Graph Picture of Linear Quantum Networks and Entanglement

Seungbeom Chin1,2, Yong-Su Kim3,4, and Sangmin Lee5

1Department of Electrical and Computer Engineering, Sungkyunkwan University, Suwon 16419, Korea
2International Centre for Theory of Quantum Technologies, University of Gdánsk, 80-308, Gdánsk, Poland
3Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Korea
4Division of Nano $\&$ Information Technology, KIST School, Korea University of Science and Technology, Seoul 02792, Korea
5College of Liberal Studies, Seoul National University, Seoul 08826, Korea

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The indistinguishability of quantum particles is widely used as a resource for the generation of entanglement. Linear quantum networks (LQNs), in which identical particles linearly evolve to arrive at multimode detectors, exploit the indistinguishability to generate various multipartite entangled states by the proper control of transformation operators. However, it is challenging to devise a suitable LQN that carries a specific entangled state or compute the possible entangled state in a given LQN as the particle and mode number increase. This research presents a mapping process of arbitrary LQNs to graphs, which provides a powerful tool for analyzing and designing LQNs to generate multipartite entanglement. We also introduce the perfect matching diagram (PM diagram), which is a refined directed graph that includes all the essential information on the entanglement generation by an LQN. The PM diagram furnishes rigorous criteria for the entanglement of an LQN and solid guidelines for designing suitable LQNs for the genuine entanglement. Based on the structure of PM diagrams, we compose LQNs for fundamental $N$-partite genuinely entangled states.

Among diverse methods to create entanglement, the indistinguishability of quantum particles is frequently employed as a resource for entanglement. A simple but inspiring example is the Hong-Ou-Mandel interference, a generation of entanglement with two identical photons through the complete path overlap. In general, a linear quantum network (LQN), a quantum system of particles that linearly evolve to arrive at multiple detectors, can carry various types of entanglement by the control of transformation operators and postselections. However, the composition procedure of a suitable LQN for a specific entangled state still lacks manifest insights on the relations between the structure of a LQN and the entanglement in it. In other words, one cannot verify the entanglement in the final postselected state of a given LQN until completing all the computation processes. We overcome this limit by introducing a mapping process of arbitrary LQNs to graphs, which provides a powerful tool for analyzing the entanglement in LQNs.

The mapping is rigorously achieved by inserting all the physically relevant variables of an LQN into the weighted and colored adjacency matrices of graphs. In the graph picture, we can consider a significantly simplified computation protocol to quantify the entanglement of the postselected state in the LQN. Furthermore, since the graph structure reveals the entanglement feature in the corresponding LQN, it provides a solid guideline to design LQNs for obtaining specific entanglement of large N subsystems. We have suggested several LQNs that carry genuinely multipartite entanglement states such as GHZ, W, and Dicke states.

Our graph theoretic method to analyze LQN opens a way to a schematic exploration of the entanglement generation with identical particles. It provides theoretical criteria for discriminating genuine entanglement from LQNs, which experiments can directly verify.

► BibTeX data

► References

[1] Horodecki, Ryszard ; Horodecki, Paweł ; Horodecki, Michał ; Horodecki, Karol: Quantum entanglement. In: Reviews of Modern Physics 81 (2009), Nr. 2, 865. http:/​/​doi.org/​10.1103/​RevModPhys.81.865.
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[2] Hong, Chong-Ki ; Ou, Zhe-Yu ; Mandel, Leonard: Measurement of subpicosecond time intervals between two photons by interference. In: Physical Review Letters 59 (1987), Nr. 18, 2044. http:/​/​doi.org/​10.1103/​PhysRevLett.59.2044.
https:/​/​doi.org/​10.1103/​PhysRevLett.59.2044

[3] Tichy, Malte C. ; Melo, Fernando de ; Kuś, Marek ; Mintert, Florian ; Buchleitner, Andreas: Entanglement of identical particles and the detection process. In: Fortschritte der Physik 61 (2013), Nr. 2-3, 225–237. http:/​/​doi.org/​10.1002/​prop.201200079.
https:/​/​doi.org/​10.1002/​prop.201200079

