Quantum non-Gaussianity witnesses in phase space

We address detection of quantum non-Gaussian states, i.e., nonclassical states that cannot be expressed as a convex mixture of Gaussian states, and present a method to derive a new family of criteria based on generic linear functionals. We then specialize this method to derive witnesses based on $s$-parametrized quasiprobability functions, generalizing previous criteria based on the Wigner function. In particular, we discuss in detail and analyze the properties of Husimi $Q$-function-based witnesses and prove that they are often more effective than previous criteria in detecting quantum non-Gaussianity of various kinds of non-Gaussian states evolving in a lossy channel.

Figure 1

We send a non-Gaussian state through a channel with loss $\epsilon$ and choose the detector with which to measure it. The $s=0$ detector corresponds to parity measurement, $s=-1$ corresponds to the probability of vacuum detection, while $s=-2$ corresponds to an inefficient vacuum detection with efficiency $\eta =2/3$.

Figure 2

Maximum value of the loss parameter ${\epsilon }_s^{\left(a\right)}$ such that the bounds are violated, as a function of the initial Fock number $m$ and for different values of $s$: yellow (dot-dashed line), $s=0$; red (dotted line), $s=-1/2$; green (dashed line), $s=-1$; blue (solid line), $s=-2$.

Figure 3

QNG witnesses ${\Delta }_s^{\left(a\right)}$ for initial Fock states (a) $|1\rangle$ and (b) $|3\rangle$ as a function of the lossy parameter $\epsilon$ and for different values of $s$: yellow (solid line), $s=0$; red (dot-dashed line), $s=-1/2$; green (dotted line), $s=-1$; blue (dashed line), $s=-2$.

Figure 4

Maximum value of the loss parameter ${\epsilon }_s^{\left(a\right)}$ such that the bounds are violated, as a function of the coherent-state parameter $\alpha$ and for different values of $s$: red (solid line), $s=0$; green (dotted line), $s=-1$; blue (dashed line), $s=-2$.

Figure 5

Loss parameters for PSS states. (a) Maximum value of the loss parameter ${\epsilon }_s^{\left(a\right)}$ such that the bounds are violated, as a function of the squeezing parameter $r$ and for different values of $s$: red (solid line), $s=0$; green (dotted line), $s=-1$; blue (dashed line), $s=-2$. (b) A comparison between the maximum values of the noise parameter ${\epsilon }_s^{\left(b\right)}$ for PSS states by using the Wigner function criterion (blue solid line) and the $Q$-function criterion (red dashed line).

Figure 6

Error on the (renormalized) bounding functions $B_s\left(n\right)$ for (from top to bottom) $s=0$ (blue line), $s=-1$ (green line), and $s=-2$ (red line).