1. Introduction

Entanglement is counterintuitive and proved to be a unique property of quantum system without any classical analogy [1]. Multipartite entangled state strengthens the deep understanding of quantum mechanics and plays as a crucial source in quantum information processing. Although its generation is always a challenging task, multipartite entanglement has been successfully implemented in many physical systems such as ion traps [2], superconducting systems [3], semiconductor quantum dots [4] and optical systems [5]. Among them, photons have particular advantages such as long coherence time and high transmission speed. But it is hard to make photons interact that the probabilistic nonlinear process and the method of post-selection are mainstream approaches for the interaction [6]. Moreover, photons carry information of multiple degrees of freedom, such as polarization [7,8], path [9], time-bin [10], orbital angular momentum [11,12] and frequency [13] etc., which can be encoded as qubits for quantum information processing. Based on bulk optics, theoretical schemes for multiphoton states have been proposed [14–16], and experiments of up to twelve photons have been demonstrated [17–21].

The well discussed multiphoton entangled states include the GHZ states [22] and W states [23] which both are widely used for testing quantum mechanics and applied in quantum technologies. They have markedly different features in enduring loss and are not interchangeable with each other via stochastic local operations and classical communication(SLOCC) [24]. Dicke states ${\left | {D_n^m} \right \rangle }$ are the extension of W states with symmetric $m$ excitations among $n$ photons [25]. It has been shown that Dicke states are able to measure the entanglement depth of the many-body system quantitively [26]. And they are optimal for quantum metrology [27] and important for studying superradiant emission dynamics [28]. Also multiphoton Dicke states should be able to excite the atomic or ion system [29–31] into Dicke states with a high-efficiency.

A common method for expanding the quantum states to more photons is by two-photon interaction [14,19,20], such as fusion operation. Recently, the striking theory for multiphoton entanglement through the path-identity was proposed in 2017 [32], and later this kind of scheme was related to the problem of perfect matching in graph theory [33]. One can explain the optical setups using undirected graph and regard down-conversion photon pairs as edges and output paths as vertices, then the perfect matchings correspond to multiphoton terms. The coherent superposition of all the perfect matchings turns into the multiphoton entanglement. Such theory has been proposed as an advantageous method for the scaling up of multiphoton states and a series of schemes for multiphoton entangled states were proposed [34,35]. This strategy is based on post-selection detection and only the states that every output path exists one and only one photon are desired [6] which is the well-known method to scale up the multiphoton state from probabilistic two-photon source. The post-selected output states have limitations in some quantum tasks [36], as they must be measured in the generation process, however, they can be used in the quantum applications without further multiphoton interaction, such as quantum key distribution [37], tests of non-locality [38] and quantum measurement-based computation [39].

Multiphoton entangled states based on perfect matchings on the graph can be implemented by bulk optics or integrated optics. For bulk optics, there lies great challenge since it suffers from both the phase fluctuation in the optical setup and the complicated alignment of massive nonlinear crystals and optical paths. Integrated optics brings scalability and stability [40] with massive miniature components [41,42] and can supply a promising strategy for overcoming the above challenges and further for large-scale quantum computation [43]. Quantum photonic chips based on $\chi ^{(3)}$ silicon-on-insulator (SOI) [44] or $\chi ^{(2)}$ lithium niobate (LN) [45] have been demonstrated to contain the on-chip spontaneous four-wave-mixing (SFWM) or spontaneous parametric down conversion (SPDC) two-photon sources for constructing edges, and also the thermal-optical or electro-optical phase-shifters for reconstructing the graph and hence for reconfiguring the multiphoton states. Recently multiphoton states from the frequency comb [46] and polarization entanglement [47] have been demonstrated from a single micro-ring or a straight waveguide on SOI chip. A reprogrammable photonic chip for generating four-qubit graph states has been reported [48] and up to eight identical photons are prepared for demonstrating Boson sampling on a chip [49]. However, the on-chip generation of widely discussed GHZ, W and Dicke states are not reported except a recent theoretical proposal which is only efficient for GHZ states [50].

