# Bidirectional quantum teleportation of even and odd coherent states through
the multipartite Glauber coherent state: Theory and implementation

Nada Ikken  LPHE-Modeling and Simulation, Faculty of Sciences, Mohammed V
University in Rabat, Rabat, Morocco.  Abdallah Slaoui  0000-0002-5284-3240
Corresponding author<EMAIL_ADDRESS>LPHE-Modeling and
Simulation, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Rachid Ahl Laamara  LPHE-
Modeling and Simulation, Faculty of Sciences, Mohammed V University in Rabat,
Rabat, Morocco.  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Lalla Btissam Drissi
0000-0002-1966-9025 LPHE-Modeling and Simulation, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Centre of Physics and
Mathematics, CPM, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  College of Physical and Chemical Sciences, Hassan II Academy of
Science and Technology, Rabat, Morocco.

###### Abstract

Quantum teleportation has become a fundamental building block of quantum
technologies, playing a vital role in the development of quantum communication
networks. Here, we present a bidirectional quantum teleportation (BQT)
protocol that enables even and odd coherent states to be transmitted and
reconstructed over arbitrary distances in two directions. To this end, we
employ the multipartite Glauber coherent state, comprising the Greenberger-
Horne-Zeilinger, ground and Werner states, as a quantum resource linking
distant partners Alice and Bob. The pairwise entanglement existing in
symmetric and antisymmetric multipartite coherent states is explored, and by
controlling the overlap and number of probes constructing various types of
quantum channels, the teleportation efficiency of teleported states in both
directions may be maximized. Besides, Alice’s and Bob’s trigger phases are
estimated to explore their roles in our protocol using two kinds of quantum
statistical speed referred to as quantum Fisher information (QFI) and Hilbert-
Schmidt speed (HSS). Specifically, we show that the lower bound of the
statistical estimation error, quantified by QFI and HSS, corresponds to the
highest fidelity from Alice to Bob and conversely from Bob to Alice, and that
the choice of the pre-shared quantum channel has a critical role in achieving
high BQT efficiency. Finally, we show how to implement the suggested scheme on
current experimental tools, where Alice can transfer her even coherent state
to Bob, and at the same time, Bob can transfer his odd coherent state to
Alice.

##  I Introduction

Quantum teleportation (QT) is an emerging technology for efficiently
transferring quantum states. Since it was proposed in 1993, QT has made great
progress, both theoretically [1, 2] and experimentally [3, 4], and has
attracted considerable attention [5, 6]. Inspired by Bennett’s work, various
QT schemes have been suggested to transfer many states and the advantage here
is that many of these processes are feasible experimentally, and a lot of them
are implemented in reality. Many demonstrations of these protocols have been
made in various physical systems such as electrons [7], atoms [8], ions [9],
photons [10], hybrid systems [11, 12, 13] and superconducting circuits [14].
Exceptional results have been obtained regarding teleportation distance with
satellite-based applications on the horizon [15, 16]. As well, some attempts
to extend this to more complex systems have been launched [17].

In fact, teleportation is understood as a fictitious method, where an object
disintegrates in one place and reconstructs itself perfectly in another.
Whereas in quantum terms, it is an ability that particles have to share
properties and information with each other through quantum systems. If a
quantum particle linked to another, regardless of the physical distance from
each other, they will be able to exchange information. This concept seems very
interesting for everyone, since teleportation is one of the great dreams of
humanity and imagined in many works of fiction. It seems that in quantum
terms, it has been achieved with Qubits, however, the areas that can benefit
from QT are logically the secure communications that are already used. A major
feature to consider here would be that the phenomenon producing QT is quantum
entanglement [18, 19, 20], known as quantum correlations [21, 22, 23]. Which
in turn, is defined as a particular interrelation between objects, i.e., the
measurement of one object immediately influences the other even though the two
remain entirely isolated and separate from each other. Therefore, the concept
of entanglement is one of the most important properties of the deep knowledge
of quantum mechanics, due to its important significance not only for QT, but
also for quantum computing [24, 25], quantum dense coding [26] and quantum key
distribution [27].

Broadly speaking, Alice (a sender) and Bob (a receiver) must share a resource
state, which must be an entangled quantum state, for the transfer of an
unknown quantum state to be performed. Throughout the teleportation procedure,
the resource state should remain entangled. For Alice to exchange measurement
results with Bob, both Alice and Bob must have a communication channel.
Superluminal communication is not allowed in this process, but it requires
classical communication and a shared entanglement resource [5]. Furthermore,
only when the resource state is maximally entangled and the system is devoid
of decoherence and dissipation can Bob employ unitary transformations to
recover an exact copy of the input state [28, 29]. In multiparty quantum
teleportation, several studies have been done. Hitherto, several QT schemes
have been successively proposed using various entangled quantum states. To the
best of our knowledge, Karlsson et al.[30] suggested the first QT between
three participants utilizing the GHZ state. Dong and his colleagues [31]
carried out a controlled communication between the three parties utilizing
imperfect Bell state measurement and GHZ state. Furthermore, Hassanpour and
his collaborators use GHZ-like states to realize a controlled quantum secure
for direct communication protocol [32]. It is essential to consider scenarios
with low entanglement and reduced efficiency because, under realistic
conditions, the effect of the environment reduces the degree of entanglement
between the system components. From this, the concept of probabilistic QT with
high probability of success was born [33]. Further, the standard QT protocol
has been extended to bidirectional (two-way) quantum teleportation (BQT),
where Alice and Bob transmit their states simultaneously to each other using
preshared entanglement, and it is an integrated feature of QT networks [34,
35]. Along these lines, Fu et al.,[36] reported a BQT scheme using a four-
qubit cluster state as a quantum channel where Alice and Bob can exchange
single-qubit states via the Hadamard operation, specified unitary operations,
and Bell basis measurement in their system concurrently. Then, Zha et al.,[37]
proposed a bidirectional controlled QT protocol where Alice and Bob can
simultaneously send unknown quantum states to one another under the
supervision of all controllers.

Recent progress has shown that coherent states are playing a significant role
in many applications of QT and communication, which are easily produced and
manipulated using optical components [38, 39, 41]. A variety of schemes have
been proposed for generating such superposition states as well as their
entangled counterparts using nonlinear interactions, photon subtraction from
squeezed vacuum states, mixing of squeeze vacuum with a coherent light [44].
There were also suggestions for creating freely moving multiparty resources
such as GHZ, W and cluster entangled coherent states [45]. Since then, the
field has continued to grow, as superposed coherent states can offer several
advantages in practice, including its robustness against noisy channels and
its status as the fastest method of information transmission [46]. In this
direction, Van-Enk et al.,[47] proposed a standard quantum teleportation
scheme of a single qubit encoded in coherent state superpositions
$|I\rangle=\epsilon_{+}|\alpha\rangle+\epsilon_{-}|-\alpha\rangle$. The
scheme’s success probability was demonstrated to be $50\%$. Nevertheless,
their suggested plan has a few flaws. For instance, mixing modes that are
located at various locations using a beam splitter, such as one mode that is
with Alice and the other mode that is with Bob. Besides, Pyrkov and Byrnes
[48] suggested a new scheme for teleporting a coherent spin state between two-
component Bose-Einstein condensates (atomic ensembles) and they found that the
error of the proposed protocol scales favorably with the number particles in
the coherent spin states, which reaches fidelities and are close to $100\%$ in
comparison with the all-classical strategy. This protocol has been generalized
by the same authors [49] to teleport a spin coherent state to another state of
the same type with an arbitrary position on the Bloch sphere. Motivated by
these works, a new BQT scheme is proposed here to transmit arbitrary even and
odd coherent states in both ways between the two users (i.e., bidirectional
teleported states). Using the quantum channel composed of multipartite Glauber
coherent states, user Alice can transmit an even coherent state to user Bob,
and simultaneously, user Bob can transmit an odd coherent state to user Alice
[50, 51, 52].

Mathematically, the efficiency of the QT process is measured by the fidelity
susceptibility between the input and output states [53, 54]. This concept is
closely related to the quantum Fisher information (QFI) such that the
expression of the fidelity susceptibility is proportional to the expression of
QFI for any teleported state [55, 83, 57]. Therefore, it is feasible to
estimate the teleported and gained parameters using the QFI which plays a
primary role in quantum estimation through the quantum Cramer-Rao bound [58,
59, 60]. Indeed, quantum metrology aims at improving the accuracy of
measurements by reducing their fundamental statistical uncertainty given by
quantum fluctuations. This is achieved by preparing the physical system in a
state that has particular quantum correlations. A fundamental tool for
evaluating and comparing the effectiveness of various metrological strategies
for quantum parameter estimates is provided by the QFI, where the higher the
QFI, the higher the precision [61, 62, 81]. Based on the knowledge that the
QFI can be derived as a quantum statistical speed from the Hellinger distance
[78], the Hilbert-Schmidt speed has in turn been used to improve the precision
of the estimated parameters that detects the lower limit of the statistical
estimation error like QFI [79]. Moreover, it has the advantage of avoiding the
diagonalization of the evolved density matrix and would be useful for high-
dimensional quantum systems.

Here, we implement BQT using two independent quantum random trigger qubits,
one of which corresponds to the even coherent state and the other to the odd
coherent state. The global state of the system and an entangled state between
the even and odd states (i.e. an entangled resource connecting Alice and Bob)
are then combined to create a circuit that was employed in the Qiskit (quantum
computer), which ultimately gives us a measurement of the probabilities. Then,
in order to overcome the traditional fidelity barrier, we derive the amount of
shared state entanglement quantified by Wootters concurrence, the
teleportation fidelity, the success probability and estimate the weight
parameters of the teleported states. We show how the trigger states and
initial states of the transported qubits affect the fidelity and QFI of
bidirectional teleportation, where both the maximum QFI and fidelity values
are the same even when predicted at distinct polarization angles. The HSS was
determined using QFI, and it was shown how well this statistical speed
quantifier can be applied to the BQT protocol. Further, by optimizing the
entanglement resources, we can achieve high-fidelity teleportation even in the
presence of noise and decoherence. The even and odd coherent states form the
basis of our protocol and enable the transmission of qubit states
bidirectionally using a single entangled state and two separate trigger
qubits. The entanglement resources are carefully chosen to ensure that the
teleported states are faithfully reproduced at the receiving end. Finally, we
conduct simulations using Qiskit to test the proposed BQT protocol in a
controlled environment and their capacity to transfer information
bidirectionally. By testing the protocol in a simulated environment, we can
identify potential issues and optimize the protocol accordingly. Once the
simulations are successful, we can move on to implementing the protocol on a
real quantum computer.

