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Double-direction quantum cyclic controlled remote state preparation of two-qubit states

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Abstract

In this article, we propose a four-party double-direction quantum cyclic controlled remote state preparation scheme, where two-qubit states can be remotely prepared cyclically among three correspondents both in clockwise and counterclockwise directions simultaneously under the control of the supervisor. Before presenting our four-party scheme, we give the quantum circuit diagram for constructing the 25-qubit quantum entangled channel. In our scheme, each correspondent merely carries out a four-qubit projective measurement and the supervisor only need to perform a single-qubit measurement in the Z-basis. After obtaining the measurement results from the other two correspondents and the supervisor, each correspondent can restore the desired states perfectly by applying proper unitary operations. The proposed four-party scheme can also be extended to the case containing \(m(m>3)\) correspondents, by using a \((8m+1)\)-qubit entangled channel. Discussions show that the success probability of both the proposed four-party and \((m+1)\)-party schemes can reach 1. We also analyze the control power of the supervisor in our scheme. Detailed analysis demonstrates that the control power of the supervisor can also be guaranteed.

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References

  1. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Vaidman, L.: Teleportation of quantum states. Phys. Rev. A 49(2), 1473–1476 (1994)

    Article  ADS  Google Scholar 

  3. Schaetz, T., Barrett, M.D., Leibfried, D., Chiaverini, J., Britton, J., Itano, W.M., Jost, J.D., Langer, C., Wineland, D.J.: Quantum dense coding with atomic qubits. Phys. Rev. Lett. 93(4), 040505 (2004)

    Article  ADS  Google Scholar 

  4. Goldenberg, L., Vaidman, L.: Quantum cryptography based on orthogonal states. Phys. Rev. Lett. 75(7), 1239–1243 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89(18), 187902 (2002)

    Article  ADS  Google Scholar 

  6. Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Lo, H.K.: Classical-communication cost in distributed quantum-information processing: a generalization of quantum-communication complexity. Phys. Rev. A 62, 012313 (2000)

    Article  ADS  Google Scholar 

  8. Bennett, C.H., DiVincenzo, D.P., Shor, P.W., Smolin, J.A.: Remote state preparation. Phys. Rev. Lett. 87(7), 077902 (2001)

    Article  ADS  Google Scholar 

  9. Pati, A.K.: Minimum classical bit for remote preparation and measurement of a qubit. Phys. Rev. A 63, 014302 (2001)

    Article  ADS  Google Scholar 

  10. Berry, D.W., Sanders, B.C.: Optimal remote state preparation. Phys. Rev. Lett. 90(5), 057901 (2003)

    Article  ADS  Google Scholar 

  11. Ye, M.Y., Zhang, Y.S., Guo, G.C.: Faithful remote state preparation using finite classical bits and a nonmaximally entangled state. Phys. Rev. A 69, 022310 (2004)

    Article  ADS  Google Scholar 

  12. Zhan, Y.B.: Remote state preparation of a Greenberger-Horne–Zeilinger class state. Commun. Theor. Phys. 43, 637–640 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  13. Dai, H.Y., Chen, P.X., Liang, M.L., Li, C.Z.: Classical communication cost and remote preparation of the four-particle GHZ class state. Phys. Lett. A 355, 285–288 (2006)

    Article  ADS  Google Scholar 

  14. Wang, Z.Y.: Highly efficient remote preparation of an arbitrary three-qubit state via a four-qubit cluster state and an EPR state. Quantum Inf. Process. 12, 1321–1334 (2013)

    Article  ADS  MATH  Google Scholar 

  15. Wei, J.H., Shi, L., Ma, L.H., Xue, Y., Zhuang, X.C., Kang, Q.Y., Li, X.S.: Remote preparation of an arbitrary multi-qubit state via two-qubit entangled states. Quantum Inf. Process. 16, 260 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Wei, J.H., Shi, L., Zhu, Y., Xue, Y., Xu, Z.Y., Jiang, J.: Deterministic remote preparation of arbitrary multi-qubit equatorial states via two-qubit entangled states. Quantum Inf. Process. 17, 70 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Zha, X.W., Wang, M.R., Jiang, R.X.: Two forms schemes of deterministic remote state preparation for four-qubit cluster-type state. Int. J. Theor. Phys. 59, 960–973 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xiang, G.Y., Li, J., Yu, B., Guo, G.C.: Remote preparation of mixed states via noisy entanglement. Phys. Rev. A 72, 012315 (2005)

