Tomography of Quantum States in Small Dimensions

We consider the problem of determining the state of a finite dimensional quantum system by a finite set of different measurements in an optimal way. The measurements can either be projective von Neumann measurements or generalized measurements (POVMs). While optimal solutions for projective measurements are only known for prime power dimensions, based on numerical solutions it is conjectured that solutions for POVMs exist in any dimension. We support this conjecture by constructing explicit algebraic solutions in small dimensions d, in particular d = 12.