Preparation of multiphoton high-dimensional GHZ state

Multipartite high-dimensional entanglement presents different physics from multipartite two-dimensional entanglement. However, how to prepare multipartite high-dimensional entanglement is still a challenge with linear optics. In this paper, a multiphoton GHZ state with arbitrary dimensions preparation protocol is proposed in optical systems. In this protocol, we use auxiliary entanglements to realize a high-dimensional entanglement gate, so that high-dimensional entangled pairs can be connected into a multipartite high-dimensional GHZ state. Specifically, we give an example of using photons' path degree of freedom to prepare a 4-particle 3-dimensional GHZ state. Our method can be extended to other degrees of freedom and can generate arbitrary GHZ entanglement in any dimension.


Introduction
Multipartite entangled states are used to test the basic problems of quantum mechanics [1]. In addition, they find application in quantum computing [2][3][4] and quantum sensing [5]. Multipartite entanglement is the core concept in many quantum processes, such as quantum teleportation [6,7], dense coding [8,9], and entanglement-based quantum key distribution [10]. Compared to two-partite, two-dimensional (2D) systems, multipartite high-dimensional systems offer superior performance in fundamental research on quantum information and quantum computing.
One special multipartite entangled state is the Greenberger-Horne-Zeilinger (GHZ) state [11], |GHZ = (|000 + |111 )/ √ 2. The generalized GHZ state is an entangled quantum state comprising > 2 subsystems. If the dimension of each system is , meaning that the local Hilbert space is isomorphic to C , then the total Hilbert space of the -partite system is H = (C ) ⊗ . This GHZ state is also called the -partite qudit GHZ state: One of the remarkable properties of the GHZ state is maximizing the entanglement monotones [12,13]; therefore, it is called maximally entangled in the multipartite sense. GHZ states find a wide range of applications. The entanglement between multiple parties is an essential feature of quantum secret sharing [14] and computational protocols [15]. The core support in completing these quantum protocols is the preparation of GHZ states, which has been well established in many experimental platforms, such as nuclear spins of a molecule [16], trapped ions [17,18], nitrogen-vacancy centers in diamond [19], superconducting circuits [20][21][22], silicon [23], and photons [24][25][26][27]. However, these studies only prepared 2D GHZ states.
Photons are the most successful physical systems for observing multipartite entanglements [28]. Therefore, many research groups have focused on optical systems to establish multipartite high-dimensional entanglements. Preliminary attempts in Refs. [29,30] for high-dimensional multiphoton produced asymmetric multipartite entanglement. To prepare high-dimensional GHZ states ( > 2), a method called "path identity" was proposed [31,32], which was extended on the Zou-Wang-Mandel experiments [33] using a computer program [34]. Recently, this method has been successfully used to prepare GHZ states |GHZ 3 4 [35]. However, "path identity" is insufficient for preparing arbitrary multiphoton high-dimensional GHZ states since "path identity" can only create arbitrarily large 2D and four-photon 3D GHZ states using graph theoretical methods till now [32]. Stimulated by reference [7], we proposed a method for preparing multiphoton high-dimensional GHZ states on optical systems. All the unwanted multiphoton terms can be canceled by polarizing beam splitters and auxiliary entangled photons to achieve a multiphoton high-dimensional GHZ state.

Preparation of multiphoton GHZ state
In any multiphoton entanglement preparation process, the photons from different two-partite entanglement sources are made indistinguishable (e.g., polarization, spatial mode, temporal mode, and frequency) by replacing the temporal determinant of the coherent window with ultrashortpulsed optical pumping [36]. This significantly reduces the coherence time requirements of the interfering photons and simplifies the establishment of multiphoton 2D entanglement.

Four-photon GHZ state
We introduce the unit of this multiphoton entanglement source, which is a post-selected fourphoton entanglement source. The four-photon preparation device is shown in Fig. 1; each Einstein-Podolsky-Rosen (EPR) source produces an entangled state.
where | and | represent the photon's horizontal and vertical polarizations, respectively. Therefore, our input state is After the photons pass through PBS1(polarizing beam splitter1 ) in Fig. 1, we select the event where each exit of the PBS has one photon, indicating that the event occurs only when photons 2 and 3 with the same polarization enter PBS. The coincidence count of the detector reveals that only the first term | 1 2 3 4 and fourth term | 1 2 3 4 remain in Eq. (3). When the Hong-Ou-Mandel (HOM) interference occurs [37], the coherent superpositioning of the two remaining terms in Eq. (3) yields the four-photon GHZ state