    Article  ADS  Google Scholar 

  19. Peters, N.A., Barreiro, J.T., Goggin, M.E., Wei, T.C., Kwiat, P.G.: Remote state preparation: arbitrary remote control of photon polarization. Phys. Rev. Lett. 94, 150502 (2005)

    Article  ADS  Google Scholar 

  20. Liu, W.T., Wu, W., Ou, B.Q., Chen, P.X., Li, C.Z., Yuan, J.M.: Experimental remote preparation of arbitrary photon polarization states. Phys. Rev. A 76, 022308 (2007)

    Article  ADS  Google Scholar 

  21. Ra, Y.S., Lim, H.T., Kim, Y.H.: Remote preparation of three-photon entangled states via single-photon measurement. Phys. Rev. A 94, 042329 (2016)

    Article  ADS  Google Scholar 

  22. Nawaz, M., Islam, R.U., Ikram, M.: Remote state preparation through hyperentangled atomic states. J. Phys. B At. Mol. Opt. Phys. 51(7), 075501 (2018)

    Article  ADS  Google Scholar 

  23. Wang, Z.Y., Liu, Y.M., Zuo, X.Q., Zhang, Z.J.: Controlled remote state preparation. Commun. Theor. Phys. 52, 235–240 (2009)

    Article  ADS  MATH  Google Scholar 

  24. Chen, X.B., Ma, S.Y., Su, Y., Zhang, R., Yang, Y.X.: Controlled remote state preparation of arbitrary two and three qubit states via the Brown state. Quantum Inf. Process. 11, 1653–1667 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Wang, C., Zeng, Z., Li, X.H.: Controlled remote state preparation via partially entangled quantum channel. Quantum Inf. Process. 14, 1077–1089 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Zhou, K.H., Shi, L., Luo, B.B., Xue, Y., Huang, C., Ma, Z.Q., Wei, J.H.: Deterministic controlled remote state preparation of real-parameter multi-qubit states via maximal slice states. Int. J. Theor. Phys. 58, 4079–4092 (2019)

    Article  MATH  Google Scholar 

  27. Nguyen, B.A.: Joint remote preparation of a general two-qubit state. J. Phys. B: At. Mol. Opt. Phys. 42, 125501 (2009)

    Article  Google Scholar 

  28. Luo, M.X., Chen, X.B., Ma, S.Y., Niu, X.X., Yang, Y.X.: Joint remote preparation of an arbitrary three-qubit state. Opt. Commun. 283, 4796–4801 (2010)

    Article  ADS  Google Scholar 

  29. Li, X.H., Ghose, S.: Optimal joint remote state preparation of equatorial states. Quantum Inf. Process. 14, 4585–4592 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Choudhury, B.S., Samanta, S.: Perfect joint remote state preparation of arbitrary six-qubit cluster-type states. Quantum Inf. Process. 17, 175 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Guan, X.W., Chen, X.B., Yang, Y.X.: Controlled-joint remote preparation of an arbitrary two-qubit state via non-maximally entangled channel. Int. J. Theor. Phys. 51, 3575–3586 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nguyen, B.A., Cao, T.B.: Perfect controlled joint remote state preparation independent of entanglement degree of the quantum channel. Phys. Lett. A 378, 3582–3585 (2014)

    Article  MATH  Google Scholar 

  33. Cao, T.B., Nguyen, V.H., Nguyen, B.A.: Flexible controlled joint remote preparation of an arbitrary two-qubit state via nonmaximally entangled quantum channels. Adv. Nat. Sci. Nanosci. Nanotechnol. 7, 025007 (2016)

    Article  ADS  Google Scholar 

  34. Sang, M.H., Yu, S.D.: Controlled joint remote state preparation of an arbitrary equatorial two-qubit state. Int. J. Theor. Phys. 58, 2910–2913 (2019)

    Article  MATH  Google Scholar 

  35. Hou, K., Wang, J., Yuan, H., Shi, S.H.: Multiparty-controlled remote preparation of two- particle state. Commun. Theor. Phys. 52, 848–852 (2009)

    Article  ADS  MATH  Google Scholar 

  36. Wang, D., Hoehn, R.D., Ye, L., Kais, S.: Efficient remote preparation of four-qubit cluster-type entangled states with multi-party over partially entangled channels. Int. J. Theor. Phys. 55, 3454–3466 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, D., Ye, L.: Multiparty-controlled joint remote state preparation. Quantum Inf. Process. 12, 3223–3237 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Chang, L.W., Zheng, S.H., Gu, L.Z., Jin, L., Yang, Y.X.: Multiparty-controlled joint remote preparation of an arbitrary four-qubit cluster-type state via two different entangled quantum channels. Int. J. Theor. Phys. 54, 2864–2880 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lv, S.X., Zhao, Z.W., Zhou, P.: Multiparty-controlled joint remote preparation of an arbitrary m-qudit state with d-dimensional Greenberger–Horne–Zeilinger states. Int. J. Theor. Phys. 54, 2864–2880 (2018)

