# Quantum Advantages of Teleportation and Dense Coding Protocols in an Open System

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## Abstract

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## 1. Introduction

## 2. The Model and Its Solution

## 3. Quantum Teleportation

## 4. Quantum Dense Coding

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Pirandola, S.; Andersen, U.L.; Banchi, L.; Berta, M.; Bunandar, D.; Colbeck, R.; Englund, D.; Gehring, T.; Lupo, C.; Ottaviani, C.; et al. Advances in quantum cryptography. Adv. Opt. Photon.
**2020**, 12, 1012. [Google Scholar] [CrossRef] [Green Version] - Wang, H.-W.; Tsai, C.-W.; Lin, J.; Huang, Y.-Y.; Yang, C.-W. Efficient and secure measure-resend authenticated semi-quantum key distribution protocol against reflecting attack. Mathematics
**2022**, 10, 1241. [Google Scholar] [CrossRef] - Zhu, Y.; Mao, L.; Hu, H.; Wang, Y.; Guo, Y. Adaptive continuous-variable quantum key distribution with discrete modulation regulative in free space. Mathematics
**2022**, 10, 4450. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G.; Crépeau, C.; Jozsa, R.; Peres, A.; Wootters, W.K. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett.
**1993**, 70, 1895. [Google Scholar] [CrossRef] [Green Version] - Horodecki, R.; Horodecki, M.; Horodecki, P. Teleportation, Bell’s inequalities and inseparability. Phys. Lett. A
**1996**, 222, 21. [Google Scholar] [CrossRef] [Green Version] - Popescu, S. Bell’s inequalities versus teleportation: What is nonlocality. Phys. Rev. Lett.
**1994**, 72, 797. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Horodecki, M.; Horodecki, P.; Horodecki, R. General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A
**1999**, 60, 1888. [Google Scholar] [CrossRef] [Green Version] - Verstraete, F.; Verschelde, H. Optimal teleportation with a mixed state of two qubits. Phys. Rev. Lett.
**2003**, 90, 097901. [Google Scholar] [CrossRef] [Green Version] - Bennett, C.H.; Wiesner, S.J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett.
**1992**, 69, 2881. [Google Scholar] [CrossRef] [Green Version] - Bareno, A.; Ekert, A. Dense coding based on quantum entanglement. J. Mod. Opt.
**1995**, 42, 1253. [Google Scholar] [CrossRef] - Mattle, K.; Weinfurter, H.; Kwiat, P.G.; Zeilinger, A. Dense coding in experimental quantum communication. Phys. Rev. Lett.
**1996**, 76, 4656. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hiroshima, T. Optimal dense coding with mixed state entanglement. J. Phys. A Math. Gen.
**2001**, 34, 6907. [Google Scholar] [CrossRef] - Bowen, G.; Bose, S. Teleportation as a depolarizing quantum channel, relative entropy, and classical capacity. Phys. Rev. Lett.
**2001**, 87, 267901. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Albeverio, S.; Fei, S.M.; Yang, W.L. Optimal teleportation based on Bell measurements. Phys. Rev. A
**2002**, 66, 012301. [Google Scholar] [CrossRef] [Green Version] - Hu, M.L. Relations between entanglement, Bell-inequality violation and teleportation fidelity for the two-qubit X states. Quantum Inf. Process.
**2013**, 12, 229. [Google Scholar] [CrossRef] [Green Version] - Horodecki, M.; Piani, M. On quantum advantage in dense coding. J. Phys. A Math. Theor.
**2012**, 45, 105306. [Google Scholar] [CrossRef] [Green Version] - Hu, M.L.; Hu, X.; Wang, J.C.; Peng, Y.; Zhang, Y.R.; Fan, H. Quantum coherence and geometric quantum discord. Phys. Rep.
**2018**, 762–764, 1–100. [Google Scholar] [CrossRef] [Green Version] - Lee, J.; Kim, M.S. Entanglement teleportation via Werner states. Phys. Rev. Lett.
