Abstract
Entanglement is a counterintuitive feature of quantum physics that is at the heart of quantum technology. Highdimensional quantum states offer unique advantages in various quantum information tasks. Integrated photonic chips have recently emerged as a leading platform for the generation, manipulation and detection of entangled photons. Here, we report a silicon photonic chip that uses interferometric resonanceenhanced photonpair sources, spectral demultiplexers and highdimensional reconfigurable circuitries to generate, manipulate and analyse pathentangled threedimensional qutrit states. By minimizing onchip electrical and thermal crosstalk, we obtain highquality quantum interference with visibilities above 96.5% and a maximally entangledqutrit state with a fidelity of 95.5%. We further explore the fundamental properties of entangled qutrits to test quantum nonlocality and contextuality, and to implement quantum simulations of graphs and highprecision optical phase measurements. Our work paves the path for the development of multiphoton highdimensional quantum technologies.
Introduction
Entanglement is a central resource for quantumenhanced technology, including quantum computation^{1}, communication^{2} and metrology^{3}. To demonstrate the advantage of quantum systems, it is necessary to generate, manipulate and detect entangled states. Quantum states span the Hilbert space with a dimensionality of d^{n}, where d is the dimensionality of a single particle and n is the number of particles in the entangled states. Most of the widely used quantum information processing protocols are based on qubits, a quantum system with d = 2. Recently, higherdimensional entangled states (qudits, d > 2) have gained substantial interest, owing to their distinguishing properties. For example, qudits provide larger channel capacity and better noise tolerance in quantum communication^{4,5,6,7}, as well as higher efficiency and flexibility in quantum computing^{8,9} and simulations^{10}. From the fundamental point of view, qudits also provide stronger violations of Bell inequalities^{11}, lower bounds for closing the fairsampling loopholes in Bell tests^{12} and possibilities to test contextuality^{13}. Recent reviews on the highdimensional entanglement can be found in refs ^{14,15}.
Highdimensional entangled photons have been realized in various degrees of freedom (DOFs), including orbital angular momentum (OAM)^{11,16}, frequency^{17,18}, path^{19,20}, temporal^{21,22} and hybrid timefrequency modes^{23}. In particular, pathentangled photon pairs have been studied with a view to quantum information processing, where they are particularly attractive due to their conceptual simplicity^{24}. However, the generation of highdimensional pathentangled photon pairs typically requires the simultaneous operation of several coherently pumped indistinguishable photonpair sources and several multipath interferometers with high phase stability^{25}. As the dimensionality increases, the phase stabilization quickly becomes a daunting task in experiments based on bulk and fibre optical elements.
Integrated photonic circuits based on silicon offer dense component integration, high optical nonlinearity and good phase stability, which are highly desirable properties for photonic quantum technology^{26,27,28,29,30,31}. Moreover, silicon photonic devices are routinely fabricated in complementary metal oxide semiconductor (CMOS) processes. Therefore, a new field called silicon quantum photonics has recently been developed and has emerged as a promising platform for largescale quantum information processing^{32}. Recent advances of onchip highdimensional entanglement have employed frequencyencoding generated from a microresonator photonpair source^{17} and pathencoding generated from meander waveguides photonpair source^{20}. Specifically, silicon waveguides with cmlength are often employed as sources to create photonpairs^{9,20,28}. However, the natural bandwidth of the photons generated from meander waveguides is about 30 nm^{20}. In order to obtain highquality photons, it is necessary to employ bandpass filters (~1 nm bandwidth in ref. ^{20}), which unavoidably reduces the photon count rate drastically.
In this work, we employ a silicon photonic chip using an advanced resonator source embedded in MachZehnder interferometers (MZIs) to generate, manipulate and characterize pathentangled qutrits (d = 3). Cavityenhanced processes and independent tuning capabilities of the coupling coefficients of pump, signal and idler photons allow us to generate highindistinguishable and highbrightness photons without using passive filtering. The bandwidth of the photon generated from our source is about 50 pm, about a factor of 600 narrower to that of ref. ^{20}. This narrowband feature not only provides highquality single photons, but also holds the promise for direct coupling with telecom quantum memory^{33}, which is not possible for the nanowire source due to the prohibitive low count rate after ~GHz bandwidth filtering. In particular, we perform the onchip test of quantum contextuality with the closed compatible loophole. Furthermore, using the entangledqutrit state, we simulate a twovertex and threeedge graph and obtain the number of the perfect matchings of this graph, which is in the #Pcomplete complexity class^{34}. Although the structure of the graph of our demonstration is simple, it can be viewed as the first step towards achieving the ambitious goal of solving #P hard problem with quantum photonic devices. We also employ our device to demonstrate the excellent phase sensitivity, exceeding both classical threepath linear interferometer and quantum secondorder nonlinear interferometer limits. To be best of our knowledge, none of these three experiments have ever been realized with an integrated chip.
