Open Access
Add to BibSonomy
Abstract
The physics associated with multipartite high-dimensional entanglement is different from that of multipartite two-dimensional entanglement. Therefore, preparing multipartite high-dimensional entanglements with linear optics is challenging. This study proposes a preparation protocol of multiphoton GHZ state with arbitrary dimensions for optical systems. Auxiliary entanglements realize a high-dimensional entanglement gate to connect the high-dimensional entangled pairs to a multipartite high-dimensional GHZ state. Specifically, we use the path degrees of freedom of photons to prepare a four-partite, three-dimensional GHZ state. Our method can be extended to other degrees of freedom to generate arbitrary GHZ entanglements in any dimension.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multipartite entangled states are used to test the basic problems of quantum mechanics [1]. In addition, they find application in quantum computing [2–4] 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], $|\text {GHZ}\rangle =(|000\rangle +|111\rangle )/\sqrt {2}$. The generalized GHZ state is an entangled quantum state comprising $n>2$ subsystems. If the dimension of each system is $d$, meaning that the local Hilbert space is isomorphic to $\mathbb {C}^d$, then the total Hilbert space of the $n$-partite system is $\mathcal {H}=(\mathbb {C} ^{d})^{\otimes n}$. This GHZ state is also called the $n$-partite qudit GHZ state:
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 [20–22], silicon [23], and photons [24–27]. 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 ($d>2$), 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 $|\text {GHZ}\rangle ^3_4$ [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 Ref. [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.
2. 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 ultrashort-pulsed optical pumping [36]. This significantly reduces the coherence time requirements of the interfering photons and simplifies the establishment of multiphoton 2D entanglement.
2.1 Four-photon GHZ state
We introduce the unit of this multiphoton entanglement source, which is a post-selected four-photon entanglement source. The four-photon preparation device is shown in Fig. 1; each Einstein-Podolsky-Rosen (EPR) source produces an entangled state.
where $|H\rangle$ and $|V\rangle$ represent the photon’s horizontal and vertical polarizations, respectively.
Fig. 1. experimental setup of $|\text {GHZ}\rangle ^2_4$: Two partite entanglement sources generate two EPR pairs to form the input state $|\psi ^{in}\rangle =|\text {EPR}\rangle _ {12} \otimes |\text {EPR}\rangle _{34}$. PBS1 post-selects photons $3$ and $4$ with the same polarization as follows: $(O_{PBS1}=|HH\rangle \langle HH|+|VV\rangle \langle VV|)$. 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.
Therefore, our input state is
2.2 Multiphoton GHZ state
The device in Fig. 1 is scaled to construct additional photonic GHZ states. For the GHZ state of $n$-photon, $\lceil n/2\rceil$ EPR entangled pairs are required to constitute the input state $|\psi ^{in}\rangle ^2_n = |\text {EPR}\rangle ^{\otimes \lceil \frac {n}{2}\rceil }$. $\lceil n/2\rceil -1$ PBSs for post-selection: The processing steps in Eqs. (3)–(4) project the input state $|\psi ^{in}\rangle ^2_n$ onto the target state—the $n$-photon GHZ state:
This multiphoton extension scheme was validated through several experiments [24–27]. 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.
3. Preparation of multiphoton high-dimensional GHZ state
A challenging problem in constructing the high-dimensional GHZ state $|\text {GHZ}\rangle ^d_n$ is how to cancel the multiphoton cross terms (MCTs) in the input state:
3.1 Four-Photon 3D GHZ state
When $d = 3$, the input state can be expanded as
Fig. 2 presents the preparation idea for $|\text {GHZ}\rangle ^3_4$, which is similar to $|\text {GHZ}\rangle ^2_4$ described previously. We cancel all terms in the input state except $|iijj\rangle$ $(i=0\cdots 2)$, which we call MCTs.
