,
Lukong Cornelius Fai,
Nguenang Jean Pierre,
Jules Carsimir Ngana,
Ateuafack Mathurin,
Georges Collince Fouokeng
We revisit the teleportation of two qubits entanglement using a single channel made of a single Majorana wire connecting two pairs of qubits each at the sending and receiving ends of the wire. The scheme is different from that use in our early work (Phys. Scr. 95 035803 (2020)) where the two qubit teleportation required two copies of the Majorana wires setup as required to implement the Lee and Kim teleportation scheme. The two pairs of qubit interact via the exchange coupling. It is in this setting that we investigate the role of the exchange coupling, the channel parameters, the spin chain parameters on the average fidelity of entanglement teleportation. We equally consider the fact that the geographically located two pairs of qubits interact differently with their respective spin chain environment. The dynamics of the average fidelity is shown to be enhanced when using a set-up with symmetric coupling between qubits and Majorana wire at both ends as compared to when using asymmetric coupling. Small values of the coupling between the qubits and spin environments should however be used. Large values of the exchange coupling between qubits in the two sub-systems should be used to have high fidelity of the teleported state. The dynamics of the average fidelity of entanglement teleportation is seen to depend on the size of the spin environment as compared to the teleportation protocol used in our earlier work where the dynamics is mildly sensitive to the size of the spin environment. Increasing the strength of the transverse magnetic field and large values of the anisotropy degree enhances the average fidelity of entanglement teleportation. The same observation is seen when considering the same values of the coupling between the two qubits sub-systems at the opposite ends of the Majorana wire and their respective spin environment. The dynamics of the system is however seen to observe almost a similar trend as that of the above cited work for the single Majorana wire setup.
| Published in | American Journal of Modern Physics (Volume 14, Issue 5) |
| DOI | 10.11648/j.ajmp.20251405.12 |
| Page(s) | 222-233 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Spin Chain Environment, Qubits, Single Majorana Wire, Entanglement Teleportation, Average Fidelity
(1) 
describes the MBSs plus the single-level two pairs of QDs and their mutual coupling as follows 
(2) 
and 
in equation (2) are the Hamiltonians for the interactions between the QDs of sub-system A and B respectively and are given as 
and 
. 
is Coulomb interaction strength 
and 
where 
are annihilation (creation) operators of the single level QDs. 
and 
are the Majorana operators associated with the two MBSs at the ends of the nano-wire. 
and 
are respectively the coupling amplitudes between Majorana end states of QD A1 and QD B1. In our studies we will consider symmetric and asymmetric coupling between the Majorana end modes and the dots. Following the transformations as reported in 
(3) 
is the single particle energy of each dot (qubit) and 
the exchange coupling along the z-axis (Ising model) and describes the interaction between the qubits of the individual sub-systems A and B. 
is the strength of the two Majorana interaction that damps exponentially with the length of the nano-wire L, 
is the superconducting coherent length. 
and 
represents the Pauli spin operators for the dots qubits and Majorana qubit respectively. In the fermionic representation of the systems Hamiltonian equation (2), we use the state basis, 
describing the possible spin configuration of sub-system A, MBSs and sub-system B, where 
for sub-system A, 
for sub-system B and 
MBSs, represents their respective spin ‘up’ or ‘down’ sates. Synonymous to the subspace with odd charge parity, we consider the following subspaces with spin configurations: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
. The Hamiltonians: 
(4) 
(5) 
the strength of their respective interactions. 
represents that of the spin environment which we consider the same for the two separate pairs of qubits, is described by the one-dimensional XY model written in the unit where 
, is given by 
(6) 
describe the spin chain of the environment. The parameter 
characterizes the intensity of the transverse magnetic field, and 
measures the anisotropy in the in-plane interaction. The XY model described by equation (6) encompasses other two spin model; the Ising spin chain with 
and the XX spin chain with 
. These different models shall be subject to our investigation. Following the result of the treatment in 
(7) 
(8) 
of the two qubits of sub-system A(B) dressed with the environment due to interaction is given by 
(9) 
correspond to the dressed eigen value due to interactions of sub-systems A(B) with their respective spin chain enviroment. 