In this paper, we propose two types of on-chip schemes for generating a variety of multiphoton entangled states which both are based on the entanglement theory on the graph [33]. Both schemes are scalable in the photon number and reconfigurable for different types of multiphoton states. Besides, for each multiphoton state, the encoded degree of freedom can be switched from the path to the polarization which will enable the on-chip quantum information processing and the off-chip free-space quantum tasks, respectively. The paper is organized as follows. In Section 2., we briefly introduce theory of entanglement on the graph. In Section 3., we develop the on-chip path identity between two-photon sources and propose a general on-chip schemes for multiphoton states. In Section 4., we present another on-chip scheme for engineering the Dicke and W states by the linear beam-splitter network which is proved to be an efficient and simplified scheme. Finally, in Section 5. we make a conclusion and give suggestions on the practical option of two schemes for different types of multiphoton entanglement.

2. Theory of multiphoton state on the graph

In this section we mainly introduce the theory of multiphoton entanglement on the graph [33–35]. Multiphoton entanglement corresponds to the coherent superposition of all the perfect matchings of the graph wherein the graph’s edges stand for the nonlinear crystals and the vertices represent the photons’ output paths. It should be noted that the graph mentioned here is independent of the graph state [51] which can also be represented by a graph. It seems interesting to further associate these two kinds of graphs beyond the scope of this work. In the following content, we prefer to choose the engineering of Dicke states as the example and then discuss its reconfigurability. As shown in Fig. 1(a), the general graph for Dicke states ${\left | {D_n^m} \right \rangle }$($0<m<n$, $m$ and $n$ are integers) which is given by Ref [35] can be divided into four parts: two complete graphs($K_m$ and $K_{n-m}$) and two complete bipartite graphs($K_{m,n-m}$ and $\tilde {K}_{m,n-m}$). The red(dark grey), black, red-black(dark grey-black), black-red(black-dark grey) edges correspond to two-photon state with ${\alpha \!\left | {00} \right \rangle }$, ${\beta \!\left | {11} \right \rangle }$, ${\gamma \!\left | {01} \right \rangle }$ and ${\delta \!\left | {10} \right \rangle }$, respectively. We define that $K_{m,n-m}$ is the graph with m vertices encoded in 1 and the other n-m vertices encoded in 0, and conversely $\tilde {K}_{m,n-m}$ with m vertices encoded in 0 and the other n-m vertices encoded in 1. In this paper, 0 and 1 represent for the horizontal and vertical polarization, or down and up path, respectively. The n-fold coincidence from the n vertices result in Dicke states ${\left | {D_n^m} \right \rangle }$. Controlling the weight of these edges, Dicke states can be optimized to be maximally entangled. For n-photon W state ${\left | {W_n} \right \rangle }$, the maximally entangled condition is $(n-1)\beta =\gamma$, and for n-photon Dicke state with two excitations, the maximally entangled condition is $\alpha \delta /2+(n-2)(n-3)\beta ^2=(n-3)\beta \gamma =2\gamma ^2$ [35]. But other symmetric Dicke states can not be generated from the graphs except ${\left | {D_n^{n/2}} \right \rangle }$ because of the reachless symmetric conditions.

 

Fig. 1. General graphs for Dicke states [35] and an instanced scheme.(a) General graph for general Dicke state ${\left | {D_n^m} \right \rangle }$($0<m<n$) which can be divided into four parts: two complete graphs($K_m$ and $K_{n-m}$) and two complete bipartite graphs($K_{m,n-m}$ and $\tilde {K}_{m,n-m}$). (b)The scheme for state ${\left | {D_6^2} \right \rangle }$ by bulk optics.