Fig. 2. Diagram of four-photon high-dimensional GHZ state preparation. The left side shows the schematic for preparing the four-photon 3D GHZ state. (a) is the input state $|\psi ^{in}\rangle ^3_4$. (b) Step. i: Control the polarization of each path as shown in the diagram and post-select the state $|\psi ^{in\ '}\rangle ^3_4$ by PBS1. (c) Step ii: Introduce auxiliary entanglement $|\xi \rangle$ and cancel all the other MCTs after Step i. After PBS1, we only retain terms $|0 0 0 0\rangle$, $|0 0 2 2\rangle$, $|1 1 1 1\rangle$, $|2 2 0 0\rangle$, and $|2 2 2 2\rangle$. MCTs $|2 2 0 0\rangle$ and $|0 0 2 2\rangle$ are filtered out using auxiliary entanglement $|\xi \rangle$. We choose the appropriate six-fold coincidence to obtain the output state $|\text {GHZ}\rangle ^3_4$ 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
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.
Second step—MCTs have two other unwanted terms, $|0 0 2 2 \rangle$ and $|2 2 0 0 \rangle$. After PBS1, we set the polarization of path $\{|0\rangle$, $|1\rangle$, $|2\rangle \}$ on $\{|0_H\rangle$, $|1_H\rangle$, $|2_V\rangle \}$ respectively, and introduce the auxiliary entanglement $|\xi \rangle =(|0_H0_H\rangle +|2_V2_V\rangle )/\sqrt {2}$ as:
Furthermore, when we choose the six-fold coincidence count with $|HV\rangle$ or $|VH\rangle$ in out5–6, we can obtain a quantum state $1/\sqrt {3}(|0 0 0 0\rangle +|1 1 1 1\rangle -|2 2 2 2\rangle )$, which is not a GHZ state. If based on the measurement results, we apply a $\pi$ phase modulation to the photons in path 2, which can be deterministically accomplished with an electro-optic feedforward [39–42], 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 $1/6$.
3.2 High-dimensional GHZ state
The protocol is scalable to arbitrary dimensions. The preparation protocol for $d\geq 2$ with four-photon GHZ states $|\text {GHZ}\rangle ^d_4$ in arbitrary dimensions can be generalized as follows:
- i) Prepare two 2-photon $d$-dimensional maximal entangled state: $|\phi \rangle ^d=(\sum _0^{d-1}|ii\rangle )/\sqrt {d}$ which encode by the path degree of freedom, and control the polarization of each path as follows: if $i$ is odd, the polarization is set at $V$; otherwise, it is set at $H$. Then, when the photons pass through PBS1 in Fig. 2, we select the events with only one photon at each exit of PBS1.
- ii) Set the polarization of photon in the path state $|i\rangle \ (|j\rangle )$ at $|i_H\rangle \ (|j_V\rangle )$. Prepare auxiliary entangled state $|\xi \rangle =(|i_Hi_H\rangle )+|j_Vj_V\rangle )/\sqrt {2}$ $(i,j=0,\ldots d-1.\ i < j)$ to cancel MCTs $|iijj\rangle$ and $|jjii\rangle$.
- iii) Repeat Step. ii until all MCTs are canceled.
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 $|iijj\rangle$, $i$ for odd numbers and $j$ for even ones, or vice versa), which have $2\lceil d/2 \rceil \times \lfloor d/2 \rfloor$ terms. Thus the success probability of Step. i in $d$ dimension is $\eta _1=1-2\lceil d/2 \rceil \times \lfloor d/2 \rfloor /d^2$. Two more MCTs will be canceled whenever Step. ii is performed. Some multiphoton terms are retained with probability $1/2$, e.g. $|iiii\rangle$ $(i=0, 1, 2)$ in the preparation of $|\text {GHZ}\rangle ^3_4$ is retained with probability $1/2$, and with the help of $\pi$ phase modulation, the probability of selecting the correct multifold coincidence is $1$. Thus, the success probability of $k$th Step. ii is:
The success probability in the preparation of $|\text {GHZ}\rangle ^d_4$ is $p=\eta _1 \times \prod ^{N^{d}_4}_{k=1}{\eta _2(k)}=1/2^{N^{d}_4}\times (\eta _1-2N^{d}_4/d^2)$, where $\eta _1-2N^{d}_4/d^2=1/d$, thus $p=1/2^{N^{d}_4}\times 1/d$. If $d=3$, then $p=1/6$. This is consistent with the result in Section 3.1.3.3 Multiphoton high-dimensional GHZ state
Figure 3 shows the diagram of the preparation of $|\text {GHZ}\rangle ^d_n$. We prepare the state $|\text {GHZ}\rangle ^d_4$ by sending the adjacent photons from the two 2-photon $d$-dimensional maximal entangled state: $|\phi \rangle ^d=(\sum _0^{d-1}|ii\rangle )/\sqrt {d}$ 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 $|\phi \rangle ^d$ and pass it through the MCTs filter, and obtain $|\text {GHZ}\rangle ^d_n$ 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.