[4] Killoran, N ; Cramer, M ; Plenio, Martin B.: Extracting entanglement from identical particles. In: Physical Review Letters 112 (2014), Nr. 15, 150501. http:/​/​doi.org/​10.1103/​PhysRevLett.112.150501.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.150501

[5] Krenn, Mario ; Hochrainer, Armin ; Lahiri, Mayukh ; Zeilinger, Anton: Entanglement by path identity. In: Physical Review Letters 118 (2017), Nr. 8, 080401. http:/​/​doi.org/​10.1103/​PhysRevLett.118.080401.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.080401

[6] Paunkovic, Nikola: The role of indistinguishability of identical particles in quantum information processing, University of Oxford, Diss., 2004. http:/​/​www.cs.math.ist.utl.pt/​ftp/​pub/​PaunkovicN/​04-P-phdthesis.pdf.
http:/​/​www.cs.math.ist.utl.pt/​ftp/​pub/​PaunkovicN/​04-P-phdthesis.pdf

[7] Franco, Rosario L. ; Compagno, Giuseppe: Indistinguishability of elementary systems as a resource for quantum information processing. In: Physical Review Letters 120 (2018), Nr. 24, 240403. http:/​/​doi.org/​10.1103/​PhysRevLett.120.240403.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.240403

[8] Chin, Seungbeom ; Huh, Joonsuk: Entanglement of identical particles and coherence in the first quantization language. In: Physical Review A 99 (2019), Nr. 5, 052345. http:/​/​doi.org/​10.1103/​PhysRevA.99.052345.
https:/​/​doi.org/​10.1103/​PhysRevA.99.052345

[9] Nosrati, Farzam ; Castellini, Alessia ; Compagno, Giuseppe ; Franco, Rosario L.: Robust entanglement preparation against noise by controlling spatial indistinguishability. In: npj Quantum Information 6 (2020), Nr. 1, 1–7. http:/​/​doi.org/​10.1038/​s41534-020-0271-7.
https:/​/​doi.org/​10.1038/​s41534-020-0271-7

[10] Barros, Mariana R. ; Chin, Seungbeom ; Pramanik, Tanumoy ; Lim, Hyang-Tag ; Cho, Young-Wook ; Huh, Joonsuk ; Kim, Yong-Su: Entangling bosons through particle indistinguishability and spatial overlap. In: Optics Express 28 (2020), Nr. 25, 38083–38092. http:/​/​doi.org/​10.1364/​OE.410361.
https:/​/​doi.org/​10.1364/​OE.410361

[11] Chin, Seungbeom ; Chun, Jung-Hoon: Taming identical particles for discerning the genuine non-locality. In: Quantum Information Processing 20 (2021), Nr. 3, 1–26. http:/​/​doi.org/​10.1007/​s11128-021-03024-0.
https:/​/​doi.org/​10.1007/​s11128-021-03024-0

[12] Yurke, Bernard ; Stoler, David: Einstein-Podolsky-Rosen effects from independent particle sources. In: Physical Review Letters 68 (1992), Nr. 9, 1251. http:/​/​doi.org/​10.1103/​PhysRevLett.68.1251.
https:/​/​doi.org/​10.1103/​PhysRevLett.68.1251

[13] Yurke, Bernard ; Stoler, David: Bell’s-inequality experiments using independent-particle sources. In: Physical Review A 46 (1992), Nr. 5, 2229. http:/​/​doi.org/​10.1103/​PhysRevA.46.2229.
https:/​/​doi.org/​10.1103/​PhysRevA.46.2229

[14] Zukowski, Marek ; Zeilinger, Anton ; Horne, Michael A. ; Ekert, Aarthur K.: " Event-ready-detectors" Bell experiment via entanglement swapping. In: Physical Review Letters 71 (1993), Nr. 26. http:/​/​doi.org/​10.1103/​PhysRevLett.71.4287.
https:/​/​doi.org/​10.1103/​PhysRevLett.71.4287

