# A secure deterministic remote state preparation via a seven-qubit entangled
channel of an arbitrary two-qubit state under the impact of quantum noise

Deepak Singh  ID<EMAIL_ADDRESS>Department of Mathematics,
Indian Institute of Technology Roorkee 247667, Uttarakhand, India  Sanjeev
Kumar  ID<EMAIL_ADDRESS>Department of Mathematics, Mehta Family
School of Data Science and Artificial Intelligence
Indian Institute of Technology Roorkee 247667, Uttarakhand, India  Bikash K.
Behera  ID<EMAIL_ADDRESS>Bikash’s Quantum (OPC) Pvt. Ltd., Balindi,
Mohanpur 741246, Nadia, West Bengal, India

###### Abstract

As one of the most prominent subfields of quantum communication research,
remote state preparation (RSP) plays a crucial role in quantum networks. Here
we present a deterministic remote state preparation scheme to prepare an
arbitrary two-qubit state via a seven-qubit entangled channel created from
Borras _et al._ state. Quantum noises are inherent to each and every protocol
for quantum communication that is currently in use, putting the integrity of
quantum communication systems and their dependability at risk. The initial
state of the system was a pure quantum state, but as soon as there was any
noise injected into the system, it transitioned into a mixed state. In this
article, we discuss the six different types of noise models namely bit-flip
noise, phase-flip noise, bit-phase-flip noise, amplitude damping, phase
damping and depolarizing noise. The impact these noises had on the entangled
channel may be seen by analysing the density matrices that have been altered
as a result of the noise. For the purpose of analysing the impact of noise on
the scheme, the fidelity between the original quantum state and the remotely
prepared state has been assessed and graphically represented. In addition, a
comprehensive security analysis is performed, demonstrating that the suggested
protocol is safe against internal and external attacks.

Quantum Communication, Remote State Preparation, Entanglement, Borras _et al._
state

##  I Introduction

Quantum communication is a very essential field in quantum mechanics, quantum
computation, and quantum information theory. Entanglement is the backbone of
quantum communication; it is considered the essential asset of overall quantum
information processing [1]. It is well known that there are some novel
phenomena in the applications of quantum communication and information theory,
which include quantum teleportation [2, 3], quantum secure direct
communication (QSDC)[4], quantum key distribution [5] and quantum dense coding
[6], quantum secret sharing [7, 8], quantum data hiding [9], quantum private
comparison (QPC) [10], quantum remote state preparation (RSP) [11, 12, 13] to
name a few. Using an entangled channel to send the quantum state securely in
all of these quantum communication protocols is an important area of research
in quantum communication. At least in theory, quantum protocols have the
potential to attain a better degree of security than their conventional
counterparts.

In a remote state preparation (RSP) protocol, the sender uses a shared
entangled channel and the right measurements to set up a known quantum state
for the faraway receiver. RSP is believed to be a more efficient method of
teleporting to a known state than the traditional method since it uses fewer
classical bits [14]. Since then, several distinct RSP algorithms [15, 16] were
put up as potential solutions. Later on, a number of research that were based
on RSP were carried out. These included joint remote state preparation (JRSP)
[17, 18, 19] and controlled remote state preparation (CRSP) [[20]. On the
other hand, the vast majority of RSP algorithms made in the past have a chance
of success that is less than 1. To raise the success probability of RSP to 1,
a new RSP algorithm called the deterministic remote state preparation (DRSP)
algorithm [21, 22] was proposed. This algorithm can set up the needed quantum
state with a one-in-one chance of success, which saves a lot of quantum
resources. In recent years, several fascinating studies about RSP have been
carried out. A unique technique for the implementation of RSP of a generic
m-qubit entangled state was suggested by Wang et al. [23] by employing the
GHZ-type states as quantum channels with a high success probability,
reducibility, and generalizability. This strategy was developed by using the
GHZ-type states. Wang et al.[24] suggested two successful measurement-based
ways for executing the RSP techniques for generic W-class entangled states for
three and four particles, employing GHZ-type states as the quantum channels in
the same year. In comparison to the previous schemes, these approaches stand a
better chance of being implemented successfully with a higher success
probability compared to the existing schemes. In addition, the suggested
methods may be successfully implemented with a total success probability of
one when the utilised channels are condensed into the most maximally entangled
versions of themselves. Wang et al. [25] proposed two optics-based
implementations for RSP and JRSP of an arbitrary single-photon pure state. The
protocols for these implementations may be achieved with a specific success
probability with the assistance of appropriate LOCC.

Every quantum protocol is practically not error-free. It has some ambiguities
in the system. These ambiguities are considered quantum noise. Quantum noise
is inevitable in implementing a quantum communication system under realistic
conditions. Quantum noise will hurt the safety and reliability of the quantum
communication system in a big way. Research on RSP [26, 27] in noisy
environments is now being conducted and explored. The JRSP method was computed
in amplitude-damping noise, and phase-damping noise by Guan et al. [28], who
also conducted an in-depth analysis of the impact that noise has on the output
state. Ma et al. [22] examined the influence of amplitude-damping and phase-
damping noise on the DJRSP method and presented a deterministic approach for
constructing distant states using a Brown et al. state as the underlying
channel. In [29] the noisy environment is studied for deterministic joint
remote state preparation of arbitrary two-qubit state. A complexity analysis
of the teleportation scheme under the influence of noise is studied in [30].

Motivated by these schemes, we have developed a protocol for the deterministic
remote state preparation of a two-qubit qubit quantum state via a maximally
entangled seven-qubit state that is derived from Borras _et al._ state. The
use of highly entangled Borras _et al._ [31] state in the remote state
preparation has not been done in any of the schemes proposed till now. This
scheme involves four participants in preparation for the remote state. Alice,
Bob, Charlie, and David share a predetermined seven-qubit entangled channel.
Alice has the first qubit, Bob has the second and third, Charlie has the
fourth and sixth, and David has the last two qubits, which are the fifth and
seventh. The first action that has to be performed is to factorize the
entangled channel into the sum of the Bell basis states that have the same
coefficients as the two-qubit state that needs to be prepared remotely. Then
Alice measures her qubit and conveys the result to Bob through a classical
channel. Charlie and David likewise communicate their outcomes to Bob via
classical channels. Finally, in order to remotely prepare the desired two
qubit-state, Bob needs to perform certain unitary operation on his qubit based
on the collapse state of other participants. All the recovery operations are
discussed in details in table 1. Next, we will study the impact of six types
of noise on the entangled channel. These six kinds of noise are named bit-flip
noise, phase-flip noise, bit-phase-flip noise, amplitude damping, phase
damping and depolarizing noise. These noise models are studied with the help
of the action of the Kraus operator on the qubit. When the Kraus operator acts
on a quantum state, it becomes a mixed state. Upon evaluation of the density
matrices, the fidelity is calculated between the initial state and the
remotely prepared state. The variation in fidelity is represented with the
help of a graph. In [20], the security analysis of the remote state
preparation protocol is performed. This study analyses the security attack
from the outside and the inside participants.

The rest of the article is structured as follows: Section II covers
introductory terminology about the underlying entangled channel and remote
state preparation procedures. In Section III, the noise analysis of the
entangled teleportation channel is described. Section V consists of the
security analysis of the protocol, and V concludes the report with a
discussion of the study’s results, followed by suggestions for further
research.