Results
Silicon quantum photonic chip and experimental setup
We have employed a scalable scheme for generating highdimensional entangled states^{25}. As shown in the conceptual scheme (Fig. 1a), entangled qutrits are generated by three coherently pumped nondegenerate spontaneous four wave mixing (SFWM) photonpair sources, in which two pump photons generate one signal photon and one idler photon with different wavelengths (Fig. 1b, c). The signal and idler photons are separated by dichroic mirrors (DMs) and then sent through reconfigurable linear optical circuits for implementing arbitrary 3D unitary operations via threedimensional multiports (3DMPs). Finally, we verify and harness the qutrit entanglement by detecting single photons at the outputs.
To obtain an efficient photonpair source, we use a dual MachZehnder interferometer microring (DMZIR) photonpair source^{35,36}. Such a DMZIR photonpair source is inspired by the design of a wavelength division multiplexer (WDM) in classical optical communication^{37} and could circumvent the tradeoff between the utilization efficiency of the pump photon and the extraction efficiency of the signal and idler photon pairs from a ring resonator^{38}. The DMZIR photonpair source was first demonstrated in ref. ^{35}, where enhanced coincidence efficiency of a single source was shown. The working principle of the DMZIR photon source is as follows: by wrapping two pulley waveguides around a ring resonator, one can construct a fourport device, as shown in Fig. 1b. We couple each waveguide at two points to the resonator, using four directional couplers. By adjusting the relative phases of two waveguides to the resonator, we can tune the coupling between waveguides to the resonator at the pump, signal and idler wavelengths independently. In the ideal case, we would like to have the pump circulate in the ring resonator to generate photon pairs, and therefore, the critical coupling condition for the pump is preferred. On the other hand, we want to extract the signal and idler photons from the resonator as soon as they are generated to minimize the propagation loss of photon pairs in the resonator. As a consequence, it is desirable to overcouple the waveguide to the resonator at the wavelengths of signal and idler photons. These two requirements can be fulfilled simultaneously by setting the free spectral range (FSR) of the MZIs to be twice that of the ring, such that every second resonance of the ring is effectively suppressed. By doing so, we can achieve the desired distinct coupling conditions for the pump, signal and idler photons and maximally utilize the pump to efficiently extract photon pairs. The ring has a radius of 15 μm and a coupling gap of 250 μm (200 μm) at the input (output) side. The length difference of the unbalanced MZI1 (MZI2) is 47.8 μm (48 μm). Optical microscopy images of the DMZIR photonpair source and the whole entangledqutrit chip are shown in Fig. 1d, e, respectively. Note that the sizes of the gaps of critical and overcritical couplings depend on the propagating loss of the photons in the ring. One should be able to obtain a higher count rate by optimizing the gap size^{35} (see Supplementary Information for a theoretical analysis and a detailed characterization of the DMZIR photon source).
Following the DMZIR photonpair source, we use an asymmetric MZI (AMZI) as an onchip WDM to separate the nondegenerate signal and idler photons. As shown in Fig. 1f, we repeat this combination of DMZIR source and WDM three times, and excite these sources coherently. When the generation rate is the same for all three sources and the relative phases of the pump are all zero, we generate a maximally entangled state of two qutrits: \(\left\Psi \right\rangle =\frac{1}{\sqrt{3}}(\left00\right\rangle +\left11\right\rangle +\left22\right\rangle )\). Note that \(\left0\right\rangle\), \(\left1\right\rangle\) and \(\left2\right\rangle\) are the individual path states of single photons.
Each qutrit can be locally manipulated by a 3DMP^{24}, which is composed of thermooptic phase shifters (PSs) and multimode interferometers (MMIs). In particular, one of the essential components, formed by a single PS and a tunable beam splitter, is realized with a MZI, consisting of two balanced MMIs and a PS.These components are used to realize R_{z}(φ_{z}) and R_{y}(θ_{y}) rotations, and thus to obtain an arbitrary SU(2) operation in the twodimensional subspace. Note that our experimental configuration is also closely related to a recent proposal on generating OAM entanglement by path identity^{39}. The collective paths and individual paths in our work correspond to the path and OAM in ref. ^{40}. After manipulating and characterizing the entangled qutrits with two 3DMPs, both the signal and idler photons are coupled out from the chip, filtered to suppress residual pumping with offchip filters, and detected with superconducting nanowire singlephoton detectors (SSPDs). The singlephoton detection events are recorded by a fieldprogrammable gate array (FPGA)based timetag unit. Then both single counts and coincidence counts CC_{ij} between path i (i = 1, 3, 5) and path j (j = 2, 4, 6) are extracted from these timetag records (see Supplementary Information for further experimental details).