[15] Zukowski, Marek ; Zeilinger, Anton ; Weinfurter, Harald: Entangling Photons Radiated by Independent Pulsed Sources a. In: Annals of the New York academy of Sciences 755 (1995), Nr. 1, 91–102. http:/​/​doi.org/​10.1111/​j.1749-6632.1995.tb38959.x.
https:/​/​doi.org/​10.1111/​j.1749-6632.1995.tb38959.x

[16] Zeilinger, Anton ; Horne, Michael A. ; Weinfurter, Harald ; Żukowski, Marek: Three-particle entanglements from two entangled pairs. In: Physical Review Letters 78 (1997), Nr. 16, 3031. http:/​/​doi.org/​10.1103/​PhysRevLett.78.3031.
https:/​/​doi.org/​10.1103/​PhysRevLett.78.3031

[17] Blasiak, Pawel ; Markiewicz, Marcin: Entangling three qubits without ever touching. In: Scientific Reports 9 (2019). http:/​/​doi.org/​10.1038/​s41598-019-55137-3.
https:/​/​doi.org/​10.1038/​s41598-019-55137-3

[18] Bellomo, Bruno ; Franco, Rosario L. ; Compagno, Giuseppe: N identical particles and one particle to entangle them all. In: Physical Review A 96 (2017), Nr. 2, 022319. http:/​/​doi.org/​10.1103/​PhysRevA.96.022319.
https:/​/​doi.org/​10.1103/​PhysRevA.96.022319

[19] Kim, Yong-Su ; Cho, Young-Wook ; Lim, Hyang-Tag ; Han, Sang-Wook: Efficient linear optical generation of a multipartite W state via a quantum eraser. In: Physical Review A 101 (2020), Nr. 2, 022337. http:/​/​doi.org/​10.1103/​PhysRevA.101.022337.
https:/​/​doi.org/​10.1103/​PhysRevA.101.022337

[20] Urbina, Juan-Diego ; Kuipers, Jack ; Matsumoto, Sho ; Hummel, Quirin ; Richter, Klaus: Multiparticle Correlations in Mesoscopic Scattering: Boson Sampling, Birthday Paradox, and Hong-Ou-Mandel Profiles. In: Physical Review Letters 116 (2016), 100401. http:/​/​doi.org/​10.1103/​PhysRevLett.116.100401.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.100401

[21] Aaronson, Scott ; Arkhipov, Alex: The Computational Complexity of Linear Optics. In: Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (2011), 333–342. https:/​/​doi.org/​10.1145/​1993636.1993682.
https:/​/​doi.org/​10.1145/​1993636.1993682

[22] Broome, Matthew A. ; Fedrizzi, Alessandro ; Rahimi-Keshari, Saleh ; Dove, Justin ; Aaronson, Scott ; Ralph, Timothy C. ; White, Andrew G.: Photonic boson sampling in a tunable circuit. In: Science 339 (2013), Nr. 6121, 794–798. http:/​/​doi.org/​10.1126/​science.1231440.
https:/​/​doi.org/​10.1126/​science.1231440

[23] Spring, Justin B. ; Metcalf, Benjamin J. ; Humphreys, Peter C. ; Kolthammer, W S. ; Jin, Xian-Min ; Barbieri, Marco ; Datta, Animesh ; Thomas-Peter, Nicholas ; Langford, Nathan K. ; Kundys, Dmytro u. a.: Boson sampling on a photonic chip. In: Science 339 (2013), Nr. 6121, 798–801. http:/​/​doi.org/​10.1126/​science.1231692.
https:/​/​doi.org/​10.1126/​science.1231692

[24] Wang, Hui ; Qin, Jian ; Ding, Xing ; Chen, Ming-Cheng ; Chen, Si ; You, Xiang ; He, Yu-Ming ; Jiang, Xiao ; You, L ; Wang, Z u. a.: Boson sampling with 20 input photons and a 60-mode interferometer in a 1 0 14-dimensional hilbert space. In: Physical Review Letters 123 (2019), Nr. 25, 250503. http:/​/​doi.org/​10.1103/​PhysRevLett.123.250503.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.250503