##  II Remote state preparation of an arbitrary two-qubit state

Suppose Alice, Bob, Charlie, and David are the participants in this system.
Alice, Charlie and David combining together, want to remotely prepare a known
quantum state at Bob’s end. The two-qubit quantum state can be parameterized
as $\ket{\xi}=\alpha\ket{00}+\beta\ket{11}$, where $\alpha,\beta\in\mathbb{C}$
such that $|\alpha|+|\beta|^{2}=1$ takes care of the normalization of the
state $\Ket{\phi}$. The seven-qubit entangled channel shared among the
participants can be prepared by using the Borras _et al._ state and an ancilla
qubit in the $\Ket{0}$ state. The Borras _et al._ state is given by

$\displaystyle\ket{\psi}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\ket{000}(\ket{0}\ket{\psi^{+}}+\ket{1}\ket{\phi^{+}})+\ket{001}(\ket{0}\ket{\phi^{-}}-\ket{1}\ket{\psi^{-}})+\ket{010}(\ket{0}\ket{\phi^{+}}-\ket{1}\ket{\psi^{+}})+\ket{011}(\ket{0}\ket{\psi^{-}}+\ket{1}\ket{\phi^{-}})$
$\displaystyle-$
$\displaystyle\ket{100}(\ket{0}\ket{\phi^{-}}+\ket{1}\ket{\psi^{-}})+\ket{101}(-\ket{0}\ket{\psi^{+}}+\ket{1}\ket{\phi^{+}})+\ket{110}(\ket{0}\ket{\psi^{-}}-\ket{1}\ket{\phi^{-}})+\ket{111}(\ket{0}\ket{\phi^{+}}+\ket{1}\ket{\psi^{+}})\Big{)}$

where $\ket{\phi^{\pm}}=\frac{1}{\sqrt{2}}(\ket{01}\pm\ket{10})$ and
$\ket{\psi^{\pm}}=\frac{1}{\sqrt{2}}(\ket{00}\pm\ket{11})$. The ancilla qubit
is entangled with the Borras _et al._ state and the final entangled channel is
given by the following equation

$\displaystyle\ket{\Psi}$ $\displaystyle=$
$\displaystyle\Big{(}\ket{\psi}_{123456}\otimes\ket{0}_{7}\Big{)}_{CX(6,7)}$
(2) $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}}\Big{(}\ket{0000000}+\ket{0000111}+\ket{0001011}+\ket{0001100}+\ket{0010011}-\ket{0010100}-\ket{0011000}+\ket{0011111}$
$\displaystyle+$
$\displaystyle\ket{0100011}+\ket{0100100}-\ket{0101000}-\ket{0101111}+\ket{0110000}-\ket{0110111}+\ket{0111011}-\ket{0111100}-\ket{1000011}$
$\displaystyle+$
$\displaystyle\ket{1000100}-\ket{1001000}+\ket{1001111}-\ket{1010000}-\ket{1010111}+\ket{1011011}+\ket{1011100}+\ket{1100000}-\ket{1100111}$
$\displaystyle-$
$\displaystyle\ket{1101011}+\ket{1101100}+\ket{1110011}+\ket{1110100}+\ket{1111000}+\ket{1111111}\Big{)}_{AB_{1}B_{2}C_{1}D_{1}C_{2}D_{2}}$

The Borras _et al._ state is a six-qubit highly entangled state, which
included two-qubit maximally entangled Bell states, $\ket{\phi^{\pm}}$ and
$\ket{\psi^{\pm}}$ [31]. Because the prepared entangled state $\ket{\Psi}$ was
generated using the C-NOT operation on a highly entangled six-qubit Borras et
al. state, which is considered to add entanglement to a quantum state, we
believe that it is indeed extremely entangled, if not maximally entangled.
Once the entangled channel is obtained, Alice reserves the first qubit for
herself and distributes the other qubits as follows: Bob receives the second
and third qubits, Charlie receives the fourth and sixth qubits, and David
receives the fifth and seventh qubits. Alice owns the qubit $A$, Bob owns the
qubits $B_{1}$ and $B_{2}$, Charlie owns the qubits $C_{1}$ and $C_{2}$, and
David owns the qubits $D_{1}$ and $D_{2}$. In order to remotely prepare the
quantum state at Bob’s end, the factorization of the quantum state plays a
crucial role. So, the known state $\ket{\Psi}$ is primarily factorized in a
certain way that only the sender utilises a basis created out of the known
parameters $\alpha$ and $\beta$. Here the factorization of $\ket{\Psi}$ can be
done in the following basis -

Figure 1:  (a) Diagram representation of the remote state preparation protocol
(b) Quantum Circuit representation of the RSP scheme. The

$\displaystyle\ket{\Psi}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}\Big{[}\ket{\Upsilon_{1}}\ket{\zeta_{1}}+\ket{\Upsilon_{2}}\ket{\zeta_{2}}\Big{]}_{AB_{1}B_{2}B_{2}C_{1}C_{2}D_{1}D_{2}}$
(3)

where the quantum state $\ket{\Upsilon_{1}}$ and $\ket{\Upsilon_{1}}$ are
given by the Eq.(4)

$\displaystyle\ket{\Upsilon_{1}}_{A}$ $\displaystyle=$
$\displaystyle(\alpha\Ket{0}+\beta\Ket{1})_{A}$
$\displaystyle\ket{\Upsilon_{2}}_{A}$ $\displaystyle=$
$\displaystyle(\alpha\Ket{1}-\beta\Ket{0})_{A}$ (4)

And the quantum states $\Ket{\zeta_{1}}$ and $\Ket{\zeta_{2}}$ are given by
the following expressions

$\displaystyle\ket{\zeta_{1}}=\dfrac{1}{\sqrt{2}}$ $\displaystyle\Big{[}$
$\displaystyle\big{(}\alpha\Ket{00}+\beta\Ket{10}+\alpha\Ket{11}-\beta\Ket{01}\big{)}_{B_{1}B_{2}}\Ket{00}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$
(5) $\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{01}+\beta\Ket{11}+\alpha\Ket{10}-\beta\Ket{00}\big{)}_{B_{1}B_{2}}\Ket{01}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}-\alpha\Ket{01}+\beta\Ket{11}-\alpha\Ket{10}-\beta\Ket{00}\big{)}_{B_{1}B_{2}}\Ket{10}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{00}-\beta\Ket{10}+\alpha\Ket{11}+\beta\Ket{01}\big{)}_{B_{1}B_{2}}\Ket{11}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}-\alpha\Ket{01}+\beta\Ket{11}+\alpha\Ket{10}+\beta\Ket{00}\big{)}_{B_{1}B_{2}}\Ket{00}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{00}+\beta\Ket{10}-\alpha\Ket{11}+\beta\Ket{01}\big{)}_{B_{1}B_{2}}\Ket{10}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{00}-\beta\Ket{10}-\alpha\Ket{11}-\beta\Ket{01}\big{)}_{B_{1}B_{2}}\Ket{01}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{01}+\beta\Ket{11}-\alpha\Ket{10}+\beta\Ket{00}\big{)}_{B_{1}B_{2}}\Ket{11}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}\Big{]}$
$\displaystyle\ket{\zeta_{2}}=\dfrac{1}{\sqrt{2}}$ $\displaystyle\Big{[}$
$\displaystyle\big{(}\alpha\Ket{10}-\beta\Ket{00}-\alpha\Ket{01}-\beta\Ket{11}\big{)}_{B_{1}B_{2}}\Ket{00}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$
(6) $\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{11}-\beta\Ket{01}-\alpha\Ket{00}-\beta\Ket{10}\big{)}_{B_{1}B_{2}}\Ket{01}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{11}+\beta\Ket{01}-\alpha\Ket{00}+\beta\Ket{10}\big{)}_{B_{1}B_{2}}\Ket{10}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}-\alpha\Ket{10}-\beta\Ket{00}+\alpha\Ket{01}-\beta\Ket{11}\big{)}_{B_{1}B_{2}}\Ket{11}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{11}+\beta\Ket{01}+\alpha\Ket{00}-\beta\Ket{10}\big{)}_{B_{1}B_{2}}\Ket{00}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{10}-\beta\Ket{00}+\alpha\Ket{01}+\beta\Ket{11}\big{)}_{B_{1}B_{2}}\Ket{10}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}-\alpha\Ket{10}-\beta\Ket{00}-\alpha\Ket{01}+\beta\Ket{11}\big{)}_{B_{1}B_{2}}\Ket{01}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}$
$\displaystyle+$
$\displaystyle\big{(}\alpha\Ket{11}-\beta\Ket{01}+\alpha\Ket{00}+\beta\Ket{10}\big{)}_{B_{1}B_{2}}\Ket{11}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}\Big{]}$

