Fixed point theorem in a partial b-metric space applied to quantum operations
Abstract
Introduction/purpose: A fixed point theorem of an order-preserving mapping on a complete partial b-metric space satisfying a contractive condition is constructed.
Methods: Extension of the results of Batsari et al.
Results: The fidelity of quantum states is used to construct the existence of a fixed quantum state.
Conclusions: The fixed quantum state is associated to an order-preserving quantum operation.
References
Aamri, M. & El Moutawakil, D. 2002. Some new common fixed point theorems under strict contractive conditions. Journal of Mathematical Analysis and Applications, 270(1), pp.181-188. Available at:
https://doi.org/10.1016/S0022-247X(02)00059-8
Agarwal, R.P., Karapinar, E., O’Regan, D. & Roldán-López-de-Hierro, A.F. 2015. Fixed point theory in metric type spaces. Springer, Cham. Available at: https://doi.org/10.1007/978-3-319-24082-4. ISBN: 978-3-319-24082-4
Arias, A., Gheondea, A. & Gudder, S. 2002. Fixed Points of Quantum Operations. Journal of Mathematical Physics, 43(12), pp.5872-5881. Available at: https://doi.org/10.1063/1.1519669
Bakhtin, I. 1989. The contraction mapping principle in quasimetric spaces. Func. An., Gos. Ped. Inst. Unianowsk, 30, pp.26-37.
Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, pp.133-181 (in French). Available at: https://doi.org/10.4064/fm-3-1-133-181
Batsari, U.Y. & Kumam, P. 2018. A Globally Stable Fixed Point in an Ordered Partial Metric Space. In: Anh, L., Dong, L., Kreinovich, V. & Thach, N. (Eds.) Econometrics for Financial Applications. ECONVN 2018. Studies in Computational Intelligence, 760, pp.360-368. Springer, Cham. Available at: https://doi.org/10.1007/978-3-319-73150-6_29
Batsari, U.Y. & Kumam, P. 2020. Some Generalised Fixed Point Theorems Applied to Quantum Operations. Symmetry, 12(5), art.ID:759. Available at: https://doi.org/10.3390/sym12050759
Batsari, U.Y., Kumam, P. & Sitthithakerngkiet, K. 2018. Some globally stable fixed points in b-metric spaces. Symmetry, 10(11), art.ID:555. Available at: https://doi.org/10.3390/sym10110555
Bourbaki, N. 1974. Topologie Generale. Paris, France: Herman. ISBN-13: 978-2705656928.
Brouwer, L.E.J. 1911. Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71, pp.97-115. Available at: https://doi.org/10.1007/BF01456931
Browder, F.E. 1959. On a generalization of the Schauder fixed point theorem. Duke Mathematical Journal, 26(2), pp.291-303. Available at: https://doi.org/10.1215/S0012-7094-59-02629-8
Bures, D. 1969. An Extension of Kakutanis Theorem on Infinite Product Measures to the Tensor Product of Semifinite w*-Algebras. Transactions of the American Mathematical Society, 135, pp.199-212. Available at:
https://doi.org/10.2307/1995012
Busch, P. & Singh, J. 1998. Lüders Theorem for Unsharp Quantum Measurements. Physics Letters A, 249)(1-2), pp.10-12. Available at: https://doi.org/10.1016/S0375-9601(98)00704-X
Chen, J-L., Fu, L., Ungar, A.A. & Zhao, X-G. 2002. Alternative fidelity measure between two states of an N-state quantum system. Physical Review A, 65(art.number:054304). Available at: https://doi.org/10.1103/PhysRevA.65.054304
Chidume, C.E. & Chidume, C.O. 2014. Foundations of Mathematical Analysis. Ibadan, Nigeria: University of Ibadan, Ibadan University Press Publishing House. ISBN: 978-978-8456-32-2
Czerwik, S. 1993. Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1), pp.5-11 [online]. Available at: https://dml.cz/handle/10338.dmlcz/120469 [Accessed: 20 March 2022].