From qubit entanglement to qutrit entanglement
To generate threedimensional (3D) pathentangled photons, it is necessary to ensure that all three coherently pumped photonpair sources are identical. This means that the emitted photon pairs from different sources should be the same in all DOFs, including polarization, spatial mode, count rate and frequency. For our chipbased system, we use singlemode waveguides, which automatically give us the same polarization states and spatial modes of photons from different sources. However, the count rate and frequency of the photons are not necessarily identical for different sources. To eliminate the count rate distinguishability, we can tune the pump power of the individual source. The last DOF is the frequency. In nonresonant broad band (~nm to tens of nm) photonpair sources, such as silicon nanowires, one can use offchip narrowband filtering to postselect identical spectra of different photons^{9,20}, which unavoidably reduces the count rate. In the resonant sources, such as our DMZIR source, we can actively tune the resonance wavelengths of each individual source with PSs. In doing so, we obtain identical photons without sacrificing the photon count rate, which is particularly important for multiphoton highdimensional experiments. However, aligning the frequency of the narrowband photons generated from resonanceenhanced sources is challenging within the submm footprint of our chip. The reasons are follows: silicon has a relatively high thermooptic coefficient. On the one hand, this feature of silicon is desirable for realizing reconfigurable photonic circuits by using thermooptic PSs with low power consumption. On the other hand, it presents an experimental challenge to stabilize the frequency of the single photons generated from resonanceenhanced photonpair sources under several distinct configurations of thermal PSs, due to thermal crosstalk^{41}.
As the first step to generate 3D entanglement, we verify the identicality of two sources with timereversed HongOuMandel (RHOM) interference^{42}. Highly indistinguishable photons produce the high visibility of RHOM interference. We investigate the indistinguishability between all pairs of the three sources by interfering signalidler photon pairs generated from S1, S2 and S3 on the top 3DMP. For instance, we set the phases S_{y1} and S_{y2} to be π and S_{y3} to be π∕2 and scan the phase of S_{z3} to obtain the RHOM interference fringe between S2 and S3. The RHOM interference fringes between S1 and S2, S1 and S3 and S2 and S3 are shown in Fig. 2a–c, respectively. The visibility of the fringe is defined as V = (CC_{max} − CC_{min})/(CC_{max} + CC_{min}), where CC_{max} and CC_{min} are the maximum and minimum of the coincidence counts. The measured visibilities are greater than 96.49% in all cases, indicating highquality spectral overlaps. All of our data are raw, and no background counts are subtracted. To obtain high interference visibilities, we have spent significant amount of efforts to eliminate thermal noise (see Supplementary Information for further experimental details). We believe that by using better designs of the thermal PSs such as reported in refs ^{43,44}, the noise can be greatly mitigated. In additional to reduce crosstalks, for reaching a visibility required for the practical applications, highfidelity quantum control is a necessity. Remarkably, previous work has shown one can achieve excellent on/off ratio (~0.5 dB), equivalent to having a PauliZ error rate of <10^{−6 }^{45} by using cascaded MZIs. Over 100 dB passband to stopband contrast filters have also been realized by cascaded microrings^{46} and AMZIs^{47}, respectively. By combining these highperformance devices, we believe integrated quantum photonics is a promising route in the development of future quantum technologies and applications^{48}.
The next step is to verify the qubit entanglement of path states between three different pairs of sources. We measure the correlation of the pathentangled states in mutually unbiased bases (MUBs). We set the measurement base of the signal photon to be the coherent superpositions of the computational base, \(\frac{1}{\sqrt{2}}(\leftj\right\rangle +\leftk\right\rangle )\), where (j,k) = (0,1), (0,2), (1,2). Then, we scan the phase φ in the quantum state of the idler photon, \(\frac{1}{\sqrt{2}}(\leftj\right\rangle +{e}^{i\varphi }\leftk\right\rangle )\) and measure the coincidence counts between the signal and idler photons. The coincidence fringes are shown in Fig. 2d–f for S1 and S2, S1 and S3 and S2 and S3, respectively. The values of various visibilities range from 94.72% ± 0.50% to 97.50% ± 0.38%, indicating highquality qubit entanglement. The phase doubling signature of RHOM fringes compared to the correlation counterparts can be seen by comparing Fig. 2a–c, and Fig. 2d–f. In the RHOM experiment, both signal and idler photons create a coherent superposition of two photons in two paths, the state that evolves under a phase shift in one of the modes then displays the phase doubling^{42}. Note that the count rate of the pathcorrelation measurement is lower than that of the RHOM measurement, mainly because the design of the onchip WDM is not optimal. Higher count rates of the correlation measurement can be achieved by optimizing the length difference between the two arms of the WDM.
Having established highquality qubit entanglement, we proceed to characterize the qutrit entangled state with complete highdimensional quantumstate tomography (QST). We use a set of all possible combinations of GellMann matrices and apply the corresponding settings to both 3DMPs^{49}. The QST method takes approximately 33 mins with a typical count rate ~100 Hz per setting. Figure 2g, h displays the real and imaginary parts of the reconstructed density matrix of the state, respectively, showing good agreement between the maximally entangled and measured quantum states with a fidelity of 95.50% ± 0.17%. The maximum matrix element of the imaginary part is smaller than 0.015. From the reconstructed density matrix, we obtain an Iconcurrence of 1.149 ± 0.002^{50}. The uncertainties in the state fidelity extracted from these density matrices are calculated using a Monte Carlo routine assuming Poissonian statistics. In the context of quantum communication, a multidimension entanglementbased Ekert91 QKD protocol^{51} was initially proposed and analyzed in refs ^{52,53}, where highdimension mutually unbiased bases correlations between two Alice and Bob can be used to generate keys. The upper bound error rate (ER) that guarantees security against coherent attacks for devicedependent QKD in three dimensions is 15.95%. For a maximally qutrit entangled state, the fidelity (F) of the state can be used to infer the ER if Alice and Bob use the same MUB^{54}; that is F = (34*ER)/3. From the fidelity we obtain, the ER is only 3.375%, which is considerably below the required bound, indicating the high quality of our qutrit state.
