On density matrix reconstruction from measured distributions

Seth Merkel, Carlos A. Riofrío, Steve Flammia +1 more

Th. Richter

The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.

Th. Richter

The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.

David A. Mazziotti

Ulf Leonhardt, Henryk Paul, Giacomo Mauro D’Ariano

We propose a general method for reconstructing directly the density matrix of a single light mode in optical homodyne tomography. In our scheme the density matrix 〈a\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}${\mathit{a}}^{\ensuremath{'}}$〉 is obtained by averaging a set of pattern functions ${\mathit{F}}_{\mathit{aa}}^{\ensuremath{'}}$(${\mathit{x}}_{\mathrm{\ensuremath{\theta}}}$,\ensuremath{\theta}) with respect to the homodyne data ${\mathit{x}}_{\mathrm{\ensuremath{\theta}}}$. The functions show the typical features of the quadrature distributions for the corresponding density-matrix elements. It is also possible to compensate the effect of detection losses which requires, however, extra effort in both experimental and numerical precision. We calculate the pattern functions for the coherent-state and Fock representations and study their properties. We believe that our method is the most efficient way for reconstructing the density matrix from homodyne measurements.

R. McWeeny