Quantum Tomography: Renormalization of Incompatible Observations

Abstract

Standard deterministic techniques for quantum state reconstruction, as for example optical homodyne tomography, photon chopping, unbalanced homodyning etc., are based on the deterministic inversion of measured data. Since the frequencies obtained in realistic experiments always differ from probabilities predicted by quantum theory due to fluctuations, imperfections and realistic restrictions, the algorithm of inversion cannot guarantee the positive definiteness of the reconstructed density matrix. Hence the estimation of the noises may appear as doubtful.

Quantum states may be successfully reconstructed within quantum and information theories using the maximum likelihood estimation. The question of deterministic schemes: “What quantum state is determined by that measurement?” is replaced by the formulation consistent with quantum theory: “What quantum state(s) seems to be most likely for that measurement?” Nonlinear equation for reconstructed state is formulated. An exact solution may be approached by subsequent iterations. Reconstruction is formulated as a problem of proper normalization of incompatible (nonorthogonal) measurements. The results obtained by this novel method may differ significantly from the standard predictions. Data are fitted better keeping the constraint of positive definiteness of reconstructed density matrix. However, this interpretation may enlarge uncertainty in prediction of quantum state in comparison with deterministic schemes, since, in general, there is a whole family of states which fit the measured data equally well. The novel technique is nonlinear and the reinterpretation of existing reconstruction schemes represents an advanced program.