Preparation of arbitrary quantum states with regular $P$ functions

We propose a quantum optical device to experimentally realize quantum processes, which perform the regularization of the—in general highly singular—Glauber–Sudarshan $P$ functions of arbitrary quantum states before their application and/or measurement. This allows us to produce a broad class of nonclassical states with regular $P$ functions, also called nonclassicality quasiprobabilities. For this purpose, the input states are combined on highly transmissive beam splitters with specific Gaussian or non-Gaussian classical states. We study both balanced and unbalanced homodyne detections for the direct sampling of the output states of the implemented processes, which requires no further regularization or state-reconstruction. By numerical simulations we demonstrate the feasibility of our approach and we outline the generalization to multimode light.

Figure 1

Logarithmic plot of the Fourier transform ${\tilde{\mathrm{\Omega }}}_w^{\left(q\right)}$ of the filters ${\mathrm{\Omega }}_w^{\left(q\right)}$ as a function of $\left|\gamma \right|$ with filter parameter $w=1.0$ and for various values of the parameter $q$.

Figure 2

Linear optical implementation of the mapping procedure. The $P$ function of the signal (SI) is converted to a regularized version $P_w$ by superimposing it on a highly transmissive beam splitter with a properly engineered classical field (ECF), to be adjusted by the parameter $w$.

Figure 3

The minimal quadrature variance of the generated state in Eq. (17) is shown as a function of the filter parameter $w$ for various values of $q$. The quadrature variance of the vacuum state (${\langle {\left[\mathrm{\Delta }\widehat{x}\right]}^2\rangle }_{\mathrm{vac}}=1$) and the minimal quadrature variance ${\langle {\left[\mathrm{\Delta }\widehat{x}\right]}^2\rangle }_{\widehat{\rho }}\approx 0.37$ of the original squeezed vacuum state $\widehat{\rho }=|\xi \rangle \langle \xi |$ is also illustrated. The vertical lines indicate the critical $w$ parameters $w_{\mathrm{crit}}\left(q\right)$ in Eq. (32), corresponding to the considered $q$ parameters.

Figure 4

Regularized $P$ function of a squeezed vacuum state with squeezing parameter $\xi =0.5$ for $w=1.3$, sampled from $3\times {10}^6$ data points $({\varphi }_j,x_j)$: (a) squeezed axis, (b) antisqueezed axis. The thin dashed lines above and below the solid line correspond to an error of one standard deviation.

Figure 5

Transmission distribution ${\mathcal{T}}_w(\tau ;{\gamma }_{\mathrm{c}})$ required for the filtering with ${\mathrm{\Omega }}_w^{\left(\infty \right)}$ in Eq. (8) for two different values of the product $w{\gamma }_{\mathrm{c}}$.

Figure 6

Systematic error $E\left(w{\gamma }_{\mathrm{c}}\right)$ due to the finite laser amplitude ${\gamma }_{\mathrm{L}}$.

Figure 7

Systematic error of the negativities of $P_w$ as a function of the cutoff amplitude ${\gamma }_{\mathrm{c}}$ for various values of the filter parameter $w$.