Skip to main content
SpringerLink
Log in

Quantum Bell nonlocality cannot be shared under a special kind of bilateral measurements for high-dimensional quantum states

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Quantum Bell nonlocality is an important quantum phenomenon. Recently, the shareability of Bell nonlocality under unilateral measurements has been widely studied. In this study, we consider the shareability of quantum Bell nonlocality under bilateral measurements. Under a specific class of projection operators, we find that quantum Bell nonlocality cannot be shared for a limited number of times, as in the case of unilateral measurements. Our proof is analytical, and our measurement strategies can be generalized to higher-dimensional cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability statement

Our manuscript has no associated data.

References

  1. Einstein, A., Podolsky, B., Rosen, N.: Can quantum mechanical description of physical reality be considered complete. Phys. Rev. 47, 777 (1935)

    Article  MATH  ADS  Google Scholar 

  2. Nielsen, M. A., Chuang, I.: Quantum computation and quantum information (2002)

  3. Macchiavello, C.: On the role of entanglement in quantum information, Physica A: statistical mechanics and its applications. In: Proceedings of the Conference A Nonlinear World: The Real World, 2nd International Conference on Frontier Science, Vol. 338, p. 68 (2004)

  4. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. Ekert, A.K.: Quantum cryptography based on bell theorem. Phys. Rev. Lett. 67, 661 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Ekert, A., Jozsa, R., Marcer, P.: Quantum algorithms: entanglement-enhanced information processing [and discussion]. Philos. Trans.: Math. Phys. Eng. Sci. 356, 1769 (1998)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)

    Article  MATH  ADS  Google Scholar 

  8. Freedman, S.J., Clauser, J.F.: Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938 (1972)

    Article  ADS  Google Scholar 

  9. Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  10. Weihs, G., Jennewein, T., Simon, C., Weinfurter, H., Zeilinger, A.: Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81, 5039 (1998)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Rowe, M.A., Kielpinski, D., Meyer, V., Sackett, C.A., Itano, W.M., Monroe, C., Wineland, D.J.: Experimental violation of a Bell’s inequality with efficient detection. Nature (London) 409, 791 (2001)

    Article  ADS  Google Scholar 

  12. Matsukevich, D.N., Maunz, P., Moehring, D.L., Olmschenk, S., Monroe, C.: Bell inequality violation with two remote atomic qubits. Phys. Rev. Lett. 100, 150404 (2008)

    Article  ADS  Google Scholar 

  13. Ansmann, M., Wang, H., Bialczak, R.C., Hofheinz, M., Lucero, E., Neeley, M., OConnell, A.D., Sank, D., Weides, M., Wenner, J., Cleland, A.N., Martinis, J.M.: Violation of Bell’s inequality in Josephson phase qubits. Nature (London) 461, 504 (2009)

    Article  ADS  Google Scholar 

  14. Hofmann, J., Krug, M., Ortegel, N., Gerard, L., Weber, M., Rosenfeld, W., Weinfurter, H.: Heralded entanglement between widely separated atoms. Science 337, 72 (2012)

    Article  ADS  Google Scholar 

  15. Giustina, M., Mech, A., Ramelow, S., Wittmann, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Nam, S.W., Ursin, R., Zeilinger, A.: Bell violation using entangled photons without the fair-sampling assumption. Nature (London) 497, 227 (2013)

    Article  ADS  Google Scholar 

  16. Christensen, B.G., McCusker, K.T., Altepeter, J.B., Calkins, B., Gerrits, T., Lita, A.E., Miller, A., Shalm, L.K., Zhang, Y., Nam, S.W., Brunner, N., Lim, C.C.W., Gisin, N., Kwiat, P.G.: Detection-loophole-free test of quantum nonlocality, and applications. Phys. Rev. Lett. 111, 130406 (2013)

    Article  ADS  Google Scholar 

  17. Stuart, T.E., Slater, J.A., Colbeck, R., Renner, R., Tittel, W.: An experimental test of all theories with predictive power beyond quantum theory. Phys. Rev. Lett. 109, 020402 (2012)