##  II Multipartite Coherent states in BQT protocol

###  II.1 Multipartite entanglement in Even and Odd coherent states

Any quantum technology attempts to exploit the non-classical properties of
light and matter to achieve supremacy and potentially exponential efficiencies
over classical deterministic approaches. There are many types of states in
this regard, the most typical of which is the coherent state. Originally, the
last state were proposed by Schrödinger for the quantum harmonic oscillator as
specific quantum states whose properties are more similar to those of their
classical counterpart [38]. Such states are regarded as quasi-classical
states, minimizing the Heisenberg uncertainty for the position and momentum
operators, and maintaining maximum localizability throughout the temporal
evolution of the system. Thence, in the 1960s, Glauber [39, 40], Sudarshan
[41] and Klauder [42, 43] launched this fruitful field by showing the
applicability and relevance of these states in quantum optics. Theoretically,
Glauber proved that these states can be defined as eigenvectors of the bosonic
annihilation operator and can be obtained by applying a displacement operator
${\cal D}\left(\eta\right)=\exp\left(\eta a^{\dagger}-\eta^{*}a\right)$ to the
vacuum state of the harmonic oscillator as

$|\eta\rangle\equiv
D(\eta)|0\rangle=\exp\left[-\frac{|\eta|^{2}}{2}\right]\sum_{n}\frac{\eta^{n}}{\sqrt{n!}}|n\rangle,$
(1)

where $\eta$ is the complex dimensionless amplitude of the coherent state and
$|n\rangle$ stands for the Fock state. When the concept is applied to a
quantized monochromatic electromagnetic wave, $n$ represents the number of
elementary quanta of energy, known as photons. On the other hand, the entire
Hilbert space is a tensor product for many systems of $n$ non-interacting
particles. This enables the superposition of tensorial products with the form
$\left|\eta_{1}\right\rangle\otimes\left|\eta_{2}\right\rangle\otimes\ldots\otimes\left|\eta_{n}\right\rangle\equiv\left|\eta_{1}\eta_{2}\ldots\eta_{n}\right\rangle$
to be used to represent any multipartite coherent state. To make our goal
clearer, we will focus mainly on the superposition of multipartite coherent
states between $\left|\eta\right\rangle$ and $\left|-\eta\right\rangle$ as

$|\psi\rangle\equiv\left|\eta,m,n\right\rangle\simeq{\cal
N}\left(|\eta,\ldots,\eta\rangle+e^{im\pi}|-\eta,\ldots,-\eta\rangle\right),$
(2)

where the normalization factor ${\cal N}$ reads

${\cal N}=\left[2+2p^{n}\cos m\pi\right]^{-1/2},$ (3)

with $m\in{\mathbb{Z}}$,
$p=\left\langle\eta\mid-\eta\right\rangle=\exp\left(-2\mid\eta\mid^{2}\right)$
is the overlap between the states, $m=1$ for antisymmetric states and $m=0$
for symmetric states. As it turns out, we will mainly investigate the two
asymptotic bounds of multipartite coherent states, with one case being
$p\rightarrow 0$ and the second case being $p\rightarrow 1$. In the first
limiting situation $p\rightarrow 0$, the two coherent states
$\left|\eta\right\rangle$ and $\left|-\eta\right\rangle$ are orthogonal and
constitute an orthogonal basis
$\left\\{\left|0\right\rangle\equiv\left|\eta\right\rangle,\left|1\right\rangle\equiv\left|-\eta\right\rangle\right\\}$.
Therefore, ${\cal N}=1/2$ and the state $\left|\eta,m,n\right\rangle$ acts as
a multipartite state of Greenberger-Horne-Zeilinger (GHZ) type such as

$\displaystyle\left|\eta,m,n\right\rangle\simeq\left|{GHZ}\right\rangle_{n}=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\left|\eta\right\rangle\otimes\left|\eta\right\rangle\otimes\ldots\otimes\left|\eta\right\rangle+e^{im\pi}\left|-\eta\right\rangle\otimes\left|-\eta\right\rangle\otimes\ldots\otimes\left|-\eta\right\rangle\right].$
(4)

Furthermore, the situations where $m=0$ and $m=1$ must be distinguished
individually in the second limiting case where $p\rightarrow 1$ (or
$\eta\rightarrow 0$). For symmetric states ($m$ even), the multipartite
coherent state (2) reduces to the ground state as

$\displaystyle\left|\eta,0,n\right\rangle\simeq\left|\eta\right\rangle\otimes\left|\eta\right\rangle\otimes\ldots\otimes\left|\eta\right\rangle,$
(5)

and for the second situation where $m$ odd (i.e., for the antisymmetric
states), the state $\left|\eta,1,n\right\rangle$ can be observed as a
multipartite state of Werner-type, i.e.,

$\displaystyle\left|\eta,1,n\right\rangle\simeq\left|W\right\rangle_{n}=\frac{1}{\sqrt{n}}$
$\displaystyle\left[\left|-\eta\right\rangle\otimes\left|\eta\right\rangle\otimes\ldots\otimes\left|\eta\right\rangle+\left|\eta\right\rangle\otimes\left|-\eta\right\rangle\otimes\ldots\otimes\left|\eta\right\rangle+\ldots+\left|\eta\right\rangle\otimes\left|\eta\right\rangle\otimes\ldots\otimes\left|-\eta\right\rangle\right].$
(6)

In other words, the symmetric states interject between the GHZ-like states
($p\rightarrow 0$) and the separable state ($p\rightarrow 1$). Conversely, we
observe that the antisymmetric states are interposed between the Werner-type
states ($p\rightarrow 1$) and the GHZ-type states ($p\rightarrow 0$). Here, we
separated our system into two parts to be able to use it in quantum
teleportation. The one generates mixed coherent states, while the other
generates exclusively pure coherent states. As a first bipartite splitting of
multipartite coherent states, we formulate Eq.(2) as follows

$|\psi\rangle=N\left(|\eta\rangle_{r}\otimes|\eta\rangle_{n-r}+e^{im\pi}|-\eta\rangle_{r}\otimes|-\eta\rangle_{n-r}\right)$
(7)

where
$|\pm\eta\rangle_{k}=|\pm\eta\rangle_{1}\otimes|\pm\eta\rangle_{2}\otimes\cdots\otimes|\pm\eta\rangle_{k}$,
and $k=r,n-r$. In fact, the total system is divided into two subsystems, one
containing any $r$ particles ($1\leq r\leq n-1$) and the other containing all
the remaining $n-r$ particles. Two logical qubits can be used to represent the
multi-particle state (2) and the implementation of this is given by

$|0\rangle_{r}=\frac{|\eta\rangle_{r}+|-\eta\rangle_{r}}{\sqrt{2\left(1+p^{r}\right)}},\hskip
28.45274pt|1\rangle_{r}=\frac{|\eta\rangle_{r}-|-\eta\rangle_{r}}{\sqrt{2\left(1-p^{r}\right)}},$
(8)

for the first subsystem in the orthogonal basis
$\left\\{|0\rangle_{r},|1\rangle_{r}\right\\}$, and likewise

$|0\rangle_{n-r}=\frac{|\eta\rangle_{n-r}+|-\eta\rangle_{n-r}}{\sqrt{2\left(1+p^{n-r}\right)}},\hskip
28.45274pt|1\rangle_{n-r}=\frac{|\eta\rangle_{n-r}-|-\eta\rangle_{n-r}}{\sqrt{2\left(1-p^{n-r}\right)}},$
(9)

for the second subsystem in the basis
$\left\\{|0\rangle_{n-r},|1\rangle_{n-r}\right\\}$. Inserting the equations
(8) and (9) into (7), the pure bipartite quantum state (7) in the
computational basis
$\left\\{|0\rangle_{r}\otimes|0\rangle_{n-r},|0\rangle_{r}\otimes|1\rangle_{n-r},|1\rangle_{r}\otimes|0\rangle_{n-r},|1\rangle_{r}\otimes|1\rangle_{n-r}\right\\}$
can be easily rewritten as

$|\psi\rangle=\sum_{\alpha=0,1}\sum_{\beta=0,1}G_{\alpha,\beta}|\alpha\rangle_{r}\otimes|\beta\rangle_{n-r},$
(10)

with the factors $G_{\alpha,\beta}$ are given by

$\displaystyle G_{0,0}={\cal N}\left(1+e^{im\pi}\right)a_{r}a_{n-r},\hskip
28.45274ptG_{0,1}={\cal N}\left(1-e^{im\pi}\right)a_{r}b_{n-r}$ $\displaystyle
G_{1,0}={\cal N}\left(1-e^{im\pi}\right)a_{n-r}b_{r},\hskip
28.45274ptG_{1,1}={\cal N}\left(1+e^{im\pi}\right)b_{r}b_{n-r},$ (11)

where the quantities $a_{k}$ and $b_{k}$ in terms of the overlap $p$ are

$\displaystyle a_{k}=\sqrt{\frac{1+p^{k}}{2}},\hskip 14.22636pt{\rm and}\hskip
14.22636ptb_{k}=\sqrt{\frac{1-p^{k}}{2}},\hskip 14.22636ptk=r,n-r.$

Further, the second bi-partition $\rho_{rk}$ can be produced from the complete
$n$-particles system by tracing all particles except those specified by the
indices $r$ and $k$. More exactly, there are $n(n-1)/2$ dissimilar density
matrices with the notation $rk$. In essence, all the reduced density matrices
$\rho_{rk}$ are the same. The next step is to establish $\rho_{12}$ and
generalize it. Such a reduced density matrix is obtained as

$\displaystyle\rho_{12}$
$\displaystyle=\operatorname{Tr}_{3,4,\ldots,n}\left(|\psi\rangle\langle\psi|\right)$
(12)
$\displaystyle=\mathcal{N}^{2}\left[|\eta,\eta\rangle\left\langle\eta,\eta\left|+e^{-im\pi}p^{n-2}\right|\eta,\eta\right\rangle\langle-\eta,-\eta|+e^{im\pi}p^{n-2}|-\eta,-\eta\rangle\langle\eta,\eta|+|-\eta,-\eta\rangle\langle-\eta,-\eta|\right].$

It’s important to notice that the states $|\pm\eta\rangle$ do not serve as an
orthogonal basis for the Fock Hilbert space $\mathcal{H}$. Moreover, the
coherent states are usually non-orthogonal; in fact, their overlap is
$|\langle\eta\mid\delta\rangle|^{2}=\exp\left(-|\eta-\delta|^{2}\right)$ for
the two coherent states $|\eta\rangle$ and $|\delta\rangle$. Nevertheless, an
orthonormal basis can be obtained by taking the even and odd coherent states
[68, 51] which are defined by

$\displaystyle\left|Even,\eta_{e}\right\rangle\equiv\left|\eta_{e}\right\rangle=N_{+}\left(\left|\eta\right\rangle+\left|-\eta\right\rangle\right),\hskip
56.9055pt\left|Odd,\eta_{o}\right\rangle\equiv\left|\eta_{o}\right\rangle=N_{-}\left(\left|\eta\right\rangle-\left|-\eta\right\rangle\right),$
(13)

where the normalization factors $N_{\pm}$ are given by

$\displaystyle N_{\pm}=\left[2\left(1\pm p\right)\right]^{-\frac{1}{2}}.$

The logical qubits involved in the two-dimensional Hilbert space are encrypted
in terms of even and odd coherent states as $\left|{\bf
0}\right\rangle_{L}\longmapsto\left|\eta_{e}\right\rangle$ and $\left|{\bf
1}\right\rangle_{L}\longmapsto\left|\eta_{o}\right\rangle$. Then one can show
that

$\displaystyle\left|\eta\right\rangle\equiv{\bf a}\left|{\bf
0}\right\rangle_{L}+{\bf b}\left|{\bf 1}\right\rangle_{L},\hskip
56.9055pt\left|-\eta\right\rangle\equiv{\bf a}\left|{\bf
0}\right\rangle_{L}-{\bf b}\left|{\bf 1}\right\rangle_{L},$ (14)

with

${\bf a}=\sqrt{\frac{1+p}{2}},\hskip 28.45274pt{\rm and}\hskip 28.45274pt{\bf
b}=\sqrt{\frac{1-p}{2}}.$ (15)

As a result, it is straightforward to verify that the superposition of
multipartite coherent states (12) can be written in the orthonormal basis (13)
formed by the even and odd coherent states
$\left\\{\left|\eta_{e}\eta_{e}\right\rangle,\left|\eta_{e}\eta_{o}\right\rangle,\left|\eta_{o}\eta_{e}\right\rangle,\left|\eta_{o}\eta_{o}\right\rangle\right\\}$
as