**2000**, 84, 4236. [Google Scholar] [CrossRef] [Green Version] - Oh, S.; Lee, S.; Lee, H.W. Fidelity of quantum teleportation through noisy channels. Phys. Rev. A
**2002**, 66, 022316. [Google Scholar] [CrossRef] [Green Version] - Jung, E.; Hwang, M.R.; Ju, Y.H.; Kim, M.S.; Yoo, S.K.; Kim, H.; Park, D.; Son, J.W.; Tamaryan, S.; Cha, S.K. Greenberger–Horne–Zeilinger versus W states: Quantum teleportation through noisy channels. Phys. Rev. A
**2008**, 78, 012312. [Google Scholar] [CrossRef] [Green Version] - Bhaktavatsala Rao, D.D.; Panigrahi, P.K.; Mitra, C. Teleportation in the presence of common bath decoherence at the transmitting station. Phys. Rev. A
**2008**, 78, 022336. [Google Scholar] - Yeo, Y.; Kho, Z.W.; Wang, L. Effects of Pauli channels and noisy quantum operations on standard teleportation. EPL
**2009**, 86, 40009. [Google Scholar] [CrossRef] - Shadman, Z.; Kampermann, H.; Macchiavello, C.; Bruß, D. Optimal super dense coding over noisy quantum channels. New J. Phys.
**2010**, 12, 073042. [Google Scholar] [CrossRef] - Quek, S.; Li, Z. Effects of quantum noises and noisy quantum operations on entanglement and special dense coding. Phys. Rev. A
**2010**, 81, 024302. [Google Scholar] [CrossRef] [Green Version] - Li, J.K.; Xu, K.; Zhang, G.F. Dense coding capacity in correlated noisy channels with weak measurement. Chin. Phys. B
**2021**, 30, 110302. [Google Scholar] [CrossRef] - Haddadi, S.; Hu, M.L.; Knedif, Y.; Dolatkhah, H.; Pourkarimi, M.R.; Daoud, M. Measurement uncertainty and dense coding in a two-qubit system: Combined effects of bosonic reservoir and dipole–dipole interaction. Results Phys.
**2022**, 32, 105041. [Google Scholar] [CrossRef] - Sun, Y.H.; Xie, Y.X. Memory effect of a dephasing channel on measurement uncertainty, dense coding, teleportation, and quantum Fisher information. Results Phys.
**2022**, 37, 105526. [Google Scholar] [CrossRef] - Yeo, Y. Teleportation with a mixed state of four qubits and the generalized singlet fraction. Phys. Rev. A
**2006**, 74, 052305. [Google Scholar] [CrossRef] [Green Version] - Yeo, Y. Local noise can enhance two-qubit teleportation. Phys. Rev. A
**2008**, 78, 022334. [Google Scholar] [CrossRef] [Green Version] - Laine, E.M.; Breuer, H.P.; Piilo, J. Nonlocal memory effects allow perfect teleportation with mixed states. Sci. Rep.
**2014**, 4, 4620. [Google Scholar] [CrossRef] [Green Version] - Guo, Y.N.; Tian, Q.L.; Zeng, K.; Chen, P.X. Fidelity of quantum teleportation in correlated quantum channels. Quantum Inf. Process.
**2020**, 19, 182. [Google Scholar] [CrossRef] - Hu, M.L.; Zhang, Y.H.; Fan, H. Nonlocal advantage of quantum coherence in a dephasing channel with memory. Chin. Phys. B
**2021**, 30, 030308. [Google Scholar] [CrossRef] - Li, Y.L.; Zu, C.J.; Wei, D.M. Enhance quantum teleportation under correlated amplitude damping decoherence by weak measurement and quantum measurement reversal. Quantum Inf. Process.
**2019**, 18, 2. [Google Scholar] [CrossRef] - Tian, M.B.; Zhang, G.F. Improving the capacity of quantum dense coding by weak measurement and reversal measurement. Quantum Inf. Process.
**2018**, 17, 19. [Google Scholar] [CrossRef] [Green Version] - Breuer, H.P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2007. [Google Scholar]
- Rivas, Á.; Huelga, S.F. Open Quantum Systems. An Introduction; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Cai, X.; Zheng, Y. Quantum dynamical speedup in a nonequilibrium environment. Phys. Rev. A
**2017**, 95, 052104. [Google Scholar] [CrossRef] - Cai, X.; Zheng, Y. Non-Markovian decoherence dynamics in nonequilibrium environments. J. Chem. Phys.
**2018**, 149, 094107. [Google Scholar] - Cai, X. Quantum dephasing induced by non-Markovian random telegraph noise. Sci. Rep.
**2020**, 10, 88. [Google Scholar] [CrossRef] [Green Version] - Czerwinski, A. Quantum communication with polarization-encoded qubits under majorization monotone dynamics. Mathematics
**2022**, 10, 3932. [Google Scholar] [CrossRef] - Breuer, H.P.; Laine, E.M.; Piilo, J. Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett.
**2009**, 103, 210401. [Google Scholar] [CrossRef] [Green Version] - Breuer, H.P.; Laine, E.M.; Piilo, J.; Vacchini, B. Colloquium: Non-Markovian dynamics in open quantum systems. Rev. Mod. Phys.
