On the quantum Rényi relative entropies and related capacity formulas

IEEE Transactions on Information Theory

Volumn 57, Issue 4, 2011, Pages 2474-2487

  Mosonyi, Milán ^a,b   Hiai, Fumio ^c  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

^c TOHOKU UNIVERSITY   (Japan)

Author keywords

capacities; generalized cutoff rates; Hoeffding distances; one shot capacities; quantum channels; R nyi relative entropies

Indexed keywords

CLASSICAL INFORMATION; CLASSICAL-QUANTUM CHANNELS; CUT-OFF RATE; GENERALIZED CUTOFF RATES; HOEFFDING DISTANCES; HOLEVO CAPACITY; ONE-SHOT CAPACITIES; OPERATIONAL INTERPRETATION; PARAMETER RANGE; QUANTUM CHANNELS; RELATIVE ENTROPY; UPPER BOUND;

INFORMATION THEORY; QUANTUM ENTANGLEMENT;

ENTROPY;

EID: 79952824089     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2011.2110050     Document Type: Article

References (42)

1. F. Hiai and D. Petz, "The proper formula for relative entropy and its asymptotics in quantum probability," Commun. Math. Phys., vol. 143, pp. 99-114, 1991.
2. T. Ogawa and H. Nagaoka, "Strong converse and Stein's lemma in quantum hypothesis testing," IEEE Trans. Inf. Theory, vol. 47, pp. 2428-2433, 2000. (Pubitemid 32090024)
3. K. M. R. Audenaert, M. Nussbaum, A. Szkola, and F. Verstraete, "Asymptotic error rates in quantum hypothesis testing," Commun. Math. Phys., vol. 279, pp. 251-283, 2008.
4. M. Hayashi, "Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding," Phys. Rev. A, vol. 76, pp. 062301-062301, 2007.
5. H. Nagaoka, "The converse part of the theorem for quantum Hoeffding bound," arXiv:quant-ph/0611289, 2006.
6. F. Hiai, M. Mosonyi, and T. Ogawa, "Error exponents in hypothesis testing for correlated states on a spin chain," J. Math. Phys., vol. 49, pp. 032112-032112, 2008.
7. A. Rényi, "On measures of entropy and information," in Proc. 4th Berkeley Symp. on Math. Statist. Probabil., Berkeley, CA, 1961, vol. 1, pp. 547-561.
8. I. Csiszár, "Generalized cutoff rates and Rényi's information measures," IEEE Trans. Inf. Theory, vol. 41, pp. 26-34, 1995.
9. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, "Quantifying entanglement," Phys. Rev. Lett., vol. 78, pp. 2275-2279, 1997. (Pubitemid 127651856)
10. A. S. Holevo, "The capacity of the quantum communication channel with general signal states," IEEE Trans. Inf. Theory, vol. 44, pp. 269-273, 1998. (Pubitemid 128737894)
11. B. Schumacher and M. D. Westmoreland, "Sending classical information via noisy quantum channels," Phys. Rev. A, vol. 56, pp. 131-138, 1997. (Pubitemid 127605032)
12. M. Ohya, D. Petz, and N. Watanabe, "On capacities of quantum channels," Prob. Mat. Stat., vol. 17, pp. 179-196, 1997. (Pubitemid 127138216)
13. R. Koenig and S. Wehner, "A strong converse for classical channel coding using entangled inputs," Phys. Rev. Lett., vol. 103, pp. 070504-070504, 2009.
14. M. Mosonyi and N. Datta, "Generalized relative entropies and the capacity of classical-quantum channels," J. Math. Phys., vol. 50, pp. 072104-072104, 2009.
15. D. Petz, "Quasi-entropies for finite quantum systems," Rep. Math. Phys., vol. 23, pp. 57-65, 1986.
16. F. Hiai, M. Mosonyi, and M. Hayashi, "Quantum hypothesis testing with group symmetry," J. Math. Phys., vol. 50, pp. 103304-103304, 2009.
17. M. Tomamichel, R. Colbeck, and R. Renner, "A fully quantum asymptotic equipartition property," IEEE Trans. Inf. Theory, vol. 55, pp. 5840-5847, 2009.