Tests of quantum nonlocality and contextuality with entangled qutrits
To benchmark the highquality qutrit entanglement and highprecision quantum control, we demonstrate experimental tests of quantum nonlocality and quantum contextuality. Violations of Bell inequalities based on local realistic theories provide evidence of quantum nonlocality. It has been demonstrated that, highdimensional correlations compatible with local realism satisfy a generalized Belltype inequality, the CollinsGisinLindenMassarPopescu (CGLMP) inequality, with I_{d} ≤ 2 for all d ≥ 2^{55}. Expression I_{3} is given by joint probabilities as
where P\(\left({A}_{{\rm{a}}}={B}_{{\rm{b}}}+k\right)\) with (a, b = 1, 2) and (k = 0, 1) represent the joint probabilities for the outcomes of A_{a} that differ from B_{b} by k. The measurement bases used to maximise the violation of Eq. (1) for the maximally entangled state \(\left\Psi \right\rangle =\sum _{j=0}^{2}{\leftj\right\rangle }_{A}\bigotimes {\leftj\right\rangle }_{B}\) are defined as
where i = \(\sqrt{1}\), α_{1} = 0, α_{2} = 1∕2, β_{1} = 1∕4 and β_{2} = −1∕4, K and L ∈ {0, 1, 2} denote Alice’s and Bob’s measurement outcomes respectively, and \(\leftj\right\rangle\) denotes the computational basis. These measurement bases can be implemented by configuring PSs in the 3DMPs. For instance, we set S_{z1} = 0.333π, S_{y1} = 0.5π, S_{z2} = 0.583π, S_{y2} = 0.392π, S_{z3} = 0.779π and S_{y3} = 0.5π to realize setting A_{1}. In the context of quantum computation, entanglement is the essential resource. For oneway quantum computation, ref. ^{23} reported the noise sensitivity of a twophoton, threelevel and fourpartite (two DOFs) cluster state with entanglement witness. CGLMP inequality is an entanglement criterion with higher correlation requirements comparing to entanglement witness. We witness the existence of entanglement between two qutrits in our experiment by using CGLMP inequality. The classical bound is violated by 51.46σ (I_{3} = 2.7307 ± 0.0142), benchmarking the resilience to errors. The experimental results for the four base settings are shown in Table 1.
Contextuality is a fundamental concept in quantum mechanics^{13,56,57,58} and an important resource for faulttolerant universal quantum computation^{59}. A single qutrit is the simplest quantum system showing the contradiction between noncontextual hiddenvariable models and quantum mechanics^{13}. However, the testability of the KochenSpecker (KS) theorem is debated due to the finite precision in a single qutrit in practical experiments^{60,61}. An approach based on maximally entangledqutrit pair has been proposed^{62}, which was recently realized with bulk optics^{63}.
The experimental setting is as follows. A pair of maximally entangled qutrits is sent to two parties, Alice (A) and Bob (B). Alice performs projective measurements, either \({D}_{1}^{A}\) or \({T}_{0}^{A}\), \({T}_{1}^{A}\), and Bob simultaneously performs measurement \({D}_{0}^{B}\), where \({D}_{1}^{A}\) and \({D}_{0}^{B}\) are dichotomic projectors with two possible outcomes, 0 or 1, and \({T}_{0}^{A}\) and \({T}_{1}^{A}\) are trichotomic projectors with three possible outcomes, a, b or c. These four projectors are defined as \({D}_{0}^{B}=\lefti\right\rangle \left\langle i\right,{D}_{1}^{A}=\leftf\right\rangle \left\langle f\right,{T}_{0}^{A}=\left{a}_{0}\right\rangle \left\langle {a}_{0}\right\) and \({T}_{1}^{A}=\left{a}_{1}\right\rangle \left\langle {a}_{1}\right\), where \(\lefti\right\rangle =(\left0\right\rangle +\left1\right\rangle +\left2\right\rangle )/\sqrt{3}\), \(\leftf\right\rangle =(\left0\right\rangle \left1\right\rangle +\left2\right\rangle )/\sqrt{3}\), \(\left{a}_{0}\right\rangle =(\left1\right\rangle \left2\right\rangle )/\sqrt{2}\) and \(\left{a}_{1}\right\rangle =(\left0\right\rangle \left1\right\rangle )/\sqrt{2}\). The noncompatibility loophole contextuality inequality can be expressed as^{62}:
where \(P({D}_{1}^{A}=1 {D}_{0}^{B}=1)\) stands for the conditional probability of Alice obtaining result 1 for \({D}_{1}^{A}\) when Bob also obtains result 1 with \({D}_{0}^{B}\). \(P({T}_{0}^{A}=1 {D}_{0}^{B}=1)\) and \(P({T}_{1}^{A}=1 {D}_{0}^{B}=1)\) are defined analogously. For our experiment, we need to reconfigure two 3DMPs according to these projectors and measure the coincidence counts to reconstruct the conditional probabilities in Eq. (4). For example, \(\leftf\right\rangle\) can be projected to port 3 by setting S_{z1} = 0, S_{y1} = 0.5π, S_{z2} = 1.25π, S_{y2} = 0.608π, S_{z3} = 0 and S_{y3} = π. We experimentally violate the noncontextuality inequality by 9.5 standard deviations (0.085 ± 0.009). The detailed experimental results are listed in Table 2. The nosignaling conditions, confirming the compatibility assumption between the measurements of Alice and Bob, are checked, as shown in Table 3. The results deviate slightly from 0 because of experimental imperfections.
