Quantum entanglement on photonic chips: a review

Abstract. Entanglement is one of the most vital properties of quantum mechanical systems, and it forms the backbone of quantum information technologies. Taking advantage of nano/microfabrication and particularly complementary metal-oxide-semiconductor manufacturing technologies, photonic integrated circuits (PICs) have emerged as a versatile platform for the generation, manipulation, and measurement of entangled photonic states. We summarize the recent progress of quantum entanglement on PICs, starting from the generation of nonentangled and entangled biphoton states, to the generation of entangled states of multiple photons, multiple dimensions, and multiple degrees of freedom, as well as their applications for quantum information processing.

photons, multiple dimensions, and multiple DoFs. Finally, we briefly review possible chip-scale applications with entangled states and discuss future challenges and opportunities.

On-Chip Encoding Single-Photon Quantum States
A photon features a broad spectrum of different DoFs that can represent the basic units of quantum information, i.e., qubit αj0i þ βj1i, where α; β are the complex amplitudes. The available DoFs include position (path), polarization, frequency (wavelength), and spatial and temporal modes. Notably, the basic quantum information unit usually is defined in the binary format as a qubit, but it can be generally defined in the d-nary format as a qudit 1 ffiffi d p P d−1 i¼0 c i jii, where jii is the logical state in the i'th mode.
In bulk-optics quantum experiments, the DoF of polarization has been extensively explored, e.g., in early seminal demonstrations. [38][39][40] Similar to the bulk-optics realization of polarized qubits by a birefringent waveplate and polarization beamsplitter (PBS) on integrated photonic chips, the polarization of photons can be manipulated by an integrated polarization rotator and PBS (i.e., birefringent waveguide or structure), which have been well developed in the silicon-photonics field. 41,42 Another commonly implemented DoF is the location or position information of photons. On integrated photonic chips, the j0i and j1i logic states can be well-defined in two modes in two separate optical waveguides that are physically phase-stabilized when manipulating the qubit states. This approach usually refers to path-encoding or dual-rail encoding in integrated quantum photonics. Pathencoded qubits can be manipulated by integrated beamsplitters and optical interferometers with high levels of fidelity, universality, dense integration, and reprogrammability, and therefore have been widely adopted in many integrated photonic quantum experiments. 24,43,44 The temporal mode of photons is also one of the available DoFs. Using fast light modulation or long optical delays, qubit states in two temporal modes or time bins can be generated. With the recent development of fast modulator and low-loss waveguides, time-bin encoded photons may provide an efficient solution for the implementation of chip-scale quantum key distributions. Moreover, optical waveguides support high-order eigenmodes, which allows the encoding of qubit or qudit states in the spatial mode DoF. The recent development of multimode silicon photonics enables mode-entanglement and its applications. 45,46 Such a mode-encoding state in optical waveguides can be reliably operated and transmitted, which is fundamentally similar to the case of optical orbital angular momentum in bulk-optics, where it is reliable. [47][48][49][50] Discrete frequency bins, usually existing in optical microresonators, can be in a coherent superposition state, thus allowing the preparation of qubits or qudits in the frequency DoF. 51 This approach recently has allowed interesting demonstrations of frequency entanglement. 52,53

Integrated Waveguide Photon-Pair Sources
High-quality single-photon sources are indispensable in photonic quantum technologies. An ideal single-photon source has to produce pure photons with high efficiency, and the photons have to be identical to those from other independent sources. Parametric nonlinear-optical sources emit photons nondeterministically, and they can be integrated into large arrays, in which each owns high purity, heralding efficiency and indistinguishability. Such parametric photon sources produce a pair of photons, and the success of detecting one of them indicates the presence of the other, referring to heralding single-photon sources. The integration of parametric nonlinear-optical sources not only provides the possibility for future multiplexing high-efficiency singlephoton sources 54 but also for the generation of different entangled states. The generation of entangled photons relies on the spontaneous parametric down conversion (SPDC) process in χ ð2Þ materials or the spontaneous four-wave mixing (SFWM) process in χ ð3Þ materials. Integrated SPDC photon-pair sources and entangled sources have been demonstrated in periodically poled lithium niobate, 26,27 gallium arsenide, 30,31 and aluminum nitride 55 waveguides and structures. Here, we focus more on the discussion of third-order silicon-based material systems, such as silicon-on-insulator and silicon nitride.
