Observation of quantum nonlocality in Greenberger-Horne-Zeilinger entanglement on a silicon chip

Nonlocality is the defining feature of quantum entanglement. Entangled states with multiple particles are of crucial importance in fundamental tests of quantum physics as well as in many quantum information tasks. One of the archetypal multipartite quantum states, Greenberger-Horne-Zeilinger (GHZ) state, allows one to observe the striking conflict of quantum physics to local realism in the so-called all-versus-nothing way. This is profoundly different from Bell's theorem for two particles, which relies on statistical predictions. Here, we demonstrate an integrated photonic chip capable of generating and manipulating the four-photon GHZ state. We perform a complete characterization of the four-photon GHZ state using quantum state tomography and obtain a state fidelity of 0.729(6). We further use the all-versus-nothing test and the Mermin inequalities to witness the quantum nonlocality of GHZ entanglement. Our work paves the way to perform fundamental tests of quantum physics with complex integrated quantum devices.

Silicon-based integrated photonic circuits promise desirable properties for photonic quantum technology.It offers dense device integration, high optical nonlinearity, and robust phase stability.Some recent works have demonstrated the ability to generate complex and programmable multi-photon quantum states with integrated optics [43][44][45][46].To study the fundamental quantum physics in more complex systems, it is crucial to realize high-fidelity operations in photonic circuitries with increasing complexity, and a larger number of photons with high dimensions.However, few of the previous demonstrations studied the quantum nonlocality of GHZ states with integrated photonics, such as all-versus-nothing nonlocal trait and Mermin conflict between quantum mechanics and local realism.They are mainly based on bulk optical setups with polarization-entangled photons [21,23,26] or photons with orbital-angular-momentum modes [42].It is therefore of crucial importance to investigate whether quantum nonlocality can be produced and verified with the scalable quantum photonic chips.In this work, we go beyond the on-chip Bell experiments with two photons, and generate and employ the three-and four-photon GHZ states to observe quantum nonlocality.We use a fully-integrated silicon chip, including path-entangled quantum light modules with advanced dual-Mach-Zehnder interferometer micro-ring sources [47][48][49], wavelength routing, quantum state parity sorter as well as single-photon tomographic modules, to generate, manipulate, and topographically reconstruct the GHZ states.Moreover, we measure the quantum nonlocality of GHZ entanglement via both the all-versus-nothing (AVN) test [19,23] and the Mermin inequalities [50,51], showing the striking conflict of quantum physics to local realism in the multiphoton scenario.Photons are encoded along the path mode of waveguides, which can be flexibly extended to high dimensions and potentially useful for future large-scale quantum information processing and quantum networks.It should be noted that a recent work by M. Pont et al. [30] has reported a powerful but different platform of a solid-state quantum-dot photon source and glass-based photonic circuits to generate high-quality GHZ entanglement and certify its quantum nonlocality via a Bell-like inequality.

Experimental setup
The integrated photonic circuit consists of two path-entangled photon-pair source modules (S1 and S2), an on-chip path-mode parity sorter (PPS), and four qubit analyzers [Fig.1].A pulsed laser is grating-coupled into the chip and split with a beam spliter (BS).Then, S1 and S2 are coherently pumped by the laser, and each emits entangled photon pairs in the Bell state, Note that |0⟩ and |1⟩ stand for different path modes.In each path-entangled photon-pair source module, we use two integrated micro-ring photon-pair sources to emit path-encoded photon pairs and asymmetric Mach-Zehnder interferometers (AMZIs) to separate them according to different wavelengths of signal and idler photons.Hence, there are four micro-rings on the chip and they are all coherently pumped to provide photon pairs for producing the four-photon GHZ state.Then, a Mach-Zehnder interferometer (MZI) and two waveguides constitute a controllable PPS to combine and interfere photon B and C.After evolution of photons at the PPS, the simultaneous detection of photons in each output A-D gives the output four-photon GHZ state (see Appendix A for a detailed description on GHZ-entanglement generation), Note that |0000⟩ and |1111⟩ stand for respectively.The post-selection probability to observe the GHZ state is 0.25 from all events of four-photon generation including the possibility that one source module S1 (or S2) generates two pairs of photons.Then, four path-mode qubit analyzers with MZIs and phase shifters are used to measure the post-selected multi-photon states.After that, all photons are coupled out with grating couplers, filtered with off-chip filters, and detected with superconducting nanowire single-photon detectors (SNSPDs).The single-photon detection events are recorded by an FPGA-based coincidence logic unit which further calculates and outputs both two-and four-fold coincidence counts between different path modes.A pulsed laser is filtered, polarized and grating-coupled into the chip.Then, the pulse coherently pumps two path-entangled photon-pair source modules, S1 and S2, which emit path-entangled photon pairs with different wavelengths (1544.5 nm and 1556.5 nm).In each source module, two micro-rings generate photon-pairs, and asymmetric Mach-Zehnder interferometers (AMZIs) separate photons.Then, on-chip path-mode parity sorter (PPS) is used to combine and sort the signal photons B and C according to their path modes.After that, photons are distributed to four qubit analyzers, coupled out, filtered, and finally detected by superconducting nanowire single-photon detectors (SNSPDs).Simultaneous detection of one photon in each outport heralds the generation of four-photon GHZ state.All on-chip phase shifters and micro-rings are controlled with a programmable multi-channel power supply.