Harnessing twoqutrit quantum correlations: quantum simulation of graphs and quantum metrology
Highorder quantum correlations are unique properties of highdimensional entangled quantum systems and are a central resource for quantum information processing. To probe the quantum correlations in the entangledqutrit system, we measure the coincidence counts between the signal and idler photons under different MUBs by tuning both the phases of the signal/idler and pump photons. Here we use the quantum correlation between two entangled qutrits to demonstrate the quantum simulation of graphs and quantum metrology based on a quantum multipath interferometer with thirdorder nonlinearity.
Quantum simulation of graphs
Graphs are mathematical structures for describing relations between objects and have been widely used in various areas, including physics, biology and information science. A graph typically consists of a set of vertices and edges connecting the vertices. A subset of the edges containing every vertex of the nvertex graph exactly once is defined as a perfect matching of the graph. To find the number of perfect matchings of a graph is a problem that lies in the #Pcomplete complexity class^{34}. To provide an algorithm to solve such a hard problem is highly desirable. Recent studies have shown that a carefully designed quantum optical experiment can be associated with an undirected graph^{64}. In Particular, the number of coherently superimposed terms of the generated highdimensional quantum state from a quantum optical experiment is exactly the number of perfect matchings in the corresponding graph. Each vertex stands for an optical path occupied by a single photon and every edge represents a photonpair source. This scheme can be viewed as a quantum simulation of graphs.
As the first step towards the quantum simulation of graphs, we use entangled qutrits to experimentally demonstrate the connection between graph theory and quantum optical experiments. Figure 3a shows a conceptual scheme of our realization. Each pair of photons generated from sources propagates along their paths, denoted by black arrows, and acquires additional mode shifts due to the mode converters between the sources. As mentioned above, the path and OAM in ref. ^{64} are equivalent to the collective and the individual paths of our integrated quantum photonic circuit, as shown in Fig. 1a. Therefore, the mode converters can be implemented by routing the individual paths of the photon on our chip. By suitably setting up the 3DMPs, we verify the resultant quantum state with coherent superimposed terms corresponding to the number of perfect matchings. We implement two experimental steps to realize this goal. First, we measure the coincidence counts between the signal and idler photons on the computational basis, S_{1}I_{1}. The experimental results are shown in Fig. 3b. It is clear that the major contributions are the 00, 11 and 22 terms. The second step is to verify the coherence between these three terms. For the entangledqutrit pair system, each individual qutrit has four MUBs. Therefore, we have to measure the correlation coefficients in all four base combinations, i.e. S_{1}I_{1}, S_{2}I_{2}, S_{3}I_{3} and S_{4}I_{4}, where S\({}_{k}={I}_{k}^{* }\), with (k = 1, 2, 3, 4). These MUBs, up to normalization, are defined as
where \(\omega ={e}^{i\frac{2\pi }{3}}\) and * stands for the complex conjugation.
The normalized coincidence counts of S_{2}I_{2}, S_{3}I_{3} and S_{4}I_{4} are shown in Fig. 3c–e, respectively, by balancing the loss for every port. The experimental correlation coefficients derived from the coincidence counts in the four MUB combinations are shown in Fig. 3f. For an ideal maximally entangledqutrit pair, the correlation coefficients should be unity. Due to the experimental imperfections, we obtain the correlation coefficients with the values of 98.17% ± 0.11%, 87.61% ± 0.27%, 91.32% ± 0.24% and 89.01 ± 0.27%, which shows correlations in all MUBs.
