phys. Rev. B 82 (2010) 161107.
  * [14] M. Kargarian, R. Jafari and A. Langari, Phys. rev. A 77 (2008) 032346.
  * [15] A. Saif, M. Hassan, B. Lari and P. S. Joag, J. Phys. A: Math. Theor. 43 (2010) 485302.
  * [16] W. K. Wootters, Quant. Info. and Comp. 1 (2001) 27-44.
  * [17] M. C. Arnesen, S. Bose and V. Vedral, Phys. Rev. Lett. 87 (2001) 017901.
  * [18] G. L. Kamta and A. F. Starace, Phys. Rev. Lett. 88 (2002) 107901.
  * [19] G. F. Zhang, Phys. Rev. A 72 (2005) 034302.
  * [20] X. Wang and P. Zanardi, Phys. Lett. A 301 (2002) 1-6.
  * [21] G. Rigolin, Int. J. Quant. Inf. 2, (2004) 393.
  * [22] G. Vidal and R. F. Werner, Phys. Rev. A 65 (2002) 032314.
  * [23] X. Wang and S. J. Gu, J. Phys. A: Math. Theor 40 (2007) 10759.
  * [24] H. A. Zad, Chin. Phys. B 25 (2016) 030303.
  * [25] H. Ollivier and W.H. Zurek, Phys. Rev. Lett. 88 (2001) 017901.
  * [26] K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral, Rev. Mod. Phys. 84 (2012) 1655.
  * [27] M. S. Sarandy, T. R. d. Oliveira and L. Amico, Int. J. Mod. Phys. B 27 (2013) 1345030.
  * [28] M. S. Sarandy, Phys. Rev. A 80 (2009) 022108.
  * [29] Z. Xi, X. M. Lu, X. Wang and Y. Li, J. Phys. A: Math. Theor 44 (2011) 375301.
  * [30] H. A. Zad, Acta Phys. Pol. B 46 (2015) 1911-1924.
  * [31] H. L. C. Grande, N. Laﬂorencie, F. Alet and A. P. Vieira, Phys. Rev. B 89 (2014) 134408.
  * [32] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Springer Verlag, Berlin, 1932.
  * [33] N. J. Cerf and C. Adami, arXiv:9605002.
  * [34] M. M. Wilde, From Classical to Quantum Shannon Theory, 2012, arXiv:110601445v4.
  * [35] I. Dzyaloshinskii, J. Phys. Chem. Solids 4 (1958) 241.
  * [36] T. Moriya, Phys. Rev. Lett. 4 (1960) 228.
  * [37] Da. C. Li and Z. L. Cao, Int. J. Quantum Inf. 7 (2009) 547.
  * [38] Da. C. Li and Z. L. Cao, Eur. Phys. J. D 50 (2008) 207.
  * [39] D. C. Li, X. P. Wang and Z. L. Cao, J. Phys. Condens. Matter 20 (2008) 325229.
  * [40] X.Y. Chen, Q. Jiang, W.Z. Shen and C.G. Zhong, Journal of Magnetism and Magnetic Materials 262 (2003) 258–263.
  * [41] S. J. Gu, H. Li, Y. Q. Li and H. Q. Lin, Phys. Rev. A 70 (2004) 052302.
  * [42] I. T. Shyiko, I. P. McCulloch, J. V. G. Sichevska and A. K. Kolezhuk, Phys. Rev. B 88 (2013) 014403.
  * [43] L.C. Kwek, Y. Takahashi and K.W. Choo,J. of Phys.: Conf. Series 143 (2009) 012014.
  * [44] N. B. Ivanov, Condens. Matt. Phys. 12 (2009) 435–447.
  * [45] S. Yamamoto and H. Hori, Phys. Rev. B 72 (2005) 054423.
  * [46] R. Jafari and A. Langari, Int. J. Quant. Inf. 9 (2011) 1057-1079.
  * [47] N. B. Ivanov, J. Richter and J. Schulenburg, Phys. Rev. B 79 (2009) 104412.
