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Additional resources for Nikolaos P. Papadakos. Quantum Information Theory and Applications to Quantum Cryptography
12: 567-594 (1964).  R. S. Ingarden. , 9: 273-282 (1965).  K. Urbanik. On the concept of information, Bull. Acad. Polon. , S´er. math. astr. , 20: 887-890 (1972). 38  P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings, 35 th Annual Symposium on Fundamentals of Computer Science, IEEE Press, Los Alamitos, CA (1994).  P. W. Shor. Polynomial time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J.
Since δ can be chosen arbitrarily small, R can be chosen to be as close to H (Y : X) as desired. Now getting the maximum over the prior probabilities of the strings Shannon’s result is found. Shannon’s noisy channel coding theorem: For a noisy channel N the capacity is given by C (N ) = max H (Y : X) , p(x) where the maximum is taken over all input distributions p (x) (a priori distributions) for X, for one use of the channel, and Y is the corresponding induced random variable at the output of the channel.
Smolin. Experimental Quantum Cryptography. J. Cryptology, 5: 3-28 (1992).  D. S. Bethune and W. P. Risk. An autocompensating quantum key distribution system using polarization splitting of light. In IQEC’98 Digest of Postdeadline Papers, pages QPD12-2, Optical Society of America, Washington, DC (1998).  Donald S. Bethune and William P. Risk. An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light. J. Quantum Electronics, 36(3): 100 (2000).  A.