    MATH  Google Scholar 

  40. Cao, T.B., Nguyen, B.A.: Deterministic controlled bidirectional remote state preparation. Adv. Nat. Sci. Nanosci. Nanotechnol. 5, 015003 (2014)

    Article  ADS  Google Scholar 

  41. Sharma, V., Shukla, C., Banerjee, S., Pathak, A.: Controlled bidirectional remote state preparation in noisy environment: a generalized view. Quantum Inf. Process. 14, 3441–3464 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Sang, Z.W.: Asymmetric bidirectional controlled remote state preparation by using a seven-particle entangled state. Int. J. Theor. Phys. 56, 3209–3212 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Li, Y.H., Qiao, Y., Sang, M.H., Nie, Y.Y.: Bidirectional controlled remote state preparation of an arbitrary two-qubit state. Int. J. Theor. Phys. 58, 2228–2234 (2019)

    Article  MATH  Google Scholar 

  44. Sun, Y.R., Chen, X.B., Xu, G., Yuan, K.G., Yang, Y.X.: Asymmetric controlled bidirectional remote preparation of two- and three-qubit equatorial state. Sci. Rep. 9, 2081 (2019)

    Article  ADS  Google Scholar 

  45. Peng, J.Y., Bai, M.Q., Mo, Z.W.: Bidirectional controlled joint remote state preparation. Quantum Inf. Process. 14, 4263–4278 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Wang, X.Y., Mo, Z.W.: Bidirectional controlled joint remote state preparation via a seven-qubit entangled state. Int. J. Theor. Phys. 56, 1052–1058 (2017)

    Article  MATH  Google Scholar 

  47. Shi, J., Zhan, Y.B.: Scheme for asymmetric and deterministic controlled bidirectional joint remote state preparation. Commun. Theor. Phys. 70, 515–520 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Wang, M.M., Yang, C., Mousoli, R.: Controlled cyclic remote state preparation of arbitrary qubit states. CMC-Comput. Mater. Contin. 55, 321–329 (2018)

    Google Scholar 

  49. Zhang, C.Y., Bai, M.Q., Zhou, S.Q.: Cyclic joint remote state preparation in noisy environment. Quantum Inf. Process. 17, 146 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Sang, Z.W.: Cyclic controlled joint remote state preparation by using a ten-qubit entangled state. Int. J. Theor. Phys. 58, 255–260 (2019)

    Article  MATH  Google Scholar 

  51. Jiang, S.X., Zhou, R.G., Xu, R.Q., Luo, G.F.: Cyclic hybrid double-channel quantum communication via Bell-state and GHZ-state in noisy environments. IEEE Access 7, 80530–80541 (2019)

    Article  Google Scholar 

  52. Sun, S.Y., Zhang, H.S.: Quantum double-direction cyclic controlled communication via a thirteen-qubit entangled state. Quantum Inf. Process. 19, 120 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  53. Yu, Y., Zhao, N.: General quantum broadcast and multi-cast communications based on entanglement. Opt. Express 26, 29296–29310 (2018)

    Article  ADS  Google Scholar 

  54. Yu, Y., Zhao, N., Pei, C.X.: Multicast-based multiparty remote state preparation schemes of two-qubit states. Quantum Inf. Process. 18, 319 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  55. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  56. Li, X.H., Ghose, S.: Control power in perfect controlled teleportation via partially entangled channels. Phys. Rev. A 90, 052305 (2014)

    Article  ADS  Google Scholar 

  57. Li, X.H., Ghose, S.: Analysis of control power in controlled remote state preparation schemes. Int. J. Theor. Phys. 56, 667–677 (2017)

    Article  MATH  Google Scholar 

  58. Monz, T., Schindler, P., Barreiro, J.T., Chwalla, M., Nigg, D., Coish, W.A., Harlander, M., Hansel, W., Hennrich, M., Blatt, R.: 14-qubit entanglement: creation and coherence. Phys. Rev. Lett. 106, 130506 (2011)

    Article  ADS  Google Scholar 

  59. Wang, Y.H., Li, Y., Yin, Z.Q., Zeng, B.: 16-qubit IBM universal quantum computer can be fully entangled. NPJ Quantum Inform. 4, 46 (2018)