18. T. Ando, "Concavity of certain maps and positive definite matrices and applications to Hadamard products," Linear Algebra Appl., vol. 26, pp. 203-241, 1979.
19. E. H. Lieb, "Convex trace functions and the Wigner-Yanase-Dyson conjecture," Adv. Math., vol. 11, pp. 267-288, 1973.
20. M. Ben-Bassat and J. Raviv, "Rényi's entropy and the probability of error," IEEE Trans. Inf. Theory, vol. 24, pp. 324-331, 1978.
21. T. Ando and F. Hiai, "Operator log-convex functions and operator means," Math. Annalen, arXiv:0911.5267, to be published.
22. F. Hiai, M. Ohya, and M. Tsukada, "Sufficiency, KMS condition and relative entropy in von Neumann algebras," Pacific J. Math., vol. 96, pp. 99-109, 1981.
23. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. Amsterdam: North-Holland and American Elsevier, 1976.
24. N. Datta, "Min-and max-relative entropies and a new entanglement measure," IEEE Trans. Inf. Theory, vol. 55, pp. 2816-2826, 2009.
25. F. Buscemi and N. Datta, "The quantum capacity of channels with arbitrarily correlated noise," IEEE Trans. Inf. Theory, vol. 56, pp. 1447-1460, 2010.
26. R. Colbeck, private communication.
27. M. Mosonyi, F. Hiai, T. Ogawa, and M. Fannes, "Asymptotic distinguishability measures for shift-invariant quasi-free states of Fermionic lattice systems," J. Math. Phys., vol. 49, pp. 072104-072104, 2008.
28. M. Mosonyi, "Hypothesis testing for Gaussian states on Bosonic lattices," J. Math. Phys., vol. 50, pp. 032104-032104, 2009.
29. M. Sion, "On general minimax theorems," Pacific J. Math., vol. 8, pp. 171-176, 1958.
30. H. Komiya, "Elementary proof for Sion's minimax theorem," Kodai Math. J., vol. 11, pp. 5-7, 1988.
31. T. Ogawa and H. Nagaoka, "Strong converse to the quantum channel coding theorem," IEEE Trans. Inf. Theory, vol. 45, pp. 2428-2433, 1999.
32. R. Sibson, "Information radius," Z. Wahrscheinlichkeitsth. Verw. Gebiete, vol. 14, pp. 149-161, 1969.
33. R. Koenig, R. Renner, and C. Schaffner, "The operational meaning of min-and max-entropy," IEEE Trans. Inf. Theory, vol. 55, pp. 4337-4347, 2009.
34. G. Kimura, T. Miyadera, and H. Imai, "Optimal state discrimination in general probabilistic theories," Phys. Rev. A, vol. 79, pp. 062306-062306, 2009.
35. A. S. Holevo, "Investigations in the general theory of statistical decisions," in Proc. Steklov Inst. Math., 1976, vol. 124, AMS Translation: 1978 Issue 3.
36. C. W. Helström, Quantum Detection and Estimation Theory. New York: Academic, 1976.
37. R. Alicki and M. Fannes, "Continuity of quantum conditional information," J. Phys. A, vol. 37, pp. L55-L57, 2004. (Pubitemid 38214163)
38. M. Horodecki and B. Synak-Radtke, "On asymptotic continuity of functions of quantum states," J. Phys. A, vol. 39, pp. L423-L437, 2006. (Pubitemid 43932264)
39. T. S. Han and K. Kobayashi, "The strong converse theorem for hypothesis testing," IEEE Trans. Inf. Theory, vol. 35, pp. 178-180, 1989.
40. M. Hayashi, Quantum Information: An Introduction. New York: Springer, 2006.
41. A. Barvinok, A Course in Convexity. New York: American Math. Soc. Graduate Studies in Math., 2002, vol. 54.
42. A. Fujiwara and H. Nagaoka, "Operational capacity and pseudoclassicality of a quantum channel," IEEE Trans. Inf. Theory, vol. 44, pp. 1071-1086, 1998. (Pubitemid 128743267)