A photonic source of heralded GHZ 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 ($547{\pm}2$ Hz) from a solid-state quantum emitter and a stable polarisation-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.

Introduction.-Quantumentanglement enables the exploration of unique phenomena absent in the classical world, such as non-locality [1][2][3] and teleportation [4][5][6]; and it ultimately provides an advantage to quantum systems over classical ones for various tasks [7,8], ranging from metrology and sensing [9][10][11][12][13], to computation [14][15][16][17].Photonics is among the leading physical platforms for creating entangled quantum systems [18], it exploits non-classical states of light and is divided mainly in two categories: continuous-variable [19,20], and discretevariable approaches [21,22].With advantages and disadvantages from both sides, continuous-variable implementations are highly sensitive to losses, degrading the quality of the quantum state.In this regard, discrete-variable encoding constitutes an appealing alternative, as even in the absence of deterministic entangling gates, there exist loss-tolerant schemes for the generation of large entangled quantum states using only probabilistic, but heralded, linear-optics quantum gate operations [23,24].
Additional approaches exist that aim to generate discrete-variable cluster states [25] directly by exploiting quantum emitters [26][27][28][29][30][31].However, this requires complex control of atomic structures, solid-state materials, and electro-magnetic fields, to name a few technological challenges.In contrast, linear-optics alone also provides a path for the scalable generation of multiphoton cluster states universal for quantum computation [24,32,33].Thereon, a ballistic (without feed-forward requirements) and loss-tolerant (where losses do not induce logical errors) model for universal quantum computing, named fusion-based quantum computation [34], exploits small resource states made up of a handful of entangled particles [35], and combines them into larger entangled states via boosted (heralded, and at the expense of ancillary photons) entangling gates called fusion operations [32].
The smallest building block in these protocols [24] is the heralded three-photon Greenberger-Horne-Zeilinger (GHZ) state [36,37].Creating them requires the quantum interference of six separable single photons [23], or a minimum of ten photons from non-linear frequency conversion processes [38,39].The efficient generation of the necessary input multiphoton states remained to date a major challenge.In this regard, semiconductor quantum dots have recently matured to a point where one handles the interference of single photons at scales of eight particles [40] and even beyond with lost-photons boosted protocols [41], thus now reaching the necessary scales for these more advanced experiments.
Here, we experimentally demonstrate a heralded threephoton polarisation-encoded GHZ state based on the interference of six single photons.We employ a 28.7% fibre-efficient quantum dot single-photon source, actively demultiplexed to produce a source of six indistinguishable photons with 547±2 Hz detected rates.The high quality of the source and interferometric apparatus enable producing heralded three-photon GHZ states at a detection rate of 0.914±0.006Hz, and fidelities up to F=0.7278±0.0106.Our results mark an important step for enabling the realisation of future fusion-based quantum computing protocols.
High-rate multiphoton source.-Wefirst describe our source of multiphoton states.An InAs/GaAs quantum dot coupled to a micropillar cavity is kept in a cryostat at ∼4 K, and is resonantly driven using a standard crossed-polarised excitation scheme.An efficient collection setup allows us to measure 19.5 MHz of single photons with simultaneous purity 1−g 2 (0)=0.981±0.003and indistinguishability [42] I=0.941±0.002,see Fig. 1, when pumped with π-pulses at 80 MHz repetition rate and using a detection system of 85% efficiency, thus corresponding to a 28.7% fibre-efficient single-photon source.
Subsequently, we utilise a time-to-space demultiplexer composed of resonant-enhanced electro-optic modulators (r-EOMs) and polarising beamsplitters (PBSs) arranged in a tree-structure [43,44], see Fig. 2(a).As a result, a number of input time bins separated from each other by 12.5 ns are deterministically routed towards, in this implementation, eight different outputs, which thereon follow suitable fibre-based temporal delays to result in a source of eight indistinguishable single photons travelling simultaneously.Figure 2(b) shows the measured multiphoton coincidence rates using eight superconducting nanowire single-photon detectors (SNSPDs) directly at the output of the demultiplexed source.In particular, the resulting six-photon source is detected at a rate of 547±2 Hz, and the eight-photon source at 15.7±0.4Hz.We note that these are the highest rates of multiphoton sources reported to date.
Heralded entanglement.-Weuse six single photons from this source as input to a polarisation-based interferometer, as depicted in Fig. 3(a), such that the detection of three photons heralds an entangled GHZ state among the other three [23].The six input photons are labelled i 1 , ...i 6 , and are first initialised in horisontal polarisation.When one, and only one, photon propagates towards each of the heralding outputs o 4 , o 5 , and o 6 , then the signal output photons o 1 , o 2 , and o 3 are left in a three-particle entangled state.Note that the successful implementation of this protocol requires that all six photons are highly indistinguishable from each other.To confirm that this is the case, we measure all 15 cases of pair-wise indis- tinguishabilities among the six input photons, and find an average indistinguishability of 0.923±0.009across all combinations, see Fig. 3(b).
The heralded generation of the entangled quantum state requires that no more than one photon is detected at each heralding output.For example, without numberresolution, a pattern with four photons among the three heralding spatial trajectories can not be discerned from another pattern with an exact number of three photons.In such cases, the state produced at signal outputs is not solely the target GHZ state, but it also contains other components with a different number of photons.Therefore, only non-heralded (post-selected) states are generated in the absence of number-resolving detection.Our implementation makes use of pseudo photonnumber resolution (PPNR) at every heralding output, o 4,h , o 4,v , o 5,h , o 5,v , o 6,h , o 6,v , by further splitting each of them into two new outputs and SNSPDs; where h and v denote horisontal and vertical polarisation, respectively.Therefore, we use 18 SNSPDs in total: six detectors to cover both polarisations of the three signal outputs, and twelve detectors for implementing the polarisation and pseudo number-resolved measurement of the three heralding outputs.At each PPNR stage (six in total), we condition a measurement such that one of the detectors clicks and the other one does not, which performs the pseudo number resolution of one, and no more, photon.
The heralded generation of the target state occurs then by imposing that each of the heralding stages measures at most one photon.
Mean value

