Entanglement Percolation in Two Types of Irregular Quantum Networks

Liuxia Han; Yaqi Zhao; Kan He

doi:10.2478/qic-2024-0004

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Entanglement Percolation in Two Types of Irregular Quantum Networks

Quantum Information & Computation

Volume 24 (2024): Issue 1 (October 2024)

By: Liuxia Han, Yaqi Zhao and Kan He

Open Access

|Jan 2025

Figures & Tables

Four examples of regular networks. (a) Bethe lattice. (b) Square lattice. (c) Honeycomb lattice. (d) Triangular lattice.

Different QN topologies between A and B. (a) Simple series QN. (b) Simple parallel QN. (c) Parallel-then-series QN. (d) Series-then-parallel QN. (e) Series-parallel QN. (f) General QN.

Table 1.

Connectivity rules of ConPT.

	ConPT
Series rule	C = c₁ c₂ ⋯
Parallel rule	$\frac{1 + \sqrt{1 - C^{2}}}{2} = max \frac{1}{2}, \frac{1 + \sqrt{1 - c_{1}^{2}}}{2} \frac{1 + \sqrt{1 - c_{2}^{2}}}{2} \dots\}$

SM transform. The left network becomes the right one after removing node R through the SM transform.

Process of the entanglement transmission on a butterfly network. (a) The initial butterfly network. (b) The butterfly network after removing node 1 through SM transform. (c) The butterfly network after removing nodes 2 and 3 in Fig. (b). (d) The butterfly network after removing node 4 in Fig. (c) through SM transform. (e) The butterfly network after removing node 5 in Fig. (d) by performing series-parallel operations. Here, links with varying line colors denote the weights of the connections, which correspond to the degree of concurrence between the two nodes.

Table 2.

Connectivity rules of CEP.

	CEP
Series rule	p = p₁p₂ ⋯
Parallel rule	1 − p = (1 − p₁)(1 − p₂) ⋯

The scatter plot of c and *C_SC* in the butterfly network under the ConPT. The black scatter plot represents the relationship graph between c and *C_SC*, the red dashed line is the entanglement percolation threshold c_th and the blue dashed line is the saturation point c_sat under the concurrence percolation.

Comparison of $P_{SC}^{ConPT}$ and P_SC in the butterfly network. The orange scatter plot represents the relationship graph between c and P_SC, the purple scatter plot also represents the relationship graph between c and $P_{SC}^{ConPT}$ .

Table 3.

The data supporting the Figure 5 and the Figure 6 in the butterfly network.

c	C_SC	$\frac{\partial^{2} C_{SC}}{\partial c^{2}}$	P_SC	$\frac{\partial^{2} P_{SC}}{\partial c^{2}}$	$P_{SC}^{ConPT}$
0.66	0.543678214	0.0013	0.0598251	7.1932	0.160706
0.68	0.584345788	0.0006	0.0733271	9.1772	0.188495
0.70	0.626347348	0.0004	0.0895879	10.3325	0.220455
0.72	0.668926005	−0.0005	0.108726	11.7100	0.256671
0.74	0.711917832	−0.00038694	0.131535	13.6725	0.297737
0.76	0.754400424	−0.0014	0.158477	15.3625	0.343585
0.78	0.796496077	−0.0021	0.190103	17.7075	0.395356
0.92	0.99929325	−0.0069	0.605555	20.2300	0.962410
0.94	1	−0.0094	0.699814	9.1200	1
0.96	1	−0.0094	0.802165	−35.4075	1

The scatter plot of c and *C_SC* in the parallel-then-series network under conditions *k_i* = 3 and i = 5. The black scatter plot represents the relationship graph between c and *C_SC*, the red dashed line is the entanglement percolation threshold c_th and the blue dashed line is the saturation point c_sat under the concurrence percolation.

Comparison of $P_{SC}^{ConPT}$ and P_SC in the parallel-then-series network under conditions *k_i* = 3 and i = 5. The orange scatter plot represents the relationship graph between c and P_SC, the purple scatter plot also represents the relationship graph between c and $P_{SC}^{ConPT}$ .

Table 4.

The data supporting the Figure 8 and the Figure 9 for the parallel-then-series network under conditions k_i = 3 and i = 5.

c	C_SC	$\frac{\partial^{2} C_{SC}}{\partial c^{2}}$	P_SC	$\frac{\partial^{2} P_{SC}}{\partial c^{2}}$	$P_{SC}^{ConPT}$
0.60	0.5551	2.8157	0.0277	4.6301	0.168216
0.62	0.6142	0.4345	0.0369	5.6487	0.210849
0.64	0.6734	−2.2899	0.0487	6.7974	0.260721
0.66	0.7317	−5.3403	0.0634	8.0677	0.318373
0.68	0.7878	−8.6893	0.0816	9.4416	0.384068
0.70	0.8405	−12.2992	0.1039	10.89	0.458188
0.72	0.8882	−16.1206	0.1308	12.3698	0.540542
0.74	0.9294	−20.0926	0.163	13.821	0.630925
0.76	0.9627	−24.1425	0.201	15.1645	0.729428
0.78	0.9863	−28.1861	0.2452	16.298	0.835038
0.80	0.9986	−32.1289	0.2962	17.0934	0.947103
0.82	1	−36.1567	0.3541	17.3919	1
0.84	1	−40.1238	0.4189	16.9994	1
0.92	1	−55.7857	0.7331	2.745	1
0.94	1	−59.7285	0.8166	−6.4183	1
0.96	1	−63.6713	0.895	−20.064	1

References

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DOI: https://doi.org/10.2478/qic-2024-0004 | Journal eISSN: 3106-0544 (formerly 1533-7146) | Journal ISSN: 1533-7146

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Language: English

Page range: 58 - 68

Submitted on: Sep 13, 2024

Accepted on: Dec 8, 2024

Published on: Jan 9, 2025

Published by: Cerebration Science Publishing Co., Limited

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

quantum network,

percolation,

entanglement transmission,

concurrence

Related subjects:

Computer sciences,

Computer sciences in the humanities,

Computer sciences in natural sciences,

Mathematics,

Applied mathematics,

Physics,

Quantum physics

© 2025 Liuxia Han, Yaqi Zhao, Kan He, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 24 (2024): Issue 1 (October 2024)

Entanglement Percolation in Two Types of Irregular Quantum Networks

Figures & Tables

Figure 1.

Figure 2.

Table 1.

Figure 3.

Figure 4.

Table 2.

Figure 5.

Figure 6.

Table 3.

Figure 7.

Figure 8.

Figure 9.

Table 4.

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