The physical laws governing photon-pair generation are the conservations of momentum and energy. The former usually refers to the phase matching condition for nonlinear-optical processes. The simplest photon-pair source is the one using straight waveguides, e.g., silicon waveguides. By optimizing the waveguide geometry such as waveguide width, thickness, or etching depth to engineer group velocity dispersion, photon pairs can be generated in a wide spectrum through SFWM. This type of straight waveguide source can be easily and reliably implemented on chip, for example, an array of 16 waveguide sources. 43 However, it only reaches high spectral purity at the expense of losing brightness and heralding efficiency. To achieve both high purity and heralding efficiency, dual-pump SFWM 56,57 and intermodal SFWM 58 have been proposed. Recently, a multimode waveguides source based on a dual-mode pump-delayed excitation scheme was demonstrated in silicon, with a spectral purity of 0.9904(6), a mutual indistinguishability of 0.987(2), and >90% heralding efficiency. 59 Another type of integrated photon-pair source is based on optical microresonators, e.g., microring and microdisk. For a simple point-coupled ring resonator, the maximum photon pair generation rate can be achieved at a critical coupling point, but the heralding efficiency of photon pairs is bounded by 0.50, implying a trade-off balance. 60 At the same time, the spectral purity relies on the quantity factor, and it shows a theoretical upper limit of the purity of 0.93. 61 An experimental demonstration of an array of microresonator-based photon-pair sources was reported with purity of 0.90, heralding efficiency of 0.50, and indistinguishability of 0.90. 62 Two approaches are proposed to improve the performance. One is to adopt two delayed pulsed lights for pumping, in which the pump spectral width can be increased, and thus the upper limit of purity can be improved up to 0.999. 63 The second is to use a dual-MZI-coupled microring resonator to independently control the linewidths of the pump and signal (idler) photons working at different coupling points, 61 and a purity of 0.95 was experimentally obtained. 64 A similar scheme has also been experimentally investigated. 65

Generation, Manipulation, and Measurement of Entanglement on Photonic Chips
The generation, manipulation, and measurement of large entanglement structure is at the heart of on-chip quantum information processing. For instance, one-way quantum computing requires large-scale cluster states to transmit coherent logical operators along the entanglement structure. 16,66 Quantum internet relies on the distribution of entanglement between quantum processors at different locations. 14 In this section, we introduce how to entangle photons such as Bell states on quantum photonic chips, and, in particular, how to generate multi-DoF, multiphoton, and multidimensional entanglement states. Integrated photonics is able to engineer multiple DoFs to encode and process quantum information. The coherent conversion between different DoFs of photons can make use of their own advantages for implementing different tasks. For example, path-to-polarization conversion allows the reliable distribution of entangled states between two separate chips, maintaining coherence and robustness both on chip and in optical fibers 62,67 [see Fig. 1(a)]. Coherent conversion of entangled states between path, polarization, and transverse mode was also reported in a silicon chip 68 [see Fig. 1(b)]. Moreover, simultaneously entangling multiple DoFs of photons, named hyperentanglement, provides an efficient way to expand the Hilbert space and enrich applications. 69,70 In bulk-optics, three different DoFs (path, polarization, and orbital angular momentum mode) of six photons have been simultaneously entangled to deliver an 18-qubit Fig. 1 On-chip conversion of multiple DoFs and multi-DoF entanglement. (a) Chip-to-chip quantum entanglement distribution by path-polarization interconversion. A pair of entangled photons, coded in path, were generated on chip A, then one photon was distributed to chip B via twodimensional grating couplers, while the other photon remained on chip A. As a result, a two-photon entangled state jϕ þ i ¼ 1 ffiffi 2 p ðj00i þ j11iÞ was distributed coherently between two separate chips. Bell-type violation of S ¼ 2.638 AE 0.039 showed the strong entanglement between two separate photons. (b) Quantum entangled state conversion between different DoFs of path, transverse mode, and polarization. Two photons were coupled into the silicon chip, and then NOON entangled state, 1 ffiffi 2 p ðj2i 0 j0i 1 þ j0i 0 j2i 1 Þ coded in path, was generated by an interference. With a mode multiplexer, the entangled state coded in path was first converted into a transverse waveguide mode TE 0 ∕TE 1 . Then, the quantum information was converted back to the path by a mode demultiplexer or to polarization by a PBS. Two-photon interference and Hong-Ou-Mandel effect measurements show the preservation of quantum coherence during the conversion between different DoFs. (c) Four-qubit hyperentanglement cluster state by entangling the path and polarization simultaneously. Quantum state tomography and genuine multipartite entanglement witness were analyzed to ensure high fidelity of the prepared four-qubit cluster state. Based on the model of one-way quantum computing, Grover's search algorithm was performed, where the average success rate of the algorithm was 0.960 AE 0.007. (d) Two photon 4-qutrit hyperentangled cluster state. Three-level time-bin entangled state with three frequency modes jψ H i was created on a micro-ring resonator. Then, hyperentanglement of time-frequency jψ C i was realized by a controlled phase gate (composed of fiber Bragg mirrors and an electro-optical phase modulator). Basic high-dimensional one way quantum computing operators were tested. entangled state. 71 The first on-chip demonstration of hyperentanglement was implemented on silica waveguide circuits fabricated by laser writing techniques 34 [see Fig. 1(c)]. Path and polarization DoFs were adopted to prepare a four-qubit cluster state, which then was used to process the Grover's search algorithm in the one-way model. A four-qutrit cluster state with hyperentanglement of frequency and time bins was created in a micoring resonator together with fiber optics 72 [see Fig. 1(d)].
