  I consider the tradeoff between the information gained about an initially
unknown quantum state, and the disturbance caused to that state by the
measurement process. I show that for any distribution of initial states, the
information-disturbance frontier is convex, and disturbance is nondecreasing
with information gain. I consider the most general model of quantum
measurements, and all post-measurement dynamics compatible with a given
measurement. For the uniform initial distribution over states, I show that the
least-disturbing way of making any measurement is with conditional dynamics
satisfying a generalization of the projection postulate, the ``square-root
dynamics.'' Thus, procedures for achieving a point on the
information-disturbance frontier may be assumed to involve such conditional
dynamics. Also, the information-disturbance frontier for the uniform ensemble
may be achieved with ``isotropic'' (unitarily covariant) dynamics. This results
in a significant simplification of the optimization problem for calculating the
tradeoff in this case, giving hope for a closed-form solution. I also show that
the discrete ensembles uniform on the d(d+1) vectors of a certain set of d+1
``mutually unbiased'' or conjugate bases in d dimensions form spherical
2-designs in CP_{d-1} when d is a power of an odd prime. This implies that many
of the results of the paper apply also to these discrete ensembles.
