Open Access
Add to BibSonomy
Abstract
Quantum correlations between identical particles are at the heart of quantum technologies. Several studies with two identical particles have shown that the spatial overlap and indistinguishability between the particles are necessary for generating bipartite entanglement. On the other hand, researches on the extension to more than two-particle systems are limited by the practical difficulty to control multiple identical particles in laboratories. In this work, we propose schemes to generate two fundamental classes of genuine tripartite entanglement, i.e., GHZ and W classes, which are experimentally demonstrated using linear optics with three identical photons. We also show that the tripartite entanglement class decays from the genuine entanglement to the full separability as the particles become more distinguishable from each other. Our results support the prediction that particle indistinguishability is a fundamental element for entangling identical particles.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Entanglement is a crucial resource of various quantum tasks such as quantum computation, quantum communication, and quantum sensing [1–6]. However, its authentic understanding and rigorous quantification still remain open issues [7,8]. In particular, the entanglement phenomena become complicated when a quantum system involves more than two parties [9–13], whereas the multipartite entanglement is beneficial in several aspects, e.g., nonlocality test [14], multi-party quantum communication [15], and quantum computation [16].
Recently, the indistinguishability of quantum particles arouse strong interest as a useful resource for generating entanglement [17–21], which was motivated by the debate on the rigorous quantification and profound origin of entanglement in identical particles [22–27]. By the exchange symmetry, the total state of identical particles always seems to have a mathematical form of entanglement. Hence, we need rigorous criteria to discard the artificial entanglement from the physically relevant one in quantum systems. In the process of quantifying the physical entanglement in identical paticles, it has become clear that the particle indistinguishability is a necessary resource for the entanglement [20,21,27–29] (see also Refs. [30–33] for the relation of particle indistinguishability and the quantum complexity).
To exploit the indistinguishability for entangling identical particles, the spatial overlap between the particles is indispensable [20,27,34]. The entanglement of identical particles should be observed at spatially distinguishable detectors, which must have nontrivial spatial coherence with the particles. There are several researches on the quantitative relation of bipartite entanglement with particle indistinguishability and spatial coherence [17,20,21,27,28]. Most notably, Ref. [21] verified theoretically and experimentally that the entanglement amount of two identical particles is a monotonically increasing function of indistinguishability and spatial coherence. There also exist some theoretical efforts for identical particle systems to generate the general $N$-partite entanglement [35–43]. However, these researches are limited by the practical difficulty to control multiple identical particles in laboratories.
In this work, we push the limit by proposing schemes to generate genuine tripartite entanglement with identical particles, which is experimentally demonstrated with three identical photons in linear optical circuits. Our schemes obtain two fundamental classes of tripartite genuinely entangled states, i.e., GHZ and W classes. Given that the GHZ and W classes are the only two genuine tripartite entanglement classes [44], our results show that all the genuine tripartite entangled states can be generated via spatial overlap among three identical particles.
Furthermore, we investigate the quantitative relation of indistinguishability with the tripartite entanglement, which is accomplished by changing the relative indistinguishability of the particles and observing how the entanglement class varies accordingly. Our study shows that the tripartite entanglement class decays from the genuine entanglement to the full separability as the particles become more distinguishable. In particular, we find that three identical particles cannot carry the genuine entanglement if one of the particles is distinguishable from the others. Our result supports the prediction that the indistinguishability is an essential resource for the entanglement of identical particles.
2. Theory
Here, we describe the general form of transformation operators for arbitrary $N$ particles that can produce entanglement. Then, we introduce specific transformations that generate two $N=3$ genuine entanglement classes, i.e., GHZ and W classes. We also analyze the relation of the tripartite entanglement class with indistinguishability.