**2016**, 88, 021002. [Google Scholar] [CrossRef] [Green Version] - Chen, H.; Han, T.; Chen, M.; Ren, J.; Cai, X.; Meng, X.; Peng, Y. Quantum state tomography in nonequilibrium environments. Photonics
**2023**, 10, 134. [Google Scholar] [CrossRef] - Werlang, T.; Souza, S.; Fanchini, F.F.; Villas Boas, C.J. Robustness of quantum discord to sudden death. Phys Rev. A
**2009**, 80, 024103. [Google Scholar] [CrossRef] [Green Version] - Xu, J.S.; Li, C.F.; Zhang, C.J.; Xu, X.Y.; Zhang, Y.S.; Guo, G.C. Experimental investigation of the non-Markovian dynamics of classical and quantum correlations. Phys Rev. A
**2010**, 82, 042328. [Google Scholar] [CrossRef] [Green Version] - Altintas, F.; Eryigit, R. Dissipative dynamics of quantum correlations in the strong-coupling regime. Phys. Rev. A
**2013**, 87, 022124. [Google Scholar] [CrossRef] [Green Version] - Czerwinski, A. Dynamics of open quantum systems–Markovian semigroups and beyond. Symmetry
**2022**, 14, 1752. [Google Scholar] [CrossRef] - Haddadi, S.; Pourkarimi, M.R.; Wang, D. Tripartite entropic uncertainty in an open system under classical environmental noise. J. Opt. Soc. Am. B
**2021**, 38, 2620. [Google Scholar] [CrossRef] - Pourkarimi, M.R.; Haseli, S.; Haddadi, S.; Hadipour, M. Scrutinizing entropic uncertainty and quantum discord in an open system under quantum critical environment. Laser Phys. Lett.
**2022**, 19, 065201. [Google Scholar] [CrossRef] - Rahman, A.U.; Haddadi, S.; Javed, M.; Kenfack, L.T.; Ullah, A. Entanglement witness and linear entropy in an open system influenced by FG noise. Quantum Inf. Process.
**2022**, 21, 368. [Google Scholar] [CrossRef] - Rahman, A.U.; Ali, H.; Haddadi, S.; Zangi, S.M. Generating non-classical correlations in two-level atoms. Alex. Eng. J.
**2023**, 67, 425. [Google Scholar] [CrossRef] - Yu, T.; Eberly, J.H. Finite-time disentanglement via spontaneous emission. Phys. Rev. Lett.
**2004**, 93, 140404. [Google Scholar] [CrossRef] [Green Version] - Almeida, M.P.; de Melo, F.; Hor-Meyll, M.; Salles, A.; Walborn, S.P.; Souto Ribeiro, P.H.; Davidovich, L. Environment-induced sudden death of entanglement. Science
**2007**, 316, 579. [Google Scholar] [CrossRef] [Green Version] - Yu, T.; Eberly, J.H. Sudden death of entanglement. Science
**2009**, 323, 5914. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bellomo, B.; Lo Franco, R.; Compagno, G. Non-Markovian effects on the dynamics of entanglement. Phys. Rev. Lett.
**2007**, 99, 160502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, M.; Chen, H.; Han, T.; Cai, X. Disentanglement Dynamics in Nonequilibrium Environments. Entropy
**2022**, 24, 1330. [Google Scholar] [CrossRef] - Czerwinski, A. Entanglement dynamics governed by time-dependent quantum generators. Axioms
**2022**, 11, 589. [Google Scholar] [CrossRef] - Kim, Y.S.; Lee, J.C.; Kwon, O.; Kim, Y.H. Protecting entanglement from decoherence using weak measurement and quantum measurement reversal. Nat. Phys.
**2012**, 8, 117. [Google Scholar] [CrossRef] [Green Version] - Lidar, D.A.; Birgitta Whaley, K. Decoherence-Free Subspaces and Subsystems. Irreversible Quantum Dynamics; Benatti, F., Floreanini, R., Eds.; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Flores, M.M.; Galapon, E.A. Two qubit entanglement preservation through the addition of qubits. Ann. Phys.
**2015**, 354, 21. [Google Scholar] [CrossRef] - Mortezapour, A.; Ahmadi Borji, M.; Lo Franco, R. Protecting entanglement by adjusting the velocities of moving qubits inside non-Markovian environments. Laser Phys. Lett.