$\displaystyle\rho_{12}$ $\displaystyle=2\mathcal{N}^{2}\left\\{\left({\bf
a}^{4}+{\bf b}^{4}\right)\left(1+p^{n-2}\cos
m\pi\right)\left[\left|\eta_{e}\eta_{e}\right\rangle\left\langle\eta_{e}\eta_{e}\right|+\left|\eta_{o}\eta_{o}\right\rangle\left\langle\eta_{o}\eta_{o}\right|\right]+{\bf
a}^{2}{\bf b}^{2}\left(1-p^{n-2}\cos m\pi\right)\right.$
$\displaystyle\left.\left[\left|\eta_{e}\eta_{o}\right\rangle\left\langle\eta_{e}\eta_{o}\right|+\left|\eta_{e}\eta_{o}\right\rangle\left\langle\eta_{o}\eta_{e}\right|+\left|\eta_{o}\eta_{e}\right\rangle\left\langle\eta_{e}\eta_{o}\right|+\left|\eta_{o}\eta_{e}\right\rangle\left\langle\eta_{o}\eta_{e}\right|\right]+{\bf
a}^{2}{\bf b}^{2}\left(1+p^{n-2}\cos
m\pi\right)\left[\left|\eta_{e}\eta_{e}\right\rangle\left\langle\eta_{o}\eta_{o}\right|+\left|\eta_{o}\eta_{o}\right\rangle\left\langle\eta_{e}\eta_{e}\right|\right]\right\\}.$
(16)

Actually, the above states (II.1) holds an intrinsic amount of entanglement
between its parts. One of the easiest ways to measure quantum entanglement is
through concurrence [67], which is defined as

$\mathcal{C}(\rho)=\max\left\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right\\},$
(17)

where $\lambda_{i}$ are the eigenvalues of the Hermition matrix
$\rho\tilde{\rho}$ in decreasing order, with
$\tilde{\rho}=\left(\sigma_{y}\otimes\sigma_{y}\right)\rho^{*}\left(\sigma_{y}\otimes\sigma_{y}\right)$
is the spin-flip density matrix. Using Wootters formula for concurrence, the
explicit expression of the concurrence entanglement in the state (II.1) can be
expressed as

$\mathcal{C}\left(\rho_{12}\right)=\max\left\\{0,\mathcal{C}_{+},\mathcal{C}_{-}\right\\},$
(18)

with

$\displaystyle\mathcal{C}_{\pm}=\frac{1-p^{2}}{2\left(1+p^{n}\cos
m\pi\right)}$ $\displaystyle\left[\mid\left(1\pm p^{n-2}\cos
m\pi\right)\mid-\left(1\mp p^{n-2}\cos m\pi\right)\right].$ (19)

$\left(a\right)$ For symmetric states

$\left(b\right)$ For antisymmetric states

Figure 1: Behavior of concurrence entanglement as a function of the coherent
state overlapping $p$ for different values of the number of probes $n$; $(a)$
for symmetric multipartite coherent states ($m$ even), $(b)$ for antisymmetric
multipartite coherent states ($m$ odd).

In order to analyze the influence of both the coherent state overlapping and
the number of probes on the entangled resource linking sender and receiver in
our BQT protocol, we plot in Fig.(1) the behavior of the concurrence
entanglement of multipartite coherent states as a function of the overlap $p$
for some values of $n$. As sketched in Fig.1($a$) for symmetric coherent
states ($m$ even), after the initial increase, the collective entanglement
between the two parts of multipartite coherent states decreases to disappear
in the limiting case $p\rightarrow 1$, in which the resource states (2) become
separable states. Moreover, we observe that the increase in the number of
particles leads to a decrease in the degree of entanglement. In Fig.1($b$), we
depict the degree of entanglement for antisymmetric multipartite coherent
states ($m$ odd). The concurrence entanglement increases with increasing
values of the overlap $p$ and reaches its maximum value in the limiting case
$p\rightarrow 1$, where the state (2) reduces to a Werner-type multipartite
state. In the limit $p\rightarrow 0$ and also for the large number of
particles, Fig.1($b$) reveals that there is no entanglement between the two
parts forming the two sub-systems of coherent states. These results indicate
that the probe numbers $n$ and the overlap $p$ of the multipartite coherent
states are factors that could be controlled to achieve pre-shared entanglement
(i.e., quantum resource connecting Alice and Bob) in our BQT protocol.

###  II.2 Description of the proposed BQT protocol using Even and Odd coherent
states

In this protocol that employs the multipartite coherent state (2) as a quantum
channel, Alice and Bob, as two legitimate users, want to bidirectionally
teleport the even and odd coherent states (13) to each other, as shown
schematically in Fig.(2). In this respect, ten qubits are used to implement
this BQT scheme, as clearly explained in the quantum circuit (see Fig.3). Two
individuals Even and Odd who each own a qubit and desire to share a state
across a quantum channel are involved in the protocol. It is essential to
mention some important steps applied in the implementation of the presented
BQT protocol. Firstly, the even and odd coherent states (13), in the Bloch
sphere representation, are expressed as follows

$\displaystyle\rho_{Even}^{(e)}=\frac{1}{2}\left(\openone_{2}+\sum_{j=x,y,z}e_{j}\hat{\sigma}_{j}^{(e)}\right),\hskip
28.45274pt\rho_{Odd}^{(o)}=\frac{1}{2}\left(\openone_{2}+\sum_{j=x,y,z}o_{j}\hat{\tau}_{j}^{(o)}\right),$
(20)

where $\openone_{2}$ is a $2\times 2$ identity operator,
$e_{j}=\operatorname{Tr}(\sigma_{j}\rho_{Even}^{(e)})$ and
$o_{j}=\operatorname{Tr}(\tau_{j}\rho_{Odd}^{(o)})$ are the correlation matrix
elements, $\hat{\sigma}_{j}^{(e)}$ and $\hat{\tau}_{j}^{(o)}$ denotes the
Pauli matrices in the Even and Odd computational basis, respectively. In fact,
these states (13) are expected to play an important role in interferometric
detection of gravitational waves by lowering the ideal intensity of the input
laser [68]. Furthermore, when performing tests to verify quantum information
technology, the size of the coherent components $\eta$ is of utmost importance
[69, 70, 71].

$\displaystyle|\psi\rangle$$\displaystyle Alice$$\displaystyle Bob$Sending Odd
stateSending Even state Grauber Coherent State
as entangled state $|Even\rangle$$|Odd\rangle$ Figure 2: BQT using
multipartite coherent states $|\psi\rangle$, Alice and Bob can communicate
their states $|Even\rangle$ and $|Odd\rangle$ at the same time, At the end,
Both sender and the receiver are given a state that has some decoherence.

In the BQT scheme, each user has two distinct types of qubits in order that
Alice and Bob can exchange even and odd states with each other, i.e. the
trigger qubits that are initially specified to be

$\displaystyle\left|T_{e}\right\rangle=\cos\left(\frac{\vartheta_{e}}{2}\right)\left|\eta_{e}\right\rangle+\sin\left(\frac{\vartheta_{e}}{2}\right)\left|-\eta_{e}\right\rangle,$
(21)
$\displaystyle\left|T_{o}\right\rangle=\cos\left(\frac{\vartheta_{o}}{2}\right)\left|\eta_{o}\right\rangle+\sin\left(\frac{\vartheta_{o}}{2}\right)\left|-\eta_{o}\right\rangle,$

with $\vartheta_{i}\in\left[0,\pi\right]$ for ($i=e,o$). These can be
represented in the Bloch vector expansion as

$\displaystyle\rho_{T_{e}}=\frac{1}{2}\left(\openone_{2}+\sum_{j=x,y,z}t_{j}\hat{\sigma}_{j}^{(e)}\right),\hskip
28.45274pt\rho_{T_{o}}=\frac{1}{2}\left(\openone_{2}+\sum_{j=x,y,z}g_{j}\hat{\tau}_{j}^{(o)}\right),$
(22)

whereas in Eq.(22), $t_{j}$ and $g_{j}$ are the Bloch vector elements of both
qubits Even’s and Odd’s trigger and those elements are calculated using the
formula $e_{j}=\operatorname{Tr}(\sigma_{j}\rho_{T_{e}})$ and
$g_{j}=\operatorname{Tr}(\tau_{j}\rho_{T_{o}})$. Besides, two storage qubits
are used in BQT, one kept by the sender and the other by the recipient. The
two parties exchange and store quantum information using these qubits;
$|S^{1}_{e}\rangle$, $|S^{2}_{e}\rangle$ for the even direction and
$|S^{1}_{o}\rangle$, $|S^{2}_{o}\rangle$ for the odd direction (see Fig.(3)).

Figure 3: The implementation of BQT with two independent triggers qubits using
Qiskit. The X-gates are applied to set the first and last qubits to the state
$\left|1\right\rangle$. Then, we perform controlled-NOT (CNOT) gates between
two pairs of qubits, these CNOT gates create entanglement between the qubits.
The application of controlled-controlled-NOT (CCNOT) gates between three
qubits are used to teleport the state of one qubit to another qubit, and the
entangled state of the entire system is generated by putting Hadamard gates
(H), Controlled-Z gates, and phase shift gates, the values of two variables,
$m$ and overlap $p$, are adjusted to achieve a high fidelity accuracy.

To perform this protocol, Alice and Bob initially share a state of the form
(II.1) and we follow the Kiktenko steps reported in [35]:

Step 1: We start by setting the users quantum state as

$\displaystyle|\xi_{s}\rangle=|T_{e},S^{1}_{e},S^{2}_{e},Even\rangle|T_{o},S^{1}_{o},S^{2}_{o},Odd\rangle\otimes|\Psi\rangle$

Step 2: Applying $X$-gates to the first and final qubits of Triggers qubits,
$|Even\rangle$ and $|Odd\rangle$, sets the two qubits to be shared.

Step 3: We apply two controlled-not (CNOT) gates to the qubits $|Even\rangle$
and $|Odd\rangle$ and the entangled state $|\Psi\rangle$, means that if
$|Even\rangle$ is in state $|1\rangle$, then the entangled state
$|\Psi\rangle$ is flipped. Similary for the entangled state $|\Psi\rangle$ and
$|Odd\rangle$ as the control qubit. The equations for these operations can be
written as

$\displaystyle|\xi_{s}\rangle^{\prime}=CNOT(|T_{e},S^{1}_{e},S^{2}_{e},Even\rangle|T_{o},S^{1}_{o},S^{2}_{o},Odd\rangle\otimes|\Psi\rangle),$

and then applies a Hadamard gate to both qubits $|Even\rangle$ and
$|Odd\rangle$ to prepare these qubits in entangled superposition state. QT can
be accomplished in this entangled state

$\displaystyle|\xi_{s}\rangle^{\prime\prime}=H$
$\displaystyle\left[CNOT\left(|T_{e},S^{1}_{e},S^{2}_{e},Even\rangle|T_{o},S^{1}_{o},S^{2}_{o},Odd\rangle\otimes|\Psi\rangle\right)\right].$
(23)

Step 4: Both Triggers qubits $|T_{e,o}\rangle$, the entangled qubit
$|\Psi\rangle$ and storage qubits $|S^{i}_{j}\rangle$ with $i=1,2$, $j=e,o$,
are all subject to the two parties perform two Controlled-CNOT $CCNOT$ qubits
on them. The entangled qubit $|\Psi\rangle$ and triggers qubits
$|T_{eo}\rangle$ are the controlled qubits, and $|S^{i}_{j}\rangle$ are the
target qubits. we express that as follow:

$\displaystyle|\xi_{s}\rangle^{\prime\prime\prime}=CCNOT$
$\displaystyle\left[H\left\\{CNOT\left(|T_{e},S^{1}_{e},S^{2}_{e},Even\rangle|T_{o},S^{1}_{o},S^{2}_{o},Odd\rangle\otimes|\Psi\rangle\right)\right\\}\right].$
(24)

Step 5: The two parties apply two Hadamard gates $H$ and two Controlled-Z gate
on $|Even\rangle$ and $|Odd\rangle$, desired to send, and they use Phase shift
gate $CP$ on the entangled state of the entire system, then use again a
Controlled-Z on $|Even\rangle$, $|Odd\rangle$ and the entangled qubit
$|\Psi\rangle$. This phase of the protocol describes the actual teleportation
of information

