Entanglement witnesses and measures for bright squeezed vacuum

Quantum entanglement is a fascinating phenomenon, especially if it is observed at the macroscopic scale. Importantly, macroscopic quantum correlations can be revealed only by accurate measurement outcomes and strategies. Here, we formulate feasible entanglement witnesses for bright squeezed vacuum in the form of the macroscopically populated polarization triplet Bell states. Their testing involves efficient photodetection and the measurement of the Stokes operators' variances. We also calculate the measures of entanglement for these states such as the Schmidt number and the logarithmic negativity. Our results show that the bright squeezed vacuum degree of polarization entanglement scales as the mean photon number squared. We analyze the applicability of an operational analog of the Schmidt number.

Figure 1

Experimental setup for testing the entanglement witnesses, Eqs. (9, 10, 11). In each beam, there is a Stokes measurement setup: DM, dichroic mirror; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; D, detectors. The signals from the two detectors in each arm are subtracted to obtain the Stokes observables.

Figure 2

Photon-number distributions for the states $|{\Psi }_{1,2}^{\pm }\rangle$.

Figure 3

The degree of polarization entanglement in the logarithmic scale for the four-mode entangled BSV state as a function of the average photon number $N_0$: the negativity $\mathcal{N}$ (solid line), the effective Schmidt number $\overline{K}$ (dashed line), and the Fedorov ratio $R_n$ (dotted line).

Figure 4

${\overline{K}}^T/d$ as a function of $\epsilon$. The solid line is for $N_0=10$, the dotted line is for $N_0=1$, and for large $N_0$ ($>$5) the curve does not change with $N_0$.