Davies, E.B. 1976. Quantum Theory of Open Systems. London, UK: Academic Press. ISBN-13: 978-0122061509.
Debnath, P., Konwar, N. & Radenović, S. 2021. Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences. Springer Verlag, Singapore. Available at: https://doi.org/10.1007/978-981-16-4896-0
Du, W-S., Karapinar, E. & He, Z. 2018. Some Simultaneous Generalizations of Well-Known Fixed Point Theorems and Their Applications to Fixed Point Theory. Mathematics, 6(7), art.ID:117. Available at: https://doi.org/10.3390/math6070117
Göhde, D. 1965. Zum Prinzip der Kontraktiveen abbildurg. Mathematische Nachrichten, 30(3-4), pp.251-258. Available at: https://doi.org/10.1002/mana.19650300312
Kannan, R. 1972. Some results on fixed points - IV. Fundamenta Mathematicae, 74, pp.181-187. Available at: https://doi.org/10.4064/fm-74-3-181-187
Khan, M.S., Swaleh, M. & Sessa, S. 1984. Fixed point theorems by altering distances between the points. Bulletin of the Australian Mathematical Society, 30(1), pp.1-9. Available at: https://doi.org/10.1017/S0004972700001659
Kirk, W. & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Springer International Publishing Switzerland. Available at: https://doi.org/10.1007/978-3-319-10927-5
Knaster, B. 1928. Un theoreme sur les fonctions densembles. Annales de la Société polonaise de mathématique, 6, pp.133-134.
Leray, J. & Schauder, J. 1934. Topologie et equations fonctionnelles. Annales scientifiques de l’École Normale Supérieure, 51, pp.45-78. Available at: https://doi.org/10.24033/asens.836
Long, L. & Zhang, S. 2011. Fixed points of commutative super-operators. Journal of Physics A: Mathematical and Theoretical, 44(9), art.ID:095201. Available at: https://doi.org/10.1088/1751-8113/44/9/095201
Lüders, G. 1950. Über die Zustandsänderung durch den Meßprozeß. Annalen der physic, 443(5-8), pp.322-328. Available at: https://doi.org/10.1002/andp.19504430510
Matthews, S.G. 1994. Partial Metric Topology. Annals of the New York Academy of Sciences, 728(1) General Topology and Applications, pp.183-197. Available at: https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
Nielsen, M.A. & Chuang, I.L. 2000. Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press. Available at: https://doi.org/10.1017/CBO9780511976667
Schauder, J. 1930. Der Fixpunktsatz in Funktionalraumen. Studia Mathematica, 2(1), pp.171-180 [online]. Available at: https://eudml.org/doc/urn:eudml:doc:217247 [Accessed: 22 March 2022].
Seevinck, M.P. 2003. Quantum Operations and Measurement, 2nd ed. Utrecht, The Netherlands: Utrecht University [online]. Available at: http://mpseevinck.ruhosting.nl/seevinck/lezingoviedo03a.pdf [Accessed: 22
March 2022].
Shukla S. 2014. Partial b-metric spaces and fixed point theorems. Mediterranean Journal of Mathematics, 11, pp.703-711. Available at: https://doi.org/10.1007/s00009-013-0327-4
Tarski, A. 1955. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5(2), pp.285-309. Available at: https://doi.org/10.2140/pjm.1955.5.285
Tarski, A.A. 1949. A fixpoint theorem for lattices and its applications (preliminary report). Bulletin of the American Mathematical Society, 1949(55), 1051-1052.
Zhang, H. & Ji, G. 2012. A Note on Fixed Points of General Quantum Operations. Reports on Mathematical Physics, 70(1), pp.111-117. Available at: https://doi.org/10.1016/S0034-4877(13)60016-6
Zhang, H. & Si, H. 2016. Fixed Points Associated to Power of Normal Completely Positive Maps*. Journal of Applied Mathematics and Physics, 4(5), pp.925-929. Available at: https://doi.org/10.4236/jamp.2016.45101
Proposed Creative Commons Copyright Notices
Proposed Policy for Military Technical Courier (Journals That Offer Open Access)
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).