We are interested in $N$ identical particles that can be distinguishable and have a two-level internal degrees of freedom. The state of a particle at position ${\psi }_i$ ($i\in \{1,2,\ldots, N\}$) is expressed as
where $s_i$ $(\in \{\uparrow,\downarrow \})$ and $d_i$ denote an internal degree of freedom that corresponds to the qubit state and particle distinguishability, respectively. Here, the particle distinguishability is a dynamical property that depends on the state of the system which can be controllable and observable. See Supplemental Material of Ref. [21] for a more detailed conceptual explanation. In our experiment in Section 3, we have realized $(s_i,d_i)$ with the polarization and optical delay.To generate entanglement with identical particles, we need to spatially overlap the particles, i.e., spatial wave functions ${\psi }_i$ of the particles evolve so that they are not orthogonal to the other. We can achieve the spatial overlap by transforming the initial state of the particles. We can generalize the transformation relation so that the internal states can change according to the transformation:
where $\phi _j$ denotes the position of $M$ detectors (modes), $s_{ij}$ $(\in \{\uparrow,\downarrow \})$ denotes the internal state of a particle that leaves ${\psi }_i$ for $\phi _j$. $T_{ij}$ is the linear transformation of the particle states that preserves the particle number, normalized as $\sum _{j}|T_{ij}|^{2} =1$. $T_{ij}$ can be considered as the entries of an $(N\times M)$ transformation matrix $T$. In addition, we can construct an $(N\times M)$ internal state distribution matrix $S$ whose entries are given by $s_{ij}$. This type of entanglement generation for $N=M=2$ is studied in Refs. [20,21]. Equation (2) shows that the internal states that make the particles (possibly) distinguishable leave marks on the past route of the particles, even if the information on the particle paths are erased by the particle identity.To analyze the effect of $T$ and $S$ to the entanglement, we use the second quantization language. Then, Eq. (2) is rewritten as
with the definitions ${\hat {a}^{\dagger }}_{i}|vac{\rangle } = |{\psi }_i,s_i,d_i{\rangle }$ and ${\hat {b}^{\dagger }}_{ij}|vac{\rangle } = |\phi _j,s_{ij},d_i{\rangle }$. Therefore, $b^{\dagger }_{ij}$ is a creation operator that includes the information of the particle that leaves the initial mode $i$ for the final mode $j$.The $N$ particle transformation is expressed as
Fig. 1. (a) The schematic diagram of a system that consists of N particles, M detectors and a transformation operator $T$. Each row (column) represents a particle (detector), and we restrict the particles to distribute in the detector region $\phi _{j}$ ($j\in \{1,2,\ldots M\}$) denoted as the gray zones. The wave amplitudes in the gray zones correspond to the spatial distribution amplitudes, and the internal states are given with arrows ($\downarrow$ and $\uparrow$). The distinguishability is depicted with different colors. (b) and (c) The transformation that generates GHZ class and W class entanglement ($T^{GHZ}$ and $T^{W}$), respectively.
In the following, we focus on $N=M=3$ cases and postselect no particle-bunching states, i.e., states in which each detector observes one particle. We analyze the tripartite entanglement according to the variations of the spatial overlap and particle distinguishability. Particularly, we show that all the genuine tripartite entangled classes, i.e., GHZ and W states, can be generated in this setup.
First, to generate the $N=3$ GHZ state, we consider the transformation relation Eq. (4) with
After the postselection of no-bunching states, the unnormalized relevant state $|{\Psi }_{GHZ}{\rangle }$ is given by
To understand the effect of distinguishablity to $|{\Psi }_{GHZ}{\rangle }$, we compute the measurable density matrix $\rho _{GHZ}$ that is obtained by tracing out the distinguishability terms of $|{\Psi }_{GHZ}{\rangle }$. Following the same method in Ref. [21], we obtain
Second, we obtain the $N=3$ W class entanglement without transforming the internal states. We set the transformation operators are given by
After the postselection, the unnormalized relevant state $|\Phi _W{\rangle }$ is given by
Table 1. Particle indistinguishability versus W state entanglement class. The state becomes more separable as the particles become more distinguishable.
To rigorously analyze the effect of distinguishability, we consider four different cases according to the relative distinguishablity of the three particles. We restrict our attention to the cases when the particles are completely indistinguishable or distinguishable with each other.
