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

Abstract

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.

Quantum process in probability representation of quantum mechanics

“…where Û mn ≡ Â † mn Û Âmn . From comparing (25) with (7) we see, that the sum mn Ûmn play here the role of the dequantizer after the operation is performed. Relationship (25), in turn, being written in terms of the scalar product of symbols becomes…”

“…In some applications, it may be attractive to realize the von Neumann measurement model, being one of the basic practical procedures to find out in what state the system was [24,25].…”

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

“…In the sequential measurement scheme, the measurement of one quadrature is followed by the measurement of its conjugate quadrature [13][14][15]. To carry out the first measurement, we use the von-Neumann measurement model, where we couple the system with a meter.…”

“…The second measurement, i.e., the measurement of the conjugate observable, is a homodyne measurement. To remove the biasedness in the order of the measurements, we divide the ensemble in two halves [15]. On the first half, we measure the Q-quadrature of the meter weakly, which renders information about the qquadrature of the system.…”

“…It has been shown that various quasiprobability distributions such as Glauber-Sudarshan distribution, Wigner distribution, and Husimi distribution [11] can be estimated using measurements of the rotated quadrature operators and has also been demonstrated experimentally [12]. One can also think of sequential measurement (SM) of a pair of conjugate observables in CV systems, where the observables are measured one after another [13][14][15]. The sequential measurement of two noncommuting observables has been employed to reconstruct the Moyal M function, which is the Fourier transform of the Wigner function [14].…”