  The Holevo bound is a keystone in many applications of quantum information
theory. We propose "weak maximal Holevo quantity" with weak measurements as the
generalization of the standard Holevo quantity which is defined as the optimal
projective measurements. The scenarios that weak measurements is necessary are
that only the weak measurements can be performed because for example the system
is macroscopic or that one intentionally tries to do so such that the
disturbance on the measured system can be controlled for example in quantum key
distribution protocols. We evaluate systematically the weak maximal Holevo
quantity for Bell-diagonal states and find a series of results. Furthermore, we
find that weak measurements can be realized by noise and project measurements.