  * [48] O. Rojas, S. M. de Souza, V. Ohanyan and M. Khurshudyan, Phys. Rev. B 83 (2011) 094430.
  * [49] N. S. Ananikian, L. N. Ananikyan, L. A. Chakhmakhchyan and O. Rojas, J. Phys.: Condens. Matt. 24 (2012) 256001.
  * [50] S. D. Han and E. Aydiner, Chin. Phys. B 23 (2014) 050305.
  * [51] L. Canova, J. Strecka and T. Lucivjansky, Condens. Matt. Phys. 12 (2009) 353.
  * [52] O. Rojas, M. Rojas, N. S. Ananikian and S. M. d. Souza, Phys. Rev. A 86 (2012) 042330.
  * [53] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo et al., Physica B 967 (2003) 329-333; H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo it et al., Phys. Rev. Lett. 94 (2005) 227201; H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo it et al., Progr. Theor. Phys. 159 (2005) 1.
  * [54] M. L. Tong, J. Wang, S. Hu, J. Solid St. Chem., 178 (2005) 1518.
  * [55] D. Rossini, V. Giovannetti and R. Fazio, Int. J. Quant. Inf. 5 (2007) 439.
  * [56] M. Demianowicz and P. Horodecki, Phys. Rev. A 74 (2006) 042336.
  * [57] D. Burgarth, V. Giovannetti and S. Bose J. Phys. A: Math. Gen. 38 (2005) 6793
  * [58] H. A. Zad, 2016 Chin. Phys. B 25 080300\.
  * [59] K. Ann and G. Gaegger, Phys. Lett. A 372 (2008) 579–583.
  * [60] T. Yu and J. H. Eberly, Science, 323 (2009) 598-601.
  * [61] S. M. Barnett, Quantum Information, Oxford university press, New York, 2009.
  * [62] C. E. Shannon, Bell Sys. Tech. J. 27 (1948) 379.
  * [63] D. Rossini, V. Giovannetti and R. Fazio, Int. J. Quant. Inf. 5 (2007) 439.
  * [64] A. S. Holevo and V. Giovannetti, Rep. Prog. Phys. 75 ( 2012) 046001.
  * [65] C. H. Bennett and P. W. Shor, IEEE Tranc. On Infor. Theor. 44 (1998) 2724.
  * [66] C. Adami and N. J. Cerf, Phys. Rev. A 56 (1997) 3470.
  * [67] S. Lloyd, Phys. Rev. Lett. 90 (2003) 167902.
  * [68] A. D. Arrigo, G. Benenti, G. Falci and C. Macchiavello, Phys. Rev. A 88 (2013) 042337.
  * [69] F. Caruso, V. Giovannetti, C. Lupo and S. Mancini, Rev. Mod. Phys. 86 (2014) 1203.
  * [70] V. Giovannetti and R. Fazio, Phys. Rev. A 71 (2005) 032314.
  * [71] D. Burgarth, Ph.D thesis, University College London, 142 2006; arXiv: 0704.1309v1.
  * [72] N. Arshed, A. H. Toor and D. A. Lidar, Phys. Rev. A 81 (2010) 062353.
  * [73] C. Macchiavello, G. M. Palma and S. Virmani, Phys. Rev. A 69 (2004) 010303.
  * [74] N. Arshed and A. H. Toor, Phys. Rev. A 73 (2006) 014304.
  * [75] V. Giovannetti, S. Lloyd, L. Maccone and J. H. Shapiro, Nature Phot. 7 (2013) 834.
  * [76] G. D. Palma, A. Mari and V. Giovannetti, Phys. Rev. A 90 (2014) 042312.
  * [77] J. Lee, S. W. Ji, J. Park and H. Nha, Phys. Rev. A 91 (2015) 042336.
  * [78] L. Ciliberti, R. Rossignoli, N. Canosa, Phys. Rev. A 82 (2010) 042316.


