Structure of backward quantum Markov chains

Luigi Accardi; El Gheted Soueidi; Abdessatar Souissi; Mohamed Rhaima; Farrukh Mukhamedov; Farzona Mukhamedova; Luigi Accardi; El Gheted Soueidi; Abdessatar Souissi; Mohamed Rhaima; Farrukh Mukhamedov; Farzona Mukhamedova

doi:10.3934/math.20241360

AIMS Mathematics

2024, Volume 9, Issue 10: 28044-28057. doi: 10.3934/math.20241360

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Research article

Structure of backward quantum Markov chains

1.
Centro Vito Volterra, Università di Roma "Tor Vergata", Roma I-00133, Italy
2.
Faculty of Sciences and Technologies, University of Nouakchott, Mauritania
3.
LR18ES45, Mathematical Physics, Quantum Modeling and Mechanical Design, University of Carthage, Sidi Bou Said, Av. of the Republic, Carthage 1054, Tunisia
4.
Department of Statistics and Operations Research, College of Sciences, King Saud University, P. O.Box, Riyadh 11451, Saudi Arabia
5.
Department of Mathematical Sciences, College of Science, UAEU, Al Ain, P.O. Box 15551, Abu Dhabi, UAE
6.
King's College London, Strand, London, WC2R 2LS, UK

Received: 10 May 2024 Revised: 12 September 2024 Accepted: 14 September 2024 Published: 27 September 2024
MSC : 46L35, 46L55

This paper extended the framework of quantum Markovianity by introducing backward and inverse backward quantum Markov chains (QMCs). We established the existence of these models under general conditions, demonstrating their applicability to a wide range of quantum systems. Our findings revealed distinct structural properties within these models, providing new insights into their dynamics and relationships to finitely correlated states. These advancements contributed to a deeper understanding of quantum processes and have potential implications for various quantum applications, including hidden quantum Markov processes.
- quantum theory,
- Markov chain,
- C^*-algebras,
- transition expectation,
- dynamics
Citation: Luigi Accardi, El Gheted Soueidi, Abdessatar Souissi, Mohamed Rhaima, Farrukh Mukhamedov, Farzona Mukhamedova. Structure of backward quantum Markov chains[J]. AIMS Mathematics, 2024, 9(10): 28044-28057. doi: 10.3934/math.20241360

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Abstract

This paper extended the framework of quantum Markovianity by introducing backward and inverse backward quantum Markov chains (QMCs). We established the existence of these models under general conditions, demonstrating their applicability to a wide range of quantum systems. Our findings revealed distinct structural properties within these models, providing new insights into their dynamics and relationships to finitely correlated states. These advancements contributed to a deeper understanding of quantum processes and have potential implications for various quantum applications, including hidden quantum Markov processes.

References

[1]	L. Accardi, Non-commutative Markov chains, Proceedings International School of Mathematical Physics, Università di Camerino, 1974. 268–295.
[2]	L. Accardi, The noncommutative markovian property, Funct. Anal. Appl., 9 (1975), 1–8. https://doi.org/10.1007/BF01078167 doi: 10.1007/BF01078167
[3]	L. Accardi, E. G. Soueidy, Y. G. Lu, A. Souissi, Algebraic Hidden Processes and Hidden Markov Processes, Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 2024. https://doi.org/10.1142/S0219025724500097
[4]	L. Accardi, E. G. Soueidy, Y. G. Lu, A. Souissi, Hidden Quantum Markov processes, Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 2024. https://doi.org/10.1142/S0219025724500073
[5]	L. Accardi, S. El Gheteb, A. Souissi, Infinite Volume Limits of Entangled States, Lobachevskii J. Math., 44 (2023), 1967–1973. https://doi.org/10.1134/S1995080223060033 doi: 10.1134/S1995080223060033
[6]	L. Accardi, A. Souissi, E. G. Soueidy, Quantum Markov chains: A unification approach, Infin. Dimens. Anal. Qu., 23 (2020), 2050016. https://doi.org/10.1142/S0219025720500162 doi: 10.1142/S0219025720500162
[7]	C. Budroni, G. Fagundes, M. Kleinmann, Memory Cost of Temporal Correlations, New J. Phys., 21 (2019), 093018. https://doi.org/10.1088/1367-2630/ab3cb4 doi: 10.1088/1367-2630/ab3cb4
[8]	A. Dhahri, F. Mukhamedov, Open quantum random walks, quantum Markov chains and recurrence, Rev. Math. Phys., 31 (2019), 1950020. https://doi.org/10.1142/S0129055X1950020X doi: 10.1142/S0129055X1950020X
[9]	A. Dhahri, F. Fagnola, Potential theory for quantum Markov states and other quantum Markov chains, Anal. Math. Phys., 13 (2023), 31. https://doi.org/10.1007/s13324-023-00790-1 doi: 10.1007/s13324-023-00790-1
[10]	M. Fannes, B. Nachtergaele, R. F. Werner, Ground states of VBS models on Cayley trees, J. Stat. Phys., 66 (1992), 939–973, https://doi.org/10.1007/BF01055710 doi: 10.1007/BF01055710
[11]	M. Fannes, B. Nachtergaele, R. F. Werner, Finitely correlated states on quantum spin chains, Commun. Math. Phys., 144 (1992), 443–490. https://doi.org/10.1007/BF02099178 doi: 10.1007/BF02099178
[12]	O. Fawzi, R. Renner, Quantum conditional mutual information and approximate Markov chains, Commun. Math. Phys., 340 (2015), 575–611. https://doi.org/10.1007/s00220-015-2466-x doi: 10.1007/s00220-015-2466-x
[13]	Y. Feng, N. Yu, M. Ying, Model checking quantum Markov chains, J. Comput. Syst. Sci., 79 (2013), 1181–1198. https://doi.org/10.1016/j.jcss.2013.04.002 doi: 10.1016/j.jcss.2013.04.002
[14]	S. Gudder, Quantum Markov chains, J. Math. Phys., 49 (2008), https://doi.org/10.1063/1.2953952
[15]	B. Ibinson, N. Linden, A. Winter, Robustness of quantum Markov chains, Comm. Math. Phys., 277 (2008), 289–304. https://doi.org/10.1007/s00220-007-0362-8 doi: 10.1007/s00220-007-0362-8
[16]	B. Kümmerer, Quantum Markov processes and applications in physics, In: Quantum independent increment processes. II, Lecture Notes in Math., 1866, Springer, Berlin, 2006,259–330. https://doi.org/10.1007/11376637_4
[17]	G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, 2003. https://doi.org/10.1017/CBO9781107360235
[18]	S. Sakai, $C^$–Algebras and $W^$–algebras, Springer, Berlin, 1971. https://doi.org/10.1007/978-3-642-61993-9
[19]	A. Souissi, F. Mukhamedov, E. G. Soueidi, M. Rhaima, F. Mukhamedova, Entangled hidden elephant random walk model, Chaos, 186 (2024), 115252. https://doi.org/10.1016/j.chaos.2024.115252 doi: 10.1016/j.chaos.2024.115252
[20]	A. Souissi, E. G. Soueidi, Entangled Hidden Markov Models, Chaos, Soliton. Fract., 174 (2023), https://doi.org/10.1016/j.chaos.2023.113804

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