Evaluation of the non-Gaussianity of two-mode entangled states over a bosonic memory channel via cumulant theory and quadrature detection

We study the properties of the cumulants of multimode boson operators and introduce the phase-averaged quadrature cumulants as the measure of the non-Gaussianity of multimode quantum states. Using this measure, we investigate the non-Gaussianity of two classes of two-mode non-Gaussian states: photon-number entangled states and entangled coherent states traveling in a bosonic memory quantum channel. We show that such a channel can skew the distribution of two-mode quadrature variables, giving rise to a strongly non-Gaussian correlation. In addition, we provide a criterion to determine whether the distributions of these states are super- or sub-Gaussian.

Figure 1

Scheme of quantum memory channel and multimode quadrature measurement. Each input mode ${\widehat{a}}_j$ interacts with the corresponding environment mode ${\widehat{b}}_k$ through a beam splitter with transmittivity $\eta \in [0,1]$, which is the modeled lossy channel. The memory effect in the quantum channel relies on the squeezing parameter $s$ of the multimode squeezed operation. Thus, we can use the parameters $\eta$ and $s$ to characterize the channel dynamics. After a BS array, a probability distribution of a linear superposition of $n$ single-mode quadratures is measured by means of standard single-mode homodyne detection. BS, beam splitter; LO, local oscillator; PD, photodetectors.

Figure 2

Phase-averaged ${\kappa }_4\left({\widehat{X}}^{\mathrm{in}}\right)$ of two-mode quadrature operators in $NNMM$ states as a function of $\mathrm{\Delta }$ for different values of $N$. In each plot the curves from top to bottom are for $N=0,10,20,30,40$. The inset shows ${\kappa }_4\left({\widehat{X}}^{\mathrm{in}}\right)$ for $\mathrm{\Delta }$ ranging between zero and two and for $N=0$ in more detail.

Figure 3

(a) The quantity ${\kappa }_4\left({\widehat{X}}^{\mathrm{out}}\right)$ is plotted versus $s$ with $\eta =0.4$; (b) the quantity ${\kappa }_4\left({\widehat{X}}^{\mathrm{out}}\right)$ is plotted versus $\eta$ with $s=0.1$. The curves from bottom to top correspond to $\mathrm{\Delta }=0,40,60,80,100$, respectively. The values of other parameters are $p=10$ and $N=5$.

Figure 4

The quantity ${\kappa }_4\left({\widehat{X}}^{\mathrm{in}}\right)$ is plotted versus ${\alpha }^2$. The dashed line stands for odd ECS and the solid line for even ECS.

Figure 5

(a) The quantity ${\kappa }_4\left({\widehat{X}}^{\mathrm{out}}\right)$ is plotted versus $s$ with $\eta =0.4$; (b) the quantity ${\kappa }_4\left({\widehat{X}}^{\mathrm{out}}\right)$ is plotted versus $\eta$ with $s=0.8$. In all subfigures, the quantum system is initially in the even ECS. The curves from top to bottom correspond to ${\alpha }^2=(1,2,2.4,2.6)$. The values of other parameters are $p=10$.