$\displaystyle|\xi_{s}\rangle^{\prime\prime\prime\prime}=$ $\displaystyle
CZ[CP[CZ\left[H\left(CCNOT\left\\{H\left(CNOT\left\\{|T_{e},S^{1}_{e},S^{2}_{e},Even\rangle|T_{o},S^{1}_{o},S^{2}_{o},Odd\rangle\otimes|\Psi\rangle\right\\}\right)\right\\}\right)\right]]].$
(25)

After following all the phases above, and perform measurements on the four
storage qubits. The final teleported states are giving by

$\displaystyle\rho^{(e)}_{out}=P_{o}\overline{P}_{e}\rho_{Odd}^{(\text{o)
}}+\left(1-P_{o}\overline{P}_{e}\right)\rho_{1},$
$\displaystyle\rho_{out}^{(o)}=P_{e}\overline{P}_{o}\rho_{Even}^{\text{(e})}+\left(1-P_{e}\overline{P}_{o}\right)\rho_{1},$
(26)

with $\overline{P}_{i}=1-P_{i}$, $P_{i}=Tr(\rho_{T_{i}}\rho_{j})$ (with
$i=e,o$ and $j=Even,Odd$) and $\rho_{1}$ denotes the reduced state of the
quantum channel (II.1), written as

$\rho_{1}={\rm
Tr}_{2}\left[\rho_{12}\right]=\frac{3+p^{2}+\left(1+3p^{2}\right)p^{n-2}\cos(m\pi)}{4\left(1+p^{n}\cos(m\pi)\right)}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}.$ (27)

Step 6: The sender applies two Pauli gates (X,Z) to the qubit $|Even\rangle$
depending on the outcomes of the measurements, and receiver applies two Pauli
gates to his qubit $|Odd\rangle$ based on the classical information he
obtained from sender’s measurements, and vice versa. The Pauli gates that Even
and Odd use are determined by the values of the classical bits retrieved from
the measurements.

##  III Optimal fidelity in the proposed BQT scheme

The fidelity of quantum teleportation refers to how closely the transported
state matches the original state. In other words, it shows us how much
information is retained throughout the transfer procedure. Fidelity is
commonly stated as a number between $0$ and $1$, with $1$ representing full
fidelity. Many factors can influence the reliability of a bidirectional
quantum teleportation for coherent states. They include the degree of
entanglement between the two participants, the quality of the quantum channel
used to transfer the state, and the measurement and reconstruction process’s
efficiency [72]. The fidelity is a crucial quantity for measuring the overlap
between the density operator $\rho_{\text{out }}$ for the teleported state
$\left|\psi_{\text{in }}\right\rangle$ and the quantum state into that is
going to be teleported, i.e.,

$\mathcal{F}\left(\eta_{e},\eta_{o}\right)=\left\langle\psi_{\text{in
}}\left|\rho_{\text{out }}\right|\psi_{\text{in
}}\right\rangle=\left(Tr\left[\sqrt{\sqrt{\rho_{in}}\rho_{out}\sqrt{\rho_{in}}}\right]\right)^{2}$
(28)

The following provides the fidelity of the teleportation between the two users
$\mathcal{F}^{A\rightarrow B}$ and $\mathcal{F}^{B\rightarrow A}$:

$\displaystyle\mathcal{F}^{A\rightarrow
B}=\left(Tr\left[\sqrt{\sqrt{\rho_{even}}\rho_{out}^{(e)}\sqrt{\rho_{even}}}\right]\right)^{2},\hskip
28.45274pt\mathcal{F}^{B\rightarrow
A}=\left(Tr\left[\sqrt{\sqrt{\rho_{odd}}\rho_{out}^{(o)}\sqrt{\rho_{odd}}}\right]\right)^{2}.$
(29)

Using Eq.(22) and Eq.(12) at an initial state setting, we are able to
determine the fidelity of the teleported state for both directions. Thus, the
final expression for the fidelity from Alice to Bob is obtained as

$\displaystyle\mathcal{F}^{A\rightarrow
B}=\frac{1}{8}\left(1+p\right)^{2}\left(1+\sin\vartheta_{e}\right)$
$\displaystyle\left[1+p+\left(1-p\right)\sin\vartheta_{o}\right]+\frac{\Lambda\left(p,n,m\right)(p^{2}-1)}{32\left[1+p^{n}\cos\left(m\pi\right)\right]}\left[\left(1+p\right)\sin\vartheta_{o}-p-3+\right.$
$\displaystyle\left.\left(1+p\right)^{2}\sin\vartheta_{e}\left(\left(p-1\right)\sin\vartheta_{o}-1-p\right)\right],$
(30)

where the function $\Lambda\left(p,n,m\right)$ is given by

$\Lambda\left(p,n,m\right)=3+p^{2}+\left(3p^{2}+1\right)p^{n-2}\cos(m\pi).$
(31)

Likewise, the analytical expression of the fidelity from Bob to Alice is given
by

$\displaystyle\mathcal{F}^{B\rightarrow
A}=\frac{p-1}{32p^{2}\left(1+p^{n}\cos(m\pi)\right)}$
$\displaystyle\left[\left(p^{2}\left(3+p^{2}\right)+p^{n}\cos(m\pi)\left(1+3p^{2}\right)\right)\right.$
$\displaystyle\left.\left(p^{2}+2p-3+\left(1+p\right)^{2}\sin\vartheta_{e}-\left(p^{2}-1\right)\left(1+\sin\vartheta_{e}\right)\sin\vartheta_{o}\right)\right].$
(32)

$\left(a\right)$: $m=0,n=3,\vartheta_{o}=0$.

$\left(b\right)$: $m=1,n=25,\vartheta_{o}=\pi$.

$\left(c\right)$: $m=0,n=3,\vartheta_{e}=0$.

$\left(d\right)$: $m=1,n=25,\vartheta_{e}=\pi$.

Figure 4: The fidelity of teleportation from Alice to Bob (Panel.($a$) and
Panel.($b$)) and from Bob to Alice (Panel.($c$) and Panel.($d$)) as a function
of E for different values of the overlap parameter $p$. The comparison is
conducted for four different scenarios, each with different values for $m=0$,
$m=1$ and $n=3$, $n=25$. The fidelity values are described for each scenario
as follows: (a) fidelity from Alice to Bob with $\vartheta_{o}=0$, (b)
fidelity from Alice to Bob with $\vartheta_{o}=\pi$, (c) fidelity from Bob to
Alice with $\vartheta_{e}=0$, and (d) fidelity from Bob to Alice with
$\vartheta_{e}=\pi$.

To show the importance of the choice of the ”preshared entanglement state” to
work with as a quantum resource on the teleportation efficiency, we depict in
the Fig.(4) the variation of the fidelity of the BQT with respect to the angle
of both sender and receiver teleported coherent states $\nu_{e,o}$ for various
value of the overlap $p$, the number of partitions $n$ and the parameter $m$.

To better appreciate this behavior, it is important to consider the limit
cases for symmetric ($m$ even) and antisymmetric ($m$ odd) states. As shown in
Fig.(4$(a)$), we observed that the fidelity increases as the coherent state
overlapping $p$ increases. We also noted that the minimum number of partitions
$n=3$ is required for this behavior to occur. This finding indicates that for
the case of symmetric state (i.e.,$m=0$), the fidelity reaches its maximum
value if the shared resource is a separable state (i.e.,$p\longrightarrow 1$).
Furthermore, for the antisymmetric state $m=1$ as represented in Fig.(4$(b)$),
we noticed that the fidelity increases and the curves to be maximum when
$p\longrightarrow 1$, and in this case the shared resource state is reduced to
a bipartite state of Werner-type which is maximally entangled. This maximum
value occurs for a maximum number of partitions, e.g. with $n=25$. Besides,
when the shared resource is in a maximally intricate state (in particular a
GHZ-state; i.e., $m=0$ and $p=0$), the maximum fidelity is achieved during
teleportation from an odd coherent state to an even coherent state (see
Fig.(4$(c)$)). Additionally, Fig.(4$(d)$) shows that the fidelity is maximized
when the shared state between Alice and Bob is a GHZ maximally entangled
state. It corresponds to the antisymmetric state m=1 and the overlap p=0 with
a maximum number of partitions $n=25$. These results provide important
insights into the optimal conditions for achieving high-fidelity quantum
teleportation between even and odd coherent states.

These findings are critical in understanding the behavior of the fidelity of
our quantum system and can be utilized to optimize the performance of the BQT
protocol. When we specifically focus on the case where $n=3$, and the overlap
parameter $p$ is set to $1$, we observe a perfect fidelity with a value of
$1$. This implies that the teleportation process is flawless, and the
teleported state is an exact replica of the original state. However, as we
vary the overlap parameter $p$ for different values of parameters $m$ and the
number of partitions $n$, we notice that the $p=0.2$ and $p=0.5$ exhibit
similar trends and shapes with different values for fidelity. Furthermore, our
analysis reveals that the fidelity of quantum teleportation tends to improve
as the number of qubits $n$ increases, particularly at $n=25$ where we observe
the interference of other parameters. $\mathcal{F}^{B\rightarrow A}$ at $p=1$
exhibits a slight curve, indicating that the quantum system is approaching the
limits of its performance in terms of fidelity.

##  IV Estimation degree of a teleported state

#### Quantum Fisher Information:

Unlike quantum state teleportation in which their credibility is gauged by
teleportation fidelity, the information transmission conveyed by a physical
parameter is generally measured using QFI [61]. In other words, the amount of
quantum information is simply the maximum QFI in a measurement of the quantum
state, optimized over all possible measurements. Moreover, the ultimate
possible precision of the estimated parameter is provided by the inverse of
the QFI [73, 74]. This provides the sensitivity of a given quantum state with
respect to changes in the relevant encoded parameter. Further, it has been
shown that by estimating the relevant parameters encoded in the teleported
state, with the efficiency quantified by means of QFI, the credibility of the
QT protocol could be assessed [75, 76]. Based on the symmetric logarithmic
derivative operator $\mathcal{L}_{\xi}$ associated with parameter $\xi$, the
corresponding QFI is expressed as
$\mathrm{F}_{\xi}=\operatorname{Tr}\left[\left(\partial_{\xi}\rho_{\xi}\right)\mathcal{L}_{\xi}\right]$,
where
$\partial_{\xi}\rho_{\xi}=\left(\mathcal{L}_{\xi}\rho_{\xi}+\rho_{\xi}\mathcal{L}_{\xi}\right)/2$
and $\partial_{\xi}=\partial/\partial\xi$. In order to derive the analytical
expression for the QFI, we write the density operator $\rho_{\xi}$ in its
spectral decomposition as
$\rho_{\xi}=\sum_{k=1}^{d}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$, where
$\lambda_{k}$ and $|\psi_{k}\rangle$ are the eigenvalues and eigenvectors of
the matrix $\rho_{\xi}$, then we find that QFI can be reformulated as

$\displaystyle\mathrm{F}_{\xi}=\sum_{k=1}^{d}\frac{\left(\partial_{\xi}\lambda_{k}\right)^{2}}{\lambda_{k}}+\sum_{k=1}^{d}\lambda_{k}\mathrm{F}_{\xi,k}-\sum_{k\neq
l}\frac{8\lambda_{k}\lambda_{l}}{\lambda_{k}+\lambda_{l}}\left|\left\langle\psi_{k}\mid\partial_{\xi}\psi_{l}\right\rangle\right|^{2},$
(33)

where $\lambda_{k}\neq 0$ and $\lambda_{k}+\lambda_{l}\neq 0$ with
$\mathrm{F}_{\xi,k}=4\left[\left\langle\partial_{\xi}\psi_{k}\mid\partial_{\xi}\psi_{k}\right\rangle-\mid\left\langle\psi_{k}\mid\partial_{\xi}\psi_{k}\right\rangle\mid^{2}\right]$
being the QFI for the pure state $\left|\psi_{k}\right\rangle$. The purpose of
this part is to estimate the phase parameters encoded in the teleported states
by means of the QFI in both directions; from Alice to Bob and from Bob to
Alice. This requires finding the QFI in both teleported even and odd coherent
states (eq.(13)). For single-qubit states like our teleported states, Zhong et
al.[77] derived a simple and explicit formula for QFI as

$\mathrm{F}_{\xi}\left(\rho_{\xi}\right)=\left\\{\begin{array}[]{cc}\left|\partial_{\xi}\vec{r}\right|^{2}+\frac{(\vec{r}\cdot\partial_{\xi}\vec{r})^{2}}{1-|\vec{r}|^{2}}&\text{
if }|\vec{r}|<1,\\\ \left|\partial_{\xi}\vec{r}\right|^{2}&\text{ if
}|\vec{r}|=1,\end{array}\right.$ (34)

with $\vec{r}=\left(r_{x},r_{y},r_{z}\right)^{T}$ is the real Bloch vector of
$\rho_{\xi}$. Indeed, QFI can also be interpreted as a measure of quantum
statistical speed; these two notions quantifying the sensitivity of an initial
state to the variations of the estimated parameter and the higher this
sensitivity is, the more the estimated parameter can be approximated with
precision and efficiency. In turn, for parametric evolutions of classical
probability distributions, each statistical distance measure is reflected by a
statistical speed. This can be evaluated by taking the derivative of the
distance, i.e. the variation of the distance caused by a small change. By
maximizing the classical statistical speed over all the quantum measurements,
we then obtain the quantum statistical speed [78, 79]. In the following, we
analyze another type of quantum statistical speed named Hilbert-Schmidt Speed,
and explore how it can be related to QFI and fidelity in our BQT protocol.