[25] Blasiak, Pawel ; Borsuk, Ewa ; Markiewicz, Marcin: On safe post-selection for Bell nonlocality: Causal diagram approach. In: Quantum 5 (2021), 575. http:/​/​doi.org/​10.22331/​q-2021-11-11-575.
https:/​/​doi.org/​10.22331/​q-2021-11-11-575

[26] Cabello, Adán ; Severini, Simone ; Winter, Andreas: Graph-theoretic approach to quantum correlations. In: Physical Review Letters 112 (2014), Nr. 4, 040401. http:/​/​doi.org/​10.1103/​PhysRevLett.112.040401.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.040401

[27] Acín, Antonio ; Fritz, Tobias ; Leverrier, Anthony ; Sainz, Ana B.: A combinatorial approach to nonlocality and contextuality. In: Communications in Mathematical Physics 334 (2015), Nr. 2, 533–628. http:/​/​doi.org/​10.1007/​s00220-014-2260-1.
https:/​/​doi.org/​10.1007/​s00220-014-2260-1

[28] Dür, Wolfgang ; Aschauer, Hans ; Briegel, H-J: Multiparticle entanglement purification for graph states. In: Physical Review Letters 91 (2003), Nr. 10, 107903. http:/​/​doi.org/​10.1103/​PhysRevLett.91.107903.
https:/​/​doi.org/​10.1103/​PhysRevLett.91.107903

[29] Hein, Marc ; Eisert, Jens ; Briegel, Hans J.: Multiparty entanglement in graph states. In: Physical Review A 69 (2004), Nr. 6, 062311. http:/​/​doi.org/​10.1103/​PhysRevA.69.062311.
https:/​/​doi.org/​10.1103/​PhysRevA.69.062311

[30] Nest, Maarten Van d. ; Dehaene, Jeroen ; De Moor, Bart: Local unitary versus local Clifford equivalence of stabilizer states. In: Physical Review A 71 (2005), Nr. 6, 062323. http:/​/​doi.org/​10.1103/​PhysRevA.75.032325.
https:/​/​doi.org/​10.1103/​PhysRevA.75.032325

[31] Hein, Marc ; Dür, Wolfgang ; Eisert, Jens ; Raussendorf, Robert ; Nest, M ; Briegel, H-J: Entanglement in graph states and its applications. In: ``Quantum computers, algorithms and chaos'' (2006), 115. http:/​/​doi.org/​10.3254/​978-1-61499-018-5-115.
https:/​/​doi.org/​10.3254/​978-1-61499-018-5-115

[32] Brádler, Kamil ; Dallaire-Demers, Pierre-Luc ; Rebentrost, Patrick ; Su, Daiqin ; Weedbrook, Christian: Gaussian boson sampling for perfect matchings of arbitrary graphs. In: Physical Review A 98 (2018), Nr. 3, 032310. http:/​/​doi.org/​10.1103/​PhysRevA.98.032310.
https:/​/​doi.org/​10.1103/​PhysRevA.98.032310

[33] Arrazola, Juan M. ; Bromley, Thomas R.: Using Gaussian boson sampling to find dense subgraphs. In: Physical Review Letters 121 (2018), Nr. 3, 030503. http:/​/​doi.org/​10.1103/​PhysRevLett.121.030503.
https:/​/​doi.org/​10.1103/​PhysRevLett.121.030503

[34] Brádler, Kamil ; Friedland, Shmuel ; Izaac, Josh ; Killoran, Nathan ; Su, Daiqin: Graph isomorphism and Gaussian boson sampling. In: Special Matrices 9 (2021), Nr. 1, 166–196. http:/​/​doi.org/​10.1515/​spma-2020-0132.
https:/​/​doi.org/​10.1515/​spma-2020-0132

[35] Krenn, Mario ; Gu, Xuemei ; Zeilinger, Anton: Quantum experiments and graphs: Multiparty states as coherent superpositions of perfect matchings. In: Physical Review Letters 119 (2017), Nr. 24, 240403. http:/​/​doi.org/​10.1103/​PhysRevLett.119.240403.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.240403