**2017**, 14, 055201. [Google Scholar] [CrossRef] [Green Version] - Haas, M.; Jentschura, U.D.; Keitel, C.H.; Kolachevsky, N.; Herrmann, M.; Fendel, P.; Fischer, M.; Udem, T.; Holzwarth, R.; Hänsch, T.W.; et al. Two-photon excitation dynamics in bound two-body Coulomb systems including ac Stark shift and ionization. Phys. Rev. A
**2006**, 73, 052501. [Google Scholar] [CrossRef] [Green Version] - Agarwal, G.S.; Pathak, P.K. dc-field-induced enhancement and inhibition of spontaneous emission in a cavity. Phys. Rev. A
**2004**, 70, 025802. [Google Scholar] [CrossRef] [Green Version] - Ghosh, B.; Majumdar, A.S.; Nayak, N. Control of atomic entanglement by the dynamic Stark effect. J. Phys. B At. Mol. Opt. Phys.
**2008**, 41, 065503. [Google Scholar] [CrossRef] [Green Version] - Baghshahi, H.R.; Tavassoly, M.K.; Faghihi, M.J. Entanglement analysis of a two-atom nonlinear Jaynes–Cummings model with nondegenerate two-photon transition, Kerr nonlinearity, and two-mode Stark shift. Laser Phys.
**2014**, 24, 125203. [Google Scholar] [CrossRef] [Green Version] - Golkar, S.; Tavassoly, M.K. Dynamics and maintenance of bipartite entanglement via the Stark shift effect inside dissipative reservoirs. Laser Phys. Lett.
**2018**, 15, 035205. [Google Scholar] [CrossRef] - Puri, R.R.; Bullough, R.K. Quantum electrodynamics of an atom making two-photon transitions in an ideal cavity. J. Opt. Soc. Am. B
**1988**, 5, 2021. [Google Scholar] [CrossRef] - Spohn, H. Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys.
**1980**, 52, 569. [Google Scholar] [CrossRef] - Dalton, B.J.; Barnett, S.M.; Garraway, B.M. Theory of pseudomodes in quantum optical processes. Phys. Rev. A
**2001**, 64, 053813. [Google Scholar] [CrossRef] [Green Version] - Yeo, Y. Teleportation via thermally entangled states of a two-qubit Heisenberg XX chain. Phys. Rev. A
**2002**, 66, 062312. [Google Scholar] [CrossRef] [Green Version] - Zhou, Y.; Zhang, G.F. Quantum teleportation via a two-qubit Heisenberg XXZ chain–effects of anisotropy and magnetic field. Eur. Phys. J. D
**2008**, 47, 227. [Google Scholar] [CrossRef] [Green Version] - Benabdallah, F.; Haddadi, S.; Arian Zad, H.; Pourkarimi, M.R.; Daoud, M.; Ananikian, N. Pairwise quantum criteria and teleportation in a spin square complex. Sci. Rep.
**2022**, 12, 6406. [Google Scholar] [CrossRef] - Liang, Y.C.; Yeh, Y.H.; Mendonça, P.E.M.F.; Teh, R.Y.; Reid, M.D.; Drummond, P.D. Quantum fidelity measures for mixed states. Rep. Prog. Phys.
**2019**, 82, 076001. [Google Scholar] [CrossRef] [Green Version] - Rahman, A.U.; Haddadi, S.; Pourkarimi, M.R.; Ghominejad, M. Fidelity of quantum states in a correlated dephasing channel. Laser Phys. Lett.
**2022**, 19, 035204. [Google Scholar] [CrossRef] - Zidan, N.; Rahman, A.U.; Haddadi, S. Quantum teleportation in a two-superconducting qubit system under dephasing noisy channel: Role of Josephson and mutual coupling energies. Laser Phys. Lett.
**2023**, 20, 025204. [Google Scholar] [CrossRef] - Pourkarimi, M.R.; Haddadi, S. Quantum-memory-assisted entropic uncertainty, teleportation, and quantum discord under decohering environments. Laser Phys. Lett.
**2020**, 17, 025206. [Google Scholar] [CrossRef] - Schumacher, B.; Westmoreland, M.D. Sending classical information via noisy quantum channels. Phys. Rev. A
**1997**, 56, 131. [Google Scholar] [CrossRef] - Holevo, A.S. The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory
**1998**, 44, 269. [Google Scholar] [CrossRef] [Green Version] - Haddadi, S.; Ghominejad, M.; Akhound, A.; Pourkarimi, M.R. Exploring entropic uncertainty relation and dense coding capacity in a two-qubit X-state. Laser Phys. Lett.
**2020**, 17, 095205. [Google Scholar] [CrossRef] - Abd-Rabbou, M.Y.; Khalil, E.M. Dense coding and quantum memory assisted entropic uncertainty relations in a two-qubit state influenced by dipole and symmetric cross interactions. Ann. Phys.
**2022**, 534, 2200204. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of two-level atoms A and B, which interact independently with their surrounding environments ${E}_{1}$ and ${E}_{2}$ respectively. There is no interaction between the two interacting subsystems $A-{E}_{1}$ and $B-{E}_{2}$.