#### Hilbert-Schmidt Speed:

Let’s start by recalling the family of distance measures such as

$|d_{\alpha}(p,q)|^{\alpha}=\frac{1}{2}\sum_{x}|p_{x}-q_{x}|^{\alpha},$ (35)

where $q=\\{q_{x}\\}_{x}$ and $p=\\{p_{x}\\}_{x}$ depend on the parameter
$\xi$ and represent the probability distributions, with $\alpha\geq 1$. To
derive the associated statistical speed from any statistical distance, we need
to quantify the distance between infinitesimally close distributions. We first
parameterize the probability distribution $p(\xi)$ and extend it to first
order in $\xi$ at $\xi_{0}$ (Taylor expansion at $\xi_{0}$), i.e.,

$p_{x}(\xi_{0}+\xi)=p_{x}(\xi_{0})+\frac{\partial
p_{x}(\xi)}{\partial\theta}|_{\xi=\xi_{0}}\xi+{\cal O}(\xi^{2}).$ (36)

Using this expansion and substituting expression (36) into equation (35), the
distance between $p(\xi_{0})$ and $p(\xi_{0}+\xi)$ is evaluated as

$d_{\alpha}(p(\xi_{0}+\xi),p(\xi_{0}))=\left(\frac{1}{2}\sum_{x}|p^{\prime}_{x}(\xi_{0})|^{\alpha}\right)^{1/\alpha}\xi+{\cal
O}(\xi^{2}),$ (37)

with $p^{\prime}_{x}(\xi_{0})=\frac{\partial
p_{x}(\xi)}{\partial\xi}|_{\xi=\xi_{0}}$. Using the above classical distance
(37), the classical statistical speed is given by

$s_{\alpha}[p(\xi_{0})]=\frac{d}{d\xi}\left[d_{\alpha}(p(\xi_{0}+\xi),p(\xi))\right]=\left(\frac{1}{2}\sum_{x}|p^{\prime}_{x}(\xi_{0})|^{\alpha}\right)^{1/\alpha}$
(38)

Taking now the quantum states $\rho$ and $\sigma$, we can expand these
classical concepts to the quantum case by taking $q_{x}={\rm
Tr}\\{E_{x}\sigma\\}$ and $p_{x}={\rm Tr}\\{E_{x}\rho\\}$ as the measurement
probabilities associated with the positive operator value measure (POVM)
satisfying $\sum_{x}E_{x}=\openone$ and $E_{x}\geq 0$. In this situation, the
quantum distance can be obtained by maximizing the Hellinger distance (38)
over all possible POVMs, that is,

$D_{\alpha}(\rho,\sigma):=\max_{\\{E_{x}\\}}d_{\alpha}(p,q)=\left(\frac{1}{2}{\rm
Tr}|\rho-\sigma|^{\alpha}\right)^{1/\alpha},$ (39)

and the statistical quantum speed is given by

$S_{\alpha}[\rho_{\xi}]=\max_{\\{E_{x}\\}}s_{\alpha}[p(\xi)]=\left(\frac{1}{2}{\rm
Tr}|\frac{d\rho_{\xi}}{d\xi}|^{\alpha}\right)^{1/\alpha}.$ (40)

In the special case where $\alpha=2$, the quantum statistical speed is reduced
to the Hilbert-Schmidt speed (HSS) and equation (40) thus becomes

${\rm HSS}[\rho_{\xi}]\equiv{\rm HSS}_{\xi}\equiv
S_{2}[\rho_{\xi}]=\sqrt{\frac{1}{2}{\rm Tr}|\frac{d\rho_{\xi}}{d\xi}|^{2}}.$
(41)

Notably, the HSS is recognized as a powerful figure of merit in refs.[80, 81,
82] to improve quantum phase estimation as well as to identify non-
Markovianity in an open quantum system consisting of $d$-qubits. More
generally, it would seem logical to explore the connections between HSS and
QFI since they are two quantum statistical speeds connected to the Hilbert-
Schmidt and Bures distances [83, 84] respectively.

To determine the analytical expression for QFI and HSS of teleported states
(II.2) from Eq.(34), it is necessary to compute the three corresponding Bloch
vector components ($e_{j}$ and $o_{j}$) given by Eq.(20). After a few
simplifications, the result is

$\displaystyle
o_{x}=2P_{e}\overline{P_{o}}+(1-P_{e}\overline{P_{o}})\frac{3+p^{2}+(1+3p^{2})p^{n-2}\cos(m\pi)}{4(1+p^{n})\cos(m\pi)},$
$\displaystyle
o_{y}=-2(1-P_{e}\overline{P_{o}})\frac{3+p^{2}+(1+3p^{2})p^{n-2}\cos(m\pi)}{4(1+p^{n})\cos(m\pi)},$
$\displaystyle o_{z}=-P_{o}\overline{P_{e}},$ (42)

for the even coherent state, and

$\displaystyle
e_{x}=2P_{o}\overline{P_{e}}+(1-P_{o}\overline{P_{e}})\frac{3+p^{2}+(1+3p^{2})p^{n-2}\cos(m\pi)}{4(1+p^{n})\cos(m\pi)},$
$\displaystyle
e_{y}=-2(1-P_{o}\overline{P_{e}})\frac{3+p^{2}+(1+3p^{2})p^{n-2}\cos(m\pi)}{4(1+p^{n})\cos(m\pi)},$
$\displaystyle e_{z}=-P_{o}\overline{P_{e}},$ (43)

for the odd coherent state with the entries $P_{e,o}$ given by

$\displaystyle
P_{e,o}=\frac{1}{2}(1-p)\left[1-2\cos\vartheta_{e,o}\sin\vartheta_{e,o}\right].$
(44)

By inserting equations (42) and (43) into equation (34) with respect to the
parameters $\vartheta_{e}$ and $\vartheta_{o}$, the QFI takes the form

$\displaystyle F_{\vartheta_{e}}^{A\rightarrow B}=$
$\displaystyle\chi(n,m,p)+\frac{F^{A\rightarrow
B}_{1}\left(n,m,p\right)}{F^{A\rightarrow B}_{2}\left(n,m,p\right)},$ (45)

from Alice to Bob and

$\displaystyle F_{\vartheta_{o}}^{B\rightarrow
A}=\Omega(n,m,p)+\frac{F^{B\rightarrow A}_{1}(n,m,p)}{F^{B\rightarrow
A}_{2}(n,m,p)},$

from Bob to Alice. Consequently, the analytical expressions of the SHSs,
associated to the estimated trigger phases of Alice $\vartheta_{e}$ and Bob
$\vartheta_{o}$, are given by

$\displaystyle{\rm HSS}_{\vartheta_{e}}^{A\rightarrow
B}=\frac{1}{2}\sqrt{F_{\vartheta_{e}}^{A\rightarrow B}},\hskip 28.45274pt{\rm
HSS}_{\vartheta_{o}}^{B\rightarrow
A}=\frac{1}{2}\sqrt{F_{\vartheta_{o}}^{B\rightarrow A}},$

with the quantities $F^{A\rightarrow B}_{i}\left(n,m,p\right)$ and
$F^{B\rightarrow A}_{i}\left(n,m,p\right)$ are

$\displaystyle F^{A\rightarrow B}_{1}\left(n,m,p\right)$
$\displaystyle=\cos^{2}\vartheta_{e}\left[\Upsilon\left(n,m,p\right)+(p-1)^{2}(1+p)(1+p^{n}\cos(m\pi))^{2}(p-1+(1+p)\sin\vartheta_{e})\left(\cos\left(\vartheta_{o}/2\right)-\sin\left(\vartheta_{o}/2\right)\right)^{4}-\right.$
$\displaystyle\left.(1-p^{2})\Theta(n,m,p)(1-\sin\vartheta_{o})\left(4+4p^{n}\cos(m\pi)-\frac{1}{4}\Theta(n,m,p)\Gamma(n,m,p)\right)\right]^{2},$
(46) $\displaystyle F^{A\rightarrow B}_{2}\left(n,m,p\right)$
$\displaystyle=256(1+p^{n}\cos(m\pi))^{4}\left[1+\frac{1}{64}\left(-4(p-1)^{2}\left(p-1+(1+p)\sin\vartheta_{e}\right)^{2}(\cos(\vartheta_{o}/2)-\sin(\vartheta_{o}/2))^{4}-\right.\right.$
$\displaystyle\left.\left.\Lambda(n,m,p)^{2}-\left(4(1-p)(p-1+(1+p)\sin\vartheta_{e})(\sin\vartheta_{o}-1)+\Lambda(n,m,p)\right)^{2}\right)\right],$
(47) $\displaystyle F^{B\rightarrow A}_{1}(n,m,p)=$
$\displaystyle(p^{2}-1)^{2}\cos(\vartheta_{o})^{2}\Xi(n,m,p)^{4}\left[4(1+p)(1+p^{n}cos(m\pi))^{2}(\Xi(n,m,p))^{2}\mu(n,m,p)\right.$
$\displaystyle\left.+((3+p^{2})p^{-2}+p^{n-4}(1+3p^{2})\cos(m\pi))^{2}\left[(1+p)\sin\vartheta_{e}\mu(n,m,p)+\kappa(n,m,p)\right]+\right.$
$\displaystyle\left.4(p^{2}-1)(1-p^{n-2}\cos(m\pi))\left[\varepsilon(n,m,p)^{2}\Xi(n,m,p)^{2}\mu(n,m,p)+\right.\right.$
$\displaystyle\left.\left.\frac{1}{4}(3+p^{2}+(3p^{n}+p^{n-2})\cos(m\pi))\left((1+p)\mu(n,m,p)\sin\vartheta_{e}+\kappa(n,m,p)\right)\right]\right]^{2},$
(48) $\displaystyle F^{B\rightarrow A}_{2}(n,m,p)=$ $\displaystyle
4096(1+p^{n}\cos(m\pi))^{4}\left[1-\Delta(n,m,p)^{2}\varsigma(n,m,p)^{2}-\frac{1}{4}\tau(n,m,p)^{2}(1-\Delta(n,m,p)\varsigma(n,m,p))^{2}\right.$
$\displaystyle\left.-4\left(\Delta(n,m,p)\varsigma(n,m,p)+\frac{1}{4}\tau(n,m,p)\left(1-\Delta(n,m,p)\varsigma(n,m,p)\right)\right)^{2}\right],$
(49)

where the functions
$\chi,\Gamma,\Theta,\Lambda,\Upsilon,\Omega,\Xi,\Delta,\varepsilon,\varsigma,\kappa,\mu,\tau,$
are given by