    Article  ADS  Google Scholar 

  60. Wang, X.L., Luo, Y.H., Huang, H.L., Chen, M.C., Su, Z.E., Liu, C., Chen, C., Li, W., Fang, Y.Q., Jiang, X., Zhang, J., Li, L., Liu, N.L., Lu, C.Y., Pan, J.W.: 18-Qubit entanglement with six photons’ three degrees of freedom. Phys. Rev. Lett. 120, 260502 (2018)

    Article  ADS  Google Scholar 

  61. Song, C., Xu, K., Li, H.K., Zhang, Y.R., Zhang, X., Liu, W.X., Guo, Q.J., Wang, Z., Ren, W.H., Hao, J., Feng, H., Fan, H., Zheng, D.N., Wang, D.W., Wang, H., Zhu, S.Y.: Observation of multi-component atomic Schrödinger cat states of up to 20 qubits. Science 365, 574–577 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61801218).

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Authors and Affiliations

  1. School of Electronics and Information, Northwestern Polytechnical University, Xi’an, 710129, China

    Shiya Sun & Huisheng Zhang

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  1. Shiya Sun
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  2. Huisheng Zhang
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Correspondence to Shiya Sun.

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Appendix

Appendix

The superposition form of \({\left| \varUpsilon \right\rangle _{{A_1}{A_2} \ldots {A_8}{B_1}{B_2} \ldots {B_8}{C_1}{C_2} \ldots {C_8}D}}\) in Eq. (16) can be represented as below