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A direct mapping of this graph to the quantum experiments will require $C_m^2+C_{n-m}^2+2m(n-m)$ two-photon sources as edges, and n output paths as vertices. The key challenge lies in how to merge the $m+n-1$ or $2n-m-1$ edges linking one vertex into 1 output path. One elegant method, namely the path-identity is suggested [32] wherein one of the paths from each two-photon source is perfectly overlapped and then two-photon sources are induced to be in a coherent superposition. Then the photons arrive in the same output path irrespectively in which sources they are created. This way of merging different paths into one output path requires the two-photon sources sharing one of their two paths to be aligned in series. In bulk optics, the noncollinear SPDC is a natural way to realize path identity enabling the two output paths of each SPDC crystals to be separated aligned and ensuring one of the path to be overlapped with the path from other crystals. We depict the optical alignment for Dicke state ${\left | {D_6^2} \right \rangle }$ in Fig. 1(b) based on bulk optics wherein the noncollinear two-photon emission is supposed. Every crystal produces a two-photon state. The label on crystal shows the two-photon state of each crystal and also the outputs of corresponding two photons. For example, the two photons produced from the crystal marked ${\left | {01} \right \rangle }_{af}$ emitting to the path $a$ in state ${\left | {0} \right \rangle }$ and path $f$ in state ${\left | {1} \right \rangle }$, respectively. When all the nonlinear crystals are pumped coherently, a multiphoton Dicke state ${\left | {D_6^2} \right \rangle }$ is achieved under six-photon coincidence of six outputs.

3. Multiphoton entanglement by on-chip path-identity and the network of anti-bunching two-photon sources

Due to the large dispersion between the TE and TM mode on the photonic chip [52] which requires delicate and complex efforts to compensate the polarization dispersion for multiphoton entanglement, path is the leading choice of degree of freedom on chips because of its high flexibility for manipulation and scalability for high-dimensional space. The path-encoded multiphoton entangled states can be transformed into the polarization-encoded states by using a path-to-polarization converter. However, the difficulty in achieving the noncollinear two-photon source comes out since the on-chip waveguide based on either SPDC or SFWM process are always collinear. We propose to adopt an equivalent noncollinear two-photon source, namely the anti-bunching two-photon source, on the chip [44,45] which is the result of reversed Hong-Ou-Mandel(RHOM) interference(detailedly described in Section 4.1). In the following design we adopt the SOI chip, and the degenerate SFWM process is considered to ensure all the converted photons are identical in frequency.

The layout of the scheme for path- and polarization-encoded Dicke state ${\left | {D_6^{2}} \right \rangle }$ is depicted in Fig. 2 in which the building block is an anti-bunching two-photon source. For the on-chip path-encoded Dicke state, each vertex has two corresponding output paths of ${\left | {0} \right \rangle }$ and ${\left | {1} \right \rangle }$ respectively, and each anti-bunching two-photon source corresponds to an edge linking two of twelve output paths. For example, a black-red(black-dark grey) edge connecting a and f can be realized by an anti-bunching source marked ${\left | {10} \right \rangle }_{af}$ whose two photons respectively emitting to $a_1$ and $f_0$ paths. When all the anti-bunching two-photon sources are coherently pumped in specific ratio, and the six-fold coincidences from the six groups of paths($p_0$ and $p_1$ is one group of paths, $p=a,b,c,d,e,f$) engenders the path-encoded Dicke state. If the path-to-polarization convertors are set, a final polarization-encoded state ${\left | {D_6^{2}} \right \rangle }$ is achieved which is transformed from the path-encoded ones.

 

Fig. 2. The layout of on-chip scheme for state ${\left | {D_6^2} \right \rangle }$ by the path-identity and the network of anti-bunching two-photon sources.