Multiphoton GHZ state
The device in Fig. 1 is scaled to construct additional photonic GHZ states. For the GHZ state ofphoton, /2 EPR entangled pairs are required to constitute the input state | 2 = |EPR ⊗ 2 .
. This operation eliminates the distinguishability of the polarization of photons 3 and 4. This device can be used as a unit of the multiphoton entanglement source. We can get an n-photon entanglement source if we repeat the facility of this device; PAS: polarization analysis system; QWP: quarter-wave plate; HWP: half-wave plate; PBS: polarizing beam splitter; APD: avalanche photodiode. | 2 onto the target state-the -photon GHZ state: where ( ) indicates the smallest (biggest) integer that is greater (less) than or equal to . The multiphoton GHZ preparation process is an extension of the preparation process of the four-photon process. In the multiphoton preparation ( > 4) process, we have to make adjacent EPRs interfere with each other by PBSs and post-select the multiphoton terms we wanted. Checking the HOM interference at each PBS ensures that all EPR sources are in coherent superposition.
This multiphoton extension scheme was validated through several experiments [24][25][26][27]. The spontaneous parametric down-conversion (SPDC) method is used for preparing multiphoton entanglement sources [38]. The real difficulty in preparing multiphoton high-dimensional GHZ states lies in the high-dimensional part generating additional multiphoton cross terms. However, these terms cannot be completely removed by PBS, thus auxiliary entanglement is required.

Preparation of multiphoton high-dimensional GHZ state
A challenging problem in constructing the high-dimensional GHZ state |GHZ is how to cancel the multiphoton cross terms (MCTs) in the input state: where and are the dimension and number of photons in the multiphoton state, respectively. The multiphoton terms that can be expressed as | are referred to as MCTs, where , = 0, 1, . . . , − 1 and ≠ . In higher dimensions, to solve this problem, we introduce auxiliary entangled sources [7]. We use | 3 as an example to illustrate our protocol.

Four-Photon 3D GHZ state
When = 3, the input state can be expanded as where | ( , = 0, 1, 2) from left to right, represents photons 1-4 in Fig. 2, respectively, here we encode the systems by the path degree of freedom of photons.

2'
3'  Step ii: Introduce auxiliary entanglement | and cancel all the other MCTs after Step i. After PBS1, we only retain terms |0000 , |0022 , |1111 , |2200 , and |2222 . MCTs |2200 and |0022 are filtered out using auxiliary entanglement | . We choose the appropriate six-fold coincidence to obtain the output state |GHZ 3 4 with probability 1/12 at out1-4. The right side represents the post-selection process for any dimension. After the post-selection of PBS1, part (b) on the left side of the figure is repeated until all MCTs are removed. Different color beams indicate different entangled photons. BD: Beam displacer. PAS: Polarization Analysis System, the structure is shown in Fig. 1 . Fig. 2 presents the preparation idea for |GHZ 3 4 , which is similar to |GHZ 2 4 described previously. We cancel all terms in the input state except | ( = 0 · · · 2), which we call MCTs. The preparation process is divided into two steps (Fig. 2 left).
i) Use PBS1 to remove part of the MCTs.
ii) Add auxiliary entangled photons to remove remaining MCTs.
Finally, the photons pass through HWP1-2 and BD1-2 to obtain the following state: We rewrite the above state as: We choose the six-fold coincidence count with | or | at out5-6 with probability 3 = 1/2 and obtain the following state: We can use the input state | 3 4 from Eq. (7) with probability 3 =1 = 1 12 to obtain the 4-photon 3D GHZ state |GHZ 3 4 . Furthermore, when we choose the six-fold coincidence count with | or | in out5-6, we can obtain a quantum state 1/ √ 3(|0000 + |1111 − |2222 ), which is not a GHZ state. If based on the measurement results, we apply a phase modulation to the photons in path 2, which can be deterministically accomplished with an electro-optic feedforward [39][40][41][42], we can obtain the same GHZ state as in Eq. (13). At this point, the success probability of obtaining the 4-photon 3D GHZ state is 1/6.

High-dimensional GHZ state
The protocol is scalable to arbitrary dimensions. The preparation protocol for ≥ 2 with four-photon GHZ states |GHZ 4 in arbitrary dimensions can be generalized as follows: i) Prepare two 2-photon -dimensional maximal entangled state: which encode by the path degree of freedom, and control the polarization of each path as follows: if is odd, the polarization is set at ; otherwise, it is set at . Then, when the photons pass through PBS1 in Fig. 2, we select the events with only one photon at each exit of PBS1.
iii) Repeat Step. ii until all MCTs are canceled.