Quantum metrology with entangled qutrits
Accurate phase measurements are at the heart of metrological science. One figureofmerit for evaluating the accuracy of phase measurement is sensitivity, S, which is defined as the derivative of the output photon number with respect to a phase change S ∝ 1∕N. In the experimental setting of classical interferometers, it is well known that increasing the number of paths of the interferometer can enhance the sensitivity^{65}. On the other hand, in the field of quantum metrology, one can further enhance the sensitivity by using entanglement^{3}. Here, we combine both traits from classical and quantum systems and employ a threedimensionentanglement thirdordernonlinearity interferometer to demonstrate the enhanced phase sensitivity compared to the classical threepath^{66} and quantum secondordernonlinearity interferometers^{19}. We send the entangledqutrit state \(\left\Psi \right\rangle =\frac{1}{\sqrt{3}}\left({e}^{i2Pz1}\left00\right\rangle +{e}^{i2Pz2}\left11\right\rangle +\left22\right\rangle \right.\) into two separate 3DMPs. We then scan the relative pump phases Pz1 and Pz2 and measure the coincidence between two qutrits (outputs 1,3,5 and 2,4,6). In total, there are nine different coincidence combinations. We quantify the qutritqutrit correlations as functions of phase settings of Pz1, Pz2. The normalized coincidences along with the theoretical results are shown in Fig. 4a–c. It is easily understood that the phase dependence is different between the second and thirdorder nonlinear interactions^{67}. In the generation of entangled photon pairs, two pump photons are involved in the thirdorder processes providing a double phase compared to the secondorder processes with the participation of only one pump photon. The measurements confirm this difference. Specifically, if the phases are chosen such that Pz1 = −Pz2, the intensity varies from maximum to minimum as the pump phase is changed. This phase setting also gives the maximal sensitivity S. The raw data are extracted from the measured results and fitted with theoretical curves as shown in Fig. 4d. The averaged phase sensitivity \(S=\frac{1}{C{C}_{ab\,max}} \frac{dC{C}_{ab}}{d\varphi }\) is 1.476 ± 0.048 rad^{−1}, more than the theoretical ideal value of 0.5 rad^{−1} for the twopath interferometer and 0.78 rad^{−1} for the ideal threepath interferometer. The reason for the increased sensitivity is that the doubled phase sensitivity of the SFWM process and the side lobes appears between the two main peaks in threepath interference patterns, which enhance the steepness of the peaks of the correlations.
Discussion
We have integrated three resonanceenhanced photonpair sources embedded in interferometers, three WDMs and two 3DMPs on a single monolithic silicon chip. We made all three sources identical without using frequency postselection and observe highvisibility quantum interference, which allowed us to prepare, manipulate and analyse the highquality pathentangledqutrit state. We violated the CGLMP inequality to confirm quantum nonlocality and the KS inequality to confirm contextuality with the entangled qutrits, verifying fundamental properties of quantum theory. Furthermore, we used twoqutrit quantum correlations to simulate graphs and identify the number of perfect matchings for a smallscale graph. Finally, by using our chip for 3D entanglement, a thirdorder nonlinearity interferometer, we improved the phase sensitivity by a factor of 2 compared to a classical threepath interferometer.
Our demonstration of finding the number of the perfect matchings of the graph could be further extended to the multiphoton and higherdimension experiments, which might be suitable to demonstrate the quantum advantage in the near or mid term. To reach this regime, one needs to develop highbrightness multiphoton sources. The source presented in this work is a promising candidate for such a source. Although there exist a few technical challenges towards full integrated silicon quantum chips, such as cryogeniccompatible photon manipulation and highefficiency photon detection, heterogeneous integrated chips are a promising approach for achieving this goal^{68,69,70}. Combined with an efficient onchip SSPD^{69} and recently demonstrated cryogenic operation of Sibarium titanate^{70}, our work could be viewed as a solid basis of future photonic quantum devices and systems for quantum information processing.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
References
 1.
Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010).
 2.
Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002).
 3.
Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011).
 4.
D’Ambrosio, V. et al. Complete experimental toolbox for alignmentfree quantum communication. Nat. Commun. 3, 961 (2012).
 5.
Graham, T. M., Bernstein, H. J., Wei, T.C., Junge, M. & Kwiat, P. G. Superdense teleportation using hyperentangled photons. Nat. Commun. 6, 7185 (2015).
 6.
Luo, Y.H. et al. Quantum teleportation in high dimensions. Phys. Rev. Lett. 123, 070505 (2019).
 7.
Hu, X.M. et al. Experimental multilevel quantum teleportation. Preprint at https://arxiv.org/abs/1904.12249 (2019).
 8.
Lanyon, B. P. et al. Simplifying quantum logic using higherdimensional Hilbert spaces. Nat. Phys. 5, 134–140 (2008).
 9.
Qiang, X. et al. Largescale silicon quantum photonics implementing arbitrary twoqubit processing. Nat. Photon. 12, 534–539 (2018).
 10.
Neeley, M. et al. Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722–725 (2009).
 11.
Dada, A. C., Leach, J., Buller, G. S., Padgett, M. J. & Andersson, E. Experimental highdimensional twophoton entanglement and violations of generalized Bell inequalities. Nat. Phys. 7, 677–680 (2011).
 12.
Vértesi, T., Pironio, S. & Brunner, N. Closing the detection loophole in bell experiments using qudits. Phys. Rev. Lett. 104, 060401 (2010).
 13.
Lapkiewicz, R. et al. Experimental nonclassicality of an indivisible quantum system. Nature 474, 490–493 (2011).
 14.
Erhard, M., Krenn, M. & Zeilinger, A. Advances in high dimensional quantum entanglement. Preprint at https://arxiv.org/abs/1911.10006 (2019).
 15.
Wang, J., Sciarrino, F., Laing, A. & Thompson, M. G. Integrated photonic quantum technologies. Nat. Photon. 1–12 (2019).
 16.
Wang, X.L. et al. Quantum teleportation of multiple degrees of freedom of a single photon. Nature 518, 516–519 (2015).
