PARITY BIT IN QUANTUM CRYPTOGRAPHY

An It-bit string is encoded as a sequence of nonorthogonal quantum sta tes. The parity bit of that n-bit string is described by one of two de nsity matrices, rho(0)((n)) and rho(1)((n)), both in a Hilbert space o f dimension 2(n). In order is to derive the parity bit the receiver mu st distinguish between the two density matrices, e.g., in terms of opt imal mutual information. In this paper we find the measurement which p rovides the optimal mutual information about the parity bit and calcul ate that information. We prove that this information decreases exponen tially with the length of the string in the case where the single bit states are almost fully overlapping. We believe this result will be us eful in proving the ultimate security of quantum cryptography in the p resence of noise.