Corresponding author: Mathias Ramm Haugland, E-mail:
<EMAIL_ADDRESS>
# Deep Learning for Improved Polyp Detection from Synthetic Narrow-Band
Imaging

Mathias Ramm Haugland  Department of Electronic Systems, Norwegian University
of Science and Technology, Trondheim, Norway  Hemin Ali Qadir  Intervention
Centre, Oslo University Hospital, Oslo, Norway  Ilangko Balasingham
Department of Electronic Systems, Norwegian University of Science and
Technology, Trondheim, Norway  Intervention Centre, Oslo University Hospital,
Oslo, Norway

###### Abstract

To cope with the growing prevalence of colorectal cancer (CRC), screening
programs for polyp detection and removal have proven their usefulness.
Colonoscopy is considered the best-performing procedure for CRC screening. To
ease the examination, deep learning based methods for automatic polyp
detection have been developed for conventional white-light imaging (WLI).
Compared with WLI, narrow-band imaging (NBI) can improve polyp classification
during colonoscopy but requires special equipment. We propose a CycleGAN-based
framework to convert images captured with regular WLI to synthetic NBI (SNBI)
as a pre-processing method for improving object detection on WLI when NBI is
unavailable. This paper ﬁrst shows that better results for polyp detection can
be achieved on NBI compared to a relatively similar dataset of WLI. Secondly,
experimental results demonstrate that our proposed modality translation can
achieve improved polyp detection on SNBI images generated from WLI compared to
the original WLI. This is because our WLI-to-SNBI translation model can
enhance the observation of polyp surface patterns in the generated SNBI
images.

###### keywords:

Synthetic Imaging, Polyp Detection, Colorectal Cancer, Narrow-Band Imaging,
Deep Learning, Artificial Intelligence, Generative Adversarial Network

##  1 INTRODUCTION

Colorectal cancer (CRC) is the third most incidental and the second deadliest
cancer type globally [1]. It often develops from growths of glandular tissue
known as polyps in the colon and rectum. In the early stage, polyps are benign
and usually do not cause symptoms until some of them turn malignant. Early
colon screening can have a great preventive effect. Colonoscopy is regarded as
the best colon screening technique for CRC, where detection and removal of
adenomatous polyps is the main goal. Colonoscopy is, however, a skill-
dependent procedure relying heavily on the experience of endoscopists.
Unfortunately, polyps might be missed or miss-classified in patients
undergoing colonoscopy. This can lead to a late diagnosis of CRC and a poor
prognosis. There have been technical efforts to reduce polyp miss rate and
miss classification by improving the capability of colonoscopy devices. For
example, narrow-band imaging (NBI)[2], a virtual chromoendoscopy technique,
has been developed to enhance the observation of the scenes and visualization
of the lesions. Compared to regular white-light imaging (WLI), which captures
images in the full visible light spectrum, NBI uses an additional special
filter to let the light of a specific narrow band of blue and green
wavelengths pass through to the camera lens. The narrow-band light makes the
thin capillary network and the thick blood vessels on the mucosal surface
appear more distinctly, making it easier for the endoscopist to classify the
polyps during colonoscopy. To discern between benign polyps, adenomas, and
cancer, the NICE (NBI International Colorectal Endoscopic)[3] classification
is often used where NBI is available.

In recent years, many computer-aided diagnoses (CAD) systems have been
proposed for automatic polyp detection [4] as a solution for polyp miss rates.
Among those, deep learning, in the form of convolutional neural networks
(CNNs), has in recent years been the most prominent approach. Most of the
proposed CNN-based automatic polyp detection algorithms are applied on WLI and
the advantages of NBI over WLI have not been fully investigated and confirmed.
CNNs also form the basis for generative adversarial networks (GANs), which are
developed for generating real-looking synthetic data [5]. GAN-based methods
have many application areas, for instance, image-to-image translation [6]. In
this paper, we first apply the state-of-the-art object detection network
EfficientDet-D0 [7] to confirm the advantages of NBI over WLI for automatic
polyp detection. The initial results show that improved performance of polyp
detection can be achieved with NBI. Therefore, we propose a GAN-based
framework to convert colonoscopy images captured in WLI to synthetic NBI
(SNBI) as a pre-processing method for the performance improvement of polyp
detection on WLI when NBI is not available.