$\displaystyle\chi\left(n,m,p\right)=\frac{(p^{2}-1)^{2}\left(p^{2}(2+p^{2})+p^{n}(1+2p^{2})\cos(m\pi)\right)^{2}\cos^{2}\vartheta_{e}(-1+\sin\vartheta_{o})^{2}}{16p^{4}\left(1+p^{n}\cos(m\pi)\right)^{2}},$
$\displaystyle\Gamma(n,m,p)=3+2p-p^{2}+(p^{2}-1)\sin\vartheta_{e}\left[\sin\vartheta_{o}-1+(p-1)^{2}\sin\vartheta_{o}\right],\hskip
14.22636pt\Theta(n,m,p)=3+p^{2}+\left(3+\frac{1}{p^{2}}\right)p^{n}\cos(m\pi),$
$\displaystyle\Lambda(n,m,p)=\frac{\left(3+p^{2}+(3p^{2}+1)p^{n-2}\cos(m\pi)\right)\left(3+2p-p^{2}+(p^{2}-1)\sin\vartheta_{e}\left(\sin\vartheta_{o}-1\right)+(p-1)^{2}\sin\vartheta_{o}\right)}{1+p^{n}\cos(m\pi)},$
$\displaystyle\Omega(n,m,p)=\frac{(p^{2}-1)^{2}\left(p^{2}\left(2+p^{2}\right)+p^{n}\left(1+2p^{2}\right)\cos(m\pi)\right)^{2}\cos(\phi)^{2}\left(\cos(\vartheta_{e}/2)+\sin(\vartheta_{e}/2)\right)^{4}}{16p^{4}\left(1+p^{n}\cos(m\pi)\right)^{2}},$
$\displaystyle\Upsilon(n,m,p)$
$\displaystyle=\frac{1}{p^{2}}(p^{2}-1)^{2}(p^{2}-p^{n}\cos(m\pi))(-1+\sin(\vartheta_{o}))$
$\displaystyle\left[(1-p)(1+p^{n}\cos(m\pi))(p-1+(1+p)\sin\vartheta_{e})(-1+\sin\vartheta_{o})+\frac{1}{4}\Theta(n,m,p)\Gamma(n,m,p)\right]$
$\displaystyle\Xi(n,m,p)$
$\displaystyle=\cos(\vartheta_{e}/2)+\sin(\vartheta_{e}/2),\hskip
56.9055pt\Delta(n,m,p)=\frac{1}{2}(1+p)\left[1+2\cos(\vartheta_{e}/2)\sin(\vartheta_{e}/2)\right],$
$\displaystyle\varepsilon(n,m,p)$
$\displaystyle=-(1+p)(1+p^{n}\cos(m\pi)),\hskip
56.9055pt\varsigma(n,m,p)=1-\frac{1}{2}(1-p)\left[1-2\cos(\vartheta_{o}/2)\sin(\vartheta_{o}/2)\right]$
$\displaystyle\kappa(n,m,p)$
$\displaystyle=(1-p)\left[3+p-(1+p)\sin(\vartheta_{o})\right],\hskip
56.9055pt\mu(n,m,p)=(p-1)\sin(\vartheta_{o})-1-p,$ $\displaystyle\tau(n,m,p)$
$\displaystyle=\frac{1}{(1+p^{n}\cos(m\pi))}\left(3+p^{2}+p^{n-2}(1+3p^{2})\cos(m\pi)\right).$

$\left(a\right)m=0,n=3,\vartheta_{o}=0$

$\left(b\right)m=1,n=25,\vartheta_{o}=\pi/6$

$\left(c\right)m=0,n=3,\vartheta_{e}=0$

$\left(d\right)m=1,n=25,\vartheta_{e}=\pi/6$

$\left(e\right)m=0,n=3,\vartheta_{o}=0$

$\left(f\right)m=1,n=25,\vartheta_{o}=\pi/6$

$\left(g\right)m=0,n=3,\vartheta_{e}=0$

$\left(h\right)m=1,n=25,\vartheta_{e}=\pi/6$

Figure 5: The top row represents the QFI, same as Fig.(4), with different
parameters: (a) from Alice to Bob with $m=0,n=3,\vartheta_{o}=0$, (b) from
Alice to Bob with $m=1,n=25,\vartheta_{o}=\pi/6$, (c) from Bob to Alice with
$m=0,n=3,\vartheta_{e}=0$, and (d) from Bob to Alice with
$m=1,n=25,\vartheta_{e}=\pi/6$. The bottom row shows the comparison between
QFI and HSS for Alice and Bob’s states.

In the top row of Fig.(5), we analyzed the behavior of the QFI as a function
of the trigger qubits phases $\vartheta_{e,o}$ under the same parameters as in
Fig.(4). Obviously, the QFI of the teleported states strongly depends on the
number of probes $n$, the parameter $m$ and the overlapping $p$ between the
coherent states, which specify the pre-shared quantum channel. In the
direction from Alice to Bob (Fig.5 panels (a)-(b)), it is obvious that the QFI
of the teleported even coherent state increases with decreasing overlapping
parameter $p$. Comparing this behavior with the above results displayed in
Fig.(4), we conclude that the maximum fidelity corresponds to the minimum
variance of the estimated parameter $\nu_{e}$. Additionally, perfect fidelity
corresponds to zero QFI, where the variance of the Alice’s trigger phase is
diverged (see the limiting case $p\longmapsto 1$). In the second direction
corresponding to Bob towards Alice, and for fixed parameters n and m, the QFI
(Fig.5 panels (c)-(d)) and the teleportation fidelity (Fig.4 panels (c)-(d))
increase with the decrease of the overlapping $p$. This indicates that the
teleportation efficiency reaches to their tops values when the estimation of
Bob’s trigger phase diverges (i.e. the variance is maximal). More importantly,
the behavior of QFI is similar in both directions, but as QFI from Alice to
Bob $F_{\vartheta_{e}}^{A\rightarrow B}$ increases, QFI from Bob to Alice
$F_{\vartheta_{o}}^{B\rightarrow A}$ decreases, which means that when the
information in Alice’s trigger hand begins to increase, the information in
Bob’s trigger hand decreases.

To compare quantitatively the QFI and HSS associated with the parameters of
the trigger phases, we plotted in the bottom row of Fig.(5) their behaviors
under the same conditions. From Fig.5(panel $(e)$ to panel $(h)$), it is
obvious that the minimum and maximum values of the HSS and QFI coincide
perfectly and both exhibit oscillatory features. These results provide a
qualitative indication that HSS can be employed to pinpoint the precise
moments when the estimated trigger qubit phases reach their best estimates.
HSS therefore performs the same role as QFI in improving the efficiency of our
BQT protocol in both directions.

This BQT protocol emerges as an exceptionally advantageous approach,
particularly suitable for scenarios necessitating simultaneous data exchange
among distributed participants. In our present study, we have deliberately
opted for the multipartite state of the entire system as our chosen quantum
channel. The utilization of multipartite Glauber coherent states as the
quantum channel for BQT brings forth numerous advantages that surpass those
associated with other types of entangled states such as GHZ states, cluster
states, or Bell states [85, 86, 87]. A punching characteristic of coherent
states is their resistance to specific types of noise and decoherence, which
separates them from far more entangled systems. This fundamental robustness
proves to be crucial, particularly in practical implementations where
environmental disturbances provide difficult challenges. The feasibility and
technical feasibility linked to the creation and control of coherent states
endorse their favorability, especially in experimental setups where
establishing and sustaining complex entanglement structures like GHZ [88, 90]
or cluster states [89, 91] could present substantial obstacles. Furthermore,
the versatility of multipartite Glauber coherent states extends beyond
teleportation, making them valuable not only for this purpose but also as
potential central components within quantum networks due to their adaptability
across a diverse range of quantum communication tasks. When compared to multi-
hop states [85], multipartite Glauber coherent states show their advantages as
a quantum channel, like in our case. Notably, when considering the impact of
noise and diminution of signals during quantum transmission, the robust
property of Multipartite Glauber Coherent States becomes even more important.
The selection of a right quantum channel shows up as a critical determinant in
the field of quantum communication. While the benefits of multi-hop states are
undisputed, the unique advantages of Multipartite Glauber Coherent States,
such as their reliability, simple encoding, integration with classical
communication, decoherence reduction, and efficient resource utilization, make
them a more interesting option for various quantum communication paradigms.
Coherent states advantages and unique properties position them not only as a
theoretical construct, but as a possible trigger in shifting the quantum
communication domain into pragmatic practical uses.

When examining the work [85], we can perceive the advantages of our BQT
protocol. In ref.[85], the teleportation process is highly reliable, with a
one hundred percent success rate in an ideal environment. This stability is a
significant advantage for secure and effective long-distance quantum
communication. We discovered that in the given protocol, the authors have
chosen the maximally entangled state (GHZ-type) alongside Bell states as a
quantum resource, which has improved the efficiency of fidelity. The protocol
can achieve a maximum fidelity of $1$ when certain conditions are satisfied.
Additionally, it demonstrates efficiency in terms of achieving perfect
teleportation fidelity under specific situations and maintains resilience with
regard to different measurement outputs. In our proposed protocol, we adopt a
distinct point of view. We treat the whole system state (including the GHZ,
ground and W type states) as a quantum resource. During our fidelity
calculations, we have demonstrated that we too may obtain maximum fidelity (as
in ref.[85]) by correctly modifying the parameter values we vary ($m$ for
symmetric and antisymmetric states, $n$ for the number of partitions, and $p$
for overlap). This proves that achieving maximum BQT efficiency is
consistently accomplished through the utilization of maximally entangled
resource states (i.e., GHZ or W states), and this will be attained in our BQT
protocol by optimizing the relevant parameters (see Fig.(4)).

Contrast between the proposed BQT protocol and prior studies
---
Protocol |  Type of Protocol |  Quantum channel |  Teleported state
Ref.[85] |  BQT |  Multi-hop (N-GHZ state and two Bell states) |  Single state
Ref.[87] |  SQT |  Single Bell state |  Single state
Ref.[88] |  SQT |  Tripartite entangled GHZ state and one Bell state |  Single state
Ref.[89] |  BQT |  Five cluster qubits |  Alice with arbitrary two-qubit state and Bob with unknown state
Ref.[90] |  BQT |  Two GHZ-States |  single-qubit states
Ref.[91] |  BQT |  Six-Qubit Cluster State |  single-qubit states
Our proposed protocol |  BQT |  Multi-partite coherent state |  even and odd coherent states
Table 1: Comparative Analysis of Seven BQT and QT Protocols in Various Studies

##  V Experimental implementation of the proposed BQT

Actually, quantum communication protocols are extremely sensitive, i.e. a
small variation in some of their parameters renders the protocol completely
useless. Countless protocols have been published over the past few decades
with supposed improvements on traditional teleportation, and in recent years,
we have had free online access to quantum computers and can implement these
protocols, and we have observed that the vast majority do not do what the
authors proclaimed. These protocols can be implemented in Quirk [92], Qiskit-
IBM-Q [93], Quantum Inspire by QuTech [94], Rigetti Forest [95], Quantum
Programming Studio [96], D-Wave [97] and more. So, to give impartiality and
credibility to the implementation, we test here our proposed BQT protocol on
Qiskit.

