Weak measurement-based state estimation of Gaussian states of one-variable quantum systems

We present a scheme to estimate Gaussian states of one-dimensional continuous variable systems, based on weak (unsharp) quantum measurements. The estimation of a Gaussian state requires us to find position ($q$), momentum ($p$) and their second order moments. We measure $q$ weakly and follow it up with a projective measurement of $p$ on half of the ensemble, and on the other half we measure $p$ weakly followed by a projective measurement of $q$. In each case we use the state twice before discarding it. We compare our results with projective measurements and demonstrate that under certain conditions such weak measurement-based estimation schemes, where recycling of the states is possible, can outperform projective measurement-based state estimation schemes.