[36] Gu, Xuemei ; Erhard, Manuel ; Zeilinger, Anton ; Krenn, Mario: Quantum experiments and graphs II: Quantum interference, computation, and state generation. In: Proceedings of the National Academy of Sciences 116 (2019), Nr. 10, 4147–4155. http:/​/​doi.org/​10.1073/​pnas.1815884116.
https:/​/​doi.org/​10.1073/​pnas.1815884116

[37] Gu, Xuemei ; Chen, Lijun ; Zeilinger, Anton ; Krenn, Mario: Quantum experiments and graphs. III. High-dimensional and multiparticle entanglement. In: Physical Review A 99 (2019), Nr. 3, 032338. http:/​/​doi.org/​10.1103/​PhysRevA.99.032338.
https:/​/​doi.org/​10.1103/​PhysRevA.99.032338

[38] Gu, Xuemei ; Chen, Lijun ; Krenn, Mario: Quantum experiments and hypergraphs: Multiphoton sources for quantum interference, quantum computation, and quantum entanglement. In: Physical Review A 101 (2020), Nr. 3, 033816. http:/​/​doi.org/​10.1103/​PhysRevA.101.033816.
https:/​/​doi.org/​10.1103/​PhysRevA.101.033816

[39] Lee, Donghwa ; Pramanik, Tanumoy ; Cho, Young-Wook ; Lim, Hyang-Tag ; Chin, Seungbeom ; Kim, Yong-Su: Entangling three identical particles via spatial overlap. In: arXiv preprint arXiv:2104.05937 (2021). https:/​/​arxiv.org/​abs/​2104.05937.
arXiv:2104.05937

[40] Kim, Yong-Su ; Lee, Jong-Chan ; Kwon, Osung ; Kim, Yoon-Ho: Protecting entanglement from decoherence using weak measurement and quantum measurement reversal. In: Nature Physics 8 (2012), Nr. 2, 117–120. http:/​/​doi.org/​10.1038/​nphys2178.
https:/​/​doi.org/​10.1038/​nphys2178

[41] Langford, Nathan K. ; Weinhold, TJ ; Prevedel, R ; Resch, KJ ; Gilchrist, Alexei ; O’Brien, JL ; Pryde, GJ ; White, AG: Demonstration of a simple entangling optical gate and its use in Bell-state analysis. In: Physical Review Letters 95 (2005), Nr. 21, 210504. http:/​/​doi.org/​10.1103/​PhysRevLett.95.210504.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.210504

[42] Kiesel, Nikolai ; Schmid, Christian ; Weber, Ulrich ; Ursin, Rupert ; Weinfurter, Harald: Linear optics controlled-phase gate made simple. In: Physical Review Letters 95 (2005), Nr. 21, 210505. http:/​/​doi.org/​10.1103/​PhysRevLett.95.210505.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.210505

[43] Okamoto, Ryo ; Hofmann, Holger F. ; Takeuchi, Shigeki ; Sasaki, Keiji: Demonstration of an optical quantum controlled-NOT gate without path interference. In: Physical Review Letters 95 (2005), Nr. 21, 210506. http:/​/​doi.org/​10.1103/​PhysRevLett.95.210506.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.210506

[44] Brualdi, Richard A. ; Harary, Frank ; Miller, Zevi: Bigraphs versus digraphs via matrices. In: Journal of Graph Theory 4 (1980), Nr. 1, 51–73. http:/​/​doi.org/​10.1002/​jgt.3190040107.
https:/​/​doi.org/​10.1002/​jgt.3190040107

[45] Fukuda, Komei ; Matsui, Tomomi: Finding all the perfect matchings in bipartite graphs. In: Applied Mathematics Letters 7 (1994), Nr. 1, 15–18. http:/​/​doi.org/​10.1016/​0893-9659(94)90045-0.
https:/​/​doi.org/​10.1016/​0893-9659(94)90045-0