Fig. 3. Schematic diagram of multiphoton high-dimensional GHZ state $|\text {GHZ}\rangle ^d_n$ preparation : We start from the state $|\text {GHZ}\rangle ^d_4$. When two adjacent photons belonging to different $|\phi \rangle ^d$ and pass through the MCTs filter, which allows filtering out the multiphoton term unrelated to the GHZ state. Then we add another $|\phi \rangle ^d$ and the MCTs filter, and after a similar post-selection process, we get the state $|\text {GHZ}\rangle ^d_6$. We prepare the state $|\text {GHZ}\rangle ^d_n=(|0\rangle ^{\otimes n}+|1\rangle ^{\otimes n}+\dots +|d-1\rangle ^{\otimes n})/\sqrt {d}$ by repeating the process of combining $|\phi \rangle ^d$ 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 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 $M^d=|m\rangle \langle m|$ for the photons of out1, where the measurement basis $|m\rangle =1/\sqrt {d}(|0\rangle +|1\rangle +\dots +|d-1\rangle )$, so that we will change with probability $1/d$ from the GHZ state of even photons to odd number. To improve the probability, we can perform a $d$-output measurement of the photon at out1. Let us take an example in three dimensions: if one of our measurements base is $M^3$, then the measurement bases of the remaining two measurement bases are: $1/\sqrt {d}(|0\rangle +e^{\frac {2\pi }{3}i}|1\rangle +e^{\frac {4\pi }{3}i}|2\rangle )$ and $1/\sqrt {d}(|0\rangle +e^{\frac {4\pi }{3}i}|1\rangle +e^{\frac {2\pi }{3}i}|2\rangle )$. We can find that this $d$-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 $e^{\frac {2\pi }{d}i}$ phases between adjacent multiphoton terms ($|iii\rangle$) 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.
The multiphoton entanglement source requires $\lceil n/2\rceil$ $|\phi \rangle ^d$ and $N^d_n=\lceil d(d-2)/4\rceil \times \lceil n/2-1\rceil$ auxiliary two-partite 2D entanglement sources to filter out the MCTs. When preparing a multiphoton high-dimensional GHZ state, the input state contains $d^{\lceil n/2\rceil }$ multiphoton terms, of which we preserve only $d$ of them (the probability is $1/d^{\lceil n/2\rceil -1}$). If auxiliary entanglement is introduced, the retained multiphoton terms will be kept with a probability of $1/2$ (e.g., multiphoton terms $|iiii\rangle$ will be retained by probability with $1/2$). Meanwhile, when we choose the correct coincident events, we should keep the PAS pairs measure $|HH\rangle$ or $|VV\rangle$ (if measure $|HV\rangle$ or $|VH\rangle$, we should add a $\pi$ 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 $1/d^{\lceil n/2\rceil -1}\times 1/2^{N^d_n}$. 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.
4. 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 high-dimensional states; the second step is to prepare additional $|\phi \rangle ^d$ and apply MCTs filters to obtain $n$-photon ($n>4$) high-dimensional GHZ states, as shown in Fig. 3. The process is illustrated using an example of a four-photon 3D GHZ state $|\text {GHZ}\rangle ^3_4$. Finally, we derived the requirement of high-dimensional entanglement sources and auxiliary entanglement sources for preparing arbitrary multiphoton high-dimensional GHZ state $|\text {GHZ}\rangle ^d_n$, 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.
