An n-bit string is encoded as a sequence of nonorthogonal quantum states. The parity bit of that n-bit string is described by one of two density matrices, ${\mathrm{\rho }}_0^{\left(\mathit{n}\right)}$ and ${\mathrm{\rho }}_1^{\left(\mathit{n}\right)}$, both in a Hilbert space of dimension $2^{\mathit{n}}$. 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 cryptography in the presence of noise. © 1996 The American Physical Society.

• Received 12 March 1996

DOI:https://doi.org/10.1103/PhysRevA.54.2675