Now Alice measure her qubit in the basis
$\\{\ket{\Upsilon_{1}},\ket{\Upsilon_{1}}\\}$, Charlie and David measure their
qubits in $\\{\Ket{00},\Ket{01},\Ket{10},\Ket{11}\\}$ basis then Bob can
easily remotely prepare the state $\Ket{\xi}$ knowing the outcome of Alice,
Charlie and David. For instance, if Alice’s measurement collapse in the state
$\Ket{\Upsilon_{1}}_{A}$, Charlie and David’s measurement collapse to the
state $\Ket{01}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$ after then, each
participant will use a conventional channel to communicate their results to
Bob. Bob will next be required to do the necessary unitary operations on his
qubit in order to remotely prepare the state at his end., in this case, Bob
will apply a controlled-not gate form “his first qubit” to the second qubit
($B_{1}$ to $B_{2}$), denoted as $CX_{1-2}$. A Hadamard gate on his first
qubit, denoted as $H(1)$, again a not operation on his second qubit, denoted
as $X_{2}$ and finally a $Z_{1}$ gate on his first qubit to flip the phase of
the first qubit. At this stage of the protocol, Bob has successfully prepared
the quantum state $\Ket{\xi}$ at his end. All the other possible cases are
given in table 1.

Alice’s Outcome | Charlie & David’s Outcome | Bob’s Collapse State | Bob’s Operations
---|---|---|---
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{00}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{00}+\beta\Ket{10}+\alpha\Ket{11}-\beta\Ket{01})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{01}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{01}+\beta\Ket{11}+\alpha\Ket{10}-\beta\Ket{00})$ | $CX_{1-2}~{}H_{1}~{}X_{2}~{}Z_{1}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{10}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(-\alpha\Ket{01}+\beta\Ket{11}-\alpha\Ket{10}-\beta\Ket{00})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}~{}Z_{2}~{}X_{2}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{11}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{00}-\beta\Ket{10}+\alpha\Ket{11}+\beta\Ket{01})$ | $CX_{1-2}~{}H_{1}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{00}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(-\alpha\Ket{01}+\beta\Ket{11}+\alpha\Ket{10}+\beta\Ket{00})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}~{}X_{1}~{}X_{2}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{10}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{00}+\beta\Ket{10}-\alpha\Ket{11}+\beta\Ket{01})$ | $CX_{1-2}~{}H_{1}~{}Z_{2}~{}X_{1}~{}CX_{2-1}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{10}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{00}-\beta\Ket{10}-\alpha\Ket{11}-\beta\Ket{01})$ | $CX_{1-2}~{}H_{1}~{}Z_{2}~{}X_{1}$
$\ket{\Upsilon_{1}}_{A}$ | $\Ket{11}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{01}+\beta\Ket{11}-\alpha\Ket{10}+\beta\Ket{00})$ | $CX_{1-2}~{}H_{1}~{}X_{2}~{}X_{1}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{00}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{10}-\beta\Ket{00}-\alpha\Ket{01}-\beta\Ket{11})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}~{}X_{1}~{}X_{2}~{}Z_{1}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{01}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{11}-\beta\Ket{01}-\alpha\Ket{00}-\beta\Ket{10})$ | $CX_{1-2}~{}H_{1}~{}Z_{2}~{}Z_{1}~{}X_{1}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{10}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{11}+\beta\Ket{01}-\alpha\Ket{00}+\beta\Ket{10})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}~{}X_{1}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{11}_{C_{1}C_{2}}\Ket{01}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(-\alpha\Ket{10}-\beta\Ket{00}+\alpha\Ket{01}-\beta\Ket{11})$ | $CX_{1-2}~{}H_{1}~{}X_{1}~{}X_{2}~{}Z_{1}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{00}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{11}+\beta\Ket{01}+\alpha\Ket{00}-\beta\Ket{10})$ | $CX_{1-2}~{}H_{1}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{10}_{C_{1}C_{2}}\Ket{10}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{10}-\beta\Ket{00}+\alpha\Ket{01}+\beta\Ket{11})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}~{}X_{2}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{10}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(-\alpha\Ket{10}-\beta\Ket{00}-\alpha\Ket{01}+\beta\Ket{11})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}~{}Z_{2}~{}X_{2}$
$\ket{\Upsilon_{2}}_{A}$ | $\Ket{11}_{C_{1}C_{2}}\Ket{11}_{D_{1}D_{2}}$ | $\frac{1}{\sqrt{2}}(\alpha\Ket{11}-\beta\Ket{01}+\alpha\Ket{00}+\beta\Ket{10})$ | $CX_{1-2}~{}H_{1}~{}Z_{1}$
Table 1: Here are the operations that Bob must do on his qubits in order to
obtain a state based on the results of Alice and Charlie/David. Here
$CX_{1-2}$ represents the CNOT gate from qubit 1 to 2, $H_{1}$ represents the
hadamard gate on qubit 1, $Z_{2}$ represents the Z gate on qubit 2, and
$X_{2}$ represents the not operation on qubit 2.

##  III Effect of noisy environment on the quantum channel

The influence that noise has on the quantum channel will be the topic of our
next discussion. Ambiguities are produced in the quantum system as a result of
the fact that a real quantum system does not operate in perfect circumstances
and that it interacts with its surrounding environment. This kind of ambiguity
is described by the phrase quantum noise. Quantum noise must be taken into
account in order to undertake precise research on a quantum communication
method.In any actual quantum experiment conducted in the real world, noise
will play a crucial role in deciding the success of the scheme. These noises
may be studied and classified into one of six categories. Bit flip, phase
flip, bit-phase flip, amplitude damping, phase damping, and depolarizing noise
are the six fundamental forms of quantum noise that may occur in quantum
channels. In this section, these six forms of quantum noise are examined.
Therefore, the most productive course of action would be to investigate the
causes and consequences of the system’s noises and to seek to reduce their
impact wherever feasible. We can analyze this by studying the evolution of the
density matrix $\rho=\ket{\psi}\bra{\psi}$ with the help of Kraus operators.
The impact of noise on the quantum channel can be done using the operator sum
representation. The action of Kraus operators $E_{k}$ on a particular qubit
$k$ described by the density matrix $\rho_{k}$ is given by the Eq. (7)

$\mathcal{G}^{r}(\rho_{k})=\sum_{j=1}^{n}\big{(}{E_{j}}\big{)}\ \rho_{k}\
\big{(}{E_{j}}\big{)}^{\dagger}$ (7)

where $r\in\\{b,w,f,a,p,d\\}$ for bit-flip, phase-flip, bit-phase-flip,
amplitude damping, phase damping and depolarizing damping respectively,
$j\in\\{0,1\\}$ for $r=b,w,f,a$, $j\in\\{0,1,2\\}$ for $r=p$ and
$j\in\\{0,1,2,3\\}$ for $r=d$.

Alice retains the first qubit and transfers the remaining qubits to Bob,
Charlie, and David. After the distribution of qubits in a noisy environment,
the shared entangled state would transform into a mixed quantum state. Bob
must perform the proper unitary operations on his qubits to remotely construct
the quantum state, therefore the final state $\rho_{out}^{r}$ may be written
as the density matrix indicated in the following Eq. (8).