Entanglement witness.
Measured mean values of the observables forming WGHZ.
The specific 3-GHZ state generated depends on the polarisation pattern that clicks at the heralding outputs.In our experiment, we start by measuring the witness whose negative value verifies the presence of genuine threeparticle entanglement for GHZ states [45].Figure 4 displays the measured mean values of the involved observables, from where we obtain ⟨W GHZ ⟩= − 0.2613±0.0335,confirming three-partite entanglement by more than seven standard deviations.
Moreover, given the high countrates of the available six-photon source, we are also able to perform overcomplete three-qubit quantum state tomography at the signal outputs, with all heralding patterns simultaneously.That is, by using 18 SNSPDs, we reconstruct both |GHZ + ⟩ and |GHZ − ⟩ from the measurement of 3 3 =27 three-qubit observables that result from all combinations of Z, X, Y Pauli matrices among the 3 signal qubits.Figure 5 shows the reconstructed density matrices of the heralded entangled states, from where we extract a fidelity of F + =0.7189±0.0109 to the |GHZ + ⟩ state when using the corresponding four heralding conditions, as well as F − =0.6995±0.0116 to |GHZ − ⟩ by using the respective other four heralding patterns.Note that small terms are present in the imaginary part of the density matrices, showing that the pre- pared states contain a small relative phase between the state basis, which can be compensated for via local unitaries.Taking this into account, we obtain fidelities of F + =0.7278±0.0106and F − =0.7083±0.0120 to the GHZ states |000⟩±e i(0.04×2π) |111⟩ / √ 2, respectively.We measure both heralded states at a combined rate of 0.914±0.006Hz.This value is expected considering: ∼80% throughput efficiency of the polarisation interferometer (affecting six photons), ∼85% throughput efficiency of the pseudo number-resolving detection setup (three photons), and ∼85% throughput given by multiple fibre mating connections (six photons).Together with a 1/32 success probability of producing both GHZ states, results in an expected rate of (547 Hz) ×0.8 6 ×0.85 3 ×0.85 6/32∼1 Hz.
Discussion.-We have experimentally demonstrated a building block for ballistic and all linear-optical photonic quantum computing: the heralded three-photon Greenberger-Horne-Zeilinger state.First, we developed a high-rate (547±2 Hz) source of six photons from a solid-state quantum emitter, with an average pairwise indistinguishability of 0.923±0.009.Subsequently, these photons propagated through a polarisation-based multimode interferometer, where the pseudo numberresolved detection of three of them heralded the generation of three-GHZ states among the remaining particles.Thanks to the high rate of the generated multiphoton source, we were able to perform three-qubit overcom-plete quantum state tomography, reaching fidelities up to F=0.7278±0.0106.
Moreover, the efficient multiphoton source presented here reached an eight-photon rate of 15.7±0.4Hz, readily enabling the implementation of more complex experiments at scales soon beyond ten photons.As a result, we anticipate that near-term further improvements in our source performance will enable a plethora of quantum photonics protocols that remained heretofore out of experimental reach, with particular emphasis on scalable all-optical entangling operations.
This research was funded in whole, or in part, from the European Union's Horizon Note added.-During the writing of this manuscript, we became aware of a similar work [46].

FIG. 2 .
FIG. 2. Multiphoton source.(a) Resonant demultiplexer.Seven synchronised r-EOMs--one driven at 40 MHz, two at 20 MHz, and four at 10 MHz-and polarising beam splitters, deterministically demultiplex eight consecutive time bins.Fibre-based delays and translation stages are used to correct the time bins' initial temporal mismatch.(b) Measured coincidence rates.Multiphoton rates at the output of the demultiplexer, up to a number of n=8 photons.

FIG. 3 .
FIG. 3. Polarisation six-photon interferometer.(a) Depiction of experimental setup.Three pairs of single photons (i1, i6) , (i2, i5) , (i3, i4) probabilistically generate three bell pairs, together with other unwanted terms, after interfering on PBSs 1,2,3.Subsequently, a non-heralded six-photon entangled state, and further unwanted states, is probabilistically generated after PBSs 4,5.The polarisation and number-resolved detection at the output of the type I and type II fusion operation among outputs o4, o5, and o6 corrects for the unwanted terms and leaves the remaining photons at outputs o1, o2, and o3 in a probabilistic but heralded three-photon GHZ state.Quater-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 pair-wise two-photon indistinguishabilities, resulting in an average value of 0.923±0.009across all combinations.

FIG. 5 .
FIG.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 o1, o2, and o3, see Fig.3(a), when the heralding patterns produce either the |GHZ + ⟩ or |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 polarisation projections among the three signal outputs, times four heralding polarisation 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 ×4×2 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 11038 six-fold coincidence counts for the |GHZ + ⟩ state, and 11178 six-folds for |GHZ − ⟩.Accordingly, each heralded state is measured at a rate of 0.454±0.004Hz and 0.460±0.004Hz, respectively, resulting in a total measured rate of 0.914±0.006Hz.
2020 and Horizon Europe research and innovation programme under grant agreement No 899368 (EPIQUS), the Marie Sk lodowska-Curie grant agreement No 956071 (AppQInfo), and the QuantERA II Programme under Grant Agreement No 101017733 (PhoMemtor); from the Austrian Science Fund (FWF) through [F7113] (BeyondC), and [FG5] (Research Group 5); from the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association.For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.