Let’s assume that the scheme (3) represents the complete transmission network.
Here, Alice and Bob are the convenient names for the sender and receiver and
they wish to send two qubits to be transmitted between them (even and odd
coherent states), and both parties have access to a conventional communication
channel. Alice’s measures it particle and one half of it entangled state in
order to transfer the quantum state, and it sends Bob the measurement result
over a conventional communication channel. Bob transfers Alice’s particle’s
quantum state to it own by performing a rectification operation on it own
particle in accordance with the measurement result. It’s crucial to remember
that quantum teleportation is probabilistic, which means it could not always
work. The success rate of quantum teleportation can, however, approach $100\%$
with enough tries.

In fact, BQT is a sort of QT that allows quantum states to be sent between two
parties as well as one way alone. BQT requires that both parties own their own
trigger qubits, which are used to store the unknown states that they wish to
share with one another. To create these states in advance, certain gates are
applied to the trigger qubits. Additionally, a traditional communication
channel and an entangled state must be shared by the parties. To transmit the
quantum states, both sides perform an entanglement measurement on their
trigger qubits and the entangled state. They then communicate the results of
this measurement across a classical communication channel. Based on the
results of the measurements, both parties perform correction operations on
their trigger qubits, thereby transferring the quantum states from one party
to the other. Below, we describe possible BQT with two random trigger qubits.
Also note that the circuit operate with $10$-qubits [98].

Qubits in the BQT circuit
---
Sender and receiver |  Qubits in Qiskit |  States
Even coherent state |  $\mathrm{qr}[0]$ |  $\left|T_{e}\right\rangle$
$\mathrm{qr}[1]$ |  $\left|S_{e}^{2}\right\rangle$
$\mathrm{qr}[2]$ |  $\left|S_{e}^{1}\right\rangle$
|  $\mathrm{qr}[3]$ |  $|Even\rangle$
Entangled state |  $\mathrm{qr}[4],\mathrm{qr}[5]$ |  $|\psi\rangle$
|  $\mathrm{qr}[6]$ |  $|Odd\rangle$
Odd coherent state |  $\mathrm{qr}[7]$ |  $\left|S_{o}^{1}\right\rangle$
|  $\mathrm{qr}[8]$ |  $\left|S_{o}^{2}\right\rangle$
|  $\mathrm{qr}[9]$ |  $\left|T_{o}\right\rangle$
Table 2: Description of the qubits in the circuit for both sender and receiver

The circuit begins by preparing the trigger qubits $\left|T_{e}\right\rangle$
and $\left|T_{o}\right\rangle$ in unknown states using X gates. Then, it
creates two entangled pairs using controlled-NOT(CNOT) gates and Hadamard (H)
gates on qubits $|Even\rangle,|Odd\rangle$ and $|\psi\rangle$ that is
initialized with the entangled state and they serve as a means of transmitting
quantum information, while $|Even\rangle$ and $|Odd\rangle$ both states that
we want to send and receive. Both qubits $|Even\rangle$ and $|Odd\rangle$ are
controlled qubits, when $|\psi\rangle$ is target qubits. Next, the circuit
performs a Controlled-Controlled Not (CCNOT) gate on each pair of trigger
qubits
$\left|T_{e}\right\rangle,\left|S_{e}^{1}\right\rangle,|\psi\rangle,\left|S_{e}^{2}\right\rangle,|Even\rangle$
on Even’s side and on qubits
$\left|T_{o}\right\rangle,\left|S_{o}^{1}\right\rangle,\left|S_{o}^{1}\right\rangle,|Odd\rangle,|\psi\rangle$
on Bob’s side, with $\left|S_{e}^{i}\right\rangle$ and
$\left|S_{o}^{i}\right\rangle$ ($i=1,2$ look back at the table) are initialize
with the state $|0\rangle$, and one half of the corresponding entangled pair.
This entangles the trigger qubits with the entangled pairs and collapses their
state into one of four possible outcomes. Applying Hadamard gates to qubits
$|Odd\rangle$ and $|\psi\rangle$, Controlled-Z gate between qubits
$|Odd\rangle$ and $|\psi\rangle$, a phase shift gate with angle $m\pi$ to
qubit $|\psi\rangle$, and Controlled-Z gate between qubits $|Odd\rangle$ and
$|\psi\rangle$. Same application was done to Alice’s side including the
entangled state of the whole system. After that, the circuit performs a
entanglement measurement on each pair of trigger qubits and one half of the
entangled pair, and sends the measurement results to the other party via
classical registers. Finally, the circuit measures the final state of the
trigger qubits and stores the results in the classical registers.

The application of Controlled-Not and Hadamard procedure are used for qubits
$|Even\rangle,|Odd\rangle$ and $|\psi\rangle$. The Controlled-Controlled-Not
gates creat kind of a ”copy” of $|Even\rangle,|Odd\rangle$ and $|\psi\rangle$
and store them in both
$\left|S_{e}^{i}\right\rangle,\left|S_{o}^{i}\right\rangle$ with ($i=1,2$) if
both triggers are $\left|T_{e}\right\rangle$ and $\left|T_{o}\right\rangle$
are in the state $|1\rangle$. the procedure is repeated in reverse so we can
measure qubits $\left|S_{e}^{i}\right\rangle$, $\left.S_{o}^{i}\right\rangle$
($i=1,2$) sent to opposite parties. The state sent From $|Even\rangle$ to
$|Odd\rangle$ and in reverse are obtained on qubit $|\psi\rangle$. The
Hadamard gate puts the qubit in a superposition state, equal probability of
being in the $\left|0\right\rangle$ state or the $\left|1\right\rangle$ state.
It is commonly used to create superposition and entanglement. The CZ gate is a
controlled phase gate that performs a phase flip to the target qubit
$|\psi\rangle$ if the control qubit $|Odd\rangle$ is in the state
$\left|1\right\rangle$. It entangles the qubits and introduces a controlled
phase shift. The phase shift gate introduces a phase shift to the state of the
qubit. In this case, the angle of the phase shift is determined by the value
of $m$ multiplied by $\pi$.

$\left(a\right)p=0,m=1$

$\left(b\right)p=0,m=1$

$\left(c\right)p=0,m=1$

$\left(d\right)p=1,m=0$

$\left(e\right)p=1,m=0$

$\left(f\right)p=1,m=0$

Figure 6: Histogram shows the output probability (outcomes) obtained by Aer
Simulator (Panels ($a$) and ($d$)), Qasm Simulator (Panels ($b$) and ($e$))
with 8192 shots, and Belem Simulator (Panels ($c$) and ($f$)) with fixed
values of $m$ and $p$ where the fidelity is theoretically maximum; The top row
(panels ($a$)-($b$)-($c$)) shows the results from Alice to Bob when $p=0$ and
$m=1$. The bottom row (panels ($d$)-($e$)-($f$)) gives the results obtained
from Bob to Alice when $p=1$ and $m=0$.

The probabilities we obtained for the output states ”$0000$”, ”$0001$”,
”$1000$”, and ”$1001$” are the result of simulating the given quantum circuit
using the qasm and aer simulators. These probabilities represent the
likelihood of measuring the quantum system in the corresponding classical
states. In the given code, after applying the gates and measurements, the
circuit has four qubits $\left|S_{e}^{i}\right\rangle$,
$\left.S_{o}^{i}\right\rangle$ ($i=1,2$) that are measured and mapped to the
classical registers. When we run the circuit on a simulator, it generates a
set of measurement outcomes based on the probabilities of different classical
states. The probabilities are determined by the amplitudes of the quantum
states after the measurements. These amplitudes are affected by the gate
operations performed on the qubits. The reason we obtained probabilities close
to $0.25\%$ for each output state is likely due to the symmetry and balanced
nature of the circuit. The Hadamard gates create superposition, resulting in
an equal probability of the qubits being in the $\left|0\right\rangle$ or
$\left|1\right\rangle$ state. The Controlled-Z gates introduce entanglement
between qubits, but since the circuit is symmetric, the probabilities of the
measured states remain balanced.

Comparison between theoretical and experimental results
---
Sender / receiver |  Input Parameters |  Phase shift Values |  Input Probabilities |  Qiskit-Qasm Simulator |  Qiskit-Aer Simulator |  Qiskit-Belem Simulator
Alice $\rightarrow$ Bob |  $p=0$, $m=1$ |  $\pi$ |  [0.25, 0.25, 0.25, 0.25] |  [0.249, 0.252] |  [0.240, 0.251] |  [0.257, 0.235]
Bob $\rightarrow$ Alice |  $p=1$, $m=0$ |  0 |  [0.25, 0.25, 0.25, 0.25] |  [0.244, 0.253] |  [0.245, 0.247] |  [0.234, 0.259]
Table 3: Comparing Theoretical and Experimental Results: Probabilities of BQT
of coherent state with varied parameters $p$ and $m$ in Phase Shift Gate using
IBM-Qasm Simulator, IBM-Aer Simulator with 8192 shots for enhanced accuracy
and quantum machine simulator (Belem Simulator), see Fig.(6).

We determine the probability of both even and odd states to facilitate a
comparison between the experimental and theoretical results. The probability
input is expressed as $P\left(x_{n}\right)=|\langle x_{n}|\psi\rangle|^{2}$,
where $|\psi\rangle$ represents the initial coherent state. As shown in Table
(3), we observe that the probability of measuring each even and odd state is
approximately $0.25\%$. This indicates an equal chance of obtaining any of the
possibilities ”$0000$,” ”$0001$,” ”$1000$,” and ”$1001$” when implementing the
BQT protocol. These probabilities serve as vital indicators of the
effectiveness and efficiency of the BQT protocol and provide valuable insights
for its optimization. To determine the angle of the phase shift, we multiply
$m$ by $\pi$. We are able to accurately control the phase shift by changing
the overlap parameter $p$ and the value of $m$ during the BQT in both
directions and at different times. Evaluating the protocol’s fidelity at
various $p$ and $m$ values allows us to determine the specific combinations
that result in the highest fidelity. We determine probabilities corresponding
to these phase shift values using IBM simulators such as Qasm Simulator, Aer
Simulator and Belem Simulator, allowing for a comparison between theoretically
predicted and experimentally observed probabilities. This comparison analysis
is critical in evaluating protocol performance and showing the agreement
between theoretical predictions and implementation. The simulation results
reveal intriguing insights into the BQT of coherent states. We show that the
probabilities obtained using the Belem, Qasm, and Aer Simulators fit high
fidelity values very well. Furthermore, these probabilities are very
influenced by specific $p$ and $m$ values, where the fidelity varies
significantly. The strong agreement between the probabilities and high
fidelity values indicates the effectiveness of the teleportation operation.
The output states are found to be very close to the initial quantum states,
with a very few errors. The simulators generate results that closely
approximate the ideal scenario of perfect teleportation, with maximum
fidelity. Finally, our study shows the importance of probabilities in the
context of coherent state teleportation and shows the implementation of the
BQT protocol using entangled quantum channels. The combination of
probabilities and high fidelity values on the Belem, Qasm, and Aer Simulators
shows the protocol’s efficacy in quantum teleportation. Our work yields
intriguing results concerning the BQT of a coherent state, discovering that
achieving perfect teleportation fidelity depends on the phase disparity
between Alice’s transmission to Bob and vice versa.