 17.
Kues, M. et al. Onchip generation of highdimensional entangled quantum states and their coherent control. Nature 546, 622–626 (2017).
 18.
Imany, P. et al. 50GHzspaced comb of highdimensional frequencybin entangled photons from an onchip silicon nitride microresonator. Opt. Express 26, 1825–1840 (2018).
 19.
Schaeff, C., Polster, R., Huber, M., Ramelow, S. & Zeilinger, A. Experimental access to higherdimensional entangled quantum systems using integrated optics. Optica 2, 523–529 (2015).
 20.
Wang, J. et al. Multidimensional quantum entanglement with largescale integrated optics. Science 360, 285–291 (2018).
 21.
Thew, R., Acín, A., Zbinden, H. & Gisin, N. Experimental realization of entangled qutrits for quantum communication. Quantum Info. Comput. 4, 93–101 (2004).
 22.
Richart, D., Fischer, Y. & Weinfurter, H. Experimental implementation of higher dimensional timeenergy entanglement. Appl. Phys. B 106, 543–550 (2012).
 23.
Reimer, C. et al. Highdimensional oneway quantum processing implemented on dlevel cluster states. Nat. Phys. 15, 148–153 (2019).
 24.
Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, P. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61 (1994).
 25.
Schaeff, C. et al. Scalable fiber integrated source for higherdimensional pathentangled photonic quNits. Opt. Express 20, 16145–16153 (2012).
 26.
Mower, J. & Englund, D. Efficient generation of single and entangled photons on a silicon photonic integrated chip. Phys. Rev. A 84, 052326 (2011).
 27.
Collins, M. J. et al. Integrated spatial multiplexing of heralded singlephoton sources. Nat. Commun. 4, 2582 (2013).
 28.
Silverstone, J. W. et al. Onchip quantum interference between silicon photonpair sources. Nat. Photon. 8, 104–108 (2014).
 29.
Harris, N. C. et al. Integrated source of spectrally filtered correlated photons for largescale quantum photonic systems. Phys. Rev. X 4, 041047 (2014).
 30.
Silverstone, J. W. et al. Qubit entanglement between ringresonator photonpair sources on a silicon chip. Nat. Commun. 6, 7948 (2015).
 31.
Feng, L.T. et al. Onchip coherent conversion of photonic quantum entanglement between different degrees of freedom. Nat. Commun. 7, 11985 (2016).
 32.
Silverstone, J. W., Bonneau, D., O’Brien, J. L. & Thompson, M. G. Silicon quantum photonics. IEEE J. Sel. Top. Quantum Electron. 22, 390–402 (2016).
 33.
Saglamyurek, E. et al. Quantum storage of entangled telecomwavelength photons in an erbiumdoped optical fibre. Nat. Photon. 9, 83 (2015).
 34.
Valiant, L. G. The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189–201 (1979).
 35.
Tison, C. C. et al. Path to increasing the coincidence efficiency of integrated resonant photon sources. Opt. Express 25, 33088–33096 (2017).
 36.
Vernon, Z. et al. Truly unentangled photon pairs without spectral filtering. Opt. Lett. 42, 3638–3641 (2017).
 37.
Barbarossa, G., Matteo, A. M. & Armenise, M. N. Theoretical analysis of triplecoupler ringbased optical guidedwave resonator. IEEE J. Lightwave Technol. 13, 148–157 (1995).
 38.
Vernon, Z., Liscidini, M. & Sipe, J. E. No free lunch: the tradeoff between heralding rate and efficiency in microresonatorbased heralded single photon sources. Opt. Lett. 41, 788–791 (2016).
 39.
Krenn, M., Hochrainer, A., Lahiri, M. & Zeilinger, A. Entanglement by path identity. Phys. Rev. Lett. 118, 080401 (2017).
 40.
Kysela, J., Erhard, M., Hochrainer, A., Krenn, M. & Zeilinger, A. Experimental highdimensional entanglement by path identity. Preprint at https://arxiv.org/abs/1904.07851 (2019).
 41.
Carolan, J. et al. Scalable feedback control of single photon sources for photonic quantum technologies. Optica 6, 335–340 (2019).
 42.
Chen, J., Lee, K. F. & Kumar, P. Deterministic quantum splitter based on timereversed HongOuMandel interference. Phys. Rev. A 76, 031804 (2007).
 43.
Harris, N. C. et al. Efficient, compact and low loss thermooptic phase shifter in silicon. Opt. Express 22, 10487–10493 (2014).
 44.
Gao, S. et al. Powerefficient thermal optical tunable grating coupler based on silicon photonic platform. IEEE Photon. Technol. Lett. 31, 537–540 (2019).
 45.
Wilkes, C. M. et al. 60 db highextinction autoconfigured machzehnder interferometer. Opt. Lett. 41, 5318–5321 (2016).
 46.
Ong, J. R., Kumar, R. & Mookherjea, S. Ultrahighcontrast and tunablebandwidth filter using cascaded highorder silicon microring filters. IEEE Photon. Technol. Lett. 25, 1543–1546 (2013).
 47.