$$\begin{aligned}&{\left| \varUpsilon \right\rangle _{{A_1}{A_2} \cdots {A_8}{B_1}{B_2} \cdots {B_8}{C_1}{C_2} \cdots {C_8}D}}\\&\quad = \frac{1}{{{{6}}4\sqrt{{2}} }}\left\{ {\left[ {{{\left| {{\varDelta _1}} \right\rangle }_{{A_1}{A_2}{A_3}{A_4}}} \otimes } \right. {{\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle + a_1^3\left| {10} \right\rangle - a_1^4\left| {11} \right\rangle } \right) }_{{B_5}{B_6}}}} \right. \\&\qquad \otimes {\left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle \left. { + a_2^3\left| {10} \right\rangle - a_2^4\left| {11} \right\rangle } \right) } \right. _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _2}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle + a_1^3\left| {10} \right\rangle - a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. {\left. { - a_2^3\left| {11} \right\rangle - a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _3}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle + a_1^3\left| {10} \right\rangle - a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad {\left. { \otimes \left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. - a_2^3\left| {00} \right\rangle + a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _4}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle + a_1^3\left| {10} \right\rangle - a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. {\left. { + a_2^3\left| {01} \right\rangle + a_2^4\left| {00} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _5}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle - a_1^3\left| {11} \right\rangle - a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. + a_2^3\left| {10} \right\rangle - a_2^4\left| {11} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _6}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle - a_1^3\left| {11} \right\rangle - a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. - a_2^3\left| {11} \right\rangle - a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _7}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle - a_1^3\left| {11} \right\rangle - a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. - a_2^3\left| {00} \right\rangle + a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _8}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle - a_1^3\left| {11} \right\rangle - a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. + a_2^3\left| {01} \right\rangle + a_2^4\left| {00} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _9}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle - a_1^3\left| {00} \right\rangle + a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad {\left. { \otimes \left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. + a_2^3\left| {10} \right\rangle - a_2^4\left| {11} \right\rangle } \right) _{{C_7}{C_8}}}\\ \end{aligned}$$
$$\begin{aligned}&\qquad + {\left| {{\varDelta _{10}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle - a_1^3\left| {00} \right\rangle + a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. - a_2^3\left| {11} \right\rangle - a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{11}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle - a_1^3\left| {00} \right\rangle + a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. {\left. { - a_2^3\left| {00} \right\rangle + a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{12}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle - a_1^3\left| {00} \right\rangle + a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. + a_2^3\left| {01} \right\rangle + a_2^4\left| {00} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{13}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle + a_1^3\left| {01} \right\rangle + a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. {\left. { + a_2^3\left| {10} \right\rangle - a_2^4\left| {11} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{14}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle + a_1^3\left| {01} \right\rangle + a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. - a_2^3\left| {11} \right\rangle - a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{15}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle + a_1^3\left| {01} \right\rangle + a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. {\left. { - a_2^3\left| {00} \right\rangle + a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{16}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle + a_1^3\left| {01} \right\rangle + a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \left. {{{\left. { \otimes \left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. + a_2^3\left| {01} \right\rangle + a_2^4\left| {00} \right\rangle } \right) }_{{C_7}{C_8}}}} \right] \\&\qquad \otimes \left[ {{{\left| {{\varSigma _1}} \right\rangle }_{{B_1}{B_2}{B_3}{B_4}}} \otimes } \right. {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle + b_1^3\left| {10} \right\rangle - b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle \left. { + b_2^3\left| {10} \right\rangle - b_2^4\left| {11} \right\rangle } \right) } \right. _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _2}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle + b_1^3\left| {10} \right\rangle - b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. {\left. { - b_2^3\left| {11} \right\rangle - b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _3}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle + b_1^3\left| {10} \right\rangle - b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad {\left. { \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. - b_2^3\left| {00} \right\rangle + b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}} \end{aligned}$$
$$\begin{aligned}&\qquad + {\left| {{\varSigma _4}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle + b_1^3\left| {10} \right\rangle - b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle } \right. {\left. { + b_2^3\left| {01} \right\rangle + b_2^4\left| {00} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _5}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle - b_1^3\left| {11} \right\rangle - b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. + b_2^3\left| {10} \right\rangle - b_2^4\left| {11} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _6}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle - b_1^3\left| {11} \right\rangle - b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. - b_2^3\left| {11} \right\rangle - b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _7}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle - b_1^3\left| {11} \right\rangle - b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. - b_2^3\left| {00} \right\rangle + b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _8}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle - b_1^3\left| {11} \right\rangle - b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle } \right. + b_2^3\left| {01} \right\rangle + b_2^4\left| {00} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _9}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle - b_1^3\left| {00} \right\rangle + b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad {\left. { \otimes \left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. + b_2^3\left| {10} \right\rangle - b_2^4\left| {11} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{10}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle - b_1^3\left| {00} \right\rangle + b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. - b_2^3\left| {11} \right\rangle - b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{11}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle - b_1^3\left| {00} \right\rangle + b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. {\left. { - b_2^3\left| {00} \right\rangle + b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{12}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle - b_1^3\left| {00} \right\rangle + b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle } \right. + b_2^3\left| {01} \right\rangle + b_2^4\left| {00} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{13}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {11} \right\rangle + b_1^2\left| {10} \right\rangle + b_1^3\left| {01} \right\rangle + b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. {\left. { + b_2^3\left| {10} \right\rangle - b_2^4\left| {11} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{14}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {11} \right\rangle + b_1^2\left| {10} \right\rangle + b_1^3\left| {01} \right\rangle + b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left. {\left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. - b_2^3\left| {11} \right\rangle - b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}} \end{aligned}$$
$$\begin{aligned}&\qquad + {\left| {{\varSigma _{15}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {11} \right\rangle + b_1^2\left| {10} \right\rangle + b_1^3\left| {01} \right\rangle + b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. {\left. { - b_2^3\left| {00} \right\rangle + b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{16}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {11} \right\rangle + b_1^2\left| {10} \right\rangle + b_1^3\left| {01} \right\rangle + b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \left. {{{\left. { \otimes \left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle } \right. + b_2^3\left| {01} \right\rangle + b_2^4\left| {00} \right\rangle } \right) }_{{A_7}{A_8}}}} \right] \\&\qquad \otimes \left[ {{{\left| {{\varOmega _1}} \right\rangle }_{{C_1}{C_2}{C_3}{C_4}}} \otimes } \right. {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle + c_1^3\left| {10} \right\rangle - c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle \left. { + c_2^3\left| {10} \right\rangle - c_2^4\left| {11} \right\rangle } \right) } \right. _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _2}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle + c_1^3\left| {10} \right\rangle - c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. {\left. { - c_2^3\left| {11} \right\rangle - c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _3}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle + c_1^3\left| {10} \right\rangle - c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad {\left. { \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. - c_2^3\left| {00} \right\rangle + c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _4}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle + c_1^3\left| {10} \right\rangle - c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle } \right. {\left. { + c_2^3\left| {01} \right\rangle + c_2^4\left| {00} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _5}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle - c_1^3\left| {11} \right\rangle - c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. + c_2^3\left| {10} \right\rangle - c_2^4\left| {11} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _6}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle - c_1^3\left| {11} \right\rangle - c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. - c_2^3\left| {11} \right\rangle - c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _7}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle - c_1^3\left| {11} \right\rangle - c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. - c_2^3\left| {00} \right\rangle + c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _8}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle - c_1^3\left| {11} \right\rangle - c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle } \right. + c_2^3\left| {01} \right\rangle + c_2^4\left| {00} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _9}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle - c_1^3\left| {00} \right\rangle + c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad {\left. { \otimes \left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. + c_2^3\left| {10} \right\rangle - c_2^4\left| {11} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{10}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle - c_1^3\left| {00} \right\rangle + c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. - c_2^3\left| {11} \right\rangle - c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{11}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle - c_1^3\left| {00} \right\rangle + c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. {\left. { - c_2^3\left| {00} \right\rangle + c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{12}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle - c_1^3\left| {00} \right\rangle + c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle } \right. + c_2^3\left| {01} \right\rangle + c_2^4\left| {00} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{13}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle + c_1^3\left| {01} \right\rangle + c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. {\left. { + c_2^3\left| {10} \right\rangle - c_2^4\left| {11} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{14}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle + c_1^3\left| {01} \right\rangle + c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left. {\left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. - c_2^3\left| {11} \right\rangle - c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{15}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle + c_1^3\left| {01} \right\rangle + c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. {\left. { - c_2^3\left| {00} \right\rangle + c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{16}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle + c_1^3\left| {01} \right\rangle + c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \left. {{{\left. { \otimes \left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle } \right. + c_2^3\left| {01} \right\rangle + c_2^4\left| {00} \right\rangle } \right) }_{{B_7}{B_8}}}} \right] \otimes {\left| 0 \right\rangle _D}\\&\qquad + \left[ {{{\left| {{\varDelta _1}} \right\rangle }_{{A_1}{A_2}{A_3}{A_4}}} \otimes } \right. {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle - a_1^3\left| {01} \right\rangle - a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle \left. { - a_2^3\left| {01} \right\rangle - a_2^4\left| {00} \right\rangle } \right) } \right. _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _2}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle - a_1^3\left| {01} \right\rangle - a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. {\left. { + a_2^3\left| {00} \right\rangle - a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _3}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle - a_1^3\left| {01} \right\rangle - a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad {\left. { \otimes \left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. + a_2^3\left| {11} \right\rangle + a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _4}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {11} \right\rangle + a_1^2\left| {10} \right\rangle - a_1^3\left| {01} \right\rangle - a_1^4\left| {00} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. {\left. { - a_2^3\left| {10} \right\rangle + a_2^4\left| {11} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _5}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle + a_1^3\left| {00} \right\rangle - a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. - a_2^3\left| {01} \right\rangle - a_2^4\left| {00} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _6}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle + a_1^3\left| {00} \right\rangle - a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. + a_2^3\left| {00} \right\rangle - a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _7}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle + a_1^3\left| {00} \right\rangle - a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. + a_2^3\left| {11} \right\rangle + a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}} \end{aligned}$$