Case I (All the particles are completely indistinguishable): In this case, Eq. (10) can be directly rewritten by omitting the distinguishabilty as
Case II (One of the two particles with internal state $|\downarrow {\rangle }$, i.e., $|{\Psi }_1{\rangle }$ or $|{\Psi }_2{\rangle }$, is distinguishable): We can set the distinguishability as $(|d_1{\rangle },|d_2{\rangle },|d_3{\rangle })=(|d_x{\rangle },|d_y{\rangle },|d_x{\rangle })$ $({\langle }d_x|d_y{\rangle } =0)$ without loss of generality. Then, we obtain the measurable density matrix ${\rho }_{W}^{II}$ as
Case III (The particle with internal state $|\uparrow \rangle$ is distinguishable) $\&$ Case IV (All the three particles are distinguishable with each other): The measurable density matrices of these two cases, ${\rho }^{III,IV}_W$, are written as
We see that the entanglement class decays from genuine entanglement to full separability as the particles become more distinguishable with each other. This tendency coincides with that of the bipartite case in Ref. [21]. Case II results in the most subtle form of bi-separable state, i.e., a mixture of all three bipartite subsystems ($12|3$, $13|2$, and $1|23$) [47]. And the fact that both Case III and Case IV provide fully separable states shows that $|{\Psi }_W{\rangle }$ cannot be genuinely entangled if one particle becomes distinguishable. This aspect is summarized in Table 1.
3. Experiment
We have experimentally investigated entangling three identical particles using single-photon states from spontaneous parametric down-conversion (SPDC). The horizontal and vertical polarization states, $|H\rangle$ and $|V\rangle$, serve as the spin states of $|\downarrow \rangle$ and $|\uparrow \rangle$, respectively. See the Supplemental Materials for details of preparation of initial three single-photon states for the following experiments.
Figure 2 presents the experimental setup to generate the GHZ class states using identical particles. The initial polarization states of the input photons $|s_1\rangle, |s_2\rangle$, and $|s_3\rangle$ are adjusted as $|D\rangle =\frac {1}{\sqrt {2}}\left (|H\rangle +|V\rangle \right )$ using sets of half- and quarter-waveplates (HWP and QWP). Then, the spatial wave function of the single photons are divided by a polarization beam displacer (PBD) which transmits and reflects the horizontal and vertical polarization states, respectively. After passing through HWPs at $\theta$ (${\rm H}_\theta$), the spatial wave functions are overlapped at PBDs such a way that the transformation relations of Eq. (5) satisfy. Note that the outputs of ${\rm PBD}_j$ correspond to the spatial wave function $|\phi _j\rangle$. The polarization states of the three qubit particles were reconstructed by quantum state tomography (QST) using sets of HWP, QWP, and polarizing beamsplitters (PBS) in front of the single-photon detectors. The four-fold coincidence counts including a trigger photon are registered by a home-made coincidence counting unit (CCU) [48,49]. The particle distinguishability is adjusted by the temporal mode of the third photon which can be controlled by the relative delay $L_3$. As investigated in theory, we have considered two cases, i) all the photons have the identical temporal modes $L_3=0$, and ii) one particle is distinguished from the others, i.e., $L_3\neq 0$. For the second case, we set $L_3=5~$mm which is much longer than the coherence length of the single photons. The experimentally reconstructed three-qubit states for all particles are indistinguishable and one particle is distinguishable are represented in Fig. 3(a) and (b), respectively.
Fig. 2. The experimental setup to generate the GHZ class states. H: half waveplate, Q: quarter waveplate, BS: beamsplitter, PBD: polarization beam displacer, PBS: polarizing beamsplitter, APD: avalanche photo detector, CCU: coincidence counting unit.
Fig. 3. The experimentally reconstructed GHZ class sates. (a) $\sigma _{GHZ}^{I}$ with $L_3=0$, and (b) $\sigma _{GHZ}^{II}$ with $L_3\neq 0$.