$\rho_{out}^{r}=Tr_{i_{1}i_{2}...i_{n-1}}\\{\mathcal{U}[\rho_{k}\otimes\xi^{r}(\rho_{l})]\mathcal{U}^{\dagger}\\}$
(8)

where $Tr_{i_{1}i_{2}...i_{n-1}}$ is the partial trace over the qubits
$i_{1},i_{2},...,i_{n-1}$ and $\mathcal{U}$ is the unitary operations, which
Bob will apply on his qubit to remotely prepare the quantum state, the
operation $\mathcal{U}$ is given by the following Eq. (9), $\mathcal{U}$

$\displaystyle\mathcal{U}$ $\displaystyle=$
$\displaystyle\\{\mathbb{I}_{1}\otimes\mathbb{I}_{2}\otimes...\mathbb{I}_{n-1}\otimes\sigma_{n}^{i_{1}i_{2}...i_{n-1}}\\}\\{\ket{\phi}_{12}\bra{\phi}_{12}\otimes\mathbb{I}_{3}...\otimes\mathbb{I}_{n}\\}\\{\mathbb{I}_{1}\otimes\mathbb{I}_{2}...U_{j_{1}j_{2}}...\otimes\mathbb{I}_{n}\\}$
(9)
$\displaystyle\\{\mathbb{I}_{1}\otimes\mathbb{I}_{2}...U_{k_{1}}...\otimes\mathbb{I}_{n}\\}$

where $\sigma_{n}^{i_{1}i_{2}...i_{n-1}}$ is the the Bob’s recovery unitary
operations after Alice has measured her qubit and communicated her result to
Bob via a classical channel, $\ket{\phi}_{12}\bra{\phi}_{12}$ is the Bell
basis measurement on the first two qubits, $U_{j_{1}j_{2}}$ represents the
C-NOT gate from qubit $j_{1}$ to $j_{2}$ and $U_{k_{1}}$ represents the
unitary gate on qubit $k_{1}$. The unitary operations vary according to the
collapse states of Alice, Charlie, and David, as given in the table 1. By
computing the fidelity between the initial two-qubit state $ket{\xi}$ and the
density matrix $\rho_{out}^{r}$, the influence of noise in the entangled
channel can now be illustrated. Fidelity reflects the proximity between two
quantum states and provides a mathematical formula for quantifying the degree
of resemblance between quantum states. The mathematical expression for nose-
effected fidelity given by Eq. (10) [32].

$\mathcal{F}=\bra{\Psi}\rho_{out}^{r}\ket{\Psi}$ (10)

Due to the fact that this communication is occurring under the influence of
noise, some quantum information may be lost during the remote state
preparation. Fidelity is the ideal statistic for measuring how much data is
lost. When the fidelity $\mathcal{F}=1$, this indicates the ideal case where
the noise hasn’t affected the communication, and no information has been lost.
Meanwhile, $\mathcal{F}=0$ implies that the noise has very badly affected the
communication and the state being communicated has been changed completely to
a different state resulting in all the information being lost. Thus, the
fidelity ranges between $0$ and $1$, i.e., $0\leq\mathcal{F}\leq 1$. We now
discuss the effect of six types of noises (bit-flip, phase-flip, bit-phase-
flip, amplitude damping, phase damping and depolarizing noise) in the next
section. The entangle channel given in Eq. (2) can be factorized in the
following manner (LABEL:eq-f)

$\displaystyle\ket{\Psi}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{(}\ket{000}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}})+\ket{001}(\ket{0}\ket{\lambda^{-}}-\ket{1}\ket{\mu^{-}})+\ket{010}(\ket{0}\ket{\lambda^{+}}-\ket{1}\ket{\mu^{+}})+\ket{011}(\ket{0}\ket{\mu^{-}}+\ket{1}\ket{\lambda^{-}})$
$\displaystyle+$
$\displaystyle\ket{100}(-\ket{0}\ket{\lambda^{-}}-\ket{1}\ket{\mu^{-}})+\ket{101}(\ket{1}\ket{\lambda^{+}}-\ket{0}\ket{\mu^{+}})+\ket{110}(\ket{0}\ket{\mu^{-}}-\ket{1}\ket{\lambda^{-}})+\ket{111}(\ket{0}\ket{\lambda^{+}}+\ket{1}\ket{\mu^{+}})\Big{)}$

where $\ket{\lambda^{\pm}}=\frac{1}{\sqrt{2}}(\ket{011}\pm\ket{100})$ and
$\ket{\mu^{\pm}}=\frac{1}{\sqrt{2}}(\ket{000}\pm\ket{111})$

###  III.1 Bit-flip noisy environment

The bit-flip noise changes the state of computational qubit $\ket{0}$ to
$\ket{1}$ and vice-versa with probability $\eta_{b}$ and the qubits remain
unchanged with the probability $(1-\eta_{b})$[33, 34]. Its operations on a
qubit can be described by Kraus operators given by the following matrices in
Eq.(12)

$\displaystyle
E_{0}^{b}=\sqrt{1-\eta_{b}}\mathbb{I}=\sqrt{1-\eta_{b}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix},~{}~{}~{}~{}E_{1}^{b}=\sqrt{\eta_{b}}\mathbb{X}=\sqrt{\eta_{b}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$ (12)

Where $\eta_{b}\in[0,1]$ represents the probability parameter for the bit-flip
error in the quantum system. The impact of bit-flip noise on the entangled
channel may be analyzed with the help of density matrices of the noisy
channel. The affected density matrix under the bit-flip noise is denoted by
$\mathcal{G}^{b}(\rho)$, given in Eq. (13).

$\displaystyle\mathcal{G}^{b}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{\\{}(1-\eta_{b})^{7}\Big{[}\Ket{\Psi}\bra{\Psi}\Big{]}+(\eta_{b})^{7}\Big{[}\ket{111}(\ket{1}\ket{\mu^{+}}+\ket{0}\ket{\lambda^{+}})+\ket{110}(\ket{0}\ket{\mu^{-}}-\ket{1}\ket{\lambda^{-}})+\ket{101}(\ket{1}\ket{\lambda^{+}}-\ket{0}\ket{\mu^{+}})$
(13) $\displaystyle-$
$\displaystyle\ket{100}(\ket{1}\ket{\mu^{-}}+\ket{0}\ket{\lambda^{-}})+\ket{011}(\ket{1}\ket{\lambda^{-}}-\ket{0}\ket{\mu^{-}})+\ket{010}(\ket{0}\ket{\lambda^{+}}-\ket{1}\ket{\mu^{+}})+\ket{001}(\ket{0}\ket{\lambda^{-}}-\ket{1}\ket{\mu^{-}})$
$\displaystyle+$
$\displaystyle\ket{000}(\ket{1}\ket{\lambda^{+}}+\ket{0}\ket{\mu^{+}})\Big{]}\times\Big{[}\bra{111}(\bra{1}\bra{\mu^{+}}+\bra{0}\bra{\lambda^{+}})+\bra{110}(\bra{0}\bra{\mu^{-}}-\bra{1}\bra{\lambda^{-}})+\bra{101}(\bra{1}\bra{\lambda^{+}}-\bra{0}\bra{\mu^{+}})$
$\displaystyle-$
$\displaystyle\bra{100}(\bra{1}\bra{\mu^{-}}+\bra{0}\bra{\lambda^{-}})+\bra{011}(\bra{1}\bra{\lambda^{-}}-\bra{0}\bra{\mu^{-}})+\bra{010}(\bra{0}\bra{\lambda^{+}}-\bra{1}\bra{\mu^{+}})+\bra{001}(\bra{0}\bra{\lambda^{-}}-\bra{1}\bra{\mu^{-}})$
$\displaystyle+$
$\displaystyle\bra{000}(\bra{1}\bra{\lambda^{+}}+\bra{0}\bra{\mu^{+}})\Big{]}\Big{\\}}$

###  III.2 Phase-flip noisy environment

The phase-flip noise affects the phase of the computational qubit. If the
system has a phase-flip noise then the computational qubit changes from
$\ket{1}$ to $-\ket{1}$, whereas it does not alter the qubit $\ket{0}$. It’s
Kraus operators [33, 34] are given by Eq. (14),