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To achieve the maximally entanglement, the ratio between different multiphoton terms should be optimized. This requires the management on the pump’s power incident on each anti-bunching source. To meet this condition, another waveguide network which is independent on the anti-bunching sources network is suggested to distribute the pump’s power and adjust the phase before each anti-bunching source. The pump can be combined with or filtered from the signal/idler photons by a wavelength-division-multiplexer (WDM) which can be implemented on chip by an asymmetric Mach-zehnder interferometer [53] that the length difference of two arms $\Delta {L}$ satisfies $(k_p-k_{s,i})\Delta {L}=(2t+1)\pi$ ($k_p$ and $k_{s,i}$ are the propagation constant of pump and the signal/idler photon respectively in the waveguide, $t$ is integer). We set two WDMs to inject pumps into the MZI for anti-bunching two-photon generation(see Section 4.1 and Fig. 4(a) for details) in order to guarantee that two new photons are separated at two exits of this source and the photons produced in previous sources keep their paths. And another two WDMs are set after the source for pump filtering. In the lower left box of Fig. 2 illustrate how the pump’s network is mixed with the anti-bunching source’s network. The box should meet two conditions. One is to generate anti-bunching two-photon state and the other is to guarantee that the anti-bunching photons generated from the previous boxes keep the their paths after passing through this box. Thus, an in-phase two sub-pumps are required to inject into the MZI’s two input ports, respectively and the phase-shifter inside the anti-bunching source should be set as $\pi$. Moreover, by configurating the network of pump, we can control each source to work or not [54], so the subgraphs of the graph are all achievable by the same chip. For example, the scheme for Dicke state ${\left | {D_6^{2}} \right \rangle }$ in Fig. 2 can also generate states ${\left | {W_4} \right \rangle }$ and ${\left | {D_4^{2}} \right \rangle }$ by reconfigurating the pump network. The pump network together with the anti-bunching two-photon source network may require plenty of waveguide components, but is still in the reach of state-of-the-art SOI chip since recently larger-scale integrated quantum chip with more than 500 components has been demonstrated [41].

This is an all-purpose method for on-chip generation of multiphoton entangled states and all the graphs can be constructed by this design such as the graphs for GHZ, W and Dicke states. However, the condition that all the anti-bunching sources are coherently pumped should be satisfied for guaranteeing the succuss of multiphoton entanglement. The arrival-time of photons to each output path should be indistinguishable despite that the photon may come from different two-photon sources. This temporal indistinguishability can be reached after carefully calculations on each optical path on chip. Moreover, by setting a phase-shifter on the pump before each two-photon source, the relative phase between any two terms is adjustable.

4. Multiphoton entanglement by the beam-splitter network

In this section, we propose a different scheme to construct the graph for the multiphoton entanglement. In this scheme the linear beam-splitter network is used for both the edges and the vertices which can greatly decrease the amount of two-photon sources and simplify the circuits.

The challenge appears when using the beam-splitter network to combine multiple path into one. Usually, for a vertex linking $n_1$ identically encoded edges, a beam-splitter network for path combination should at least be $log_{2} n_1$ layers. This $n_1$-to-1 combination lead to a big loss of photons and cause the involved edges’ efficiency dropping down to $1/n_1$. For a certain graph, there is positive correlation between the efficiency of edges and total success rate of multiphoton entanglement. The efficiency of detecting the perfect matching or multiphoton decreases at a complexity of $1/(n_{1})^{n}$ wherein the total vertex number is $n$ and each vertex has the same degree of $n_{1}$. In the graph for GHZ states, the degree of every vertex defined as $n_0+n_1$ with $n_0=n_1=1$($n_0$ and $n_1$ are the number of edges linking this vertex with $\left | {0} \right \rangle$ and $\left | {1} \right \rangle$, respectively), thus this scheme is equal to the above path-identity scheme and also the scheme for GHZ states in [50]. While for Dicke states, the efficiencies of edges are decreased severely especially when the state scales up to more photons [50].

