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Additional resources for Nikolaos P. Papadakos. Quantum Information Theory and Applications to Quantum Cryptography
Example text
1. Moreover Alice and Bob need a classical line to discuss their result, as described by the same protocol, and send the encrypted message. The encryption and the decryption of the message is done using a string K, the key, which should be of equal length to the message, M. Then by applying the modulo 2 addition ⊕ for each bit of the strings, the encrypted message is E = M ⊕ K. Finally the message is decrypted by further addition of the key, (M ⊕ K) ⊕ K = M ⊕ (K ⊕ K) = M ⊕ 0 = M. 2: The setup of BB84 quantum key distribution protocol.
Is exhibited by d † Vi↔j BVi↔j d d † (Vi↔j )kl Blm Vi↔j = l=1 m=1 = † (Vi↔j )kl bl Vi↔j l=1 nl δ kl , k or l = i, j δ nl , n or l = i, j 1, k = j and l = i 1, n = j and l = i 1, k = i and l = j bl 1, n = i and l = j l=1 0, else 0, else bnδ kn, k = i, j or n = i, j bi, k = j and n = j = diag {. . , bj , . . , bi, . . } . bj , k = i and n = i 0, else d = mn = Defining now V23···d1 V1↔2V2↔3 · · · Vd−1↔d Vd↔1 , which is unitary matrix as a multiplication of unitary † matrices, then V23···d1U (A) AU (A)† V23···d1 is a diagonal matrix where the in the first diagonal place is the second 35 diagonal element of A, in the second diagonal place the third diagonal element of A, and so on until the final diagonal place where the first diagonal element of A stands.
A(d−1)(d−1) = tr (A) I. 2 Projective measurements are equivalent to an addition of unitary operations Let P be a projector and Q I−P the complementary projector, in this appendix it will be sawn that there exist unitary matrices U1 , U2 and a probability p such that for all ρ, P ρP + QρQ = pU1ρU1† + (1 − p) U2 ρU2† . Choose p 21 , U1 Q−P and U2 Q+P I. It is obvious that U1 U1† = (Q − P ) (Q − P ) = QQ+QP −P Q−+P P = Q + 0 + P = I, and U2 U2† = II = I. Now it is easy to check that 1 1 U1 ρU1† + U2 ρU2† 2 2 = = = 1 1 (Q − P ) ρ (Q − P ) + (Q + P ) ρ (Q + P ) 2 2 1 1 (QρQ − QρP − P ρQ + P ρP ) + (QρQ + QρP + P ρQ + P ρP ) 2 2 P ρP + QρQ.