Experimentally feasible generation protocol for polarized hybrid entanglement

We propose a scheme to generate polarized hybrid entanglement in the form of $|H\rangle |\alpha \rangle +|V\rangle |-\alpha \rangle$ by interference between a superposition of coherent states and a polarization entangled photon pair. In the scheme two sets of interferences with perpendicular polarizations are designed to erase “which path” information. The joint output fields of the two sets of interferences are used to herald the generation of the polarized hybrid entanglement. The advantage of the protocol is its experimental feasibility since it not only realizes effective interference but also retains the polarization information in the output fields of interferences. We also analyze the effect of imperfect experimental conditions on the generated hybrid state, including inefficiency on-off single-photon detectors with dark counts and available approximate resource states. The theoretical analysis shows that the fidelity of the heralded hybrid state with small amplitude can be maintained at a high level under actual experimental conditions.

Figure 1

Generation schematic diagram for polarized hybrid entanglement. ${|\chi \rangle }_{12}$, polarization entangled photon pairs; $|\mathrm{SC}{\mathrm{S}}_{\phi }\left({\alpha }_i\right){\rangle }_3$, superposition of coherent states; BS1, beam splitter with a small reflectivity; BS2-4, 50/50 beam splitters; PBS, polarized beam splitter; $D_H\left(\sqrt{\frac{R}{2}}{\alpha }_i\right)$ and $D_V\left(\sqrt{\frac{R}{2}}{\alpha }_i\right)$ are displacement operations with H and V polarizations, respectively; $\frac{\lambda }{2}$, half wave plate; SPDs, single-photon detectors.

Figure 2

Success probability of the polarized hybrid state in terms of its amplitude ${\alpha }_f$ and BS1 transmissivity $T$.

Figure 3

(a), (c) Fidelity and (b), (d) success probability as a function of hybrid state size ${\alpha }_f$ and detection efficiency $\eta$. The photon-number-resolving and on-off detectors are used in the upper and lower figures, respectively.

Figure 4

(a) Fidelity and (b) success probability as a function of equivalent input amplitude ${\alpha }_i$ of approximate SCS. A photon-subtracted squeezed state is used to approximate a SCS. The black solid line in (a) represents maximized fidelities between SCSs and photon-subtracted squeezed states. The BS1 transmissivities are 0.99 (red dots), 0.95 (blue triangles), and 0.90 (pink squares), respectively.

Figure 5

Effective fidelity as a function of interaction strength ${\lambda }^2$ for different hybrid state size ${\alpha }_f$ (a) and for different detection efficiency $\eta$ (b). (a) The hybrid state size ${\alpha }_f\approx 0.7$, $1.0$, $1.2$, and $1.5$ for the lines i, ii, iii, and iv, respectively. (b) The detection efficiency $\eta =0.9$, $0.7$, $0.5$, and $0.3$ starting from the top, respectively. The BS1 transmissivity is 0.99 for both (a) and (b).

Figure 6

Effective fidelity as a function of dark count rate for different squeezing parameters and BS1 transmissivities. The detection efficiency is 0.7.