are the Bogoliubov-transformed annihilation (creation) fermion operators arising from the interactions between the two qubits and the environment, defined as 
(10) 
is given by 
, with 
. 
with angles 
satisfying, 
. We note that the energy spectrum 
carries information about all the possible excitations in the spin chain. Interaction between the spin chain and qubits induces overlap between these excitations and thus quantum fluctuations. 
(11) 
. 
is the density matrix of sub-system A, MBS and sub-system B in the initial state. 
is the initial prepared state of the system Hamiltonian assumed to be in the general form: 
(12) 
(13) 
describes the density matrix of the spin chain in the initial state 
. Tracing the environmental degrees of freedom leads to the following equation of the time evolving density matrix: 
(14) 
and 
. 
and 
are respectively the eigen value and eigen state of the system Hamiltonian equation (3). The absolute value of the decoherence factor 
written in the form as given in 
(15) 
written in the basis: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
are those of the interaction Hamiltonians as discussed earlier of sub-system A and B interacting with their spin environment. The corresponding eigen values are: 
, 
,, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
is given as 
(16) 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
. 
and 
with 
and 
. 
represent the amplitude and 
the phase of the input state. The density matrix related to the input state is 
. The output state can be obtained by applying joint measurements and local unitary transformations on the input state 
. Following the teleportation protocol proposed by Lee and Kim the output state of the two qubit teleportation can be written as 
, where 
and 
is the identity matrix, 
, 
, 
are the Pauli matrices. The probabilities 
satisfy the condition 
, with 
and 
being the reduced density matrix of the channel corresponding to system B. The projective measurements 
are defined from the four maximally entangle Bell states as 
, 
, 
, 
, where 
and 
. 
. To efficiently measure the quality of the protocol, we calculate the average fidelity between the input and output states by averaging over all possible input states. Under the influence of the environment and channel parameters, the maximally entangled component can vary with time. Therefore, the best quality of the teleportation can be obtained by the optimal estimation of the projective measurements. Thus the average fidelity will be a function of time. The average fidelity is defined from the fidelity as 
. From straight evaluation we obtain the fidelity and average fidelity for the input state 
respectively as: 
(17) 
.(18) 
we have the fidelity and average fidelity respectively as: 
(19) 
(20) 
and 
with corresponding average fidelities 
and 
given in equations (18) and (20) respectively, in Figure 2. The same trend in the dynamics is observed in both cases however with enhancement when the entangled input state, 
is used. Figure 2(b) is the scenario when the values of the coupling between the spin chain and the two qubits sub-systems are interchanged. 
with average fidelity given by 
. 
(
red curve) and 
(
blue curve), a) 
,
b) 
,
other parameters are set to: 
,
, 
, 
,
, 
. with average fidelity (blue solid curve symmetric coupling, blue dash curve asymmetric coupling) b) with average fidelity (red solid curve symmetric coupling, red dash curve asymmetric coupling) other parameters are set to: ,, ,, 
, , .
setting b) for different exchange coupling strength
, setting . The other parameters are set to: ,, ,, ,
, .
, showing enhancement in the dynamics of teleportation after a short period of time. This scenario has already been reported in 
. Large values of the anisotropy parameter strengthens the interactions between the atomic spin environment and coherent excitations having less decoherence effect prevails in the spin environment, as compared to small values of the anisotropy parameter that makes the spin environment prone to quantum fluctuations and increasing decoherence effects on the dynamics of the teleported states. 
setting b) for different transverse magnetic field strength 
, setting 
,
, 
. The other parameters are set to: ,,
, , .
