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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access January 15, 2013

Notes on entropic characteristics of quantum channels

  • Alexey Rastegin EMAIL logo
From the journal Open Physics


One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.

[1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000) Search in Google Scholar

[2] I. Bengtsson, K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006) in Google Scholar

[3] A. Rényi, In: J. Neyman (Ed.), Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. I (University of California Press, Berkeley, 1961) 547 Search in Google Scholar

[4] C. Tsallis, J. Stat. Phys. 52, 479 (1988) in Google Scholar

[5] X. Hu, Z. Ye, J. Math. Phys. 47, 023502 (2006) in Google Scholar

[6] A.E. Rastegin, J. Stat. Phys. 143, 1120 (2011) in Google Scholar

[7] B. Schumacher, Phys. Rev. A 54, 2614 (1996) in Google Scholar

[8] K. Zyczkowski, I. Bengtsson, Open Sys. Inf. Dyn. 11, 3 (2004) in Google Scholar

[9] W. Roga, Z. Puchała, Ł. Rudnicki, K. Zyczkowski, arXiv:1206.2536 [quant-ph] Search in Google Scholar

[10] S. Wehner, A. Winter, New J. Phys. 12, 025009 (2010) in Google Scholar

[11] I. Bialynicki-Birula, Ł. Rudnicki, Entropic Uncertainty Relations in Quantum Physics, In: K. D. Sen (Ed.), Statistical Complexity, 1 (Springer, Berlin, 2011) 10.1007/978-90-481-3890-6-1 in Google Scholar

[12] W. Roga, M. Fannes, K. Zyczkowski, Int. J. Quant. Inf. 9, 1031 (2011) in Google Scholar

[13] P.W. Shor, Commun. Math. Phys. 246, 453 (2004) in Google Scholar

[14] F.G.S.L. Brandão, M. Horodecki, Open Sys. Inf. Dyn. 17, 31 (2010) in Google Scholar

[15] A.E. Rastegin, J. Phys. A: Math. Theor. 45, 045302 (2012) in Google Scholar

[16] G. Lindblad, In: C. Bendjaballah, O. Hirota, S. Reynaud (Eds.), Lect. Notes Phys. 378, 71 (1991) Search in Google Scholar

[17] J. Havrda, F. Charvát, Kybernetika 3, 30 (1967) Search in Google Scholar

[18] E.M.F. Curado, C. Tsallis, J. Phys. A: Math. Gen. 24, L69 (1991) in Google Scholar

[19] V. Majerník, E. Majerníková, S. Shpyrko, Cent. Eur. J. Phys. 3, 393 (2003) in Google Scholar

[20] S. Wehner, A. Winter, J. Math. Phys. 49, 062105 (2008) in Google Scholar

[21] A.E. Rastegin, J. Phys. A: Math. Theor. 43, 155302 (2010) in Google Scholar

[22] A.E. Rastegin, J. Phys. A: Math. Theor. 44, 095303 (2011) in Google Scholar

[23] A.E. Rastegin, Phys. Scr. 84, 057001 (2011) in Google Scholar

[24] A.E. Rastegin, Int. J. Theor. Phys. 51, 1300 (2012) in Google Scholar

[25] G. Wilk, Z. WŁodarczyk, Cent. Eur. J. Phys. 10, 568 (2012) in Google Scholar

[26] G.A. Raggio, J. Math. Phys. 36, 4785 (1995) in Google Scholar

[27] J. Watrous, Theory of Quantum Information (Lecture notes for CS 798, University of Waterloo, 2008) Search in Google Scholar

[28] A. JamioŁkowski, Rep. Math. Phys. 3, 275 (1972) in Google Scholar

[29] M.-D. Choi, Linear Algebra Appl. 10, 285 (1975) in Google Scholar

[30] J.A. Miszczak, Int. J. Mod. Phys. C 22, 897 (2011) in Google Scholar

[31] W. Roga, M. Fannes, K. Zyczkowski, Phys. Rev. Lett. 105, 040505 (2011) in Google Scholar PubMed

[32] A.E. Rastegin, Quantum Inf. Process., DOI:10.1007/s11128-011-0347-6 10.1007/s11128-011-0347-6Search in Google Scholar

[33] K. Audenaert, J. Math. Phys. 48, 083507 (2007) in Google Scholar

[34] S. Furuichi, J. Math. Phys. 47, 023302 (2006) in Google Scholar

[35] J. Preskill, Quantum Computation and Information (Lecture notes for Physics 229, California Institute of Technology, 1998) Search in Google Scholar

[36] E.A. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course (Lecture course given at “Entropy and the Quantum”, Tucson, Arizona, 2009) 10.1090/conm/529/10428Search in Google Scholar

[37] J.-C. Bourin, F. Hiai, Int. J. Math. 22, 1121 (2011) in Google Scholar

[38] A.E. Rastegin, J. Stat. Phys. 148, 1040 (2012) in Google Scholar

[39] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities (Cambridge University Press, London, 1934) Search in Google Scholar

[40] K. Fan, Proc. Nat. Acad. Sci. USA 35, 652 (1949) in Google Scholar PubMed PubMed Central

[41] R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985) 10.1017/CBO9780511810817Search in Google Scholar

Published Online: 2013-1-15
Published in Print: 2013-1-1

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