Remote preparation of three-photon entangled states via single-photon measurement

Remote state preparation (RSP) provides an indirect way of transferring quantum information based on the nonlocal effect of quantum measurement. Although RSP has been demonstrated in recent years to remotely prepare multiphoton states, quantum measurement on the same number of additional photons was required, i.e., to prepare $N$-photon states via RSP, quantum measurement on the other $N$-photons was required, hence significantly limiting practicality and applicability of RSP. Here we report an experimental demonstration of remote preparation of three-photon entangled states by measuring only a single-photon entangled with the three photons. We further generalize our protocol to prepare multiphoton entangled states with arbitrary photon number and purity and to prepare a genuinely three-partite entangled state, both via single-photon measurement. As our RSP scheme relies on the nonlinearity induced by single-photon measurement, it enables quantum state engineering of multiphoton entangled states beyond the linear optical limit. Our results are expected to have significant impacts on quantum metrology and quantum information processing.

Figure 1

Experimental setup. At the input of a nonpolarizing beam splitter ${\mathrm{BS}}_1$ (T, 80%, R, 20%), the quantum state of four photons is $|2_H,2_V\rangle$, and when a single photon is reflected and the other three photons are transmitted, Alice and Bob share the entangled state in Eq. (1). Alice measures the single photon by projection onto $cos2\gamma |1_H,0_V{\rangle }_A+e^{i\theta }sin2\gamma {|0_H,1_V\rangle }_A$ using a phase shifter ${\mathrm{PS}}_1\left(\theta \right)$, a half-wave plate ${\mathrm{H}}_2\left(\gamma \right)$, a polarizing beam splitter ${\mathrm{PBS}}_2$, and a single-photon avalanche detector ${\mathrm{D}}_1$ (Perkin-Elmer SPCM-AQRH-13). As a consequence, the three-photon entangled state in Eq. (2) is prepared at Bob, and he measures the three photons by projection onto $cos2\delta |2_H,1_V{\rangle }_B+e^{i\varphi }sin2\delta {|1_H,2_V\rangle }_B$ using ${\mathrm{BS}}_2,{\mathrm{PS}}_2\left(\varphi \right),{\mathrm{H}}_3\left(\delta \right),{\mathrm{PBS}}_3,{\mathrm{PBS}}_4$, and coincidence detection on ${\mathrm{D}}_2,{\mathrm{D}}_3$, and ${\mathrm{D}}_4$. RSP of a three-photon mixed state [Eq. (9)] requires the use of a partial polarizer (PP) instead of ${\mathrm{PBS}}_2$ at Alice [14]. The phase shifters ${\mathrm{PS}}_1\left(\theta \right)$ and ${\mathrm{PS}}_2\left(\varphi \right)$ are implemented by rotating a half-wave plate between two quarter-wave plates set at the angle of $\pi /4$. BBO, $\beta \text{−}{\mathrm{BaB}}_2{\mathrm{O}}_4$ crystal; $s$, signal; $i$, idler; IF, interference filter; SMF, single-mode fiber.

Figure 2

Measurement of remotely prepared three-photon states. (a)–(c) Remotely prepared three-photon entangled states having different phases, in Eq. (6), are measured by projection onto $\frac{1}{\sqrt{2}}{\left(\right|2_H,1_V\rangle }_B+e^{i\varphi }|1_H,2_V{\rangle }_B)$. (d)–(f) Remotely prepared three-photon states having different amplitudes, in Eq. (7), are measured by projection onto $cos2\delta |2_H,1_V{\rangle }_B+sin2\delta {|1_H,2_V\rangle }_B$. Black dots are experimental data, and red solid lines are sinusoidal fittings to the experimental data. Error bars represent one standard deviation by assuming Poissonian counting statistics. Visibilities of the fittings are (a) $93.8\pm 1.7\%$, (b) $97.8\pm 3.1\%$, (c) $97.4\pm 1.6\%$, (d) $90.8\pm 1.3\%$, (e) $95.7\pm 1.0\%$, and (f) $92.7\pm 1.9\%$.

Figure 3

Contributions of $|3_H,0_V{\rangle }_B,|2_H,1_V{\rangle }_B,|1_H,2_V{\rangle }_B$, and $|0_H,3_V{\rangle }_B$ in experimentally generated three-photon states. The maximum contribution from $|3_H,0_V{\rangle }_B$ and $|0_H,3_V{\rangle }_B$ is below 0.03. This result also justifies the choice the Hilbert space dimension $2\otimes 2$ as contributions from $|3_H,0_V{\rangle }_B$ and $|0_H,3_V{\rangle }_B$ are negligible. Error bars represent one standard deviation, estimated from Poissonian statistics of obtained coincidence counts.