$$\begin{aligned}&\qquad + {\left| {{\varDelta _8}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {10} \right\rangle - a_1^2\left| {11} \right\rangle + a_1^3\left| {00} \right\rangle - a_1^4\left| {01} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. - a_2^3\left| {10} \right\rangle + a_2^4\left| {11} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _9}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle + a_1^3\left| {11} \right\rangle + a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad {\left. { \otimes \left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. - a_2^3\left| {01} \right\rangle - a_2^4\left| {00} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{10}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle + a_1^3\left| {11} \right\rangle + a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. + a_2^3\left| {00} \right\rangle - a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{11}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle + a_1^3\left| {11} \right\rangle + a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\otimes {\left. {\left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. + a_2^3\left| {11} \right\rangle + a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{12}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {01} \right\rangle + a_1^2\left| {00} \right\rangle + a_1^3\left| {11} \right\rangle + a_1^4\left| {10} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. - a_2^3\left| {10} \right\rangle + a_2^4\left| {11} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{13}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle - a_1^3\left| {10} \right\rangle + a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {11} \right\rangle + a_2^2\left| {10} \right\rangle } \right. - a_2^3\left| {01} \right\rangle - a_2^4\left| {00} \right\rangle } \right) _{{C_7}{C_8}}} \end{aligned}$$
$$\begin{aligned}&\qquad + {\left| {{\varDelta _{14}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle - a_1^3\left| {10} \right\rangle + a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {10} \right\rangle - a_2^2\left| {11} \right\rangle } \right. + a_2^3\left| {00} \right\rangle - a_2^4\left| {01} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{15}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle - a_1^3\left| {10} \right\rangle + a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes {\left. {\left( {a_2^1\left| {01} \right\rangle + a_2^2\left| {00} \right\rangle } \right. + a_2^3\left| {11} \right\rangle + a_2^4\left| {10} \right\rangle } \right) _{{C_7}{C_8}}}\\&\qquad + {\left| {{\varDelta _{16}}} \right\rangle _{{A_1}{A_2}{A_3}{A_4}}} \otimes {\left( {a_1^1\left| {00} \right\rangle - a_1^2\left| {01} \right\rangle - a_1^3\left| {10} \right\rangle + a_1^4\left| {11} \right\rangle } \right) _{{B_5}{B_6}}}\\&\qquad \otimes \left. {{{\left. {\left( {a_2^1\left| {00} \right\rangle - a_2^2\left| {01} \right\rangle } \right. - a_2^3\left| {10} \right\rangle + a_2^4\left| {11} \right\rangle } \right) }_{{C_7}{C_8}}}} \right] \\&\qquad \otimes \left[ {{{\left| {{\varSigma _1}} \right\rangle }_{{B_1}{B_2}{B_3}{B_4}}} \otimes } \right. {\left( {b_1^1\left| {{{11}}} \right\rangle + b_1^2\left| {{{10}}} \right\rangle - b_1^3\left| {01} \right\rangle - b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle \left. { - b_2^3\left| {01} \right\rangle - b_2^4\left| {00} \right\rangle } \right) } \right. _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _2}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {{{11}}} \right\rangle + b_1^2\left| {{{10}}} \right\rangle - b_1^3\left| {01} \right\rangle - b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. {\left. { + b_2^3\left| {00} \right\rangle - b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _3}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {{{11}}} \right\rangle + b_1^2\left| {{{10}}} \right\rangle - b_1^3\left| {01} \right\rangle - b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad {\left. { \otimes \left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. + b_2^3\left| {11} \right\rangle + b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _4}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {{{11}}} \right\rangle + b_1^2\left| {{{10}}} \right\rangle - b_1^3\left| {01} \right\rangle - b_1^4\left| {00} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. {\left. { - b_2^3\left| {10} \right\rangle + b_2^4\left| {11} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _5}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle + b_1^3\left| {00} \right\rangle - b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle \left. { - b_2^3\left| {01} \right\rangle - b_2^4\left| {00} \right\rangle } \right) } \right. _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _6}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle + b_1^3\left| {00} \right\rangle - b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. {\left. { + b_2^3\left| {00} \right\rangle - b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _7}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle + b_1^3\left| {00} \right\rangle - b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad {\left. { \otimes \left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. + b_2^3\left| {11} \right\rangle + b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _8}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {10} \right\rangle - b_1^2\left| {11} \right\rangle + b_1^3\left| {00} \right\rangle - b_1^4\left| {01} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. {\left. { - b_2^3\left| {10} \right\rangle + b_2^4\left| {11} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _9}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle + b_1^3\left| {11} \right\rangle + b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle \left. { - b_2^3\left| {01} \right\rangle - b_2^4\left| {00} \right\rangle } \right) } \right. _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{10}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle + b_1^3\left| {11} \right\rangle + b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. {\left. { + b_2^3\left| {00} \right\rangle - b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}} \end{aligned}$$