$\displaystyle
E_{0}^{w}=\sqrt{1-\eta_{w}}\mathbb{I}=\sqrt{1-\eta_{w}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{},~{}~{}~{}~{}E_{1}^{w}=\sqrt{\eta_{w}}\mathbb{Z}=\sqrt{\eta_{w}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ (14)

where $\eta_{w}\in[0,1]$ represents the probability parameter for the phase-
flip error in the quantum system. The effect of phase-flip noise on the
entangled channel can be studied by using the density matrices of the noisy
phase-affected channel, which is given by $\mathcal{G}^{w}(\rho)$

$\displaystyle\mathcal{G}^{w}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{\\{}(1-\eta_{w})^{7}\Big{[}\Ket{\Psi}\bra{\Psi}\Big{]}+(\eta_{w})^{7}\Big{[}\ket{000}(\ket{0}\ket{\mu^{-}}-\ket{1}\ket{\lambda^{-}})-\ket{001}(\ket{0}\ket{\lambda^{+}}+\ket{1}\ket{\mu^{+}})-\ket{010}(\ket{0}\ket{\lambda^{-}}+\ket{1}\ket{\mu^{-}})$
(15) $\displaystyle-$
$\displaystyle\ket{011}(\ket{0}\ket{\mu^{+}}-\ket{1}\ket{\lambda^{+}})-\ket{100}(\ket{1}\ket{\mu^{+}}-\ket{0}\ket{\lambda^{+}})-\ket{101}(+\ket{0}\ket{\mu^{-}}+\ket{1}\ket{\lambda^{-}})+\ket{110}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}})$
$\displaystyle-$
$\displaystyle\ket{111}(\ket{0}\ket{\lambda^{-}}-\ket{1}\ket{\mu^{-}})\Big{]}\times\Big{[}\bra{000}(\bra{0}\bra{\mu^{-}}-\bra{1}\bra{\lambda^{-}})-\bra{001}(\bra{0}\bra{\lambda^{+}}+\bra{1}\bra{\mu^{+}})-\bra{010}(\bra{0}\bra{\lambda^{-}}+\bra{1}\bra{\mu^{-}})$
$\displaystyle-$
$\displaystyle\bra{011}(\bra{0}\bra{\mu^{+}}-\bra{1}\bra{\lambda^{+}})-\bra{100}(\bra{1}\bra{\mu^{+}}-\bra{0}\bra{\lambda^{+}})-\bra{101}(+\bra{0}\bra{\mu^{-}}+\bra{1}\bra{\lambda^{-}})+\bra{110}(\bra{0}\bra{\mu^{+}}+\bra{1}\bra{\lambda^{+}})$
$\displaystyle-$
$\displaystyle\bra{111}(\bra{0}\bra{\lambda^{-}}-\bra{1}\bra{\mu^{-}})\Big{]}\Big{\\}}$

###  III.3 Bit-phase-flip noisy environment

The combination of a phase flip and a bit flip is referred to as a bit–phase
flip, and it is represented by the Kraus operators given in Eq. (16) [33, 34],

$\displaystyle
E^{f}=\sqrt{1-\eta_{f}}\mathbb{I}=\sqrt{1-\eta_{f}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{},~{}~{}~{}~{}E^{f}=\sqrt{\eta_{f}}\mathbb{Y}=\sqrt{\eta_{f}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}$ (16)

where $\eta_{f}\in[0,1]$ denotes the bit-phase-flip noise parameter
probability, which defines the likelihood of an error in the quantum state
arising owing to a computational qubit. The impact of bit-phase flip noise on
the entangled channel can be determined by calculating the noise affected
density matrix. The impacted density matrix under bit-phase flip noise is
denoted by $\mathcal{G}^{f}(\rho)$, which is provided by Eq. (17)

$\displaystyle\mathcal{G}^{f}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{\\{}(1-\eta_{f})^{7}\Big{[}\Ket{\Psi}\bra{\Psi}\Big{]}-(\eta_{f})^{7}\Big{[}\ket{111}(-\ket{1}\ket{\mu^{-}}+\ket{0}\ket{\lambda^{-}})-\ket{110}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}})+\ket{101}(\ket{1}\ket{\lambda^{-}}+\ket{0}\ket{\mu^{-}})$
(17) $\displaystyle+$
$\displaystyle\ket{100}(\ket{1}\ket{\mu^{+}}-\ket{0}\ket{\lambda^{+}})-\ket{011}(\ket{0}\ket{\mu^{+}}-\ket{1}\ket{\lambda^{+}})+\ket{010}(\ket{0}\ket{\lambda^{-}}+\ket{1}\ket{\mu^{-}})+\ket{001}(\ket{0}\ket{\lambda^{+}}+\ket{1}\ket{\mu^{+}})$
$\displaystyle-$
$\displaystyle\ket{000}(\ket{0}\ket{\lambda^{-}}+\ket{0}\ket{\mu^{-}})\Big{]}\times\Big{[}\bra{111}(-\bra{1}\bra{\mu^{-}}+\bra{0}\bra{\lambda^{-}})-\bra{110}(\bra{0}\bra{\mu^{+}}+\bra{1}\bra{\lambda^{+}})+\bra{101}(\bra{1}\bra{\lambda^{-}}+\bra{0}\bra{\mu^{-}})$
$\displaystyle+$
$\displaystyle\bra{100}(\bra{1}\bra{\mu^{+}}-\bra{0}\bra{\lambda^{+}})-\bra{011}(\bra{0}\bra{\mu^{+}}-\bra{1}\bra{\lambda^{+}})+\bra{010}(\bra{0}\bra{\lambda^{-}}+\bra{1}\bra{\mu^{-}})+\bra{001}(\bra{0}\bra{\lambda^{+}}+\bra{1}\bra{\mu^{+}})$
$\displaystyle-$
$\displaystyle\bra{000}(\bra{0}\bra{\lambda^{-}}+\bra{0}\bra{\mu^{-}})\Big{]}\Big{\\}}$

###  III.4 Effect of amplitude damping (AD) noise

The amplitude damping plays a vital role since it is responsible for
characterising the energy loss of a system, which is the effect that occurs
the most frequently in open systems. The idea of amplitude damping is crucial
to the modelling of energy dissipation in a variety of quantum systems, and
the matrices that follow supply the Kraus operators for this process [33, 34].

$\displaystyle E_{0}^{a}=\begin{pmatrix}1&0\\\
0&\sqrt{1-\eta_{a}}\end{pmatrix}~{},\quad
E_{1}^{a}=\begin{pmatrix}0&\sqrt{\eta_{a}}\\\ 0&0\end{pmatrix}$ (18)

Where $\eta_{a}\in[0,1]$ signifies the decoherence rate of amplitude damping,
which specifies the likelihood of quantum state inaccuracy associated with
computational qubits. The influence of amplitude damping on the entangled
channel may be detected by analysing the channel’s noise-affected density
matrix. This matrix is represented by $\mathcal{G}^{a}$, given in Eq. (19)

