Determining an unknown state of a high-dimensional quantum system with a single, factorized observable

One can uniquely identify an unknown state of a quantum system S by measuring a "quorum" consisting of a complete set of noncommuting observables. It is also possible to determine the quantum state by repeated measurements of a single and factorized observable, when the system S is coupled to an assistant system A whose initial state is known. This is because that the redistribution of the information about the unknown quantum state into the composite system A + S results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. Here we focus on quantum state tomography of high-dimensional quantum systems (e.g., a spin greater than 1/2) via a single observable. We determine the condition for the best determination and the upper bound to achieve the most robust measurements. From the experimental view we require a suitable interaction Hamiltonian to maximize the measure efficiency. For this we numerically investigate a three-level system. Moreover, the error analysis for the different-dimensional quantum states shows that the present measurement method is still very effective in determining an unknown state of a high-dimensional quantum system.