##  2 MATERIALS AND METHODS

###  2.1 Method

Figure 1 presents our proposed framework for the performance improvement of
polyp detection in SNBI mode. The proposed framework consists of two networks:
a network to perform an unpaired image-to-image translation and a detection
network. For the image-to-image translation, we adapt CycleGAN [6], and for
the detection model, we adapt EfficientDet-D0 [7]. The goal of CycleGAN is to
learn patterns that connect two image domains, in this case WLI ($X$) and NBI
($Y$). CycleGAN is made up of two generators, G and F, and two discriminators,
$D_{X}$ and $D_{Y}$. The aim is to learn the mapping $G:X\rightarrow Y$. For
every training iteration, $G$ generates an SNBI image from a WLI image and
$D_{Y}$ tries to discern between the generated SNBI image and a real NBI
image. To ensure that the content of the SNBI image is not altered, F converts
each generated SNBI $\hat{y}=G(x)$ back to $X$. The difference between x and
$\hat{x}=F(\hat{y})=F(G(x))$ is then calculated as what is called the forward
cycle-consistency loss, which, along with the backward cycle-consistency loss
and the two adversarial losses, makes up the loss function that trains both
the generators and discriminators,

$\displaystyle\mathcal{L}(G,F,D_{X},D_{Y})$
$\displaystyle=\mathcal{L}_{GAN}(G,D_{Y},X,Y)+\mathcal{L}_{GAN}(F,D_{X},Y,X)+\lambda\mathcal{L}_{cyc}(G,F).$
(1)

$\mathcal{L}_{cyc}$ is the sum of the forward and the backward cycle-
consistency losses, calculated as mean absolute error, and $\lambda$ is a
weighting factor. During training, CycleGAN is given a WLI and a NBI image in
each iteration. Because multiple datasets of different quality and origin are
used for training and testing our models we propose a ”semi-pairing” of the
training data to improve model performance. This means that for each
iteration, the two images are taken from the same video clip of the same
colon.

We use EfficientDet-D0 to first evaluate the performance of polyp detection in
the WLI modality against NBI, and second to assess the usefulness of the
generated SNBI images for the improvement of automatic polyp detection. In an
effort to have a fair modality comparison, the NBI and WLI datasets contain an
equal number of similar images of the same polyps.

Figure 1: The proposed framework. CycleGAN for modality translation (left),
followed by EfficientDet-D0 for polyp detection (right).

###  2.2 Datasets

In this study, three public and one private dataset of still images and videos
of hyperplastic (benign) and adenomatous polyps were used:

  1. a)

PICCOLO [8]: This is a dataset of 76 lesions from 40 patients. The dataset
contains both NBI and WLI of most polyps, NICE classification, and clinically
annotated segmentation masks. All type 3 (deep submucosal invasive cancer)
polyps were removed from the dataset. Then, all images were cropped on the
sides, removing the black canvases and leaving the informative part of the
image. Because both the EfficientDet-D0 and CycleGAN resize the images to
512x512, using more square-shaped images were believed to preserve the
information in the images better.

  2. b)

OUS-NBI-ColonVDB: This is a private dataset containing 21 high-quality polyp
videos: eleven hyperplastic and ten adenomatous polyps. The videos were
recorded at Oslo University Hospital (OUS) in both NBI and WLI modes. The
polyps are annotated by clinicians as binary segmentation masks in every
frame.

  3. c)

KUMC [9]: This is a mixed dataset of NBI and WLI frames obtained from 80
videos. Each frame is manually classified into adenoma or hyperplastic and
annotated with a bounding box for each polyp present.

  4. d)

Mesejo Videos [10]: This dataset comprises 76 polyp videos recorded in both
NBI and WLI modes. The polyp distribution is as follows: 15 serrated, 21
hyperplastic, and 40 adenomatous.

The videos used were converted to still images and blurry images were manually
removed. Binary segmentation masks were converted to bounding boxes.

###  2.3 Evaluation metrics

For the evaluation of polyp detection performance, we derive recall and
precision from the well-known medical terms true positive (TP), false positive
(FP), and false negative (FN) as follows:

$Precision=\frac{\sum_{i}TP_{i}}{\sum_{i}TP_{i}+\sum_{i}FP_{i}}=\frac{\sum_{i}TP_{i}}{all\
detections},$ (2)
$Recall=\frac{\sum_{i}TP_{i}}{\sum_{i}TP_{i}+\sum_{i}FN_{i}}=\frac{\sum_{i}TP_{i}}{all\
ground\ truths}.$ (3)

We identify a detection output as TP if the intersection over union (IoU)
between the detected bounding box and the corresponding ground-truth bounding
box is $>0.5$ and the confidence score is above a threshold. This threshold
was found independently in each model based on the precision-recall curve from
the validation data, keeping the trade-off between recall and precision.