We now present the experimental setup to generate the W class states, see Fig. 4. The necessary transformation for the W state generation, Eq. (9) was achieved using an optical fiber based $3\times 3$ symmetrical multiport (or a tritter) with two horizontally and one vertically polarized photon inputs. [51] The various Case I–IV are implemented with the relative delay of the second and third photons, $L_2$ and $L_3$. For non-zero temporal delay, we set the values $|L_2|=|L_3|=5~$mm which is much larger than the coherence length of the single photon. At the outputs of the tritter, HWP and QWP compensate the polarization mode dispersion during the optical fiber transmission. Then, sets of HWP, QWP, and PBS are placed for three-qubit QST. The single-photons are detected by superconducting nanowire single-photon detectors (SNSPD). The experimentally reconstructed three-qubit density matrices are presented in in Fig. 5.
Fig. 6 shows the theoretical and experimental fidelity results between each case of $\sigma _{GHZ}~(\sigma _{W})$ and ideal GHZ (W) state with red (blue) color bars. The fidelity value of $F(\rho _{GHZ},\sigma _{GHZ})=\langle \Psi _{GHZ}|\sigma _{GHZ}^{I}|\Psi _{GHZ}\rangle =0.72\pm 0.03>0.5$ verifies that $\sigma _{GHZ}^{I}$ possesses the genuine GHZ class three-partitie entanglement [53]. On the other hand, $\sigma _{GHZ}^{II}$ resembles classical mixture of $|HHH\rangle$ and $|VVV\rangle$. The fidelity from the ideal GHZ state of $F(\rho _{GHZ},\sigma _{GHZ})=0.43\pm 0.02<0.5$ does not show the genuine GHZ class tripartitie entanglement. The experimental results agrees well with our theoretical investigation.
Fig. 4. The experimental setup to generate W class states. H: half waveplate, Q: quarter waveplate, BS: beamsplitter, PBS: polarizing beamsplitter, SNSPD: superconducting nanowire single photon detector, CCU: coincidence counting unit.
Fig. 5. The experimentally reconstructed W class sates. (a) $\sigma _{GHZ}^{I}$ with $L_3=0$, (b) $\sigma _{GHZ}^{II}$ with $L_3\neq 0$, (c) $\sigma _W^{III}$ with $L_2=0,~L_3\neq 0$, and (d) $\sigma _W^{IV}$ with $L_2\neq 0,~L_3\neq 0$.
Fig. 6. The theoretical and experimental fidelity values with ideal GHZ (red bars) and W states (blue bars). The horizontal lines of $F_{GHZ}=\frac {1}{2}$ and $F_W=\frac {2}{3}$ present the genuine tripartite entanglement bound of GHZ and W classes, respectively.
In the W class, when all the photons are indistinguishable (Case I), the maximized fidelity between the generalized W state, $\vert \Psi _{W}^{I}\rangle =\dfrac {1}{\sqrt {3}}(\vert HHV \rangle +e^{i\phi _1}\vert HVH \rangle +e^{i\phi _{2}}\vert VHH \rangle )$, and $\sigma ^{I}_W$ is $F(|\Psi _W^{I}\rangle \langle \Psi _W^{I}|,\sigma ^{I}_W)=0.81\pm 0.02$ for $\phi _1=-0.21\pi$ and $\phi _2=0.28\pi$. Note that the non-zero relative phase $\phi _1$ and $\phi _2$ can be induced during the tritter channel transmission [50–52]. The genuine W class tripartite entanglement of $\sigma ^{I}_W$ can be witnessed by the fidelity with an ideal W state higher than $F=\frac {2}{3}$ [53]. Note also that, in Fig. 5(a), the reconstructed density matrix $\sigma ^{I}_W$ clearly shows non-zero off-diagonal elements which correspond to the coherence among $|HHV\rangle$, $|HVH\rangle$, and $|VHH\rangle$ states. When one of two $|H\rangle$ photons is distinguishable with $L_2\neq 0$ and $L_3=0$, the fidelity from an ideal W state of $F(|\Psi _W^{I}\rangle \langle \Psi _W^{I}|,\sigma ^{II}_W)=0.63\pm 0.02<\frac {2}{3}$ shows that it does not possess the genuine W class three-party entanglement. However, its high fidelity with $\rho _W^{II}$, $F(\rho _W^{II},\sigma _W^{II})=0.92\pm 0.01$ presents that the experimentally reconstructed state $\sigma ^{II}_W$ is close to the bi-separable state $\rho _W^{II}$. In Fig. 5(b), it clearly shows smaller off-diagonal elements comparing to $\sigma ^{I}_W$ of the Case I. As expected, both states of $\sigma ^{III}_W$ and $\sigma ^{IV}_W$ have low fidelities with an ideal W state where $F(|\Psi _W^{I}\rangle \langle \Psi _W^{I}|,\sigma ^{III}_W)=0.43\pm 0.04$ and $F(|\Psi _W^{I}\rangle \langle \Psi _W^{I}|,\sigma ^{IV}_W)=0.39\pm 0.02$ present that they have limited overlap with an ideal W state. On the other hand, they are closed to the classical mixture of $\rho _{W}^{mix}=\frac {1}{3}\left (|HHV\rangle \langle HHV|+|HVH\rangle \langle HVH|+|VHH\rangle \langle VHH|\right )$ as witnessed by high fidelities, $F(\rho _{W}^{mix},\sigma _W^{III})=0.92\pm 0.02$ and $F(\rho _{W}^{mix},\sigma _W^{IV})=0.94\pm 0.01$, respectively.