$$\begin{aligned}&\qquad + {\left| {{\varSigma _{11}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle + b_1^3\left| {11} \right\rangle + b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad {\left. { \otimes \left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. + b_2^3\left| {11} \right\rangle + b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{12}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {01} \right\rangle + b_1^2\left| {00} \right\rangle + b_1^3\left| {11} \right\rangle + b_1^4\left| {10} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. {\left. { - b_2^3\left| {10} \right\rangle + b_2^4\left| {11} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{13}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle - b_1^3\left| {10} \right\rangle + b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes {\left( {b_2^1\left| {11} \right\rangle + b_2^2\left| {10} \right\rangle \left. { - b_2^3\left| {01} \right\rangle - b_2^4\left| {00} \right\rangle } \right) } \right. _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{14}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle - b_1^3\left| {10} \right\rangle + b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left( {b_2^1\left| {10} \right\rangle - b_2^2\left| {11} \right\rangle } \right. {\left. { + b_2^3\left| {00} \right\rangle - b_2^4\left| {01} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{15}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle - b_1^3\left| {10} \right\rangle + b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad {\left. { \otimes \left( {b_2^1\left| {01} \right\rangle + b_2^2\left| {00} \right\rangle } \right. + b_2^3\left| {11} \right\rangle + b_2^4\left| {10} \right\rangle } \right) _{{A_7}{A_8}}}\\&\qquad + {\left| {{\varSigma _{16}}} \right\rangle _{{B_1}{B_2}{B_3}{B_4}}} \otimes {\left( {b_1^1\left| {00} \right\rangle - b_1^2\left| {01} \right\rangle - b_1^3\left| {10} \right\rangle + b_1^4\left| {11} \right\rangle } \right) _{{C_5}{C_6}}}\\&\qquad \otimes \left. {\left( {b_2^1\left| {00} \right\rangle - b_2^2\left| {01} \right\rangle } \right. {{\left. { - b_2^3\left| {10} \right\rangle + b_2^4\left| {11} \right\rangle } \right) }_{{A_7}{A_8}}}} \right] \\&\qquad \otimes \left[ {{{\left| {{\varOmega _1}} \right\rangle }_{{C_1}{C_2}{C_3}{C_4}}} \otimes } \right. {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle - c_1^3\left| {01} \right\rangle - c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle \left. { - c_2^3\left| {01} \right\rangle - c_2^4\left| {00} \right\rangle } \right) } \right. _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _2}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle - c_1^3\left| {01} \right\rangle - c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. {\left. { + c_2^3\left| {00} \right\rangle - c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _3}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle - c_1^3\left| {01} \right\rangle - c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad {\left. { \otimes \left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. + c_2^3\left| {11} \right\rangle + c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _4}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {11} \right\rangle + c_1^2\left| {10} \right\rangle - c_1^3\left| {01} \right\rangle - c_1^4\left| {00} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. {\left. { - c_2^3\left| {10} \right\rangle + c_2^4\left| {11} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _5}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle + c_1^3\left| {00} \right\rangle - c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle \left. { - c_2^3\left| {01} \right\rangle - c_2^4\left| {00} \right\rangle } \right) } \right. _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _6}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle + c_1^3\left| {00} \right\rangle - c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. {\left. { + c_2^3\left| {00} \right\rangle - c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _7}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle + c_1^3\left| {00} \right\rangle - c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad {\left. { \otimes \left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. + c_2^3\left| {11} \right\rangle + c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _8}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {10} \right\rangle - c_1^2\left| {11} \right\rangle + c_1^3\left| {00} \right\rangle - c_1^4\left| {01} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. {\left. { - c_2^3\left| {10} \right\rangle + c_2^4\left| {11} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _9}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle + c_1^3\left| {11} \right\rangle + c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle \left. { - c_2^3\left| {01} \right\rangle - c_2^4\left| {00} \right\rangle } \right) } \right. _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{10}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle + c_1^3\left| {11} \right\rangle + c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. {\left. { + c_2^3\left| {00} \right\rangle - c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{11}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle + c_1^3\left| {11} \right\rangle + c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad {\left. { \otimes \left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. + c_2^3\left| {11} \right\rangle + c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{12}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {01} \right\rangle + c_1^2\left| {00} \right\rangle + c_1^3\left| {11} \right\rangle + c_1^4\left| {10} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. {\left. { - c_2^3\left| {10} \right\rangle + c_2^4\left| {11} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{13}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle - c_1^3\left| {10} \right\rangle + c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes {\left( {c_2^1\left| {11} \right\rangle + c_2^2\left| {10} \right\rangle \left. { - c_2^3\left| {01} \right\rangle - c_2^4\left| {00} \right\rangle } \right) } \right. _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{14}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle - c_1^3\left| {10} \right\rangle + c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \otimes \left( {c_2^1\left| {10} \right\rangle - c_2^2\left| {11} \right\rangle } \right. {\left. { + c_2^3\left| {00} \right\rangle - c_2^4\left| {01} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{15}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle - c_1^3\left| {10} \right\rangle + c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad {\left. { \otimes \left( {c_2^1\left| {01} \right\rangle + c_2^2\left| {00} \right\rangle } \right. + c_2^3\left| {11} \right\rangle + c_2^4\left| {10} \right\rangle } \right) _{{B_7}{B_8}}}\\&\qquad + {\left| {{\varOmega _{16}}} \right\rangle _{{C_1}{C_2}{C_3}{C_4}}} \otimes {\left( {c_1^1\left| {00} \right\rangle - c_1^2\left| {01} \right\rangle - c_1^3\left| {10} \right\rangle + c_1^4\left| {11} \right\rangle } \right) _{{A_5}{A_6}}}\\&\qquad \left. {\left. { \otimes \left( {c_2^1\left| {00} \right\rangle - c_2^2\left| {01} \right\rangle } \right. {{\left. { - c_2^3\left| {10} \right\rangle + c_2^4\left| {11} \right\rangle } \right) }_{{B_7}{B_8}}}} \right] \otimes {{\left| 1 \right\rangle }_D}} \right\} . \end{aligned}$$

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Sun, S., Zhang, H. Double-direction quantum cyclic controlled remote state preparation of two-qubit states . Quantum Inf Process 20, 211 (2021). https://doi.org/10.1007/s11128-021-03149-2

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