Displaced photon-number entanglement tests

Based on correlations of coherently displaced photon numbers, we derive entanglement criteria for the purpose of verifying non-Gaussian entanglement. Our construction method enables us to verify bipartite and multipartite entanglement of complex states of light. An important advantage of our technique is that the certified entanglement persists even in the presence of arbitrarily high, constant losses. We exploit experimental correlation schemes for the two-mode and multimode scenarios, which allow us to directly measure the desired observables. To detect entanglement of a given state, a genetic algorithm is applied to optimize over the infinite set of our constructed witnesses. In particular, we provide suitable witnesses for several distinct two-mode states. Moreover, a mixed non-Gaussian four-mode state is shown to be entangled in all possible nontrivial partitions.

Figure 1

Outline of an experimental scheme to measure $\langle \widehat{L}\rangle$. The two modes of the incident light field, labeled “a” and “b,” are displaced, which is indicated by the displacement operators $\widehat{D}\left(\alpha \right)$ and $\widehat{D}\left(\beta \right)$. The desired coherent displacements ${\left\{({\alpha }_k,{\beta }_k)\right\}}_{k=1,\cdots ,m}$ are randomly realized according to the distribution $p(\alpha ,\beta )$, which is defined through the discrete probabilities ${\left\{{\lambda }_k\right\}}_{k=1,...,m}$.

Figure 2

Inseparability criteria of (a) Simon and (b) Duan et al. for state (25), with $\epsilon \in \mathbb{C}$ and $0<\left|\epsilon \right|\le 1$. The negative region, bounded by the dashed red curve, successfully probes the entanglement. For nonnegative values, such as for the example $\epsilon =e^{i(3/4)\pi }$ (blue circle), no entanglement is identified by these criteria.

Figure 3

Displacements ${\alpha }_k$ (larger, orange circles) and ${\beta }_k$ (smaller, blue circles) with $k=1,2,3$ for witnessing entanglement of state (25) are shown in the complex plane. Re and Im denote the real and imaginary axes, respectively. Small filled red circles depict $\left\{Q_1,Q_2,Q_3\right\}$.

Figure 4

Possible displacements ${\alpha }_k$ (larger, orange circles) and ${\beta }_k$ (smaller, blue circles) with $k=1,2,3$ for witnessing entanglement of the two-mode squeezed-vacuum state $|\xi \rangle$ are shown in the complex plane. Re and Im denote the real and imaginary axes, respectively.

Figure 5

(a) The critical radius $r_{\mathrm{crit}}$ (solid blue line) [Eq. (33)] as a function of the amount of squeezing $\xi$. The shaded area corresponds to possible radii which successfully certify entanglement. The dashed orange line shows the radius $r_{max}$ [Eq. (34)] for which $R$ [Eq. (35)] is maximal. (b) $R$ on a logarithmic scale for $r_{max}\left(\xi \right)$.

Figure 6

Experimental setup to measure the observable, (43), for the example of the tripartition $\{1,2,3\}:\left\{4\right\}:\{5,6\}$. Compare also to the bipartite case in Fig. 1. The displacements of the individual modes are randomly realized. The measurement of the resulting displaced photon-number statistics is combined in a weighted sum for each subsystem—using the weights $q^{\left(1\right)},...,q^{\left(6\right)}$—and is then correlated.

Figure 7

Expectation value $\langle L\rangle$ for different $K$ partitions, ${\mathcal{I}}^{\left(1\right)}:...:{\mathcal{I}}^{\left(K\right)}$ as a function of the noise parameter $\sigma$ (solid blue curves). (a) $\left\{1\right\}:\left\{2\right\}:\left\{3\right\}:\left\{4\right\}$; (b) $\left\{1\right\}:\{2,3\}:\left\{4\right\}$; (c) $\{1,2\}:\{3,4\}$; (d) $\left\{1\right\}:\{2,3,4\}$. The bounds for $K$ separability, $g_{min}^{{\mathcal{I}}^{\left(1\right)}:\cdots :{\mathcal{I}}^{\left(K\right)}}$, are represented as dashed orange lines. Entanglement is verified for $\langle L\rangle <g_{min}^{{\mathcal{I}}^{\left(1\right)}:...:{\mathcal{I}}^{\left(K\right)}}$; see entanglement condition (44).