Abstract
Following Schumacher and Westmoreland, we address the problem of the capacity of a quantum wiretap channel. We first argue that, in the definition of the so-called “quantum privacy,” Holevo quantities should be used instead of classical mutual informations. The argument actually shows that the security condition in the definition of a code should limit the wiretapper’s Holevo quantity. Then we show that this modified quantum privacy is the optimum achievable rate of secure transmission.
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REFERENCES
Holevo, A.S., Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel, Probl. Peredachi Inf., 1973, vol. 9, no. 3, pp. 3–11 [Probl. Inf. Trans. (Engl. Transl.), 1973, vol. 9, no. 3, pp. 177–183].
von Neumann, J., Thermodynamik quantenmechanischer Gesamtheiten, Nachr. der Gesellschaft der Wiss. Gött., 1927, pp. 273–294.
Shannon, C.E., A Mathematical Theory of Communication, Bell Syst. Tech. J., 1948, vol. 27, no. 3, pp. 379–423; no. 4, pp. 623–656.
Holevo, A.S., The Capacity of a Quantum Channel with General Signal States, IEEE Trans. Inform. Theory, 1998, vol. 44, no. 1, pp. 269–273.
Schumacher, B. and Westmoreland, M.D., Sending Classical Information via Noisy Quantum Channels, Phys. Rev. A, 1997, vol. 56, no. 1, pp. 131–138.
Winter, A., Coding Theorem and Strong Converse for Quantum Channels, IEEE Trans. Inform. Theory, 1999, vol. 45, no. 7, pp. 2481–2485.
Holevo, A.S., Coding Theorems for Quantum Channels, LANL e-print quant-ph/9809023, 1998.
Schumacher, B. and Westmoreland, M.D., Relative Entropy in Quantum Information Theory, LANL e-print quant-ph/0004045, 2000.
Shannon, C.E., Communication Theory of Secrecy Systems, Bell Syst. Tech. J., 1949, vol. 28, no. 4, pp. 656–715.
Wyner, A.D., The Wire-tap Channel, Bell Syst. Tech. J., 1975, vol. 54, no. 8, pp. 1355–1387.
Csiszàr, I. and Körner, J., Broadcast Channels with Confidential Messages, IEEE Trans. Inform. Theory, 1978, vol. 24, no 3, pp. 339–348.
Ahlswede, R. and Csiszàr, I., Common Randomness in Information Theory and Crytography—Part I: Secret Sharing, IEEE Trans. Inform. Theory, 1993, vol. 39, no. 4, pp. 1121–1132.
Maurer, U.M., Secret Key Agreement by Public Discussion Based on Common Information, IEEE Trans. Inform. Theory, 1993, vol. 39, no. 3, pp. 733–742.
Cai, N. and Lam, K.Y., How to Broadcast Privacy: Secret Coding for Derministic Broadcast Channels, Numbers, Information, and Complexity, Althöfer, I., Cai, N., Dueck, G., Khachatrian, L., Pinsker, M., Sarkozy, A., Wegener, I., and Zhang, Z., Eds., Boston: Kluwer, 2000, pp. 353–368.
Schumacher, B. and Westmoreland, M.D., Quantum Privacy and Quantum Coherence, Phys. Rev. Lett., 1998, vol. 80, no. 25, pp. 5695–5697.
DiVincenzo, D.P., Shor, P.W., and Smolin, J.A., Quantum-Channel Capacity of Very Noisy Channels, Phys. Rev. A, 1998, vol. 57, no. 2, pp. 830–839.
Nielsen, M.A. and Chuang, I.L., Quantum Computation and Quantum Information, Cambridge: Cambridge Univ. Press, 2000.
Csiszàr, I. and Körner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems, Budapest: Akademiai Kiado, 1981. Translated under the title Teoriya informatsii: teoremy kodirovaniya dlya diskretnykh sistem bez pamyati, Moscow: Mir, 1985.
Cover, T.M. and Thomas, J.A., Elements of Information Theory, New York: Wiley, 1991.
Yeung, R.W., A First Course in Information Theory, New York: Kluwer, 2002.
DiVincenzo, D.P., Horodecki, M., Leung, D.W., Smolin, J.A., and Terhal, B.M., Locking Classical Correlation in Quantum States, LANL e-print quant-ph/0303088, 2003.
Löber, P., Quantum Channels and Simultaneous ID Coding, Doctoral Dissertation, Bielefeld: Universität Bielefeld, 1999. Available at http://archiv.ub.uni-bielefeld.de/disshabi/mathe.htm.
Ahlswede, R. and Dueck, G., Identification via Channels, IEEE Trans. Inform. Theory, 1989, vol. 35, no. 1, pp. 15–29.
Ahlswede, R. and Winter, A., Strong Converse for Identification via Quantum Channels, IEEE Trans. Inf. Theory, 2002, vol. 48, no. 3, pp. 569–579. Addendum: IEEE Trans. Inf. Theory, 2003, vol. 49, no. 1, p. 346.
Elias, P., List Decoding for Noisy Channels, 1957 IRE Wescon Convention Record, Part 2, 1957, pp. 94–104.
Ahlswede, R., Channel Capacities for List Codes, J. Appl. Probab., 1973, vol. 10, no. 4, pp. 824–836.
Arikan, E., An Inequality on Guessing and Its Application to Sequential Decoding, IEEE Trans. Inform. Theory, 1996, vol. 42, no. 1, pp. 99–105.
Csiszàr, I, Almost Independence and Secrecy Capacity, Probl. Peredachi Inf., 1996, vol. 32, no. 1, pp. 48–57 [Probl. Inf. Trans. (Engl. Transl.), 1996, vol. 32, no. 1, pp. 40–47].
Fannes, M., A Continuity Property of the Entropy Density for Spin Lattice Systems, Comm. Math. Phys., 1973, vol. 31, pp. 291–294.
Devetak, I., The Private Classical Information Capacity and Quantum Information Capacity of a Quantum Channel, LANL e-print quant-ph/0304127, 2003.
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Translated from Problemy Peredachi Informatsii, No. 4, 2004, pp. 26–47.
Original Russian Text Copyright © 2004 by Cai, Winter, Yeung.
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Cai, N., Winter, A. & Yeung, R.W. Quantum privacy and quantum wiretap channels. Probl Inf Transm 40, 318–336 (2004). https://doi.org/10.1007/s11122-005-0002-x
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DOI: https://doi.org/10.1007/s11122-005-0002-x