    Article  ADS  Google Scholar 

  18. Dutta, S., Mukherjee, A., Banik, M.: Operational characterization of multipartite nonlocal correlations. Phys. Rev. A 102, 052218 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  19. Barrett, J., Hardy, L., Kent, A.: No signalling and quantum key distribution. Phys. Rev. Lett. 95, 010503 (2005)

    Article  ADS  Google Scholar 

  20. Acin, A., Brunner, N., Gisin, N., Massar, S., Pironio, S., Scarani, V.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007)

    Article  ADS  Google Scholar 

  21. Colbeck, R.: Quantum and relativistic protocols for secure multi- party computation, Ph.D. thesis, University of Cambridge, also available as arXiv:0911.3814 (2007)

  22. Colbeck, R., Kent, A.: Private randomness expansion with untrusted devices. J. Phys. A 44, 095305 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  23. Pironio, S., Acin, A., Massar, S., Boyer de la Giroday, A., Matsukevich, D.N., Maunz, P., Olmschenk, S., Hayes, D., Luo, L., Manning, T.A., Monroe, C.: Random numbers certified by Bell’s theorem. Nature (London) 464, 1021 (2010)

    Article  ADS  Google Scholar 

  24. Colbeck, R., Renner, R.: Free randomness can be amplified. Nat. Phys. 8, 450 (2012)

    Article  Google Scholar 

  25. Silva, R., Gisin, N., Guryanova, Y., Popescu, S.: Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements. Phys. Rev. Lett. 114, 250401 (2015)

    Article  ADS  Google Scholar 

  26. Mal, S., Majumdar, A., Home, D.: Sharing of nonlocalityof a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the otherwing. Mathematics 4, 48 (2016)

    Article  MATH  Google Scholar 

  27. Das, D., Ghosal, A., Sasmal, S., Mal, S., Majumdar, A.S.: Facets of bipartite nonlocality sharing by multiple observers via sequential measurements. Phys. Rev. A 99, 022305 (2019)

    Article  ADS  Google Scholar 

  28. Ren, C., Feng, T., Yao, D., Shi, H., Chen, J., Zhou, X.: Passive and active nonlocality sharing for a two-qubit system via weak measurements. Phys. Rev. A 100, 052121 (2019)

    Article  ADS  Google Scholar 

  29. Peter, J.: Brown and Roger Colbeck, Arbitrarily many independent observers can share the nonlocality of a single maximally entangled qubit pair. Phys. Rev. Lett. 125, 090401 (2020)

    Article  MathSciNet  Google Scholar 

  30. Zhang, T., Fei, S.M.: Sharing quantum nonlocality and genuine nonlocality with independent observables. Phys. Rev. A 103, 032216 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  31. Cheng, S., Liu, L., Baker, T.J., Hall, M.J.W.: Limitations on sharing Bell nonlocality between sequential pairs of observers. Phys. Rev. A. 104, L060201 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  32. Cheng, S., Liu, L., Baker, T.J., Hall, M.J.W.: Recycling qubits for the generation of Bell nonlocality between independent sequential observers. Phys. Rev. A. 105, 022411 (2022)

    Article  MathSciNet  ADS  Google Scholar 

  33. Zhu, J., Hu, M.J., Guo, G.C., Li, C.F., Zhang, Y.S.: Einstein–Podolsky–Rosen steering in two-sided sequential measurements with one entangled pair. Phys. Rev. A 105, 032211 (2022)

    Article  MathSciNet  ADS  Google Scholar 

  34. Saha, S., Das, D., Sasmal, S., Sarkar, D., Mukherjee, K., Roy, A., Bhattacharya, S.S.: Sharing of tripartite nonlocality by multiple observers measuring sequentially at one side. Quant. Inf. Process. 18, 42 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  35. Hall, M.J.W., Cheng, S.: Generalising the Horodecki criterion to nonprojective qubit observables. J. Phys. A: Math. Theor. 55, 045301 (2022)

    Article  MathSciNet  ADS  Google Scholar 

  36. Hou, W., Liu, X., Ren, C.: Network nonlocality sharing via weak measurements in the extended bilocal scenario. Phys. Rev. A 105, 042436 (2022)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgements

