Gaussian states under coarse-grained continuous variable measurements

The quantum-to-classical transition of a quantum state is a topic of great interest in fundamental and practical aspects. A coarse-graining in quantum measurement has recently been suggested as its possible account in addition to the usual decoherence model. We here investigate the reconstruction of a Gaussian state (single mode and two modes) by coarse-grained homodyne measurements. To this aim, we employ two methods, the direct reconstruction of the covariance matrix and the maximum likelihood estimation (MLE), respectively, and examine the reconstructed state under each scheme compared to the state interacting with a Gaussian (squeezed thermal) reservoir. We clearly demonstrate that the coarse-graining model, though applied equally to all quadrature amplitudes, is not compatible with the decoherence model by a thermal (phase-insensitive) reservoir. Furthermore, we compare the performance of the direct reconstruction and the MLE methods by investigating the fidelity and the nonclassicality of the reconstructed states and show that the MLE method can generally yield a more reliable reconstruction, particularly without information on a reference frame (phase of input state).

Figure 1

Illustration of the coarse-graining process in Eq. (3). A Gaussian probability distribution (red solid line) is transformed to a piecewisely flat distribution (black dashed line) under the coarse-graining of size $\sigma =0.5$.

Figure 2

(a) Original squeezed state (b) reconstructed state with coarse-graining $\sigma =0.1$ (c) difference in squeezing angle between an input state and its estimated state as a function of the coarse-graining size $\sigma$ and the input squeezing angle ${\varphi }_i$. In all plots, the input state has the parameters $(\overline{n},r)=(0,1)$.

Figure 3

The fraction $y$ in Eq. (28) to make equal the state estimation process and the decoherence program with an isotropic (thermal) reservoir as a function of coarse-graining size $\sigma$. The input squeezed thermal states are characterized by $(\overline{n},r)=(0,1)$ (red solid line), $(\overline{n},r)=(1,1)$ (orange dotted line), and $(\overline{n},r)=(0,2)$ (brown dot-dashed line).

Figure 4

(a) Fidelity $F$ between an input state and its reconstructed state and (b) nonclassical squeezing $r_{\text{nc}}$ of the reconstructed state as functions of the coarse-graining size $\sigma$, for the input squeezed thermal states with $(\overline{n},r)=(0,1)$ [green dot-dashed line, red solid and blue dashed lines, the second curves from the top for (a) and (b)], $(\overline{n},r)=(1,1)$ [pink dot-dashed line, orange solid and purple dashed lines, the first curves from the top for (a) and the third curves from the top for (b)], and $(\overline{n},r)=(0,2)$ (gray dot-dashed line, brown solid and black dashed lines, the third curves from the top for (a) and the first curves from the bottom for (b)]. Solid curves represent the case of the MLE method, dot-dashed (dashed) curves the direct reconstruction method with (without) information on the input phase, respectively. For the plots of dashed curves, each point represents an averaged value over the whole range of the input squeezing angles. See main text.

Figure 5

(a) Fidelity $F$ between an input state and its reconstructed state and (b) logarithmic negativity $E_{\mathcal{N}}$ of the reconstructed state as functions of the coarse-graining size $\sigma$, for the input squeezed thermal states with $(\overline{n},r)=(0,1)$ [green dot-dashed line, red solid and blue dashed lines, the second curves from the top for (a) and (b)], $(\overline{n},r)=(1,1)$ [pink dot-dashed line, orange solid and purple dashed lines, the first curves from the top for (a) and the third curves from the top for (b)], and $(\overline{n},r)=(0,2)$ [gray dot-dashed line, brown solid and black dashed lines, the third curves from the top for (a) and the first curves from the bottom for (b)]. For simplicity, we assume that the thermal photon number of two modes are the same, ${\overline{n}}_1={\overline{n}}_2=\overline{n}$. Solid curves represent the case of the MLE method, dot-dashed (dashed) curves the direct reconstruction method with (without) information on the input phase, respectively. For the plots of dashed curves, each point represents an averaged value over the whole range of the input squeezing angles.