Heralded preparation of polarization entanglement via quantum scissors

Quantum entanglement is at the heart of quantum information sciences and quantum technologies. In the optical domain, the most common type of quantum entanglement is polarization entanglement, which is usually created in a postselection manner involving destructive photon detection and thus hindering further applications which require readily available entanglement resources. In this work, we propose a scheme to prepare multipartite entangled states of polarized photons in a heralded manner, i.e., without postselection. We exploit the quantum scissors technique to truncate a given continuous-variable entanglement into the target entangled states which are of hybrid discrete-continuous or solely discrete types. We consider two implementations of the quantum scissors: one modified from the original quantum scissors [Pegg et al., Phys. Rev. Lett. 81, 1604 (1998)] using single photons and linear optics and the other designed here using a type-II two-mode squeezer. We clarify the pros and cons of these two implementations as well as discussing practical aspects of the entanglement preparation. Our work illustrates an interface between various types of optical entanglement and the proposed quantum scissors techniques could serve as alternative methods for heralded generation of polarization entanglement.

Figure 1

(a) Schematic setup to prepare the CV polarization entanglement ${|\mathrm{\Xi }\rangle }_{12}$ in Eq. (1). Here $|{\mathrm{Cat}}_H\rangle =N_0\left(\right|{\delta }_H\rangle +e^{i\phi }|-{\delta }_H\rangle )$ with $N_0={\left[2(1+cos\left(\phi \right)e^{-2{\delta }^2})\right]}^{1/2}$, BBS is balanced beam splitter, PBS polarizing beam splitter, $\mathrm{BS}\left(t_0\right)$ beam splitter with transmissivity $t_0$, and HWP half-wave plate. (b) Splitting of mode 2 in the state ${|\mathrm{\Xi }\rangle }_{12}$ by $\mathrm{BS}\left(t_1\right)$ prepares the tripartite CV polarization entanglement ${|\mathrm{\Lambda }\rangle }_{12\cdots n}$ with $n=3$ in Eq. (7).

Figure 2

(a) Black-box representation with the label $\mathrm{QS}\left(t\right)$ and physical setup of the quantum scissors proposed in Refs. [51, 55]. Here $t$ denotes the transmissitivity of the BS, which characterizes the performance of the scissors. The input $|\psi \rangle ={\sum }_{n=0}^{\infty }{c}_n|n\rangle$, going through the $\mathrm{QS}\left(t\right)$, is truncated into the (unnormalized) output state $\sqrt{1-t}{c}_0|0\rangle \pm \sqrt{t}{c}_1|1\rangle$, conditioned by detection of a single photon at detector D1 and no photons at detector D2 or no photons at detector D1 and a single photon at detector D2. Choosing $t$ properly this output can be made very close to a single-photon state. (b) Linear-optics-based implementation of a nonideal PQS with the black-box labeled as $\mathrm{PQS}1\left(t\right)$, comprising two modules of the quantum scissors in panel (a). The $\mathrm{PQS}1\left(t\right)$ truncates a general polarized state $|{\psi }_{\mathrm{p}}\rangle ={\sum }_{n,m=0}^{\infty }{c}_{nm}|n_H,m_V\rangle$ into an output state that is very close to the (unnormalized) state $|{\psi }_{\mathrm{p}}^{\left(1\right)}\rangle =c_{10}|1_H,0_V\rangle +c_{01}|1_V,0_H\rangle$. The box with the label ${\mathrm{QS}}_H\left(t\right)$ (${\mathrm{QS}}_V\left(t\right)$) represents the physical setup in panel (a) with the ancilla single photon being horizontally (vertically) polarized.

Figure 3

Two-mode-squeezer-based implementation of a nonideal PQS with the black box labeled as $\mathrm{PQS}2\left(\mathrm{\Gamma }\right)$ using a type-II two-mode squeezer represented by a SPDC crystal [58]. Here $\mathrm{\Gamma }$ is the squeezing parameter characterizing the performance of the scissors. The input state $|{\psi }_{\mathrm{p}}\rangle ={\sum }_{n,m=0}^{\infty }{c}_{nm}|n_H,m_V\rangle$ is injected into the signal ($s$) mode of the squeezer while the idle ($i$) mode is in the vacuum. Co-detection of single photons at both detectors D1 and D2 heralds the truncated output at the idle mode which is very close to the desired state $|{\psi }_{\mathrm{p}}^{\left(1\right)}\rangle =c_{10}|1_H,0_V\rangle +c_{01}|1_V,0_H\rangle$. A pump, denoted by $p$, stimulates squeezing in the SPDC crystal.

