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By Samuel J. Lomonaco, Jr.

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Rodney Loudon, David T. D. Phoenix, Communication using quantum states, Journal of Modern Optics, Vol. 41, No. 12, 1994, pp 2351 - 2373. D. Phoenix, Informationtheoretic limits to quantum cryptography, Physical Review A, vol. 48, no. 1, July 1993, 1050-2947. , Physics, 1, (1964), pp 195 - 200. [5] Benioff, Paul, Quantum mechanical models of Turing machines that dissipate no energy, Physical Review Letters, Vol. 48, No. 23, 7 June 1982, pp 1581 - 1585. , Fran¸cois Bessette, Gilles Brassard, Louis Salvail, and John Smolin, Experimental quantum cryptography, J.

Lipton, Quantum cryptanalysis of hidden linear functions, pp 424 - 437 in “Advances in Cryptology - CRYPTO’95: Proceedings of the 15th Annual International Cryptology Conference Santa Barbara, California, USA, Agust 1995, edited by Don Coppersmith, Springer-Verlag, NY (1995). , John M. Meyers, And Samuel J. , Aspects of entangled translucent eavesdropping in quantum cryptography, Phys. Rev. A, Vol. 56, No. 6, December 1997, pp. 4456 - 4465. , Positive operator valued measure in quantum information processing, preprint.

An operator O such that O† = O . An eigenvalue a of an operator A is a complex number for which there is a ket |label such that A |label = a |label . The ket |label is called an eigenket of A corresponding to the eigenvalue a. An important theorem about observables is given below: Theorem 5 The eigenvalues ai of an observable A are all real numbers. Moreover, the eigenkets for distinct eigenvalues of an observable are orthogonal. 37 Definition 6 An eigenvalue is degenerate if there are at least two linearly independent eigenkets for that eigenvalue.

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