##  3 Results and Discussion

###  3.1 Results on videos

In this experiment, we trained our CycleGAN on the Mesejo dataset for 100
epochs with a constant learning rate of 0.0002. The detection model was
trained three times: a) the WLI model was trained on the original WLI images
of the PICCOLO dataset, b) the SNBI model was trained on the corresponding
SNBI images obtained by our trained CycleGAN applied to the same WLI images,
and c) the NBI model was trained on the original NBI images of the PICCOLO
dataset. The NBI training set had an equal ammount of images of each lesion as
the WLI/SNBI sets.

We split OUS-NBI-ColonVDB into a validation set (4 videos) and a test set (17
videos). To create as similar as possible datasets of both NBI and WLI videos,
we removed several WLI and NBI frames so that there were equal numbers of
similar video frames in both imaging modalities. Table 1 shows our results on
the 17 test videos.

Table 1: Results on the 17 test videos of OUS-NBI-ColonVDB. | NBI | WLI | SNBI | SNBIx
---|---|---|---|---
Precision | 0.757 | 0.55 | 0.634 | 0.457
Recall | 0.594 | 0.52 | 0.537 | 0.426
F1 | 0.666 | 0.535 | 0.581 | 0.441

The results show that polyps can be detected more efficiently with NBI than
with WLI. This is supported by the results from the SNBI images obtained from
the WLI, which also show that the proposed transform improves polyp detection.
The results of SNBIx were obtained when the NBI model was tested on the SNBI
images. The SNBIx results are a good indicator that our CycleGAN generates
SNBI images that resemble real NBI.

###  3.2 Results on still images

In this experiment, we first trained our CycleGAN on the Mesejo dataset for
100 epochs with a constant learning rate of 0.0002. The model was further
trained on OUS-NBI-ColonVDB for 30 epochs with the same learning rate,
followed by a linearly decaying learning rate for another 30 epochs. The
detection model was trained on the PICCOLO dataset after being split into a
training set (70%) and a validation set (30%). We trained two detection
models: the WLI model was only trained on images in WLI mode, while the SNBI
model was trained on the corresponding SNBI obtained by applying our trained
CycleGAN on the same WLI images.

We manually created our test set by handpicking images of 154 hyperplastic and
154 adenomatous polyps from the KUMC dataset. Table 2 shows the results
obtained on our test images from the WLI and the SNBI models.

Table 2: Results on the KUMC dataset. | WLI | SNBI
---|---|---
Precision (hyperplastic) | 0.648 | 0.675
Recall (hyperplastic) | 0.669 | 0.677
F1 (hyperplastic) | 0.658 | 0.676
Precision (adenomas) | 0.734 | 0.742
Recall (adenomas) | 0.734 | 0.747
F1 (adenomas) | 0.734 | 0.744

As seen in Table 2, the SNBI model shows slightly better detection performance
than the corresponding WLI model for both hyperplastic and adenomatous polyps.
These results confirm that the detection model benefits from the enhanced SNBI
polyps generated by our CycleGAN.

Figure 2 shows an example of the WLI-to-SNBI conversion performed by our
CycleGAN. Compared to the original WLI, the surface patterns on the SNBI can
be better observed. Figure 3 demonstrates that our detection model misses the
same polyp in the WLI, while it can detect it with a high confidence value and
precise bounding box in the corresponding SNBI. This result clearly shows the
benefits of our CycleGAN in generating synthetic NBI images for improved
automatic polyp detection.

|  |
---|---|---
(a) | (b) | (c)
Figure 2: The same polyp (from the PICCOLO set) is shown in (a) original WLI, (b) original NBI, and (c) SNBI generated from (a) by our CycleGAN. |  |
---|---|---
(a) Ground truth. | (b) Detection in WLI. | (c) Detection in SNBI.
Figure 3: An output detection shows that the detection model misses a
hyperplastic polyp in WLI but detects it in SNBI.