4. Conclusion
We have provided schemes to construct two fundamental classes of tripartite genuinely entangled states, i.e., GHZ and W classes, and have experimentally implemented using identical photons. Our results verify that the particle indistinguishability plays an essential role for the entanglement of identical particles. We remark that the concepts of particle identity and spatial overlap forms fundamental aspects in quantum science, and thus, our photonic demonstration can be reproduced and extended by any quantum systems such as trapped atoms and ions [54], and solid state circuit quantum electrodynamics [55].
Our current work on the tripartite entanglement of identical particles can extend to general $N$ identical particle systems with more complicated spatial coherence. For instance, our entanglement generation schemes can be interpreted under the viewpoint of linear quantum networks (LQNs), which one can map into bipartite graphs [42]. Therefore, we believe that our work initiates a new research avenue to generate multipartite entangled states with identical particles, which are essential for quantum information processing.
Funding
Korea Institute of Science and Technology (2E31021); MSIP/IITP (2020-0-00947, 2020-0-00972); National Research Foundation of Korea (2019M3E4A1079777, 2019R1A2C2006381, 2019R1I1A1A01059964, 2021R1C1C1003625).
Disclosures
The authors declare no conflicts of interest.
Data Availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935). [CrossRef]
2. C. Monroe, “Quantum information processing with atoms and photons,” Nature 416(6877), 238–246 (2002). [CrossRef]
3. D. Gross and J. Eisert, “Novel schemes for measurement-based quantum computation,” Phys. Rev. Lett. 98(22), 220503 (2007). [CrossRef]
4. V. Vedral, “Quantum entanglement,” Nat. Phys. 10(4), 256–258 (2014). [CrossRef]
5. S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, “Advances in quantum teleportation,” Nat. Photonics 9(10), 641–652 (2015). [CrossRef]
6. S. Hong, J. Rehman, Y.-S. Kim, Y.-W. Cho, S.-W. Lee, H. Jung, S. Moon, S.-W. Han, and H.-T. Lim, “Quantum enhanced multiple-phase estimation with multi-mode N00N states,” Nat. Commun. 12(1), 5211 (2021). [CrossRef]
7. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865–942 (2009). [CrossRef]
8. B. Regula and G. Adesso, “Entanglement quantification made easy: Polynomial measures invariant under convex decomposition,” Phys. Rev. Lett. 116(7), 070504 (2016). [CrossRef]
9. H. Mikami, Y. Li, K. Fukuoka, and T. Kobayashi, “New high-efficiency source of a three-photon W state and its full characterization using quantum state tomography,” Phys. Rev. Lett. 95(15), 150404 (2005). [CrossRef]
10. S. Szalay, “Multipartite entanglement measures,” Phys. Rev. A 92(4), 042329 (2015). [CrossRef]
11. M. Walter, D. Gross, and J. Eisert, “Multipartite entanglement,” Quantum Information: From Foundations to Quantum Technology Applications (Wiley, 2016) pp. 293–330.