Results
The silicon photonic chip was manufactured at the Advanced Micro Foundry (AMF), with 500 nm×220 nm fully-etched SOI waveguides and TiN heaters placed 2 µm above waveguides.We use a picosecond pulsed pump laser of ∼ 4 mW power, 60 MHz repetition rate, and 1550.5 nm wavelength with ∼ 0.8 nm bandwidth.The laser is coupled into the chip by grating couplers with a coupling loss of ∼ 5 dB/facet.After the laser propagating on chip and splitting with MZIs, four micro-ring sources (R1, R2 in S1 and R3, R4 in S2) are coherently pumped with ∼ 0.2 mW power on each micro-ring.Then, photon pairs with different wavelengths (1544.5 nm and 1556.5 nm) are generated via spontaneous four wave mixing (SFWM) process and routed according to their wavelengths with AMZIs.The measured pair-generation rate of each micro-ring ranges from 1400 Hz to 2400 Hz, and the four-fold coincidence count rate for the four-photon GHZ state is 380 ± 32 per hour.First, we verify the fidelity of the two-photon Bell state |Ψ 2 ⟩ generated from each path-entangled photon-pair source module (S1 or S2) by measuring the path-correlation in mutually unbiased bases (MUBs) [52].We project idler photons into the basis 1/ deviation of visibilities is due to multi-pair generation events and the spectral distinguishability of photons from different micro-rings.From the relationship between fidelity and visibility in a Werner state [53],  = (1 + 3)/4, the estimated fidelities for two-photon entangled states are 0.952 ± 0.002 for S1 and 0.953 ± 0.002 for S2, indicating high quality of qubit-entanglement of our sources.
Next, we investigate the indistinguishability between source modules S1 and S2 via four-photon heralded Hong-Ou-Mandel (HOM) interference [54].In this case, signal photons from S1 and S2 are combined and interfered by a tunable MZI in the PPS.By scanning the splitting ratio of the MZI, we obtain a visibility of 0.814 ± 0.038 from the interference fringe [Fig.2(c)].Here, the visibility of heralded-HOM fringe is defined as    = (  /2 −   ) /(  /2) (see Appendix C for details).The reduced visibilities are mainly due to the large accidentals from multi-pair generation events and the spectral impurity of micro-ring photon sources.
Having established the high-quality GHZ entanglement, we first demonstrate its quantum nonlocality via the AVN test.The AVN test is based on perfect correlations in multipartite systems and offers a logical contradiction between local-hidden-variable models and quantum mechanics directly without any inequality [19,21,23].The AVN test for 4-photon GHZ state is based on the following equations: = −1.
Note that  i ( i ) denotes the value, +1 or -1, obtained when measuring  ()  ( ()  ) on photon i.Quantum mechanics predicts that all equations can be satisfied, while local realistic theories can reproduce only 6 out of 8 predictions [Fig3(a), (b)].For example, when we take the products of the equations (3), ( 5), (6), and (7), each value  i ( i ) appears twice while the product on the right is -1, which causes the AVN contradiction.For the fact that we cannot get perfect 1 or -1 due to experimental noise, a certain bound, error rate , is introduced to identify the nonlocality [26,57].The error rate , which describes the degree to which we get the wrong events, gives quantitative estimates of the inconsistency between quantum mechanics and local realism.To get the AVN contradiction, the error rate  needs to be smaller than 1/4 (the bound for 4-photon and 3-photon GHZ states).Experimentally, the outcomes of measurement settings in Eqs.(3)(4)(5)(6)(7)(8)(9)(10) are shown in Fig. 3(c) with the average error rate ε = 0.191 ± 0.021 and the largest error rate  max = 0.218 ± 0.021, revealing the violation of local realism.Error rate for 3-photon GHZ states is also below the bound of 1/4, showing in Tab.1.
Mermin inequality is another witness for GHZ nonlocality, using the expression below, For four-photon GHZ states, quantum mechanics predicts the maximum possible value of 8