Because colonoscopy equipment cannot capture NBI and WLI images from the same
place at the same time, a fair comparison between the two modalities is
challenging. Despite that our effort to compare the two neither was perfect,
one can argue that NBI at least will be able to perform as well as the SNBI
does. This is because NBI data has been used to create the SNBI transforms.
The SNBI features that make polyps easier to detect than in the original WLI
come arguably from NBI, implying the advantage of NBI over WLI in polyp
detection.

##  4 CONCLUSION AND FUTURE WORK

This paper has shown that better results for automatic polyp detection can be
achieved on NBI compared to a relatively similar dataset of WLI. We,
therefore, proposed a method based on unpaired image-to-image translation for
converting colonoscopy images taken with WLI to SNBI. Our polyp detection
results confirm that, compared to the original WLI, improved performance can
be obtained on SNBI images. This shows the effectiveness of the proposed WLI-
to-SNBI translation method and the advantage of using synthetic NBI images in
automatic polyp detection. In our future work, we plan to extend the proposed
method for automatic polyp classification based on the NICE classification. If
the method can be confirmed, more polyps could be classified during
colonoscopy. Thus, only the potentially dangerous polyps will be removed,
reducing the time and labor-consuming biopsies and examinations under the
microscope.

###### Acknowledgements.

This work is derived from the master’s thesis Deep Learning for Polyp
Detection from Narrow-Band Imaging [11] by Mathias Ramm Haugland, submitted in
June 2022 at the Norwegian University of Science and Technology in Trondheim,
Norway.

## References

  * [1] Sung, H., Ferlay, J., Siegel, R. L., Laversanne, M., Soerjomataram, I., Jemal, A., and Bray, F., “Global cancer statistics 2020: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries,” CA: a cancer journal for clinicians 71(3), 209–249 (2021).
  * [2] Gono, K., “Narrow band imaging: technology basis and research and development history,” Clinical Endoscopy 48(6), 476–480 (2015).
  * [3] Iwatate, M., Hirata, D., and Sano, Y., “NBI international colorectal endoscopic (NICE) classification,” in [Endoscopy in Early Gastrointestinal Cancers, Volume 1 ], 69–74, Springer Singapore (Oct. 2020).
  * [4] Hassan, C., Spadaccini, M., Iannone, A., Maselli, R., Jovani, M., Chandrasekar, V. T., Antonelli, G., Yu, H., Areia, M., Dinis-Ribeiro, M., et al., “Performance of artificial intelligence in colonoscopy for adenoma and polyp detection: a systematic review and meta-analysis,” Gastrointestinal endoscopy 93(1), 77–85 (2021).
  * [5] Goodfellow, I. J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y., “Generative adversarial networks,” (June 2014).
  * [6] Zhu, J.-Y., Park, T., Isola, P., and Efros, A. A., “Unpaired image-to-image translation using cycle-consistent adversarial networks,” (Mar. 2017).
  * [7] Tan, M., Pang, R., and Le, Q. V., “EfficientDet: Scalable and efficient object detection,” in [2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) ], IEEE (June 2020).
  * [8] Sánchez-Peralta, L. F., Pagador, J. B., Picón, A., Calderón, Á. J., Polo, F., Andraka, N., Bilbao, R., Glover, B., et al., “PICCOLO white-light and narrow-band imaging colonoscopic dataset: A performance comparative of models and datasets,” Applied Sciences 10, 8501 (Nov. 2020).
  * [9] Li, K., Fathan, M. I., Patel, K., Zhang, T., Zhong, C., et al., “Colonoscopy polyp detection and classification: Dataset creation and comparative evaluations,” PLOS ONE 16, e0255809 (Aug. 2021).
  * [10] Mesejo, P., Pizarro, D., Abergel, A., Rouquette, O., Beorchia, S., Poincloux, L., and Bartoli, A., “Computer-aided classification of gastrointestinal lesions in regular colonoscopy,” IEEE Transactions on Medical Imaging 35, 2051–2063 (Sept. 2016).
  * [11] Haugland, M. R., Deep Learning for Polyp Detection from Narrow-Band Imaging, master’s thesis, NTNU – Norwegian University of Science and Technology (June 2022).