$\displaystyle\mathcal{G}^{a}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{\\{}\Big{[}\ket{000}(\ket{0}\ket{\Xi^{+}}+\sqrt{1-\eta_{a}}\ket{1}\ket{\Omega^{+}})+\sqrt{1-\eta_{a}}\ket{001}(\ket{0}\ket{\Omega^{-}}-\sqrt{1-\eta_{a}}\ket{1}\ket{\Xi^{-}})$
(19) $\displaystyle+$
$\displaystyle\sqrt{1-\eta_{a}}\ket{010}(\ket{0}\ket{\Omega^{+}}-\sqrt{1-\eta_{a}}\ket{1}\ket{\Xi^{+}})+(1-\eta_{a})\ket{011}(\ket{0}\ket{\Xi^{-}}+\sqrt{1-\eta_{a}}\ket{1}\ket{\Omega^{-}})$
$\displaystyle-$
$\displaystyle\sqrt{1-\eta_{a}}\ket{100}(\ket{0}\ket{\Omega^{-}}+\sqrt{1-\eta_{a}}\ket{1}\ket{\Xi^{-}})+(1-\eta_{a})\ket{101}(\sqrt{1-\eta_{a}}\ket{1}\ket{\Omega^{+}}-\ket{0}\ket{\Xi^{+}})$
$\displaystyle+$
$\displaystyle(1-\eta_{a})\ket{110}(\ket{0}\ket{\Xi^{-}}-\sqrt{1-\eta_{a}}\ket{1}\ket{\Omega^{-}})+\sqrt{(1-\eta_{a})^{3}}\ket{111}(\ket{0}\ket{\Omega^{+}}+\sqrt{1-\eta_{a}}\ket{1}\ket{\Xi^{+}})\Big{]}$
$\displaystyle\times$
$\displaystyle\Big{[}\bra{000}(\bra{0}\bra{\Xi^{+}}+\sqrt{1-\eta_{a}}\bra{1}\bra{\Omega^{+}})+\sqrt{1-\eta_{a}}\bra{001}(\bra{0}\bra{\Omega^{-}}-\sqrt{1-\eta_{a}}\bra{1}\bra{\Xi^{-}})$
$\displaystyle+$
$\displaystyle\sqrt{1-\eta_{a}}\bra{010}(\bra{0}\bra{\Omega^{+}}-\sqrt{1-\eta_{a}}\bra{1}\bra{\Xi^{+}})+(1-\eta_{a})\bra{011}(\bra{0}\bra{\Xi^{-}}+\sqrt{1-\eta_{a}}\bra{1}\bra{\Omega^{-}})$
$\displaystyle-$
$\displaystyle\sqrt{1-\eta_{a}}\bra{100}(\bra{0}\bra{\Omega^{-}}+\sqrt{1-\eta_{a}}\bra{1}\bra{\Xi^{-}})+(1-\eta_{a})\bra{101}(\sqrt{1-\eta_{a}}\bra{1}\bra{\Omega^{+}}-\bra{0}\bra{\Xi^{+}})$
$\displaystyle+$
$\displaystyle(1-\eta_{a})\bra{110}(\bra{0}\bra{\Xi^{-}}-\sqrt{1-\eta_{a}}\bra{1}\bra{\Omega^{-}})+\sqrt{(1-\eta_{a})^{3}}\bra{111}(\bra{0}\bra{\Omega^{+}}+\sqrt{1-\eta_{a}}\bra{1}\bra{\Xi^{+}})\Big{]}$
$\displaystyle+$
$\displaystyle(\eta_{a})^{7}\Ket{1111111}\bra{1111111}\Big{\\}}$

where the quantum states
$\ket{\Omega^{\pm}}=\dfrac{\sqrt{1-\eta_{a}}\big{(}\sqrt{1-\eta_{a}}\Ket{011}\pm\Ket{100}\big{)}}{\sqrt{2}}$
and
$\ket{\Xi^{\pm}}=\dfrac{\big{(}\Ket{000}\pm\sqrt{(1-\eta_{a})^{3}}\Ket{111}\big{)}}{\sqrt{2}}$

###  III.5 Phase damping noisy environment

The phenomenon of phase-damping results in the loss of information regarding
the relative phases of a quantum state. During phase damping, the fundamental
quantum system becomes entangled with the surrounding environment [35]. The
Kraus operators {$E_{0}^{p},E_{1}^{p},E_{2}^{p}$} for phase-damping noise are
given in Eq. (20) [33, 34].

$\displaystyle E_{0}^{p}=\sqrt{1-\eta_{p}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{},\quad E_{1}^{p}=\begin{pmatrix}\sqrt{\eta_{p}}&0\\\
0&0\end{pmatrix}~{},\quad~{}~{}~{}E_{2}^{p}=\begin{pmatrix}0&0\\\
0&\sqrt{\eta_{p}}\end{pmatrix}$ (20)

where $\eta_{p}\in[0,1]$ denotes the phase-damping decoherence rate, which
specifies the likelihood of an error occurring in the quantum state associated
with the computational qubit. After the noise has been introduced into the
channel, the affected density matrix can be used to figure out how phase
damping affected the entangled channels. The affected density matrix under the
phase damping noise is denoted by $\mathcal{G}^{p}$.

$\displaystyle\mathcal{G}^{p}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{\\{}(1-\eta_{b})^{7}\Big{[}\Ket{\Psi}\bra{\Psi}\Big{]}+(\eta_{p})^{7}\Big{[}\ket{0000000}\bra{0000000}+\Ket{1111111}\bra{1111111}\Big{]}\Big{\\}}$
(21)

###  III.6 Depolarizing Noisy Environment

When exposed to a depolarizing noisy environment, the quantum state’s qubits
are depolarized with a probability of $\eta_{d}$, and the qubits are left with
an invariant probability of $(1-\eta_{d})$. The following matrices give the
Kraus operators for depolarizing noise [33, 34].

$\displaystyle E_{0}^{d}=\sqrt{1-\eta_{d}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{},\quad
E_{1}^{d}=\sqrt{\frac{\eta_{d}}{3}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}~{},~{}~{}~{}E_{2}^{d}=\sqrt{\frac{\eta_{d}}{3}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}~{},\quad
E_{3}^{d}=\sqrt{\frac{\eta_{d}}{3}}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$
(22)

After introducing noise into a channel, the impact of depolarizing noise on
the entangled channel may be determined by analysing the affected density
matrix, the noise affected density matrix under the depolarizing noise is
denoted by $\mathcal{G}^{D}$