We thank Shao-Ming Fei, Naihuan Jing, and Yuan-Hong Tao for their helpful discussions. This work was supported by Hainan Provincial Natural Science Foundation of China under Grant No.121RC539 and the National Natural Science Foundation of China under Grant Nos.12126314,12126351,11861031. This project is also supported by the specific research fund of the Innovation Platform for Academicians of Hainan Province under Grant No.YSPTZX202215.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, China

    Tinggui Zhang, Qiming Luo & Xiaofen Huang

  2. Key Laboratory of Data Science and Smart Education, Ministry of Education, Hainan Normal Universit, Haikou, 571158, China

    Tinggui Zhang & Xiaofen Huang

Authors
  1. Tinggui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiming Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaofen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tinggui Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Proof of Lemma 2

By straightforward calculation, we have

$$\begin{aligned} \rho _{A^{1}B^{2}}= & {} {\frac{2+{\sqrt{1-\gamma _1^2}}}{4}}\rho _{A^{1}B^{1}}+{\frac{1}{4}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) }\rho _{A^{1}B^{1}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) } \\&\quad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{4}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) \right) }\rho _{A^{1}B^{1}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) \right) } \end{aligned}$$

and

$$\begin{aligned} \rho _{A^{2}B^{2}}= & {} {\frac{1}{2}}{\sum \limits _{a,x}}{(\sqrt{A_{a|x}}{\otimes }I)}\rho _{A^{1}B^{2}}{(\sqrt{A_{a|x}}{\otimes }I)}\\= & {} {\frac{1}{8}}\left( \left[ I_4+\left( \begin{array}{cc} {\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \rho _{A^{1}B^{2}}\left( \left[ I_4+\left( \begin{array}{cc} {\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \\&+{\frac{1}{8}}\left( \left[ I_4-\left( \begin{array}{cc} {\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \rho _{A^{1}B^{2}}\left( \left[ I_4-\left( \begin{array}{cc} {\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \\&+{\frac{1}{8}}\left( \left[ I_4+\left( \begin{array}{cc} {\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \rho _{A^{1}B^{2}}\left( \left[ I_4+\left( \begin{array}{cc} {\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \\&+{\frac{1}{8}}\left( \left[ I_4-\left( \begin{array}{cc} {\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \rho _{A^{1}B^{2}}\left( \left[ I_4-\left( \begin{array}{cc} {\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \right] \otimes {I}\right) \\= & {} {\frac{1}{2}}\rho _{A^{1}B^{2}}+{\frac{1}{4}}\left( \left( \begin{array}{cc} {\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \rho _{A^{1}B^{2}}\left( \left( \begin{array}{cc} {\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \\&+{\frac{1}{4}}\left( \left( \begin{array}{cc} {\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \rho _{A^{1}B^{2}}\left( \left( \begin{array}{cc} {\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) . \end{aligned}$$

Set \(P={\cos (\theta )}\sigma _3+{\sin (\theta )}\sigma _1\) and \(Q={\cos (\theta )}\sigma _3-{\sin (\theta )}\sigma _1\). \(\rho _{A^{2}B^{2}}\) can be expressed as

$$\begin{aligned} \rho _{A^{2}B^{2}}= & {} {\frac{2+{\sqrt{1-\gamma _1^2}}}{8}}\rho _{A^{1}B^{1}}\\&+{\frac{1}{8}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) }\rho _{A^{1}B^{1}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) }\\&+{\frac{1-{\sqrt{1-\gamma _1^2}}}{8}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) \right) }\rho _{A^{1}B^{1}}{\left( I\otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) \right) }\\&+{\frac{1}{16}}\left( \left( \begin{array}{cc} P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) }\right) \\&+{\frac{2+{\sqrt{1-\gamma _1^2}}}{16}}\left( \left( \begin{array}{cc} P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \\&+{\frac{1-{\sqrt{1-\gamma _1^2}}}{16}}\left( \left( \begin{array}{cc} P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {{\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) }}\right) \rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {{\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) }}\right) \\&+{\frac{1}{16}}\left( \left( \begin{array}{cc} Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) }\right) \rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) }\right) \\&+{\frac{2+{\sqrt{1-\gamma _1^2}}}{16}}\left( \left( \begin{array}{cc} Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {I}\right) \\&+{\frac{1-{\sqrt{1-\gamma _1^2}}}{16}}\left( \left( \begin{array}{cc} Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {{\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) }}\right) \rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes {{\left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _1 \\ \end{array} \right) }}\right) , \end{aligned}$$

where \(\rho _{A^{1}B^{1}}=|\varphi \rangle \langle \varphi |\).