A multiphoton (three-photon) Greenberger-Horne-Zeilinger (GHZ) entangled state was proposed and experimentally demonstrated in 1997 with strong incompatibility to local realism, 73 and it then became a key resource for quantum computing 74 and communication. 75 In a bulk-optics system, to date, an up to 12-photon GHZ entangled state has been reported. 76 In photonic chips, the number of GHZ entangled photons has been limited to 4 to date, 62,77 due to a far less-optimized photon source and relatively high loss of the chip. Multiphoton quantum interference or multiphoton nonentangled states have been prepared on chip. [78][79][80][81] The first on-chip demonstrations of fourphoton GHZ entanglement were reported on two silicon chips, one enabled to be reconfigured to generate both the GHZ state and graph state 77 [see Fig. 2(b)], and the other chip with microring resonant sources to create and verify genuine GHZ entanglement and teleportation 62 [see Fig. 2(a)]. Recently, fourphoton eight-qubit graph entangled states were generated on a silicon photonic chip by remapping the qudit state into qubits. 82 It was reconfigured to implement the one-way quantum computing model and to implement error-corrected qubits [see Fig. 2(c)].
Going beyond the two-level qubit systems, multilevel quantum dit (qudit) systems offer unique properties and new capacities. 83 It not only leads to the expansion of Hilbert space, but brings in new physics and applications, such as stronger Bell violation, 84 noise-robustness in quantum communication, 85 and Fig. 2 On-chip generation of multiphoton entanglement. (a) Generation of four-photon four-qubit genuine GHZ entangled states on a silicon photonic chip. Two photon-pairs with high purity and indistinguishability were produced by an array of microring resonators. Then, a reconfigurable fusion entangling operator was performed on two indistinguishable photons, generating the genuine four-photon GHZ entangled state. Entanglement witness confirmed the genuine multiphoton entanglement. (b) Generation of four-photon four-qubit cluster states on a silicon photonic chip. An entangling gate could be tuned to perform either a fusion operator or controlled-Z operator, by which the linear-and star-cluster states were created. Thanks to the highly reconfigurable photonics chip, all types of four-photon graph states were prepared. Nonlocality of the multiphoton state was verified by the Mermin test. (c) Generation of four-photon eight-qubit graph states on a silicon photonic chip, in which each of the four-dimensional qudits was remapped into two qubits. High-dimensional entangling gates were ultilized to generate four-photon four-dimensional entanglement from two pairs of two-photon four-dimensional entangled states. Error-corrected qubits were used to implement quantum computational algorithms. When running the phase-estimation algorithm with error protection, the success rate would raise from 62.5% to 95 high efficiency in quantum computing. 86 Photonics naturally allows the preparation of multidimensional entanglement in various DoFs. Utilizing an integrated microring, multidimensional entangled states with a 10 frequency-bin have been demonstrated [see Fig. 3(a)], 51 in which arbitrary operation is performed by telecommunication fiber optical components. Qudit states encoded in multiple paths can be arbitrarily and reconfigurably manipulated by integrated quantum photonic circuits. 87,88 A 15-dimensional entangled state has been demonstrated on a large-scale silicon chip, 43 allowing the generation, manipulation, and measurement of entanglement [see Fig. 3(b)]. Genuine multidimensional entanglement was verified by quantum state tomography and experimental violation of generalized Bell inequality and steering inequality. A three-dimensional entangled state was obtained by a similar method. 89 In addition, generalized multipath wave-particle duality, multipath coherence, and multimode quantizations were confirmed on a largescale integrated quantum chip, providing the basics for multidimensional quantum technologies [see Fig. 3(c)]. 90

Outlook and Conclusion