Funding
National Key Research and Development Program of China (2021YFE0113100); National Natural Science Foundation of China (11904357, 12174367, 12204458); Innovation Program for Quantum Science and Technology (2021ZD0301200); Fundamental Research Funds for the Central Universities; USTC Tang Scholarship; Science and Technological Fund of Anhui Province for Outstanding Youth (2008085J02); China Postdoctoral Science Foundation (2021M700138); China Postdoctoral for Innovative Talents (BX2021289); USTC Center for Micro- and Nanoscale Research and Fabrication; the Shanghai Municipal Science and Technology Fundamental Project (21JC1405400).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
References
1. D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Multiparticle Interferometry and the Superposition Principle,” Phys. Today 46(8), 22–29 (1993). [CrossRef]
2. D. Gottesman and I. L. Chuang, “Quantum Teleportation is a Universal Computational Primitive,” Nature 402(6760), 390–393 (1999). [CrossRef]
3. R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer,” Phys. Rev. Lett. 86(22), 5188–5191 (2001). [CrossRef]
4. S. Nezami and M. Walter, “Multipartite Entanglement in Stabilizer Tensor Networks,” Phys. Rev. Lett. 125(24), 241602 (2020). [CrossRef]
5. L.-Z. Liu, Y.-Z. Zhang, Z.-D. Li, R. Zhang, X.-F. Yin, Y.-Y. Fei, L. Li, N.-L. Liu, F. Xu, Y.-A. Chen, and J.-W. Pan, “Distributed quantum phase estimation with entangled photons,” Nat. Photonics 15(2), 137–142 (2021). [CrossRef]
6. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993). [CrossRef]
7. X.-M. Hu, C. Zhang, B.-H. Liu, Y. Cai, X.-J. Ye, Y. Guo, W.-B. Xing, C.-X. Huang, Y.-F. Huang, C.-F. Li, and G.-C. Guo, “Experimental High-Dimensional Quantum Teleportation,” Phys. Rev. Lett. 125(23), 230501 (2020). [CrossRef]
8. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992). [CrossRef]
9. Y. Guo, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Advances in quantum dense coding,” Adv. Quantum Technol. 2(5-6), 1900011 (2019). [CrossRef]
10. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]
11. D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going Beyond Bell’s Theorem,” in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, M. Kafatos, ed. (Springer Netherlands, Dordrecht, 1989), pp. 69–72.
12. G. Vidal, “Entanglement monotones,” J. Mod. Opt. 47(2-3), 355–376 (2000). [CrossRef]
13. E. Chitambar and G. Gour, “Quantum resource theories,” Rev. Mod. Phys. 91(2), 025001 (2019). [CrossRef]
14. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999). [CrossRef]
15. M. Ben-Or and A. Hassidim, “Fast quantum byzantine agreement,” in Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, (Association for Computing Machinery, New York, NY, USA, 2005), STOC ’05, p. 481–485.