Piekarek, M. et al. Highextinction ratio integrated photonic filters for silicon quantum photonics. Opt. Lett. 42, 815–818 (2017).
 48.
Rudolph, T. Why i am optimistic about the siliconphotonic route to quantum computing. APL Photonics 2, 030901 (2017).
 49.
Thew, R. T., Nemoto, K., White, A. G. & Munro, W. J. Qudit quantumstate tomography. Phys. Rev. A 66, 012303 (2002).
 50.
Fedorov, M., Volkov, P., Mikhailova, J. M., Straupe, S. & Kulik, S. Entanglement of biphoton states: qutrits and ququarts. New J. Phys. 13, 083004 (2011).
 51.
Ekert, A. K. Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991).
 52.
Cerf, N. J., Bourennane, M., Karlsson, A. & Gisin, N. Security of quantum key distribution using dlevel systems. Phys. Rev. Lett. 88, 127902 (2002).
 53.
Bruß, D. & Macchiavello, C. Optimal eavesdropping in cryptography with threedimensional quantum states. Phys. Rev. Lett. 88, 127901 (2002).
 54.
Zhu, H. & Hayashi, M. Optimal verification and fidelity estimation of maximally entangled states. Phys. Rev. A 99, 052346 (2019).
 55.
Collins, D., Gisin, N., Linden, N., Massar, S. & Popescu, S. Bell inequalities for arbitrarily highdimensional systems. Phys. Rev. Lett. 88, 040404 (2002).
 56.
Kochen, S. & Specker, E. P. The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967).
 57.
Klyachko, A. A., Can, M. A., Binicioğlu, S. & Shumovsky, A. S. Simple test for hidden variables in spin1 systems. Phys. Rev. A 101, 020403 (2008).
 58.
Kirchmair, G. et al. Stateindependent experimental test of quantum contextuality. Nature 460, 494–497 (2009).
 59.
Howard, M., Wallman, J., Veitch, V. & Emerson, J. Contextuality supplies the ‘magic’ for quantum computation. Nature 510, 351–355 (2014).
 60.
Meyer, D. A. Finite precision measurement nullifies the KochenSpecker theorem. Phys. Rev. Lett. 83, 3751 (1999).
 61.
Kent, A. Noncontextual hidden variables and physical measurements. Phys. Rev. Lett. 83, 3755 (1999).
 62.
Cabello, A. & Cunha, M. T. Proposal of a twoqutrit contextuality test free of the finite precision and compatibility loopholes. Phys. Rev. Lett. 106, 190401 (2011).
 63.
Hu, X.M. et al. Experimental test of compatibilityloopholefree contextuality with spatially separated entangled qutrits. Phys. Rev. Lett. 117, 170403 (2016).
 64.
Krenn, M., Gu, X. & Zeilinger, A. Quantum experiments and graphs: multiparty states as coherent superpositions of perfect matchings. Phys. Rev. Lett. 119, 240403 (2017).
 65.
Sheem, S. K. Optical fiber interferometers with [3 × 3] directional couplers: analysis. J. Appl. Phys. 52, 3865–3872 (1981).
 66.
Weihs, G., Reck, M., Weinfurter, H. & Zeilinger, A. Allfiber threepath MachZehnder interferometer. Opt. Lett. 21, 302–304 (1996).
 67.
Reimer, C. et al. Generation of multiphoton entangled quantum states by means of integrated frequency combs. Science 351, 1176–1180 (2016).
 68.
He, M. et al. Highperformance hybrid silicon and lithium niobate machzehnder modulators for 100 gbit s^{−1} and beyond. Nat. Photon. 13, 359 (2019).
 69.
Ferrari, S., Schuck, C. & Pernice, W. Waveguideintegrated superconducting nanowire singlephoton detectors. Nanophotonics 7, 1725–1758 (2018).
 70.
Eltes, F. et al. An integrated cryogenic optical modulator. Preprint at https://arxiv.org/abs/1904.10902 (2019).
Acknowledgements
We thank M. Erhard, X. Gu, M. Krenn, A. Zeilinger and J. Wang for fruitful discussions. This research is supported by the National Key Research and Development Program of China (2017YFA0303704, 2019YFA0308704), National Natural Science Foundation of China (Grant Nos. 11674170, 11690032, 11321063, 11804153), NSFCBRICS (No. 61961146001), NSF Jiangsu Province (No. BK20170010), the program for Innovative Talents and Entrepreneur in Jiangsu, and the Fundamental Research Funds for the Central Universities.
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L.L., L.X., Z.C. and X.M. designed and performed the experiment. L.C., T.Y., T.T., Y.P. and X.C. provided experimental assistance and suggestions. W.M. provided theoretical assistance. L.L., L.X. and X.M. analysed the data. L.L. and X.M. wrote the manuscript with input from all authors. Y.L., S.Z. and X.M. supervised the project. L.L, L.X. and Z.C. contributed equally to this work.
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Lu, L., Xia, L., Chen, Z. et al. Threedimensional entanglement on a silicon chip. npj Quantum Inf 6, 30 (2020). https://doi.org/10.1038/s415340200260x
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