To overcome the problem above, we figure out a solution for Dicke states with a maintained high success rate. The design can be explained by two steps. In the first step, as shown in Fig. 3(a), complete graphs $K_m$ and $K_{n-m}$ can be constructed by one single collinear two-photon source passing through a 1-to-$m$ beam-splitter and 1-to-($n\!-\!m$) beam-splitter, respectively. The post-selection probability of our graph-theoretical schemes includes two parts: the photon-pair efficiency and the probability of perfect matchings. The photon-pair efficiency is the proportion of the valid edges which is the goal of our optimization. The probability of perfect matchings is the intrinsic post-selection characteristic of graph-based experiments. Although photon pair has the probability of shooting into the same detector which brings about some loss, the total number of required photon sources for $K_m$ and $K_{n-m}$ decreased from $C_m^2+C_{n-m}^2$ into 2. The photon-pair efficiency to realize complete graph $K_m$, i.e. the amplitude proportion of all edges is calculated following $2C_m^2/(2C_m^2+m)=(m-1)/m$ which converges to 1 as m increases to infinity. Complete bipartite graphs $K_{m,n-m}$ and $\tilde {K}_{m,n-m}$ are respectively constructed by one single anti-bunching two-photon source and 1-to-$m$ and 1-to-($n\!-\!m$) equiprobable beam-splitter networks. The original anti-bunching two-photon amplitude is divided into $m\times (n-m)$ two-photon amplitudes which correspond to the $m\times (n-m)$ edges in graph $K_{m,n-m}$ or $\tilde {K}_{m,n-m}$. The photon-pair efficiency for two complete bipartite graphs is 1 without any theoretical loss. Then edge merging from four separated graphs will introduce huge loss by beam-combination process. However, we find that the establishments of three graphs $K_m$, $K_{n-m}$ and $K_{m,n-m}$ require the same size of beam-splitter network, so it is possible to establish three graphs simultaneously via the same beam-splitter network. Then the edge merging from three graphs is completed with no beam-combination loss. So in step 2, we superpose the quantum source for graphs $K_m$, $K_{n-m}$ and $K_{m,n-m}$ into one single quantum source(see in Fig. 3(b)). This single source produces a superposition of bunching state and anti-bunching state which can be achieved by modulating the phase in the RHOM interference. Then graphs $K_m$, $K_{n-m}$ and $K_{m,n-m}$ are constructed simultaneously wherein the bunching and anti-bunching sources are sharing optical paths, thus it is more efficient than the beam-combination method. Although the merged quantum source brings a new condition between the edge weights, it doesn’t break the symmetric conditions. The final edge merging between the aggregation of $K_m$, $K_{n-m}$, $K_{m,n-m}$ and $K_{n-m,m}$ adopts the path-to-polarization convertor such as two-dimensional grating for transforming the up and down paths into vertical and horizontal polarizations, respectively, with high fidelity and high extinction ratio for cross conversion. This combination between orthogonal modes is lossless theoretically. So before and after the path-to-polarization converter, we obtain an on-chip path-encoded Dicke state and off-chip polarization Dicke state, respectively, in an efficient way.

 

Fig. 3. The layout of on-chip scheme for multiphoton entanglement by the beam-splitter network. (a) Optical circuits for separate graphs $K_m$, $K_{n-m}$, $K_{m,n-m}$ and $\tilde {K}_{m,n-m}$) and (b) a simplified circuits for simultaneously construction of them.

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In the following, we describe detailedly how to implement the Dicke states on the photonic chip. Moreover, we give the configuration of different multiphoton entangled states and analyze the photon pair efficiency for each state.

4.1 Elements on a quantum photonic chip

Firstly we briefly introduce the key components on the quantum photonic chip which support our schemes. Figure 4(a) shows one component which is capable of supplying all the required quantum sources such as the anti-bunching two-photon source and the superposition of bunching and anti-bunching states. It is composed of two multi-mode interferometers(MMIs), two collinear sources based on degenerate SFWM and a phase shifter wherein each SFWM photon-pair should be non-correlated in frequency to ensure the high distinguishability between different photons. The spectrally non-correlated SFWM can be realized but not limited by the reported methods [49,55–57], such as the spiralled waveguide and the micro-ring resonator. The pump power should be set reasonably to make sure that the possibility of higher-photon-number states is low enough to be neglected. The RHOM process [44,45] produces the state

 

Fig. 4. The key elements of optical quantum chip. (a) Two-photon reversed Hong-Ou-Mandel interference. (b) The Mach-Zehnder Interferometer. (c) Spiralled waveguide or micro-ring resonator for on-chip SFWM. (d) Two-dimensional grating coupler for path-to-polarization conversion.

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On-chip MMI is used for beam splitting or combining. The Mach-Zehnder interferometers(MZI) can split the beam into two paths with arbitrary splitting ratio. The MZI contains two 50:50 MMIs and a phase shifter between them(see in Fig. 4(b)), whose evolution matrix is

Different from conventional grating coupler, two-dimensional grating [58,59] allows the photons from two modes to pass through as shown in Fig. 4(d). Photons from the upper path carry vertical polarization information to the exit of 2D grating, and from lower path, horizontal photons output. The 2D grating coupler provides a path-to-polarization transformational interface.

