Estimation of Wigner distribution of single mode Gaussian states: a comparative study

In this work, we consider the estimation of single mode Gaussian states using four different measurement schemes namely: i) homodyne measurement, ii) sequential measurement, iii) Arthurs-Kelly scheme, and iv) heterodyne measurement, with a view to compare their relative performance. To that end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out involved calculations. We show that the optimal performance of the Arthurs-Kelly scheme and the sequential measurement is equal to the heterodyne measurement. While the heterodyne measurement outperforms the homodyne measurement in the mean estimation of squeezed state ensemble, the homodyne measurement outperforms the heterodyne measurement for variance estimation of squeezed state ensemble up to a certain range of squeezing parameter. We then modify the Hamiltonian in the Arthurs-Kelly scheme, such that the two meters can have correlations and show that the optimal performance is achieved when the meters are uncorrelated. We expect that the results will be useful in analyzing various quantum information and quantum communication protocols.