. We see that as the values of the coupling between the MBSs state increases, the average fidelity of teleportation decreases. Large values of 
corresponds to very small wire length that lead to overlap between the Majorana end modes which may intend lifting of the topological phase that harbour’s the Majorana bound state. In the opposite scenario, small values imply long wire length and enhancement in the teleportation state. This owes explanation from non-locality of the MBSs. We recollect that in 
(very long wire length), a remarkable teleportation feature known as superluminal holds in electron transport. 
setting other parameters to: ,, ,, , , .MBSs | Majorana Bound States |
QD | Quantum Dot |
SMW | Single Majorana Wire |
DMWs | Double Majorana Wires |
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APA Style
Afuoti, N. E., Fai, L. C., Pierre, N. J., Ngana, J. C., Mathurin, A., et al. (2025). Average Fidelity of Two Qubits Entanglement Teleportation Mediated Via a Single Majorana Wire. American Journal of Modern Physics, 14(5), 222-233. https://doi.org/10.11648/j.ajmp.20251405.12
ACS Style
Afuoti, N. E.; Fai, L. C.; Pierre, N. J.; Ngana, J. C.; Mathurin, A., et al. Average Fidelity of Two Qubits Entanglement Teleportation Mediated Via a Single Majorana Wire. Am. J. Mod. Phys. 2025, 14(5), 222-233. doi: 10.11648/j.ajmp.20251405.12
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Download@article{10.11648/j.ajmp.20251405.12,
author = {Ngwa Engelbert Afuoti and Lukong Cornelius Fai and Nguenang Jean Pierre and Jules Carsimir Ngana and Ateuafack Mathurin and Georges Collince Fouokeng},
title = {Average Fidelity of Two Qubits Entanglement Teleportation Mediated Via a Single Majorana Wire
},
journal = {American Journal of Modern Physics},
volume = {14},
number = {5},
pages = {222-233},
doi = {10.11648/j.ajmp.20251405.12},
url = {https://doi.org/10.11648/j.ajmp.20251405.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20251405.12},
abstract = {We revisit the teleportation of two qubits entanglement using a single channel made of a single Majorana wire connecting two pairs of qubits each at the sending and receiving ends of the wire. The scheme is different from that use in our early work (Phys. Scr. 95 035803 (2020)) where the two qubit teleportation required two copies of the Majorana wires setup as required to implement the Lee and Kim teleportation scheme. The two pairs of qubit interact via the exchange coupling. It is in this setting that we investigate the role of the exchange coupling, the channel parameters, the spin chain parameters on the average fidelity of entanglement teleportation. We equally consider the fact that the geographically located two pairs of qubits interact differently with their respective spin chain environment. The dynamics of the average fidelity is shown to be enhanced when using a set-up with symmetric coupling between qubits and Majorana wire at both ends as compared to when using asymmetric coupling. Small values of the coupling between the qubits and spin environments should however be used. Large values of the exchange coupling between qubits in the two sub-systems should be used to have high fidelity of the teleported state. The dynamics of the average fidelity of entanglement teleportation is seen to depend on the size of the spin environment as compared to the teleportation protocol used in our earlier work where the dynamics is mildly sensitive to the size of the spin environment. Increasing the strength of the transverse magnetic field and large values of the anisotropy degree enhances the average fidelity of entanglement teleportation. The same observation is seen when considering the same values of the coupling between the two qubits sub-systems at the opposite ends of the Majorana wire and their respective spin environment. The dynamics of the system is however seen to observe almost a similar trend as that of the above cited work for the single Majorana wire setup.