[46] Tassa, Tamir: Finding all maximally-matchable edges in a bipartite graph. In: Theoretical Computer Science 423 (2012), 50–58. http:/​/​doi.org/​10.1016/​j.tcs.2011.12.071.
https:/​/​doi.org/​10.1016/​j.tcs.2011.12.071

[47] Uno, Takeaki: Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs. In: International Symposium on Algorithms and Computation (1997), 92–101. http:/​/​doi.org/​10.1007/​3-540-63890-3_11.
https:/​/​doi.org/​10.1007/​3-540-63890-3_11

[48] Uno, Takeaki: A fast algorithm for enumerating bipartite perfect matchings. In: International Symposium on Algorithms and Computation (2001), 367–379. http:/​/​doi.org/​10.1007/​3-540-45678-3_32.
https:/​/​doi.org/​10.1007/​3-540-45678-3_32

[49] Ausiello, Giorgio ; Crescenzi, Pierluigi ; Gambosi, Giorgio ; Kann, Viggo ; Marchetti-Spaccamela, Alberto ; Protasi, Marco: Complexity and approximation: Combinatorial optimization problems and their approximability properties. Springer Science & Business Media, 2012 http:/​/​doi.org/​10.1007/​978-3-642-58412-1.
https:/​/​doi.org/​10.1007/​978-3-642-58412-1

[50] Seevinck, Michael ; Uffink, Jos: Partial separability and entanglement criteria for multiqubit quantum states. In: Physical Review A 78 (2008), Nr. 3, 032101. http:/​/​doi.org/​10.1103/​PhysRevA.78.032101.
https:/​/​doi.org/​10.1103/​PhysRevA.78.032101

[51] Szalay, Szilárd ; Kökényesi, Zoltán: Partial separability revisited: Necessary and sufficient criteria. In: Physical Review A 86 (2012), Nr. 3, 032341. http:/​/​doi.org/​10.1103/​PhysRevA.86.032341.
https:/​/​doi.org/​10.1103/​PhysRevA.86.032341

[52] Briegel, Hans J. ; Raussendorf, Robert: Persistent entanglement in arrays of interacting particles. In: Physical Review Letters 86 (2001), Nr. 5, 910. http:/​/​doi.org/​10.1103/​PhysRevLett.86.910.
https:/​/​doi.org/​10.1103/​PhysRevLett.86.910

[53] M Cunha, Márcio ; Fonseca, Alejandro ; O Silva, Edilberto: Tripartite entanglement: Foundations and applications. In: Universe 5 (2019), Nr. 10, 209. http:/​/​doi.org/​10.3390/​universe5100209.
https:/​/​doi.org/​10.3390/​universe5100209

Cited by

[1] Donghwa Lee, Tanumoy Pramanik, Seongjin Hong, Young-Wook Cho, Hyang-Tag Lim, Seungbeom Chin, and Yong-Su Kim, "Entangling three identical particles via spatial overlap", Optics Express 30 17, 30525 (2022).

[2] Carlos Ruiz-Gonzalez, Sören Arlt, Jan Petermann, Sharareh Sayyad, Tareq Jaouni, Ebrahim Karimi, Nora Tischler, Xuemei Gu, and Mario Krenn, "Digital Discovery of 100 diverse Quantum Experiments with PyTheus", Quantum 7, 1204 (2023).

[3] Sebastian Horvat and Borivoje Dakić, "Accessing inaccessible information via quantum indistinguishability", New Journal of Physics 25 11, 113008 (2023).

[4] Pawel Blasiak, Ewa Borsuk, and Marcin Markiewicz, "Arbitrary entanglement of three qubits via linear optics", Scientific Reports 12 1, 21596 (2022).

[5] Donghwa Lee, Tanumoy Pramanik, Young-Wook Cho, Hyang-Tag Lim, Seungbeom Chin, and Yong-Su Kim, "Entangling three identical particles via spatial overlap", arXiv:2104.05937, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 11:18:48) and SAO/NASA ADS (last updated successfully 2024-03-28 11:18:49). The list may be incomplete as not all publishers provide suitable and complete citation data.