Experimentally Determining Bloch Parameters of Hamiltonian for Single-Qubit Systems

By considering the identification problem of unknown but fixed Hamiltonian H=S₀σ₀+ ∑_i=x,y,zS_iσ_i where σ_i (i=x,y,z) are pauli matrices and σ₀=I, weexplore the feasibility and limitation of empirically determining the Hamiltonian parameters for quantum systems under experimental conditions imposed by projective measurements and initialization procedures. It may be unsurprising to physicists that one can not obtain the knowledge of S₀ no matter what kind of projective measurements and initialization are permitted, but the observation draws our attention to the importance of the parameter identifiability under different experimental condition. It has also been revealed that one can obtain the knowledge of |S_z| and S_x²+S_y² at most when only the projective measurement |0><0|,| 1><1| is permitted to perform on and initialize the qubit. Further more, we demonstrated that it is feasible to distinguish |S_x|, |S_y|, and |S_z| even without any a priori information about Hamiltonian if at least two kinds of projective measurement or initialization procedures are permitted. It should be emphasized that both projective measurements and initialization procedures play an important role in quantum system identification.