$\displaystyle\mathcal{G}^{d}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\Big{\\{}(1-\eta_{d})^{7}\Big{[}\Ket{\Psi}\bra{\Psi}\Big{]}+(\eta_{b})^{7}\Big{[}\ket{111}(\ket{1}\ket{\mu^{+}}+\ket{0}\ket{\lambda^{+}})+\ket{110}(\ket{0}\ket{\mu^{-}}-\ket{1}\ket{\lambda^{-}})+\ket{101}(\ket{1}\ket{\lambda^{+}}-\ket{0}\ket{\mu^{+}})$
(23) $\displaystyle-$
$\displaystyle\ket{100}(\ket{1}\ket{\mu^{-}}+\ket{0}\ket{\lambda^{-}})+\ket{011}(\ket{1}\ket{\lambda^{-}}-\ket{0}\ket{\mu^{-}})+\ket{010}(\ket{0}\ket{\lambda^{+}}-\ket{1}\ket{\mu^{+}})+\ket{001}(\ket{0}\ket{\lambda^{-}}-\ket{1}\ket{\mu^{-}})$
$\displaystyle+$
$\displaystyle\ket{000}(\ket{1}\ket{\lambda^{+}}+\ket{0}\ket{\mu^{+}})\Big{]}\times\Big{[}\bra{111}(\bra{1}\bra{\mu^{+}}+\bra{0}\bra{\lambda^{+}})+\bra{110}(\bra{0}\bra{\mu^{-}}-\bra{1}\bra{\lambda^{-}})+\bra{101}(\bra{1}\bra{\lambda^{+}}-\bra{0}\bra{\mu^{+}})$
$\displaystyle-$
$\displaystyle\bra{100}(\bra{1}\bra{\mu^{-}}+\bra{0}\bra{\lambda^{-}})+\bra{011}(\bra{1}\bra{\lambda^{-}}-\bra{0}\bra{\mu^{-}})+\bra{010}(\bra{0}\bra{\lambda^{+}}-\bra{1}\bra{\mu^{+}})+\bra{001}(\bra{0}\bra{\lambda^{-}}-\bra{1}\bra{\mu^{-}})$
$\displaystyle+$
$\displaystyle\bra{000}(\bra{1}\bra{\lambda^{+}}+\bra{0}\bra{\mu^{+}})\Big{]}-(\eta_{f})^{7}\Big{[}\ket{111}(-\ket{1}\ket{\mu^{-}}+\ket{0}\ket{\lambda^{-}})-\ket{110}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}})+\ket{101}(\ket{1}\ket{\lambda^{-}}+\ket{0}\ket{\mu^{-}})$
$\displaystyle+$
$\displaystyle\ket{100}(\ket{1}\ket{\mu^{+}}-\ket{0}\ket{\lambda^{+}})-\ket{011}(\ket{0}\ket{\mu^{+}}-\ket{1}\ket{\lambda^{+}})+\ket{010}(\ket{0}\ket{\lambda^{-}}+\ket{1}\ket{\mu^{-}})+\ket{001}(\ket{0}\ket{\lambda^{+}}+\ket{1}\ket{\mu^{+}})$
$\displaystyle-$
$\displaystyle\ket{000}(\ket{0}\ket{\lambda^{-}}+\ket{0}\ket{\mu^{-}})\Big{]}\times\Big{[}\bra{111}(-\bra{1}\bra{\mu^{-}}+\bra{0}\bra{\lambda^{-}})-\bra{110}(\bra{0}\bra{\mu^{+}}+\bra{1}\bra{\lambda^{+}})+\bra{101}(\bra{1}\bra{\lambda^{-}}+\bra{0}\bra{\mu^{-}})$
$\displaystyle+$
$\displaystyle\bra{100}(\bra{1}\bra{\mu^{+}}-\bra{0}\bra{\lambda^{+}})-\bra{011}(\bra{0}\bra{\mu^{+}}-\bra{1}\bra{\lambda^{+}})+\bra{010}(\bra{0}\bra{\lambda^{-}}+\bra{1}\bra{\mu^{-}})+\bra{001}(\bra{0}\bra{\lambda^{+}}+\bra{1}\bra{\mu^{+}})$
$\displaystyle-$
$\displaystyle\bra{000}(\bra{0}\bra{\lambda^{-}}+\bra{0}\bra{\mu^{-}})\Big{]}+(\eta_{w})^{7}\Big{[}\ket{000}(\ket{0}\ket{\mu^{-}}-\ket{1}\ket{\lambda^{-}})-\ket{001}(\ket{0}\ket{\lambda^{+}}+\ket{1}\ket{\mu^{+}})-\ket{010}(\ket{0}\ket{\lambda^{-}}+\ket{1}\ket{\mu^{-}})$
$\displaystyle-$
$\displaystyle\ket{011}(\ket{0}\ket{\mu^{+}}-\ket{1}\ket{\lambda^{+}})-\ket{100}(\ket{1}\ket{\mu^{+}}-\ket{0}\ket{\lambda^{+}})-\ket{101}(+\ket{0}\ket{\mu^{-}}+\ket{1}\ket{\lambda^{-}})+\ket{110}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}})$
$\displaystyle-$
$\displaystyle\ket{111}(\ket{0}\ket{\lambda^{-}}-\ket{1}\ket{\mu^{-}})\Big{]}\times\Big{[}\bra{000}(\bra{0}\bra{\mu^{-}}-\bra{1}\bra{\lambda^{-}})-\bra{001}(\bra{0}\bra{\lambda^{+}}+\bra{1}\bra{\mu^{+}})-\bra{010}(\bra{0}\bra{\lambda^{-}}+\bra{1}\bra{\mu^{-}})$
$\displaystyle-$
$\displaystyle\bra{011}(\bra{0}\bra{\mu^{+}}-\bra{1}\bra{\lambda^{+}})-\bra{100}(\bra{1}\bra{\mu^{+}}-\bra{0}\bra{\lambda^{+}})-\bra{101}(+\bra{0}\bra{\mu^{-}}+\bra{1}\bra{\lambda^{-}})+\bra{110}(\bra{0}\bra{\mu^{+}}+\bra{1}\bra{\lambda^{+}})$
$\displaystyle-$
$\displaystyle\bra{111}(\bra{0}\bra{\lambda^{-}}-\bra{1}\bra{\mu^{-}})\Big{]}\Big{\\}}$

Figure 2: The plot of fidelity against the noise parameter $\eta$ for all the
six kinds of noise models.

##  IV Security analysis

Quantum communication protocols, in comparison to their classical
counterparts, often have the characteristic of a superior level of security.
Our protocols are protected not just from attacks coming from the outside, but
also from those coming from inside the system from the dishonest participant.
Here, we provide two different types of security analysis about the process of
remotely preparing a state. The first of these is an outside attack from an
eavesdropper, attempting to learn the state that is being remotely prepared.
And the second threat is an attempt from a non-authorized participant who is
interested in finding out the quantum state that is being prepared remotely.

###  IV.1 Outside attack

Before the remote state preparation scheme is put into action, the seven-quit
entangled channel is to be created from the six-qubit Borras _et al._ state
and then distributed among the participants. Without loss of generality,
suppose Alice prepares the state $\Ket{\Psi}_{1234567}$ and send the qubits
$(2,3)$ to Bob, qubits $(4,6)$ to Charlie and $(5,7)$ to David. Alice includes
a predetermined number of decoy-state particles in the transmission which are
randomly distributed in one of the following four quantum states
$\\{\Ket{0},\Ket{1},\Ket{+},\Ket{-}\\}$. After it has been determined that all
three participants have received the particles, Alice will then proceed to
make the statement about the placements of the decoy particles as well as the
measurement basis. Then, participants Bob, Charlie, and David measure their
qubits in accordance with the provided basis and declare their results. Next,
Alice makes a comparison between the results of the measurement and the
initial states of the decoy particles. The fact that any eavesdropping leaves
a trace in the outcomes of the decoy sampling photons [36], enables the
security checking process to detect multiple types of attacks coming from an
outside attacker Eve. These attacks include several attacks mentioned in [20]
named as an intercept-resend attack, a measurement-resend attack, an
entanglement-measure attack, and a denial-of-service attack. During the
security checking, the impact of these attacks will be detected with a
probability larger than zero. This verification approach is based on the
notion of the BB84 QKD protocol [37], which has been shown to be completely
safe by a number of different researchers [38]. Moreover, the particles used
to construct the quantum channel do not transmit any concealed information. As
a result, if an eavesdropper is present, she is not only identifiable but also
incapable of gathering any relevant information during the security screening
process. After the completion of the security checks, the entangled channel
that is sufficiently safe will be distributed among the participants. If there
is evidence of eavesdropping, the participants will abandon this procedure and
begin over. After three different parties have validated the safety of the
quantum channel, an eavesdropper from the outside can no longer attack the
protocol since no qubits are being sent at this point. During the
implementation of the protocol, only classical information is sent, which has
no relevance to the secrets. Therefore, our remote state preparation protocol
is robust against an attack from an outsider.

###  IV.2 Inside attack

It is possible for a participant to carry out his attack by entangling an
auxiliary particle with his own particle. Let’s suppose that Alice is the
participant interested in finding out about Bob’s state, being remotely
prepared in an unethical way. She can prepare the auxiliary state
$\Ket{\varepsilon}$ and entangle it using the local unitary operation
$\mathcal{\hat{U}}$, which is defined as follows:

$\displaystyle\mathcal{\hat{U}}(\Ket{0}_{1}\Ket{\varepsilon}_{E})$
$\displaystyle=$
$\displaystyle\Ket{0}_{1}\Ket{\varepsilon_{00}}+\Ket{1}_{1}\Ket{\varepsilon_{01}}$
$\displaystyle\mathcal{\hat{U}}(\Ket{1}_{1}\Ket{\varepsilon}_{E})$
$\displaystyle=$
$\displaystyle\Ket{0}_{1}\Ket{\varepsilon_{10}}+\Ket{1}_{1}\Ket{\varepsilon_{11}}$
(24)