12. Y. Zhou, Q. Zhao, X. Yuan, and X. Ma, “Detecting multipartite entanglement structure with minimal resources,” npj Quantum Inf. 5(1), 83 (2019). [CrossRef]
13. S. Nezami and M. Walter, “Multipartite entanglement in stabilizer tensor networks,” Phys. Rev. Lett. 125(24), 241602 (2020). [CrossRef]
14. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86(2), 419–478 (2014). [CrossRef]
15. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999). [CrossRef]
16. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010). [CrossRef]
17. M. C. Tichy, F. de Melo, M. Kuś, F. Mintert, and A. Buchleitner, “Entanglement of identical particles and the detection process,” Fortschr. Phys. 61(2-3), 225–237 (2013). [CrossRef]
18. N. Killoran, M. Cramer, and M. B. Plenio, “Extracting entanglement from identical particles,” Phys. Rev. Lett. 112(15), 150501 (2014). [CrossRef]
19. M. Krenn, A. Hochrainer, M. Lahiri, and A. Zeilinger, “Entanglement by path identity,” Phys. Rev. Lett. 118(8), 080401 (2017). [CrossRef]
20. R. L. Franco and G. Compagno, “Indistinguishability of elementary systems as a resource for quantum information processing,” Phys. Rev. Lett. 120(24), 240403 (2018). [CrossRef]
21. M. R. Barros, S. Chin, T. Pramanik, H.-T. Lim, Y.-W. Cho, J. Huh, and Y.-S. Kim, “Entangling bosons through particle indistinguishability and spatial overlap,” Opt. Express 28(25), 38083–38092 (2020). [CrossRef]
22. F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement and squeezing with identical particles: ultracold atom quantum metrology,” J. Phys. B 44(9), 091001 (2011). [CrossRef]
23. F. Benatti, R. Floreanini, and U. Marzolino, “Bipartite entanglement in systems of identical particles: the partial transposition criterion,” Ann. Phys. 327(5), 1304–1319 (2012). [CrossRef]
24. A. P. Balachandran, T. R. Govindarajan, A. R. de Queiroz, and A. Reyes-Lega, “Entanglement and particle identity: a unifying approach,” Phys. Rev. Lett. 110(8), 080503 (2013). [CrossRef]
25. R. L. Franco and G. Compagno, “Quantum entanglement of identical particles by standard information-theoretic notions,” Sci. Rep. 6(1), 20603 (2016). [CrossRef]
26. A. C. Lourenço, T. Debarba, and E. I. Duzzioni, “Entanglement of indistinguishable particles: A comparative study,” Phys. Rev. A 99(1), 012341 (2019). [CrossRef]
27. S. Chin and J. Huh, “Entanglement of identical particles and coherence in the first quantization language,” Phys. Rev. A 99(5), 052345 (2019). [CrossRef]
28. K. Sun, Y. Wang, Z.-H. Liu, X.-Y. Xu, J.-S. Xu, C.-F. Li, G.-C. Guo, A. Castellini, F. Nosrati, G. Compagno, and R. L. Franco, “Experimental quantum entanglement and teleportation by tuning remote spatial indistinguishability of independent photons,” Opt. Lett. 45(23), 6410–6413 (2020). [CrossRef]
29. S. Chin and J.-H. Chun, “Taming identical particles for discerning the genuine non-locality,” Quantum Inf. Process. 20(3), 86 (2021). [CrossRef]
30. V. Tamma and S. Laibacher, “Multiboson correlation interferometry with arbitrary single-photon pure states,” Phys. Rev. Lett. 114(24), 243601 (2015). [CrossRef]
31. V. S. Shchesnovich, “Partial indistinguishability theory for multiphoton experiments in multiport devices,” Phys. Rev. A 91(1), 013844 (2015). [CrossRef]
32. P. P. Rohde, “Boson sampling with photons of arbitrary spectral structure,” Phys. Rev. A 91(1), 012307 (2015). [CrossRef]
33. M. C. Tichy, “Sampling of partially distinguishable bosons and the relation to the multidimensional permanent,” Phys. Rev. A 91(2), 022316 (2015). [CrossRef]
34. N. Paunkovic, “The role of indistinguishability of identical particles in quantum information processing,” Ph.D. thesis, Citeseer (2004).
