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

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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.

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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", arXiv:2104.05937, Optics Express 30 17, 30525 (2022).

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