Therefore,

$$\begin{aligned}&Tr[\rho _{A^{2}B^{2}}((A_0+A_1){\otimes }B_0^2)]\\&\quad =2Tr\left[ \rho _{A^{2}B^{2}}\left( \left( \begin{array}{cc} \cos (\theta )\sigma _3 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \right] \\&\quad =2Tr\bigg [{\frac{2+{\sqrt{1-\gamma _1^2}}}{8}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \cos (\theta )\sigma _3 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1}{8}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \cos (\theta )\sigma _3 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{8}}\rho _{AB^{1}}\left( \left( \begin{array}{cc} \cos (\theta )\sigma _3 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P\cos (\theta )\sigma _3P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{2+{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P\cos (\theta )\sigma _3P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P\cos (\theta )\sigma _3P &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q\cos (\theta )\sigma _3Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{2+{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q\cos (\theta )\sigma _3Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q\cos (\theta )\sigma _3Q &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\sigma _3 \\ \end{array} \right) \right) \bigg ]\\&\quad =Tr\bigg [{\frac{3+{\sqrt{1-\gamma _1^2}}}{2}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \cos ^3(\theta )\sigma _3 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \sigma _3 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{2}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \cos ^3(\theta )\sigma _3 &{} 0 \\ 0 &{} I_{d-2} \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\sigma _3 \\ \end{array} \right) \right) \bigg ]\\&\quad =2\cos ^3(\theta )c_1^2-2\cos ^3(\theta )c_2^2 -(1+{\sqrt{1-\gamma _1^2}})c_d^2\\&\qquad +(1+{\sqrt{1-\gamma _1^2}})[c_3^2+c_4^2+\cdots +c_{d-1}^2]. \end{aligned}$$

Since \(2\cos ^3(\theta )\le 2\), \(-2\cos ^3(\theta )\le 2\), \((1+{\sqrt{1-\gamma _1^2}})\le 2\) and \((1+{\sqrt{1-\gamma _1^2}})\le 2\), where \( 0<\gamma _1<1, 0<\theta \le {\frac{\pi }{4}}\), and \(\sum _{i=1}^dc_i^2=1\) we have \(Tr[\rho _{A^{2}B^{2}}((A_0+A_1){\otimes }B_0)]\le 2.\)

Similarly, we have

$$\begin{aligned}&Tr[\rho _{A^{2}B^{2}}((A_0-A_1){\otimes }B_1)]\\&\quad =2Tr\left[ \rho _{A^{2}B^{2}}\left( \left( \begin{array}{cc} \sin (\theta )\sigma _1 &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \right] \\&\quad =2Tr\bigg [{\frac{2+{\sqrt{1-\gamma _1^2}}}{8}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \sin (\theta )\sigma _1 &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{1}{8}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \sin (\theta )\sigma _1 &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{8}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} \sin (\theta )\sigma _1 &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{1}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P\sin (\theta )\sigma _1P &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{2+{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P\sin (\theta )\sigma _1P &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} P\sin (\theta )\sigma _1P &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{1}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q\sin (\theta )\sigma _1Q &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} -\gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{2+{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q\sin (\theta )\sigma _1Q &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \\&\qquad +{\frac{1-{\sqrt{1-\gamma _1^2}}}{16}}\rho _{A^{1}B^{1}}\left( \left( \begin{array}{cc} Q\sin (\theta )\sigma _1Q &{} 0 \\ 0 &{} 0 \\ \end{array} \right) \otimes \left( \begin{array}{cc} I_{d-2} &{} 0 \\ 0 &{} \gamma _1\sigma _1 \\ \end{array} \right) \right) \bigg ]\\&\quad =0. \end{aligned}$$

\(\square \)

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, T., Luo, Q. & Huang, X. Quantum Bell nonlocality cannot be shared under a special kind of bilateral measurements for high-dimensional quantum states. Quantum Inf Process 21, 350 (2022). https://doi.org/10.1007/s11128-022-03699-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-022-03699-z

Keywords

Search

Navigation