##  VI Concluding remarks

To summarize, we proposed a bidirectional quantum teleportation scheme to
simultaneously teleport even and odd coherent states in two directions through
the multipartite Glauber coherent state. First, we examined the amount of
entanglement in the shared resource state that can be maximally entangled,
non-maximally entangled or separated depending on specific values of
parameters such as overlap $p$, $m$ (symmetric or antisymmetric state) and the
number of partitions $n$. We found that concurrence entanglement increases
with higher values of overlap $p$ and reaches its maximum in the limiting
case. This suggests that controlling the number of partitions and the
overlapping of multi-partite coherent states can achieve pre-shared
entanglement. Then, we evaluated the efficiency of the protocol via fidelity.
We found that a minimal number of probes (e.g.$n=3$) is required for maximum
fidelity. For the symmetrical state (i.e.$m=0$), fidelities in both directions
reach their maximum when the shared resource is a entangled state ($p=1$).
Conversely, for the anti-symmetrical state (i.e.$m=1$), fidelities increase
and curve towards a maximum and in this situation, the state of the shared
resource linking Alice to Bob corresponds to a Werner state ($p=1$). We have
also observed that the fidelities reach their maximum when the shared resource
is a maximally entangled state (GHZ-type with $p=0$ and $m=0$ or $p=0$ and
$m=1$). Additionally, we compared the QFI with HSS associated with the trigger
phase parameters and found that HSS plays a similar role to QFI in enhancing
the efficiency of our BQT protocol in both directions. Then, we established a
relationship between QFI and fidelity in both directions. In the case of the
direction from Alice to Bob, we found that the variance of the estimated
parameter is proportional to the fidelity. This implies that higher fidelity
leads to a more accurate estimation of Alice’s trigger phase parameter,
denoted as $\vartheta_{e}$. It’s important to note that this relationship is
well-established in the literature within the context of standard quantum
teleportation. Conversely, an inverse behavior was observed in the other
direction, from Bob to Alice. In this scenario, lower fidelity actually
results in a more accurate estimation of Bob’s trigger phase parameter,
$\vartheta_{o}$. These intriguing behaviors have been confirmed by Hilbert-
Schmidt speed and offer valuable insights that go beyond the established
knowledge of standard quantum teleportation. To determine the performance of
the protocol and validate its predictions, probabilities obtained from IBM
simulators are compared with the theoretically predicted probabilities. This
enables a comparison between the expected and experimentally observed
probabilities, providing insights into the agreement between theory and
practice.

Ultimately, maximizing and minimizing the fidelities of teleported states in
both orientations depends on the Alice’s and Bob’s trigger states as well as
the type of teleported information. Furthermore, the proposed BQT scheme is
based on the multipartite coherent state as a resource generally influenced by
environmental effects. So it is important to control the impact of the
environment on the various entries parameters during the teleportation
process, and to see how to realize this scheme in noisy channels as well as
testing other quantum resource kinds beyond entanglement. We look forward to
reporting on these issues in the next step.

Disclosures: The authors declare no conflicts of interest or personal
relationships that could have appeared to influence this work, and no Data
associated in the manuscript.

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A_{2}}(A_{1}A_{2}A_{3})=A_{1}^{T}A_{3}^{T}\,.$

Moreover, for any scalar $c\in\mathbb{R}$, $c=tr(c)=c^{T}$.

We then derive the following:

$\displaystyle\begin{split}&\quad~{}\frac{\partial}{\partial
X}\mathcal{L}(X,D_{1},D_{2},U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3})\\\
&=\frac{\partial}{\partial
X}tr\Big{(}(U_{1}^{T}E-U_{2}^{T}+\mathbbm{1}_{N_{S}}\vec{h}_{1}^{T}+\mathbbm{1}_{N_{S}}\vec{h}_{2}^{T}E_{+})X\Big{)}\\\
&=E^{T}U_{1}-U_{2}+(\vec{h}_{1}+E_{+}^{T}\vec{h}_{2})\mathbbm{1}_{N_{S}}^{T}\,.\end{split}$
(50)

Similarly,

$\displaystyle\begin{split}&\quad~{}\frac{\partial}{\partial
D_{1}}\mathcal{L}(X,D_{1},D_{2},U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3})\\\
&=\frac{\partial}{\partial
D_{1}}(-\mathbbm{1}^{T}_{N_{S}}D_{1}^{T}\mathbbm{1}_{N_{\mathcal{V}}}+\vec{h}_{3}^{T}D_{1}^{T}\mathbbm{1}_{N_{\mathcal{V}}})+\frac{\partial}{\partial
D_{1}}tr(-U_{1}^{T}D_{1}+U_{3}^{T}D_{1}-U_{5}^{T}D_{1})\\\
&=\frac{\partial}{\partial
D_{1}}tr(-\mathbbm{1}^{T}_{N_{\mathcal{V}}}D_{1}\mathbbm{1}_{N_{S}}+\mathbbm{1}_{N_{\mathcal{V}}}^{T}D_{1}\vec{h}_{3})+\frac{\partial}{\partial
D_{1}}tr(-U_{1}^{T}D_{1}+U_{3}^{T}D_{1}-U_{5}^{T}D_{1})\\\
&=-\mathbbm{1}_{N_{\mathcal{V}}}\mathbbm{1}^{T}_{N_{S}}+\mathbbm{1}_{N_{\mathcal{V}}}\vec{h}_{3}^{T}-U_{1}+U_{3}-U_{5}\,,\end{split}$
(51)

and

$\displaystyle\begin{split}&\quad~{}\frac{\partial}{\partial
D_{2}}\mathcal{L}(X,D_{1},D_{2},U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3})\\\
&=\frac{\partial}{\partial
D_{2}}tr(-U_{1}^{T}D_{2}-U_{4}^{T}D_{2}+U_{6}^{T}D_{2})+\frac{\partial}{\partial
D_{2}}\vec{h}_{3}^{T}D_{2}^{T}\mathbbm{1}_{N_{\mathcal{V}}}\\\
&=\mathbbm{1}_{N_{\mathcal{V}}}\vec{h}_{3}^{T}-U_{1}-U_{4}+U_{6}\,.\end{split}$
(52)

Since the Lagrangian (LABEL:eq:lagrangian) is a linear function of
$X,D_{1},D_{2}$, then the minimum of Lagrangian subject to the variables
$X,D_{1},D_{2}$ is equal to the Lagrangian if and only if
(LABEL:eq:lagrangian) is not changing with values of $X,D_{1},D_{2}$, that is,
$G(U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3})=\mathcal{L}(U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3})$
if and only if $\frac{\partial}{\partial A}\mathcal{L}(X,D_{1},D_{2},\\\
U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3})=0$, for
$A=X,D_{1},D_{2}$. We then convert the dual program (49) into the following
equivalently:

$\displaystyle\begin{split}\max_{X,D_{1},D_{2},U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3}}&-\vec{h}_{1}^{T}C^{\mathcal{E}}-\vec{h}_{2}^{T}C^{\mathcal{V}}-tr(U_{3}^{T}\Delta_{1}M)+tr(U_{4}^{T}\Delta_{2}M)\\\
\text{s.t.}\quad&E^{T}U_{1}-U_{2}+(\vec{h}_{1}+E_{+}^{T}\vec{h}_{2})\mathbbm{1}_{N_{S}}^{T}=0\\\
&-\mathbbm{1}_{N_{\mathcal{V}}}\mathbbm{1}^{T}_{N_{S}}+\mathbbm{1}_{N_{\mathcal{V}}}\vec{h}_{3}^{T}-U_{1}+U_{3}-U_{5}=0\\\
&\mathbbm{1}_{N_{\mathcal{V}}}\vec{h}_{3}^{T}-U_{1}-U_{4}+U_{6}=0\\\
&U_{2},U_{3},U_{4},U_{5},U_{6}\geq 0,\quad\vec{h}_{1},\vec{h}_{2}\geq
0\end{split}$ (53)

We then conclude that, $X,D_{1},D_{2}$ (resp.,
$U_{1},U_{2},\cdots,U_{6},\vec{h}_{1},\vec{h}_{2},\vec{h}_{3}$) are optimal
solutions to the original problem (7) (resp., the dual problem (53)) if and
only if the primal and dual feasibility conditions (8)–(13) and the zero
duality gap optimality condition (14) are satisfied.

##  Appendix B Additional Case Study Results

Table 2: Information on the median and worst case of travel alternative diversity and maximum landing distance vs. budget level for all three city cases Budget Level | Travel Alternative Diversity | Maximum Landing Distance
---|---|---
Milwaukee | Atlanta | Dallas–Fort Worth | Milwaukee | Atlanta | Dallas–Fort Worth
Median | Min | Median | Min | Median | Min | Median | Max | Median | Max | Median | Max
0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 7.3 | 9.1 | 6.3 | 10.0 | 10.9 | 15.9
5 | 2.0 | 1.0 | 2.0 | 1.0 | 2.0 | 1.0 | 7.3 | 9.1 | 6.3 | 10.0 | 10.5 | 15.9
10 | 2.0 | 1.0 | 2.0 | 1.0 | 3.0 | 1.0 | 6.7 | 8.1 | 6.1 | 9.7 | 10.0 | 15.9
15 | 2.0 | 1.0 | 3.0 | 1.0 | 3.0 | 1.0 | 6.6 | 8.1 | 6.3 | 10.0 | 10.0 | 15.9
20 | 2.5 | 1.0 | 4.0 | 2.0 | 4.0 | 1.0 | 6.0 | 8.0 | 6.3 | 10.0 | 9.8 | 15.9
25 | 3.0 | 1.0 | 4.0 | 2.0 | 4.5 | 1.0 | 5.0 | 8.0 | 5.4 | 9.7 | 9.8 | 14.4
30 | 3.0 | 2.0 | 5.0 | 2.0 | 4.5 | 1.0 | 5.0 | 7.9 | 5.1 | 9.7 | 9.4 | 14.4
35 | 3.0 | 2.0 | 6.0 | 3.0 | 4.5 | 1.0 | 5.6 | 7.9 | 5.1 | 9.7 | 8.6 | 14.4
40 | 3.0 | 2.0 | 6.0 | 2.0 | 5.5 | 1.0 | 4.8 | 6.7 | 4.8 | 9.7 | 8.6 | 14.4
45 | 3.0 | 2.0 | 6.0 | 2.0 | 6.0 | 1.0 | 4.4 | 6.9 | 4.6 | 9.7 | 8.3 | 14.4
50 | 3.5 | 2.0 | 7.0 | 3.0 | 6.0 | 1.0 | 4.1 | 6.9 | 4.6 | 9.7 | 8.3 | 14.4
55 | 4.0 | 2.0 | 8.0 | 3.0 | 6.5 | 1.0 | 4.1 | 5.2 | 4.6 | 9.7 | 7.9 | 14.4
60 | 4.5 | 3.0 | 9.0 | 3.0 | 7.5 | 1.0 | 4.1 | 5.2 | 4.5 | 8.8 | 7.6 | 14.4
65 | 4.5 | 3.0 | 8.0 | 4.0 | 8.0 | 1.0 | 4.1 | 5.2 | 4.9 | 9.7 | 7.6 | 14.4
70 | 4.5 | 3.0 | 9.0 | 4.0 | 8.5 | 1.0 | 4.1 | 5.2 | 4.6 | 9.7 | 7.4 | 14.2
75 | 4.5 | 3.0 | 10.0 | 5.0 | 8.5 | 1.0 | 4.1 | 5.2 | 4.5 | 8.8 | 7.4 | 13.7
80 | 4.5 | 3.0 | 10.0 | 4.0 | 9.0 | 1.0 | 4.1 | 5.2 | 4.6 | 8.8 | 7.0 | 13.7
85 | 4.5 | 3.0 | 11.0 | 5.0 | 10.0 | 1.0 | 4.1 | 5.2 | 4.4 | 8.8 | 6.9 | 13.7
90 | 4.5 | 3.0 | 11.0 | 5.0 | 10.0 | 1.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.9 | 9.2
95 | 4.5 | 3.0 | 12.0 | 5.0 | 10.0 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 7.1 | 9.2
100 | 4.5 | 3.0 | 12.0 | 5.0 | 11.0 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 7.0 | 10.5
105 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.3 | 9.2
110 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.3 | 9.2
115 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.3 | 9.2
120 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.8 | 9.2
125 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.3 | 9.2
130 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.8 | 9.2
135 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.3 | 9.2
140 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.8 | 9.2
145 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.8 | 9.2
150 | 4.5 | 3.0 | 12.0 | 5.0 | 10.5 | 2.0 | 4.1 | 5.2 | 4.3 | 8.8 | 6.8 | 9.2