Photonic Source of Heralded Greenberger-Horne-Zeilinger States

Generating large multiphoton entangled states is of main interest due to enabling universal photonic quantum computing and all-optical quantum repeater nodes. These applications exploit measurement-based quantum computation using cluster states. Remarkably, it was shown that photonic cluster states of arbitrary size can be generated by using feasible heralded linear optics fusion gates that act on heralded three-photon Greenberger-Horne-Zeilinger (GHZ) states as the initial resource state. Thus, the capability of generating heralded GHZ states is of great importance for scaling up photonic quantum computing. Here, we experimentally demonstrate this required building block by reporting a polarisation-encoded heralded GHZ state of three photons, for which we build a high-rate six-photon source ($547\pm 2\mathrm{Hz}$) from a solid-state quantum emitter and a stable polarization-based interferometer. The detection of three ancillary photons heralds the generation of three-photon GHZ states among the remaining particles with fidelities up to $\mathcal{F}=0.7278\pm 0.0106$. Our results initiate a path for scalable entangling operations using heralded linear-optics implementations.

Figure 1

Single-photon quality. (a) Normalized second-order autocorrelation function $g^{(2)}(\mathrm{\Delta }t)$ from a Hanbury Brown–Twiss setup, and (b) at the output of a Hong-Ou-Mandel experiment at $\pi$-pulse excitation. We measure the single-photon purity $1-g^{(2)}(0)=0.981\pm 0.003$, and two-photon indistinguishability $\mathcal{I}=0.941\pm 0.002$.

Figure 2

Multiphoton source. (a) Resonant demultiplexer. Seven synchronized r-EOMs—one driven at 40 MHz, two at 20 MHz, and four at 10 MHz—and polarizing beam splitters, deterministically demultiplex eight consecutive time bins. Fiber-based delays and translation stages are used to correct the initial temporal mismatch of the time bins. (b) Measured coincidence rates. Multiphoton rates at the output of the demultiplexer, up to a number of $n=8$ photons.

Figure 3

Polarization six-photon interferometer. (a) Depiction of experimental setup. Three pairs of single photons $(i_1,i_6),(i_2,i_5),(i_3,i_4)$ probabilistically generate three bell pairs, together with other unwanted terms, after interfering on PBSs 1,2,3. Subsequently, a nonheralded six-photon entangled state, and further unwanted states, is probabilistically generated after PBSs 4,5. The polarization and number-resolved detection at the output of the type I and type II fusion operation among outputs $o_4$, $o_5$, and $o_6$ corrects for the unwanted terms and leaves the remaining photons at outputs $o_1$, $o_2$, and $o_3$ in a probabilistic but heralded three-photon GHZ state. Quarter-wave (QWP) and half-wave plates (HWP) together with extra PBSs are used to perform three-qubit quantum state tomography. (b) Photons’ indistinguishability. We use the same six-photon interferometer to measure all 15 pairwise two-photon indistinguishabilities, resulting in an average value of $0.923\pm 0.009$ across all combinations.

Figure 4

Entanglement witness. Measured mean values of the observables forming ${\mathcal{W}}_{\mathrm{GHZ}}$. We obtain $⟨X_1{X}_2{X}_3⟩=0.5629\pm 0.0252$, $⟨Z_1{Z}_2⟩=0.7971\pm 0.0166$, $⟨Z_2{Z}_3⟩=0.8151\pm 0.0191$, and $⟨Z_1{Z}_3⟩=0.7846\pm 0.0199$.

Figure 5

Quantum state tomography of the heralded GHZ states. (a),(b) Real (left) and imaginary part (right) of the reconstructed density matrices at the signal outputs $o_1$, $o_2$, and $o_3$, see Fig. 3, when the heralding patterns produce either the $|{\mathrm{GHZ}}^{+}⟩$ or $|{\mathrm{GHZ}}^{-}⟩$ state. We collect 27 observables for an overcomplete quantum state tomography. To obtain the mean value of a particular observable, we measure simultaneously $2^3=8$ polarization projections among the three signal outputs, times four heralding polarization combinations that produce the same target state, times $2^3=8$ PPNR configurations for number-resolving the three heralding outputs. That is, a total of $2^3\times 4\times 2^3=256$ cases of six-photon coincidence patterns collected simultaneously among 18 output SNSPDs reconstruct a single three-qubit observable of one heralded state. Accordingly, 512 six-photon coincidence patterns are measured simultaneously to obtain the same observable for both heralded states. Each observable is measured for 900 s, which produced density matrices with a total of 11 038 sixfold coincidence counts for the $|{\mathrm{GHZ}}^{+}⟩$ state, and 11 178 sixfolds for $|{\mathrm{GHZ}}^{-}⟩$. Accordingly, each heralded state is measured at a rate of $0.454\pm 0.004$ and $0.460\pm 0.004\mathrm{Hz}$, respectively, resulting in a total measured rate of $0.914\pm 0.006\mathrm{Hz}$.