Discussion and Conclusion
In this work, we have presented a silicon photonic chip for generating, manipulating, and analyzing multi-photon GHZ states.Advanced micro-ring sources provide high purity of photons without extra spectral filtering.Thus, high visibilities of path-correlation and heralded-HOM interference fringes are obtained.To witness the quantum nonlocality, we have shown the first AVN test and the Mermin inequality on an integrated photonic chip with both four-and three-photon GHZ states, demonstrating the reliability of current photonic integration techniques for testing fundamental properties of quantum systems.In contrast to the previous demonstrations of AVN and Mermin tests which are mainly based on photons generated from nonlinear bulk crystals [21,23,26,42], the same spatial mode of single-mode waveguides and good interference easily-performed on MMIs guarantee high quality of path-entangled photons on chip.It makes path-mode encoding be an ideal degree of freedom which can be easily extended to high dimensions [46,58,59].Besides, chip-based GHZ-entanglement devices with photons at telecom band can easily interface with fiber quantum network, showing potential for applications in multipartite quantum communication, such as quantum cryptographic conferencing [60][61][62][63][64] and quantum secret sharing [65][66][67][68][69].Our work paves the way for employing integrated quantum

Fig. 1 .
Fig.1.Experimental setup for generating and manipulating the four-photon GHZ state.A pulsed laser is filtered, polarized and grating-coupled into the chip.Then, the pulse coherently pumps two path-entangled photon-pair source modules, S1 and S2, which emit path-entangled photon pairs with different wavelengths (1544.5 nm and 1556.5 nm).In each source module, two micro-rings generate photon-pairs, and asymmetric Mach-Zehnder interferometers (AMZIs) separate photons.Then, on-chip path-mode parity sorter (PPS) is used to combine and sort the signal photons B and C according to their path modes.After that, photons are distributed to four qubit analyzers, coupled out, filtered, and finally detected by superconducting nanowire single-photon detectors (SNSPDs).Simultaneous detection of one photon in each outport heralds the generation of four-photon GHZ state.All on-chip phase shifters and micro-rings are controlled with a programmable multi-channel power supply.

√ 2 (√ 2 Fig. 2 .
Fig. 2. Experimental results of four-photon GHZ state.(a), (b) Two-photon interference fringes of source modules S1 and S2, indicating high-quality path correlation of qubit entanglement.The measured path-correlation visibilities are 0.935 ± 0.004 and 0.938 ± 0.003, respectively.(c) Heralded-HOM interference between micro-rings R1 and R3, showing indistinguishability of signal photons from S1 and S2.The heralded-HOM visibility is 0.814 ± 0.038.(d), (e) Real and imaginary part of the density matrices of four-photon GHZ states.Experimental results are shown with colored bars.Theoretical predictions are shown with the wire grid.The quantum-state fidelity is 0.729 ± 0.006.Uncertainties are obtained from Monte Carlo simulations with Poisson counting statistics.

√ 2 ,
while the bound of local realism is 4. Note that the maximum violation is reached when the GHZ state is rotated with a phase and into the form: |0000⟩ + e 3 /4 |1111⟩.The measured Mermin value of our experiment is ⟨ 4 ⟩ = 6.98 ± 0.16, showing a violation of local realism by 18.7 standard deviations.Projecting one photon in mode 1/ √ 2 (|0⟩ + |1⟩), we get 3-photon GHZ states and measure 3-photon Mermin inequalities by using  3 = |   − ( +   +  )|, where the quantum value is 4 while the classical bound is 2. The average result for 3-photon GHZ states is ⟨ 3 ⟩ = 2.50 ± 0.13 with 3.8 standard deviations violating the Mermin inequality [Tab.1].

Fig. 3 .
Fig. 3. All-versus-nothing test for four-photon GHZ states.(a) Predictions of quantum mechanics are based on perfect correlations of GHZ states.(b) Local realism contradict quantum mechanics on at least two predictions.(c) Experimental results are in agreement with the quantum mechanics and in conflict with the local realism.