Abstract
The notion of ‘quantum family of maps’ (QFM) has been defined by Piotr Sołtan as a noncommutative analogue of ‘parameterized family of continuous maps’ between locally compact spaces. A QFM between C*-algebras B, A, is given by a pair \((C,\phi )\) where C is a C*-algebra and \(\phi :B\rightarrow A\check{\otimes }C\) is a \(*\)-morphism. The main goal of this note, is to introduce the notion of ‘random quantum map’ (RQM), which is a noncommutative analogue of ‘random continuous map’ between compact spaces. We define a RQM between B, A, to be given by a triple \((C,\phi ,\nu )\) where \((C,\phi )\) is a QFM and \(\nu \) a state (normalized positive linear functional) on C. Our first application of RQMs takes place in theory of completely positive maps (CPM): RQMs give rise canonically to a class of CPMs which we call implemented CPMs. We consider some partial results about the natural and important problem of characterization of implemented CPMs. For instance, using Stinespring’s Theorem, we show that any CPM from B to A is implemented if A is finite-dimensional. Our second application of RQMs takes place in theory of quantum stochastic processes: We show that iterations of any RQM with \(B=A\), gives rise to a quantum Markov chain in a sense introduced by Luigi Accardi.
Similar content being viewed by others
Data availability
Not applicable.
References
Accardi, L.: Noncommutative Markov chains. In: Proceedings of the International School of Mathematical Physics, pp. 268–295 (1974)
Accardi, L.: Nonrelativistic quantum mechanics as a noncommutative Markov process. Adv. Math. 20, 329–366 (1976)
Accardi, L.: Quantum stochastic processes. In: Fritz, J., Jaffe, A., Szász, D. (eds.) Statistical Physics and Dynamical Systems, pp. 285–302. Birkhäuser, Boston (1985)
Accardi, L., Frigerio, A., Lewis, J.T.: Quantum stochastic processes. Publ. Res. Inst. Math. Sci. 18(1), 97–133 (1982)
Accardi, L., Souissi, A., Soueidy, E.G.: Quantum Markov chains: a unification approach. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23(02), 2050016 (2020). arXiv:1811.00500 [math.OA]
Baraquin, I.: Random walks on finite quantum groups. J. Theor. Probab. 33(3), 1715–1736 (2020). arXiv:1812.06862 [math.QA]
Blumenthal, R.M., Corson, H.H.: On continuous collections of measures. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 33–40 (1972)
Bochniak, A., Kasprzak, P., Sołtan, P.M.: Quantum correlations on quantum spaces. Int. Math. Res. Not. rnac139 (2022). https://doi.org/10.1093/imrn/rnac139. arXiv:2105.07820 [math.OA]
Effros, E.G., Ruan, Z.-J.: Operator Spaces. London Mathematical Society Monographs: New Series 23, Clarendon Press, Oxford (2000)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics, Springer, New York (2003)
Jost, J., Kell, M., Rodrigues, C.S.: Representation of Markov chains by random maps: existence and regularity conditions. Calc. Var. Partial Differ. Equ. 54(3), 2637–2655 (2015). arXiv:1207.5003 [math.DS]
Kifer, Y.: Random Perturbations of Dynamical Systems. Progress in Probability and Statistics, vol. 16. Birkhäuser, Boston (1988)
Kifer, Y.: Ergodic Theory of Random Transformations, vol. 10. Springer, Berlin (2012)
Lindsay, J.M., Skalski, A.G.: Quantum random walk approximation on locally compact quantum groups. Lett. Math. Phys. 103(7), 765–775 (2013). arXiv:1110.3990 [math.OA]
Phillips, N.C.: Inverse limits of C*-algebras and applications. In: Operator Algebras and Applications. London Mathematical Society Lecture Note Series 135, vol. 1. Cambridge University Press, pp. 127–185 (1988)
Rebowski, R.: A note on integral representation of Feller kernels. Ann. Polon. Math. 56, 93–96 (1991)
Sadr, M.M.: On the quantum groups and semigroups of maps between noncommutative spaces. Czechoslov. Math. J. 67(1), 97–121 (2017). arXiv:1506.06518 [math.QA]
Sadr, M.M.: Path-connected components of affine schemes and algebraic K-theory, preprint. arXiv:1911.04204 [math.KT]
Skoufranis, P.: Completely positive maps. https://pskoufra.info.yorku.ca (2014)
Sołtan, P.M.: Quantum families of maps and quantum semigroups on finite quantum spaces. J. Geom. Phys. 59, 354–368 (2009). arXiv:math/0610922 [math.OA]
Acknowledgements
The author would like to express his sincere gratitude to the anonymous referee for very valuable comments on the early version of this manuscript.
Funding
No funding was received to assist with the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
Both authors contributed to the study conception and design. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
There is no conflict of interests or competing interests for this study. The authors have no relevant financial or non-financial interests to disclose. Both authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sadr, M.M., Ganji, M.B. Random quantum maps and their associated quantum Markov chains. Positivity 27, 18 (2023). https://doi.org/10.1007/s11117-023-00969-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11117-023-00969-7