Abstract
The classical capacity of a quantum channel with arbitrary Markovian correlated noise is evaluated. For the general case of a channel with long-term memory, which corresponds to a Markov chain which does not converge to equilibrium, the capacity is expressed in terms of the communicating classes of the Markov chain. For an irreducible and aperiodic Markov chain, the channel is forgetful, and one retrieves the known expression (Kretschmann and Werner in Phys. Rev. A 72:062323, 2005) for the capacity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlswede, R.: The weak capacity of averaged channels. Z. Wahrscheinlichk. Verwandte Geb. 11, 61–73 (1968)
Bjelaković, I., Boche, H.: Ergodic classical-quantum channels: structure and coding theorems. quant-ph/0609229
Bjelaković, I., Boche, H.: Classical capacities of compound and averaged quantum channels. arXiv:0710.3027
Bowen, G., Mancini, S.: Quantum channels with a finite memory. Phys. Rev. A 69, 01236 (2004)
Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)
Datta, N., Dorlas, T.C.: A quantum version of Feinstein’s lemma and its application to channel coding. In: Proc. of Int. Symp. Inf. Theor. ISIT 2006, Seattle, pp. 441–445 (2006)
Datta, N., Dorlas, T.C.: The coding theorem for a class of channels with long-term memory. J. Phys. A, Math. Theory 40, 8147–8164 (2007)
Feinstein, A.: A new basic theorem of information theory. IRE Trans. PGIT 4, 2–22 (1954)
Hayashi, M., Nagaoka, H.: General formulas for capacity of classical-quantum channels. IEEE Trans. Inf. Theory 49, 1753–1768 (2003)
Khinchin, A.I.: Mathematical Foundations of Information Theory. Part II: On the Fundamental Theorems of Information Theory. Dover, New York (1957), Chap. IV
Helstrøm, C.W.: Quantum Detection and Estimation Theory. Mathematics in Science and Engineering, vol. 123. Academic Press, London (1976)
Hiai, F., Petz, D.: The proper formula for the relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 257–281 (1991)
Holevo, A.S.: The capacity of a quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998)
Jacobs, K.: Almost periodic channels. In: Colloqium on Comb. Methods in Prob. Theory. Aarhus (1962)
Kretschmann, D., Werner, R.F.: Quantum channels with memory. Phys. Rev. A 72, 062323 (2005). quant-ph/0502106
McMillan, B.: The basic theorems of information theory. Ann. Math. Stat. 24, 196–219 (1953)
Macchiavello, C., Palma, G.M.: Entanglement-enhanced information transmission over a quantum channel with correlated noise. Phys. Rev. A 65, 050301 (2002)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Norris, J.R.: Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1997)
Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Berlin (1993)
Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997)
Shannon, C.E.: A mathematical theory of communication. Part I. Bell Syst. Tech. J. 27, 379–423 (1948)
Shannon, C.E.: A mathematical theory of communication. Part II. Bell Syst. Tech. J. 27, 623–656 (1948)
Winter, A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inf. Theory 45, 2481–2485 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
T. Dorlas is on leave from the Dublin Institute of Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland.
Rights and permissions
About this article
Cite this article
Datta, N., Dorlas, T. Classical Capacity of Quantum Channels with General Markovian Correlated Noise. J Stat Phys 134, 1173–1195 (2009). https://doi.org/10.1007/s10955-009-9726-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-009-9726-0