After
Step. i, the MCTs between the odd-and even-numbered paths are canceled; therefore, in Step. ii, , is either odd or even. There are /2 paths encoded as even and /2 encoded as odd. When we repeat Step. ii C 2 /2 ( C 2 /2 ) times, we can cancel MCTs between odd-(even-) numbered paths. We conclude that 4 = C 2 /2 + C 2 /2 = ( − 2)/4 auxiliary entanglements are required to prepare four-photon GHZ states in dimension. It should be noted that before any post-selection by PBSs, the HOM interference on the PBS must be checked to ensure the coherent superposition of the auxiliary entangled photons and the | photons. The post-selection in Step. i filter out all MCTs between the odd and even paths (i.e, the MCTs which can be written as | , for odd numbers and for even ones, or vice versa), which have 2 /2 × /2 terms. Thus the success probability of Step. i in dimension is 1 = 1 − 2 /2 × /2 / 2 . Two more MCTs will be canceled whenever Step. ii is performed. Some multiphoton terms are retained with probability 1/2, e.g. | ( = 0, 1, 2) in the preparation of |GHZ 3 4 is retained with probability 1/2, and with the help of phase modulation, the probability of selecting the correct multifold coincidence is 1. Thus, the success probability of th Step. ii is: The success probability in the preparation of |GHZ 4 is = 1 × 4 =1 2 ( ) = 1/2 4 × ( 1 − 2 4 / 2 ), where 1 − 2 4 / 2 = 1/ , thus = 1/2 4 × 1/ . If = 3, then = 1/6. This is consistent with the result in Section 3.1. Fig. 3 shows the diagram of the preparation of |GHZ . We prepare the state |GHZ 4 by sending the adjacent photons from the two 2-photon -dimensional maximal entangled state: | = ( −1 0 | )/ √ through the MCTs filter (shown as the first blue line in Fig. 3.), which filters out multiphoton terms unrelated to the GHZ state. We proceed to prepare the entanglement source | and pass it through the MCTs filter, and obtain |GHZ state. The post-selection processes by PBSs and auxiliary entanglements are included in the MCTs filter. The MCTs filter is implemented in two parts and contains all the post-selection processes. The first part includes the post-selection using PBS1, shown in Fig. 2(a). The second part contains the post-selection part using several auxiliary entanglements as shown in the right of Fig. 2.

Multiphoton high-dimensional GHZ state
The number of photons in our input state is always an even number. When we want to prepare a GHZ state with an odd number of photons, we need to perform the measurement = | | for the photons of out1, where the measurement basis | = 1/ √ (|0 + |1 + · · · + | − 1 ), so that we will change with probability 1/ from the GHZ state of even photons to odd number. To improve the probability, we can perform a -output measurement of the photon at out1. Let us take an example in three dimensions: if one of our measurements base is 3 , then the measurement bases of the remaining two measurement bases are: 1/ √ (|0 + We can find that this -output measurement corresponds to a complete Fourier base measurement performed on the photon at out1. Each measurement output represents the photon performing a Fourier basis measurement that would make it have 2 phases between adjacent multiphoton terms (| ) of our final state. If we apply the phase modulation device from the previous section, we make it so that the photon in out1 is capable of preparing a GHZ state regardless of which measurement output it is detected by.   Fig. 3. Schematic diagram of multiphoton high-dimensional GHZ state |GHZ preparation : We start from the state |GHZ 4 . When two adjacent photons belonging to different | and pass through the MCTs filter, which allows filtering out the multiphoton term unrelated to the GHZ state. Then we add another | and the MCTs filter, and after a similar post-selection process, we get the state |GHZ 6 . We prepare the state |GHZ = (|0 ⊗ + |1 ⊗ + · · · + | − 1 ⊗ )/ √ by repeating the process of combining | and the MCTs filters. The MCTs filter contains two post-selection processes. The first process is post-selection using PBS (as illustrated in Fig .2(a)), while the second involves auxiliary entangled photons (as illustrated on the right side of Fig .2).
The multiphoton entanglement source requires /2 | and = ( − 2)/4 × /2 − 1 auxiliary two-partite 2D entanglement sources to filter out the MCTs. When preparing a multiphoton high-dimensional GHZ state, the input state contains /2 multiphoton terms, of which we preserve only of them (the probability is 1/ /2 −1 ). If auxiliary entanglement is introduced, the retained multiphoton terms will be kept with a probability of 1/2 (e.g., multiphoton terms | will be retained by probability with 1/2). Meanwhile, when we choose the correct coincident events, we should keep the PAS pairs measure | or | (if measure | or | , we should add a phase modulation in path 2). So we have a probability of 1 to obtain our target state. As a result, the success probability of preparing the multiphoton high-dimensional GHZ state using our protocol is 1/ /2 −1 × 1/2 . The step-by-step analysis becomes complex since the preparation procedure demands more post-selection processes. We determine the final preparation success probability by comparing the number of multiphoton terms in the output and input states, which is comparable to the preceding section's method.

Conclusion
Our protocol is feasible using current experimental techniques. The key is selecting the target state through the joint action of PBSs and auxiliary entanglements [7] after multiple post-selections and HOM interferences.
The protocol is divided into two steps. The first step is the preparation of 4-photon highdimensional states; the second step is to prepare additional | and apply MCTs filters to obtain -photon ( > 4) high-dimensional GHZ states, as shown in Fig. 3. The process is illustrated using an example of a four-photon 3D GHZ state |GHZ 3 4 . Finally, we derived the requirement of high-dimensional entanglement sources and auxiliary entanglement sources for preparing arbitrary multiphoton high-dimensional GHZ state |GHZ , and give the success probability of the process.
However, an important question remains on how to efficiently reduce auxiliary entanglement and improve the success probability. The solution will facilitate the scalability and stability of the dimensions of the experimental setup used in real experiments, and improve the feasibility of experimental protocols.
In contrast to the "path identity", our protocol successfully solves the problem of preparing arbitrary multiphoton high-dimensional GHZ states in optical systems. With the development of quantum information technology, this protocol makes many quantum protocols have better application prospects in optical systems. Although our protocol is assumed to be performed in a bulk optical system setup using path encoding, it can be applied to other degrees of freedom of photons and integrated optical system.