4.2 Multiphoton Dicke states with one or two excitations

The on-chip scheme for ${\left | {D_n^m} \right \rangle }$ with $m=1\left ({\left | {W_n} \right \rangle }\right )$ or $m=2\left ({\left | {D_n^2} \right \rangle }\right )$ is shown in Fig. 5. The chip can be divided into three regions: photons generation, distribution and transcoding. In the first region, pump lasers are tunably divided and degenerate SFWM occurs in four resonators. After two RHOM processes, the two-photon state turns into a superposition:

 

Fig. 5. The practical on-chip scheme for generating states ${\left | {D_n^m} \right \rangle }$ with m=1$\left ({\left | {W_n} \right \rangle }\right )$ or m=2$\left ({\left | {D_n^2} \right \rangle }\right )$.

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In the second region, according to the target state and the corresponding graph, these photon pairs are distributed to construct the graph by the MZI network. Taking the ${ \left | {D_6^{2}} \right \rangle }$ as the example, the upper anti-bunching source constructs the complete bipartite graph $\tilde {K}_{2,4}$(complete connection between $\left \{a_0, b_0\right \}$ and $\left \{c_1, d_1, e_1, f_1\right \}$), while the lower superposition source establishes $K_2$, $K_4$, $K_{2,4}$ simultaneously (complete connection within $\left \{a_1, b_1\right \}$ and $\left \{c_0, d_0, e_0, f_0\right \}$ and complete connection between $\left \{a_1, b_1\right \}$ and $\left \{c_0, d_0, e_0, f_0\right \}$). The parameters $x,y$ associated with edge’s weights $\gamma$ and $\beta$ respectively and $z$ determine both of the weights $\alpha$ and $\delta$. By setting parameters $x,y,z$ the weights of the edges can be modified. So far, path-encoded state is achieved. We explain the state evolution of four-photon case for facilitating the understanding. After the second region, the four-photon state can be calculated as following:

In Eq. (4), the first ten two-photon terms are valid representing ten edges. The last four terms are invalid, which can not contribute to the four-photon event in the later post-selection process. The photon pair efficiency is ${\left | {x} \right |}^2+{\left | {y} \right |}^2+{\left | {z} \right |}^2$, i.e. the probability of valid terms. If the post-selection detection acts on the $a,b,c,d$ vertices with each vertex has one and only photon, the terms in Eq. (4) are truncated with a post-selection operator and then the four-fold coincidence can be written as

In the third region, the path-encoded state converts to polarization-encoded via 2D grating array.

Figure 6 shows the graphs for all possible multiphoton entangled states allowed by the designed chip in Fig. 5 and also shows the set of vertices involved with different target states, such as $\left \{b, c, d, e\right \}$ for ${\left | {W_4} \right \rangle }$, $\left \{a, b, c, d, e, f\right \}$ for ${\left | {D_6^2} \right \rangle }$, etc.. Table 1 lists the parameters. For example, if the six-photon Dicke state ${\left | {D_6^2} \right \rangle }$ is desired, one should set the parameters into: $x={3}/{\sqrt {23}}, y={2}/{\sqrt {23}}, z=\sqrt {5}/{\sqrt {23}}$, and distribute the photon pairs to vertices a,b,c,d,e,f in the second region. By simply deepening the MZI networks and distributing photons to more paths, we can extend W states and Dicke states to higher photon numbers. Thus, the designed chip is reconfigurable for a variety of multiphoton entangled states as listed in Table 1. For n-photon W state ${\left | {W_n} \right \rangle }$, the photon pair efficiency converges to 3/4 as n increases to infinity, while for n-photon Dicke state ${\left | {D_n^2} \right \rangle }$, the photon pair efficiency converges to 1 as n increases to infinity.

 

Fig. 6. The corresponding graphs of the states which can be generated by the chip in Fig. 5.

