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

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