Pure-state informationally complete and “really” complete measurements

I construct a positive-operator-valued measure (POVM) which has $2d$ rank-1 elements and which is informationally complete for generic pure states in $d$ dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph∕0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in $d$ dimensions, it must have at least $3d-2$ elements. I also show that, in a POVM which is informationally complete for all pure states in $d$ dimensions, for any vector there must be at least $2d-1$ POVM elements which do not annihilate that vector.