Reconstructing the discrete Wigner function and some properties of the measurement bases

We derive a direct reconstruction algorithm for the discrete Wigner function through different types of measurements. For a system described in a Hilbert space of dimension ${N=N}_1\dots N_p,$ where the numbers $N_i$ are prime, the reconstruction is accomplished with ${(N}_1+1)\dots {(N}_p+1)$ factorable (local) von Neumann measurements. For the special case where the dimension is a power of a prime, the reconstruction can be performed in a much more efficient way using $N+1\mathrm{}$ complementary measurements. If the system is composed of a number of smaller subsystems, these measurements will then in general be nonseparable.