Generation and control of entanglement with integrated quantum photonics could enable profound applications in quantum communication, computing, and simulations. The first chip-to-chip entanglement distribution 67 and quantum teleportation 62 were demonstrated between two programmable photonic chips. The path-polarization conversion technique was invented to ensure the stability and coherence of the chip-to-chip system. Integrated optics may lead to low-cost, compact, fast, and portable chip-scale quantum communication chips. Integrated quantum photonics could provide a reliable, programmable, and scalable system to generate largely entangled cluster states, which is the key for the implementation of measurement-based quantum computing. 34,77 Four-photon four-qubit GHZ states 62 and cluster states 77 have been generated on silicon photonic chips. Two-photon four-qubit cluster states 34 and four-photon eightqubit graph states 82 have been demonstrated on photonic chips. Such controllable quantum devices may find near-term applications in the simulations of complex physical and chemical systems. For example, phase estimation, 91 variational eigenvalue solver, 92 and their combined algorithm 35 were demonstrated to calculate the ground state energy of molecules. Together with machine learning techniques, integrated photonic chips could be adopted to validate the Hamiltonian model and verify the simulating device. 36 The functionality and capability of integrated quantum photonics rely on the ability to generate, control, and analyze complex entanglement. It thus requires further and continuous Each single-photon source would generate a pair of highly indistinguishable photons. By this means, this chip can create, control, and measure 15 × 15 entangled states. Genuine multidimensional entanglement was verified by quantum state tomography, and experimental violation of generalized Bell inequality and steering inequality. (c) Multipath wave-particle duality experiment on a large-scale silicon chip. The state-process entanglement technique was adopted to implement the quantum-controlled generalized Hadamard operations. The process (wave/particle) of the target photon going through would be coherently controlled by the state of the control photon. The generalized multipath wave-particle duality was demonstrated qualitatively and quantitatively. development of on-chip multi-DoF entanglement with many different DoFs, multiphoton entanglement with a large number of single photons, and multidimensional entanglement with a large number of spatial modes.
Thanks to a mature silicon-on-insulator CMOS fabrication process, 93 a large quantity of quantum photonic components can be integrated monolithically on a single silicon chip. 43,44,90,94 Meanwhile, CMOS-compatible platforms based on other material systems, like silicon nitride and high-index doped silica, 95 are also explored widely to make use of their intrinsic optical properties. The negligible two-photon absorption gives silicon nitride a huge advantage over silicon, while the intrinsic ultralow loss makes silica a strong candidate for next generation of integrated quantum photonics platform. Lithium niobate 96 is the first choice for ultra-fast on-chip modulators. The key technical challenge is how to integrate various materials monolithically on a single chip by means of hybrid integration. An integrated chip with multiple superior performances may become the next development direction of integrated photonics quantum technology.
the School of Physics of Peking University. His current research focuses on the integrated silicon photonic platform for quantum information, especially in multidimensional quantum technology for fundamental sciences and applications.
Zhaorong Fu is a senior undergraduate student majoring in physics at Peking University. He joined the PKU Q-chip Lab in 2019. His research focuses on integrated photonics and quantum information.
Qihuang Gong is currently the Boya Chair Professor and Cheung Kong Professor of Physics at Peking University, Beijing, China. His current research interests include ultrafast optics and spectroscopy, nonlinear and quantum photonics, and mesoscopic optical devices for applications in optical information processing and communication. He is an academician of the Chinese Academy of Sciences, a member of the World Academy of Sciences, president of the Chinese Optical Society, and vice president of the Chinese Physical Society. He is a standing committee member of China Association for Science and Technology, vice president for International Commission for Optics, and vice chair for IUPAP C17.
Jianwei Wang received his bachelor's and master's degrees from Zhejiang University, Hangzhou, China, in 2008 and 2011, respectively, and his PhD in physics from the University of Bristol, Bristol, UK, in 2016. He is currently an assistant professor at the School of Physics, Peking University, Beijing, China. He was a postdoc at the University of Bristol. His current research focuses on quantum information science and technologies with photons, in both fundamental physics and advanced applications.