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Sharing Genuine Entanglement of Generalized Tripartite States by Multiple Sequential Observers

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Abstract

Genuine entanglement serves as a valuable resource for multipartite quantum information processing. Recently, the ability to share the genuine entanglement among a long sequence of independent observers has gained considerable attention, especially when the state preparation faces significant limitations. Firstly, this paper focuses on investigating the sharing ability of genuine tripartite entanglement under unilateral sequential measurements. We propose a measurement strategy for generalized tripartite Greenberger-Horne-Zeilinger (GHZ) states to enable arbitrarily long sequence of independent observers to detect the genuine entanglement using entanglement witnesses. Moreover, when the sharpness parameters of each sequential observer’s measurement settings are equal, we give the correlation between the controllable angle of generalized GHZ state and the maximum number of sequential observers capable of detecting genuine entanglement. In particular, we identify the controllable angle range within which the same number of such observers can be achieved for generalized GHZ states as with the maximally entangled GHZ state. We also determine the controllable angle range for generalized tripartite \(\varvec{W}\) states, where the number of sequential observers can be same as that with maximally entangled \(\varvec{W}\) state. Furthermore, we design a measurement strategy for the trilateral sequential scenario, demonstrating that the genuine tripartite entanglement of any generalized GHZ state can be shared by arbitrarily many groups of observers.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 62171056, 61973021) and Henan Key Laboratory of Network Cryptography Technology (LNCT2022-A03).

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Yao Xiao & Fenzhuo Guo

  2. Henan Key Laboratory of Network Cryptography Technology, Zhengzhou, 450001, China

    Yao Xiao & Fenzhuo Guo

  3. State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Yao Xiao, Fenzhuo Guo & Qiaoyan Wen

  4. School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing, 100191, China

    Haifeng Dong

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  1. Yao Xiao
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  2. Fenzhuo Guo
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  3. Haifeng Dong
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  4. Qiaoyan Wen
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Correspondence to Fenzhuo Guo.

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Appendix

Appendix

The entanglement witness operator \(\mathcal {W}_{\textsf{GHZ}}\) given by (3) can be decomposed as [4]

$$\begin{aligned} \mathcal {W}_{\textsf{GHZ}}= & {} \dfrac{1}{8}\bigg [3\mathbb {I}\otimes \mathbb {I}\otimes \mathbb {I}-\mathbb {I}\otimes \sigma _z\otimes \sigma _z-\sigma _z\otimes \mathbb {I}\otimes \sigma _z-\sigma _z\otimes \sigma _z\otimes \mathbb {I}\nonumber \\{} & {} -2\sigma _x^{\otimes 3}+\sqrt{2}\left( \dfrac{\sigma _x+\sigma _y}{\sqrt{2}}\right) ^{\otimes 3}+\sqrt{2}\left( \dfrac{\sigma _x-\sigma _y}{\sqrt{2}}\right) ^{\otimes 3}\bigg ], \end{aligned}$$
(44)

with four correlations \(\sigma _z^{\otimes 3}\), \(\sigma _x^{\otimes 3}\), \(((\sigma _x+\sigma _y)/\sqrt{2})^{\otimes 3}\) and \(((\sigma _x-\sigma _y)/\sqrt{2})^{\otimes 3}\), where \(\sigma _x\), \(\sigma _y\), \(\sigma _z\) are the Pauli matrices.

The decomposition of entanglement witness operator \(\mathcal {W}_{W}\) given by (4) can be written as