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Table 1. The key parameters and photon pair efficiency for different multiphoton states from chip of Fig. 5

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4.3 Multiphoton Dicke states with half of subsystems excitations

We also propose a general design for constructing the graph for Dicke state with half of subsystems excited ${\left | {D_n^{n/2}} \right \rangle }$ whose layout is shown in Fig. 7(a). In the general graph for ${\left | {D_n^{n/2}} \right \rangle }$, vertices are connected to each other by both red-black(dark grey-black) edge and black-red(black-dark grey) edge which correspond to the state ${\left | {01} \right \rangle }+{\left | {10} \right \rangle }$ [35]. The weights of edges are required to be equal for maximally entanglement. The graph can be efficiently constructed by only one anti-bunching two-photon source. And each of two photons equiprobably splits into n paths before path-to-polarization convertors.

 

Fig. 7. (a) The layout and (b) the on-chip practical scheme for constructing the graph of Dicke state with half of subsystems excited ${\left | {D_n^{n/2}} \right \rangle }$ .

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As shown in Fig. 7(b), the on-chip scheme for generating Dicke states ${\left | {D_n^{n/2}} \right \rangle }$ also includes three regions i.e. photons generation, distribution and transcoding. An anti-bunching two-photon source is prepared first, and then each photon equiprobably distributes to n paths by a MZI network. After beam-splitting, a path-encoded multiphoton state is got and it will transform to be polarization-encoded by 2D grating couplers. For example of n=6, a Dicke state ${\left | {D_6^{3}} \right \rangle }$ is got when six detectors click simultaneously.

Also one can easily extends the scheme to generate the ${\left | {D_n^{n/2}} \right \rangle }$ state with more photons by simply deepening the MZI network to ensure more distributed paths, without changing the region of quantum source. The photon-pair efficiency is $(n-1)/n$, which converges to 1 as n increases to infinity. This generating process is similar to the experiments in [18,21], and have the same theoretic efficiency, but it is an on-chip version which is scaled and reconfigured more easily.

5. Conclusion

The above multiphoton states with n-1 photons can be directly obtained from the n-photon state by special operations. For an n-photon W state, if one qubit triggers, the rest of qubits forms a new W state with n-1 photons. For an n-photon Dicke state ${\left | {D_n^2} \right \rangle }$, if one qubit is projected to ${ \left | {1} \right \rangle }$ before detected, the rest of qubits is a W state with n-1 photons [21]; and if one qubit is projected in ${ \left | {0} \right \rangle }$ before detected, the rest of qubits is a Dicke state ${\left | {D_{n-1}^2} \right \rangle }$ [60]. Thus, all the Dicke states within six photons can be generated by our on-chip schemes.

The matches of graph other than perfect matchings contribute amplitude to invalid multiphoton terms and will be discarded by post selection of each detector receiving one and only one photon. Experiments with higher success rate may be realized referring to the theory between quantum experiments and hypergraphs [16,61] if nonlinear source can produce a correlation of more than two photons [62]. Besides, in-waveguide propagation loss and imperfections in real devices such as the asymmetric MMI and the modes crosstalk of grating coupler, will lower down the multiphoton success rate and even lead to error multiphoton terms. Rapid development of on-chip devices and other low-loss waveguide based on different substrate material will be helpful to reduce these effects.

In practical experiments for relatively small graphs such as the graphs for W, Dicke states with few photons and GHZ states, the first scheme based on the path-identity and the network of anti-bunching two-photon sources is suggested. Otherwise, for complex graphs for W and Dicke states with large number of photons, the second scheme is more suitable which only requires a small number of two-photon sources and simple beam-splitter networks.

In conclusion, we propose two different schemes for generating multiphoton entangled states on a quantum photonic chip which both are based on the theory of entanglement on the graph. Both schemes can be reconfigured to generate several target states. These results supply a systematic solution for the on-chip generation of a variety of multiphoton entanglement which is superior to bulk optics by mass-production, low-cost, stability, scalability and high-efficiency. All of the proposed chips can be realized within existing integrated photonic technologies which will promote the practical development of multiphoton quantum technologies.