  An $n$-bit string is encoded as a sequence of non-orthogonal quantum states.
The parity bit of that $n$-bit string is described by one of two density
matrices, $\rho_0^{(n)}$ and $\rho_1^{(n)}$, both in a Hilbert space of
dimension $2^n$. In order to derive the parity bit the receiver must
distinguish between the two density matrices, e.g., in terms of optimal mutual
information. In this paper we find the measurement which provides the optimal
mutual information about the parity bit and calculate that information. We
prove that this information decreases exponentially with the length of the
string in the case where the single bit states are almost fully overlapping. We
believe this result will be useful in proving the ultimate security of quantum
crytography in the presence of noise.
