Non-Gaussianity and entropy-bounded uncertainty relations: Application to detection of non-Gaussian entangled states

We suggest an improved version of the Robertson–Schrödinger uncertainty relation for canonically conjugate variables by taking into account a pair of characteristics of states: non-Gaussianity and mixedness quantified by using fidelity and entropy, respectively. This relation is saturated by both Gaussian and Fock states and provides a strictly improved bound for any non-Gaussian states or mixed states. For the case of Gaussian states, it is reduced to the entropy-bounded uncertainty relation derived by Dodonov. Furthermore, we consider readily computable measures of both characteristics and find a weaker but more readily accessible bound. With its generalization to the case of two-mode states, we show applicability of the relation to detect entanglement of non-Gaussian states.

Figure 1

Plots of $\sqrt{detV}$ and the bounds of the NE, PG, and RS uncertainty relations for (a) even and (b) odd cat states against the amplitude $\left|\alpha \right|$.

Figure 2

Plot of $\sqrt{detV}$ and the bounds of the NE, PG, and RS uncertainty relations for photon-added coherent states $|{\psi }_{\mathrm{pacs}}\rangle$ against $\left|\alpha \right|$.

Figure 3

Enhanced restrictions on symplectic eigenvalues $\nu +\ge {\nu }_{-}$ according to the bound in inequality (15) denoted by $B=S\left({\rho }^{AB}\right)+N\left({\rho }^{AB}\right)$. With the increase of $B$ from 0 to 5 distinguished by the contrast of blue, available regions of ${\nu }_{\pm }$ become more confined.

Figure 4

Plots of entanglement conditions for (a) $\alpha =0$ and (b) $\alpha =1$ detected by the Simon–Duan (light blue) and the non-Gaussianity–based criteria (dashed blue and red) (24) against the squeezing parameter $r$ and mean photon number $\overline{n}$ of a thermal state ${\widehat{\tau }}^B\left(\overline{n}\right)$.