Figure 4

(a) Applying the PQS on $j$ modes of the CV entanglement ${|\mathrm{\Lambda }\rangle }_{12\cdots n}$ in Eq. (7) produces the entanglement ${|\mathrm{\Omega }\rangle }_{12\cdots n}$ in Eq. (9). For $n=2$ and $j=1$, panel (a) reduces to panel (b), which shows the preparation of the hybrid entanglement $|{\mathrm{\Phi }}^{\mathrm{DV}\text{−}\mathrm{CV}}\rangle$ in Eq. (39) from the CV entanglement ${|\mathrm{\Xi }\rangle }_{12}$ in Eq. (1). For $n=2$ and $j=2$, panel (a) reduces to panel (c) which shows the preparation of the DV PBP $|{\mathrm{\Phi }}^{\mathrm{DV}\text{−}\mathrm{DV}}\rangle$ in Eq. (40) from the CV entanglement ${|\mathrm{\Xi }\rangle }_{12}$. Here the PQS can be realized by either the PQS1 or the PQS2.

Figure 5

(a), (b) Success probability $P_{\mathrm{PQS}1}^{\mathrm{DV}\text{−}\mathrm{CV}}$ and fidelity $F_{\mathrm{PQS}1}^{\mathrm{DV}\text{−}\mathrm{CV}}$ given in Eqs. (44) and (45), respectively, as functions of the input amplitude $\delta$ and the transmissivity $t$ for the preparation of the hybrid entanglement $|{\mathrm{\Phi }}^{\mathrm{DV}\text{−}\mathrm{CV}}\rangle$ in Eq. (39) using the PQS1 developed in Sec. 3a. (c), (d) Success probability $P_{\mathrm{PQS}2}^{\mathrm{DV}\text{−}\mathrm{CV}}$ and fidelity $F_{\mathrm{PQS}2}^{\mathrm{DV}\text{−}\mathrm{CV}}$ given in Eqs. (47) and (48), respectively, as functions of the input amplitude $\delta$ and the squeezing parameter $\left|\mathrm{\Gamma }\right|$ for the preparation of the same entanglement but using the PQS2 developed in Sec. 3b. In plotting these figures $\varphi =0$ and $t_0=0.5$ were chosen.

Figure 6

(a), (b) Success probability $P_{\mathrm{PQS}1}^{\mathrm{DV}\text{−}\mathrm{DV}}$ and fidelity $F_{\mathrm{PQS}1}^{\mathrm{DV}\text{−}\mathrm{DV}}$ given in Appendix as functions of the input amplitude $\delta$ and the transmissivity $t$ for the preparation of the PBP $|{\mathrm{\Phi }}^{\mathrm{DV}\text{−}\mathrm{DV}}\rangle$ in Eq. (40) using the PQS1 developed in Sec. 3a. (c), (d) Success probability $P_{\mathrm{PQS}2}^{\mathrm{DV}\text{−}\mathrm{DV}}$ and fidelity $F_{\mathrm{PQS}2}^{\mathrm{DV}\text{−}\mathrm{DV}}$ given in Appendix as functions of the input amplitude $\delta$ and the squeezing parameter $\left|\mathrm{\Gamma }\right|$ for the preparation of the same entanglement but using the PQS2 developed in Sec. 3b. In plotting these figures $\varphi =0$ and $t_0=0.5$ were chosen.

Figure 7

(a) Reduction of the number of photodetectors in the PQS1 implementation. In the left, two spatially separated photons in modes 1 and 2 of the same horizontal $\left(H\right)$ polarization are detected by two detectors D1 and D2. In the right, employing a HWP, a delay circle, a mirror, and a PBS makes the two photons detected by only one detector D but at adjustably separated arrival times. (b) Similar to (a) but for the PQS2 implementation with two detected photons initially having orthogonal polarizations.