Inverse spin’s portrait and representation of qudit states by single probability vectors

Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distri-bution. Within the framework of the approach proposed, any quantum spin-j state is associated with the (2j + 1)(4j + 1)-dimensional probability vector whose components are labeled by spin projections and points on the sphere S2. Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the 2j(4j + 3) simplex, with the boundary being illustrated for qubits (j = 1/2) and qutrits (j = 1). A relation to the (2j + 1)2- and (2j + 1)(2j + 2)-dimensional probability vectors is established in terms of spin-s portraits. We also address an auxiliary problem of the optimum reconstruction of qudit states, where the optimality implies a minimum rela-tive error of the density matrix due to the errors in measured probabilities