},
year = {2025}
}
TY - JOUR T1 - Average Fidelity of Two Qubits Entanglement Teleportation Mediated Via a Single Majorana Wire AU - Ngwa Engelbert Afuoti AU - Lukong Cornelius Fai AU - Nguenang Jean Pierre AU - Jules Carsimir Ngana AU - Ateuafack Mathurin AU - Georges Collince Fouokeng Y1 - 2025/10/27 PY - 2025 N1 - https://doi.org/10.11648/j.ajmp.20251405.12 DO - 10.11648/j.ajmp.20251405.12 T2 - American Journal of Modern Physics JF - American Journal of Modern Physics JO - American Journal of Modern Physics SP - 222 EP - 233 PB - Science Publishing Group SN - 2326-8891 UR - https://doi.org/10.11648/j.ajmp.20251405.12 AB - We revisit the teleportation of two qubits entanglement using a single channel made of a single Majorana wire connecting two pairs of qubits each at the sending and receiving ends of the wire. The scheme is different from that use in our early work (Phys. Scr. 95 035803 (2020)) where the two qubit teleportation required two copies of the Majorana wires setup as required to implement the Lee and Kim teleportation scheme. The two pairs of qubit interact via the exchange coupling. It is in this setting that we investigate the role of the exchange coupling, the channel parameters, the spin chain parameters on the average fidelity of entanglement teleportation. We equally consider the fact that the geographically located two pairs of qubits interact differently with their respective spin chain environment. The dynamics of the average fidelity is shown to be enhanced when using a set-up with symmetric coupling between qubits and Majorana wire at both ends as compared to when using asymmetric coupling. Small values of the coupling between the qubits and spin environments should however be used. Large values of the exchange coupling between qubits in the two sub-systems should be used to have high fidelity of the teleported state. The dynamics of the average fidelity of entanglement teleportation is seen to depend on the size of the spin environment as compared to the teleportation protocol used in our earlier work where the dynamics is mildly sensitive to the size of the spin environment. Increasing the strength of the transverse magnetic field and large values of the anisotropy degree enhances the average fidelity of entanglement teleportation. The same observation is seen when considering the same values of the coupling between the two qubits sub-systems at the opposite ends of the Majorana wire and their respective spin environment. The dynamics of the system is however seen to observe almost a similar trend as that of the above cited work for the single Majorana wire setup. VL - 14 IS - 5 ER -
Department of Energy and Thermal Engineering, University of Douala, Douala, Cameroon
Department of Physics, University of Dschang, Dschang, Cameroon
Department of Physics, University of Douala, Douala, Cameroon
Department of Physics, University of Buea, Buea, Cameroon
Department of Electrical and Electronic Engineering, Univesity of Buea, Buea, Cameroon
Department of Physics, University of Dschang, Dschang, Cameroon
Figure 1. Schematic setup of the system to carry out the two qubit entanglement teleportation. The setup constitutes two sub-systems A and B each containing two coupled qubits submerge in spin chain environments A and B. The two sub-systems are in contact via the Mojorana wire that harbours the Majorana bound states represented by white dots.
Figure 2. Plot of the dynamics of the average fidelity for two different input states: ( red curve) and ( blue curve), a) , b) , other parameters are set to: ,, , ,, .
Figure 3. Plot of the dynamics of the average fidelity for two different input states: a) with average fidelity (blue solid curve symmetric coupling, blue dash curve asymmetric coupling) b) with average fidelity (red solid curve symmetric coupling, red dash curve asymmetric coupling) other parameters are set to: ,, ,, , , .
Figure 4. Plot of the dynamics of the average fidelity (a) for different numbers of atoms setting b) for different exchange coupling strength, setting . The other parameters are set to: ,, ,, ,, .
Figure 5. Plot of the dynamics of the average fidelity (a) for different coupling strength between spin chain environment and qubits setting b) for different transverse magnetic field strength , setting ,, . The other parameters are set to: ,,, , .
Figure 6. Plot of the dynamics of the average fidelity (a) for different anisotropy values b) for different spin model; , XX spin model (blue solid curve), , arbitrary XY spin model (red solid curve), , Ising spin model (blue dash curve), other parameters are set to: , , , , , , ..
Figure 7. Plot of the dynamics of the average fidelity for different values of the coupling between Majorana bound states, setting other parameters to: ,, ,, , , .