35. B. Bellomo, R. L. Franco, and G. Compagno, “N identical particles and one particle to entangle them all,” Phys. Rev. A 96(2), 022319 (2017). [CrossRef]
36. S. Laibacher and V. Tamma, “Symmetries and entanglement features of inner-mode-resolved correlations of interfering nonidentical photons,” Phys. Rev. A 98(5), 053829 (2018). [CrossRef]
37. P. Blasiak and M. Markiewicz, “Entangling three qubits without ever touching,” Sci. Rep. 9(1), 20131 (2019). [CrossRef]
38. M. Karczewski, R. Pisarczyk, and P. Kurzyński, “Genuine multipartite indistinguishability and its detection via the generalized Hong-Ou-Mandel effect,” Phys. Rev. A 99(4), 042102 (2019). [CrossRef]
39. L. Ju, M. Yang, N. Paunković, W.-J. Chu, and Z.-L. Cao, “Creating photonic GHZ and W states via quantum walk,” Quantum Inf. Process. 18(6), 176 (2019). [CrossRef]
40. A. Castellini, R. L. Franco, and G. Compagno, “Effects of indistinguishability in a system of three identical qubits,” Proceedings 12(1), 23 (2019). [CrossRef]
41. Y.-S. Kim, Y.-W. Cho, H.-T. Lim, and S.-W. Han, “Efficient linear optical generation of a multipartite W state via a quantum eraser,” Phys. Rev. A 101(2), 022337 (2020). [CrossRef]
42. S. Chin, Y.-S. Kim, and S. Lee, “Graph picture of linear quantum networks and entanglement,” Quantum 5, 611611 (2021). [CrossRef]
43. P. Blasiak, E. Borsuk, M. Markiewicz, and Y.-S. Kim, “Efficient linear-optical generation of a multipartite W state,” Phys. Rev. A 104(2), 023701 (2021). [CrossRef]
44. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62(6), 062314 (2000). [CrossRef]
45. W. Dür and J. I. Cirac, “Classification of multiqubit mixed states: Separability and distillability properties,” Phys. Rev. A 61(4), 042314 (2000). [CrossRef]
46. M. Seevinck and J. Uffink, “Partial separability and entanglement criteria for multiqubit quantum states,” Phys. Rev. A 78(3), 032101 (2008). [CrossRef]
47. S. Szalay and Z. Kökényesi, “Partial separability revisited: Necessary and sufficient criteria,” Phys. Rev. A 86(3), 032341 (2012). [CrossRef]
48. B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54(15), 4727–4731 (2015). [CrossRef]
49. B. K. Park, Y.-S. Kim, Y.-W. Cho, S. Moon, and S.-W. Han, “Arbitrary configurable 20-channel coincidence counting unit for multi-qubit quantum experiment,” Electronics 10(5), 569 (2021). [CrossRef]
50. G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “Two-photon interference in optical fiber multiports,” Phys. Rev. A 54(1), 893–897 (1996). [CrossRef]
51. N. Spagnolo, C. Vitelli, L. Aparo, P. Mataloni, F. Sciarrino, A. Crespi, R. Ramponi, and R. Osellame, “Three-photon bosonic coalescence in an integrated tritter,” Nat. Commun. 4(1), 1606 (2013). [CrossRef]
52. I. Kim, D. Lee, S. Hong, Y.-W. Cho, K. J. Lee, Y.-S. Kim, and H.-T. Lim, “Implementation of a 3×3 directionally-unbiased linear optical multiport,” Opt. Express 29(18), 29527–29540 (2021). [CrossRef]
53. M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett. 92(8), 087902 (2004). [CrossRef]
54. R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, “Measuring entanglement entropy in a quantum many-body system,” Nature 528(7580), 77–83 (2015). [CrossRef]
55. Y. Y. Gao, B. J. Lester, Y. Zhang, C. Wang, S. Rosenblum, L. Frunzio, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, “Programmable interference between two microwave quantum memories,” Phys. Rev. X 8(2), 021073 (2018). [CrossRef]