16. C. Negrevergne, T. S. Mahesh, C. A. Ryan, M. Ditty, F. Cyr-Racine, W. Power, N. Boulant, T. Havel, D. G. Cory, and R. Laflamme, “Benchmarking Quantum Control Methods on a 12-Qubit System,” Phys. Rev. Lett. 96(17), 170501 (2006). [CrossRef]
17. H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438(7068), 643–646 (2005). [CrossRef]
18. H. Kaufmann, T. Ruster, C. T. Schmiegelow, M. A. Luda, V. Kaushal, J. Schulz, D. von Lindenfels, F. Schmidt-Kaler, and U. G. Poschinger, “Scalable Creation of Long-Lived Multipartite Entanglement,” Phys. Rev. Lett. 119(15), 150503 (2017). [CrossRef]
19. P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite Entanglement Among Single Spins in Diamond,” Science 320(5881), 1326–1329 (2008). [CrossRef]
20. J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007). [CrossRef]
21. L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010). [CrossRef]
22. M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010). [CrossRef]
23. K. Takeda, A. Noiri, T. Nakajima, J. Yoneda, T. Kobayashi, and S. Tarucha, “Quantum tomography of an entangled three-qubit state in silicon,” Nat. Nanotechnol. 16(9), 965–969 (2021). [CrossRef]
24. C. Zhang, Y.-F. Huang, Z. Wang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental Greenberger–Horne–Zeilinger-Type six-photon quantum nonlocality,” Phys. Rev. Lett. 115(26), 260402 (2015). [CrossRef]
25. Y.-F. Huang, B.-H. Liu, L. Peng, Y.-H. Li, L. Li, C.-F. Li, and G.-C. Guo, “Experimental generation of an eight-photon Greenberger–Horne–Zeilinger state,” Nat. Commun. 2(1), 546 (2011). [CrossRef]
26. X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117(21), 210502 (2016). [CrossRef]
27. H.-S. Zhong, Y. Li, W. Li, et al., “12-Photon Entanglement and Scalable Scattershot Boson Sampling with Optimal Entangled-Photon Pairs from Parametric Down-Conversion,” Phys. Rev. Lett. 121(25), 250505 (2018). [CrossRef]
28. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). [CrossRef]
29. M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016). [CrossRef]
30. X.-M. Hu, W.-B. Xing, C. Zhang, B.-H. Liu, M. Pivoluska, M. Huber, Y.-F. Huang, C.-F. Li, and G.-C. Guo, “Experimental creation of multi-photon high-dimensional layered quantum states,” npj Quantum Inf. 6(1), 88 (2020). [CrossRef]
31. M. Krenn, A. Hochrainer, M. Lahiri, and A. Zeilinger, “Entanglement by Path Identity,” Phys. Rev. Lett. 118(8), 080401 (2017). [CrossRef]
32. M. Krenn, X. Gu, and A. Zeilinger, “Quantum Experiments and Graphs: Multiparty States as Coherent Superpositions of Perfect Matchings,” Phys. Rev. Lett. 119(24), 240403 (2017). [CrossRef]
33. X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. 67(3), 318–321 (1991). [CrossRef]
34. M. Krenn, M. Malik, R. Fickler, R. Lapkiewicz, and A. Zeilinger, “Automated Search for new Quantum Experiments,” Phys. Rev. Lett. 116(9), 090405 (2016). [CrossRef]
35. J. Bao, Z. Fu, T. Pramanik, et al., “Very-large-scale integrated quantum graph photonics,” Nat. Photonics 17(7), 573–581 (2023). [CrossRef]
36. M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sourcesa,” Ann. N. Y. Acad. Sci. 755(1), 91–102 (1995). [CrossRef]
37. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987). [CrossRef]
38. C. Zhang, Y.-F. Huang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Spontaneous Parametric Down-Conversion Sources for Multiphoton Experiments,” Adv. Quantum Technol. 4(5), 2000132 (2021). [CrossRef]
39. F. Sciarrino, M. Ricci, F. De Martini, R. Filip, and L. Mišta, “Realization of a minimal disturbance quantum measurement,” Phys. Rev. Lett. 96(2), 020408 (2006). [CrossRef]
40. R. Prevedel, P. Walther, F. Tiefenbacher, P. Böhi, R. Kaltenbaek, T. Jennewein, and A. Zeilinger, “High-speed linear optics quantum computing using active feed-forward,” Phys. Rev. Lett. 445, 65–69 (2007). [CrossRef]
41. X.-S. Ma, S. Zotter, N. Tetik, A. Qarry, T. Jennewein, and A. Zeilinger, “A high-speed tunable beam splitter for feed-forward photonic quantum information processing,” Opt. Express 19(23), 22723 (2011). [CrossRef]
42. V. Švarc, J. Hloušek, M. Nováková, J. Fiurášek, and M. Ježek, “Feedforward-enhanced fock state conversion with linear optics,” Opt. Express 28(8), 11634 (2020). [CrossRef]
