Symmetric informationally complete quantum measurements

We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d(2) rank-one operators all of whose operator inner products are equal. Such a set is called a "symmetric, informationally complete" POVM (SIC-POVM) and is equivalent to a set of d(2) equiangular lines in C-d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim. (C) 2004 American Institute of Physics.