$$\begin{aligned} \mathcal {W}_{W}= & {} \dfrac{1}{24}\Bigg [13\mathbb {I}\otimes \mathbb {I}\otimes \mathbb {I}+3\sigma _z\otimes \mathbb {I}\otimes \mathbb {I}+3\mathbb {I}\otimes \sigma _z\otimes \mathbb {I}+3\mathbb {I}\otimes \mathbb {I}\otimes \sigma _z+5\mathbb {I}\otimes \sigma _z\otimes \sigma _z\nonumber \\{} & {} +5\sigma _z\otimes \mathbb {I}\otimes \sigma _z+5\sigma _z\otimes \sigma _z\otimes \mathbb {I}+7\sigma _z\otimes \sigma _z\otimes \sigma _z-\sqrt{2}\Bigg (\mathbb {I}\otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \nonumber \\{} & {} +\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}+\mathbb {I}\otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \nonumber \\{} & {} +\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}+\mathbb {I}\otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \nonumber \\{} & {} +\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}+\mathbb {I}\otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \nonumber \\{} & {} +\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}\Bigg )-2\sqrt{2}\Bigg (\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) ^{\otimes 3}\nonumber \\{} & {} +\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) ^{\otimes 3}+\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) ^{\otimes 3}+\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) ^{\otimes 3}\Bigg )\nonumber \\{} & {} -2\Bigg (\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}+\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\nonumber \\{} & {} +\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}+\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\Bigg )\Bigg ], \end{aligned}$$
(45)

with five correlations \(\sigma _z^{\otimes 3}\), \(((\sigma _z+\sigma _x)/\sqrt{2})^{\otimes 3}\), \(((\sigma _z-\sigma _x)/\sqrt{2})^{\otimes 3}\), \(((\sigma _z+\sigma _y)/\sqrt{2})^{\otimes 3}\) and \(((\sigma _z-\sigma _y)/\sqrt{2})^{\otimes 3}\).

The decomposition of witness operator \(\mathcal {W}_{W}\) in the case of the scenario in Section 3.2 becomes

$$\begin{aligned} \mathcal {W}_{W}^{\lambda _{{k}}}= & {} \dfrac{1}{24}\Bigg [13\mathbb {I}\otimes \mathbb {I}\otimes \mathbb {I}+3\sigma _z\otimes \mathbb {I}\otimes \mathbb {I}+3\mathbb {I}\otimes \sigma _z\otimes \mathbb {I}+3\mathbb {I}\otimes \mathbb {I}\otimes \lambda _{{k}}\sigma _z+5\mathbb {I}\otimes \sigma _z\otimes \lambda _{{k}}\sigma _z\nonumber \\{} & {} +5\sigma _z\otimes \mathbb {I}\otimes \lambda _{{k}}\sigma _z+5\sigma _z\otimes \sigma _z\otimes \mathbb {I}+7\sigma _z\otimes \sigma _z\otimes \lambda _{{k}}\sigma _z\nonumber \\{} & {} -\sqrt{2}\Bigg (\mathbb {I}\otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) +\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}\nonumber \\{} & {} +\mathbb {I}\otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) +\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}\nonumber \\{} & {} +\mathbb {I}\otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) +\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}\nonumber \\{} & {} +\mathbb {I}\otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) +\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}+\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \mathbb {I}\Bigg )\nonumber \\{} & {} -2\sqrt{2}\lambda _{{k}}\Bigg (\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) ^{\otimes 3}+\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) ^{\otimes 3}+\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) ^{\otimes 3}+\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) ^{\otimes 3}\Bigg )\nonumber \\{} & {} -2\Bigg (\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \lambda _{{k}}\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z+\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}+\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \lambda _{{k}}\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z-\sigma _x}{\sqrt{2}}\right) \otimes \mathbb {I}\nonumber \\{} & {} +\mathbb {I}\otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \lambda _{{k}}\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z+\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}+\mathbb {I}\otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \lambda _{{k}}\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \nonumber \\{} & {} +\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\otimes \lambda _{{k}}\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) +\left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \left( \dfrac{\sigma _z-\sigma _y}{\sqrt{2}}\right) \otimes \mathbb {I}\Bigg )\Bigg ], \end{aligned}$$
(46)

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Xiao, Y., Guo, F., Dong, H. et al. Sharing Genuine Entanglement of Generalized Tripartite States by Multiple Sequential Observers. Int J Theor Phys 63, 21 (2024). https://doi.org/10.1007/s10773-023-05539-x

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