**Figure 2.**Average fidelity of teleportation (23) as a function of the scaled time $\tau ={\gamma}_{0}t$ when $\mu =1/\sqrt{2}$ for different values of Stark shift parameter $\beta $. (

**a**) Non-Markovian regime with $\lambda =0.1{\gamma}_{0}$ and (

**b**) Markovian regime with $\lambda =10{\gamma}_{0}$.

**Figure 3.**Average fidelity of teleportation in terms of the scaled time $\tau ={\gamma}_{0}t$ and Stark shift parameter $\beta $ for (

**a**) non-Markovian $\lambda =0.1{\gamma}_{0}$ and (

**b**) Markovian $\lambda =10{\gamma}_{0}$ regimes with $\mu =1/\sqrt{2}$.

**Figure 4.**Average fidelity of teleportation versus the scaled time $\tau ={\gamma}_{0}t$ and parameter $\mu $ for different values of the Stark shift parameter $\beta $ in the Markovian regime [(

**a**) $\beta =0$, (

**b**) $\beta =10{\gamma}_{0}$, and (

**c**) $\beta =15{\gamma}_{0}$] with $\lambda =10{\gamma}_{0}$ and in non-Markovian regime [(

**d**) $\beta =0$, (

**e**) $\beta =0.5{\gamma}_{0}$, and (

**f**) $\beta ={\gamma}_{0}$] with $\lambda =0.1{\gamma}_{0}$.

**Figure 5.**Dense coding capacity (26) as a function of the scaled time $\tau ={\gamma}_{0}t$ when $\mu =1/\sqrt{2}$ for different values of Stark shift parameter $\beta $. (

**a**) Non-Markovian regime with $\lambda =0.1{\gamma}_{0}$ and (

**b**) Markovian regime with $\lambda =10{\gamma}_{0}$.

**Figure 6.**Dense coding capacity in terms of the scaled time $\tau ={\gamma}_{0}t$ and Stark shift parameter $\beta $ for (

**a**) non-Markovian $\lambda =0.1{\gamma}_{0}$ and (

**b**) Markovian $\lambda =10{\gamma}_{0}$ regimes with $\mu =1/\sqrt{2}$.

**Figure 7.**Dense coding capacity versus scaled time $\tau ={\gamma}_{0}t$ and parameter $\mu $ for different values of Stark shift parameter $\beta $ in Markovian regime [(

**a**) $\beta =0$, (

**b**) $\beta =10{\gamma}_{0}$, and (

**c**) $\beta =15{\gamma}_{0}$] with $\lambda =10{\gamma}_{0}$ and in non-Markovian regime [(

**d**) $\beta =0$, (

**e**) $\beta =0.5{\gamma}_{0}$, and (

**f**) $\beta ={\gamma}_{0}$] with $\lambda =0.1{\gamma}_{0}$.

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**MDPI and ACS Style**

Haddadi, S.; Hadipour, M.; Haseli, S.; Rahman, A.U.; Czerwinski, A.
Quantum Advantages of Teleportation and Dense Coding Protocols in an Open System. *Mathematics* **2023**, *11*, 1407.
https://doi.org/10.3390/math11061407

**AMA Style**

Haddadi S, Hadipour M, Haseli S, Rahman AU, Czerwinski A.
Quantum Advantages of Teleportation and Dense Coding Protocols in an Open System. *Mathematics*. 2023; 11(6):1407.
https://doi.org/10.3390/math11061407

**Chicago/Turabian Style**

Haddadi, Saeed, Maryam Hadipour, Soroush Haseli, Atta Ur Rahman, and Artur Czerwinski.
2023. "Quantum Advantages of Teleportation and Dense Coding Protocols in an Open System" *Mathematics* 11, no. 6: 1407.
https://doi.org/10.3390/math11061407