Where
$\braket{\varepsilon_{00}}{\varepsilon_{00}}+\braket{\varepsilon_{01}}{\varepsilon_{01}}=1$
and
$\braket{\varepsilon_{10}}{\varepsilon_{10}}+\braket{\varepsilon_{11}}{\varepsilon_{11}}=1$.
Suppose Charlie and David simultaneously measure their qubits and broadcast
their results to Bob through a classical channel. Without compromising
generality, assume that the Charlie and David’s measurement results are
$\Ket{00}_{C_{1}C_{2}}\Ket{00}_{D_{1}D_{2}}$. From table 1, Alice’s
measurement result is $\ket{\Upsilon_{1}}_{A}$ or $\ket{\Upsilon_{2}}_{A}$.
After completing the unitary operations (24) on her state, Alice entangled an
auxiliary qubit on it, and the resulting state becomes

$\displaystyle\ket{\Psi}_{AEB_{1}B_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\mathcal{\hat{U}}\Ket{\Upsilon_{1}}_{A}\Ket{\varepsilon}_{E}\Big{)}\otimes\Big{[}\frac{1}{\sqrt{2}}(\alpha\Ket{00}+\beta\Ket{10}+\alpha\Ket{11}-\beta\Ket{01})\Big{]}$
(25) $\displaystyle+$
$\displaystyle\frac{1}{4}\Big{(}\mathcal{\hat{U}}\Ket{\Upsilon_{2}}_{A}\Ket{\varepsilon}_{E}\Big{)}\otimes\Big{[}\frac{1}{\sqrt{2}}(\alpha\Ket{10}-\beta\Ket{00}-\alpha\Ket{01}-\beta\Ket{11})\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\Big{[}\alpha(\Ket{0}\Ket{\varepsilon_{00}}+\Ket{1}\Ket{\varepsilon_{01}})+\beta(\Ket{0}\Ket{\varepsilon_{10}}+\Ket{1}\ket{\varepsilon_{11}})\Big{]}\Big{[}\alpha\Ket{\psi^{+}}-\beta\Ket{\phi^{-}}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{1}{4}\Big{[}\alpha(\Ket{0}\Ket{\varepsilon_{10}}+\Ket{1}\Ket{\varepsilon_{11}})-\beta(\Ket{0}\Ket{\varepsilon_{00}}+\Ket{1}\ket{\varepsilon_{01}})\Big{]}\Big{[}-\beta\Ket{\psi^{+}}-\alpha\Ket{\phi^{-}}\Big{]}$

At this point, Alice must follow the rules for preparing the remote state and
tell Bob the result of her measurement. Bob will then do the unitary
operations on his qubit. Now, we will concentrate on the quantum system of
particles $AE$, which represents a portion of the entire system.

$\displaystyle\rho_{AE}$ $\displaystyle=$ $\displaystyle
tr_{B_{1}B_{2}}\big{(}\rho_{AEB_{1}B_{2}}\big{)}$ (26) $\displaystyle=$
$\displaystyle\dfrac{(\alpha^{4}+\beta^{4})}{16}\Big{[}(\Ket{0}\Ket{\varepsilon_{00}}+\Ket{1}\Ket{\varepsilon_{01}})(\bra{0}\bra{\varepsilon_{00}}+\bra{1}\bra{\varepsilon_{01}})+(\Ket{0}\Ket{\varepsilon_{10}}+\Ket{1}\Ket{\varepsilon_{11}})(\bra{0}\bra{\varepsilon_{10}}+\bra{1}\bra{\varepsilon_{11}})\Big{]}$
$\displaystyle=$
$\displaystyle\dfrac{(\alpha^{4}+\beta^{4})}{16}\Big{(}(\Ket{0\varepsilon_{00}}+\Ket{1\varepsilon_{01}})(\bra{0\varepsilon_{00}}+\bra{1\varepsilon_{01}})+(\Ket{0\varepsilon_{10}}+\Ket{1\varepsilon_{11}})(\bra{0\varepsilon_{10}}+\bra{1\varepsilon_{11}})\Big{)}$

Now, we will check the purity of the quantum state by evaluating the trace of
the square of the density matrix, given by

$\displaystyle tr(\rho^{2}_{AE})$ $\displaystyle=$
$\displaystyle\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}^{2}+\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}^{2}+2\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}|\braket{\varepsilon_{00}}{\varepsilon_{10}}+\braket{\varepsilon_{01}}{\varepsilon_{11}}|^{2}$
(27) $\displaystyle\leq$ $\displaystyle
2\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}^{2}\Big{[}1+(|\varepsilon_{00}|\varepsilon_{10}|+|\varepsilon_{10}|\varepsilon_{11}|)^{2}]$
$\displaystyle\leq$ $\displaystyle
2\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}^{2}\Big{[}1+(\sqrt{\braket{\varepsilon_{00}}{\varepsilon_{00}}\braket{\varepsilon_{10}}{\varepsilon_{10}}}+\sqrt{\braket{\varepsilon_{01}}{\varepsilon_{01}}\braket{\varepsilon_{11}}{\varepsilon_{11}}})^{2}\Big{]}$
$\displaystyle\leq$ $\displaystyle
2\Big{(}\dfrac{\alpha^{4}+\beta^{4}}{16}\Big{)}^{2}\Big{[}1+\Big{(}\dfrac{\braket{\varepsilon_{00}}{\varepsilon_{00}}+\braket{\varepsilon_{10}}{\varepsilon_{10}}}{2}+\dfrac{\braket{\varepsilon_{01}}{\varepsilon_{01}}+\braket{\varepsilon_{11}}{\varepsilon_{11}}}{2}\Big{)}^{2}\Big{]}$

Simplifying the expression using the Cauchy–Schwarz inequality and the
triangular inequality, we get

$\displaystyle tr(\rho^{2}_{AE})<1$ (28)

If this is the case, the quantum state of the particle $AE$ will be in a mixed
state, and Alice will not be able to extract any information from the system.
That is to say, the inner attack initiated by Alice cannot be considered valid
and Alice will not be able to steal any information from the attack.

##  V Discussion and Conclusion

In this study, the deterministic remote state preparation is studied via the
seven-qubit entangled channel, created from the maximally entangled Borras _et
al._ state under the effect of six different kinds of noises. First, the
seven-qubit entangled channel was constructed from the highly entangled Borras
state by taking an ancillary qubit and then performing a C-NOT operation from
the terminal qubit of the Borras state to the ancillary qubit. This was done
in order to establish the highly entangled Borras state. It generates an
entangled channel in order to facilitate the remote state preparation of a
two-qubit quantum state. The qubits that make up the entangled channels are
split up among the four participants. Alice is the one who is sending the
information, while Bob is the one who is participating and preparing the state
at his end remotely. After Alice and the other two players, Charlie and David,
have sent their measurement results to Bob using a classical channel, Bob then
performs the relevant operation to his qubit in order to remotely prepare the
state. The operations in the different scenarios are presented in table 1. In
addition, the impacts of six distinct types of noise are investigated during
the course of this study. The noise may be modelled using the Kraus operators
by applying them to the entangled channel. We have evaluated the density
matrices for all the noise channels and calculated the fidelity of the
remotely prepared state with respect to the noise-affected channel. For
various noise channels, it is possible to discern how the system loses
information due to decoherence when it interacts with its environment. A
fidelity metric has been used to calculate the amount of information that is
lost due to each type of noise, for which variation of fidelity with respect
to the noise parameter $\eta$ is plotted in figure 2. The plot shows that as
the noise parameter increases in the range $[0,1]$, the fidelity shows
different trends in different types of noise. The graph reveals that
depolarizing noise has the most negative effect on the channel, hence reducing
its fidelity and causing information loss. In addition, the phase flip noise
loses the least amount of information compared to all others. Moreover, the
security analysis is conducted for a protocol intruder. In the event of an
external attack, an eavesdropper may attempt to attack the protocol, tamper
with the process, and determine the prepared quantum state. However, it
appears that our protocol is secure against these types of threats, as
outsiders cannot steal information. Again, in the event of an inside attack, a
dishonest player may attempt to steal information by entangling an auxiliary
qubit $\Ket{\varepsilon}$ with his qubit in order to determine the state being
prepared. Attempting to do so, however, will result in the quantum state
transforming into a mixed state, resulting in the loss of information.

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