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Analytic expression of quantum correlations in qutrit Werner states undergoing local and nonlocal unitary operations

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Abstract

Quantum correlations of a qutrit pair in Werner states undergoing local and nonlocal unitary operations are quantified in terms of two different methods, i.e., quantum discord (Ollivier and Zurek in Phys Rev Lett 88:017901, 2001) and measurement-induced disturbance (Luo in Phys Rev A 77:022301, 2008). Analytic expressions of the two kinds of quantum correlations in the system are worked on and found to be completely the same. By virtue of investigations and discussions, we expose some distinct features of the correlations and their underlying physics. More importantly, we found that both local and nonlocal unitary operations cannot increase quantum correlation in a qutrit Werner state for the two quantifiers which are concerned.

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References

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

    Article  ADS  MATH  Google Scholar 

  2. Ekert, A.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. 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  ADS  MATH  MathSciNet  Google Scholar 

  4. Bennett, C.H., Brassard, G., Crepeau, C., et al.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. Deng, F.G., Long, G.L., Liu, X.S.: Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block. Phys. Rev. A 68, 042317 (2003)

    Article  ADS  Google Scholar 

  7. Xiao, L., Long, G.L., Deng, F.G., Pan, J.W.: Efficient multiparty quantum-secret-sharing schemes. Phys. Rev. A 69, 052307 (2004)

    Article  ADS  Google Scholar 

  8. Zhang, Z.J., Man, Z.X.: Multiparty quantum secret sharing of classical messages based on entanglement swapping. Phys. Rev. A 72, 022303 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  9. Zhang, Z.J., Liu, Y.M.: Perfect teleportation of arbitrary n-qudit states using different quantum channels. Phys. Lett. A 372, 28 (2007)

    Article  ADS  MATH  Google Scholar 

  10. Cheung, C.Y., Zhang, Z.J.: Criterion for faithful teleportation with an arbitrary multiparticle channel. Phys. Rev. A 80, 022327 (2009)

    Article  ADS  Google Scholar 

  11. Yu, C.S., Song, H.S., Wang, Y.H.: Remote preparation of a qudit using maximally entangled states of qubits. Phys. Rev. A 73, 022340 (2006)

    Article  ADS  Google Scholar 

  12. Ekert, A., Jozsa, R.: Quantum computation and Shor’s factoring algorithm. Rev. Mod. Phys. 68, 733 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  13. Vedral, V., Plenio, M.B.: Basics of quantum computation. Prog. Quantum Electron. 22, 1 (1998)

    Article  ADS  Google Scholar 

  14. Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672 (1998)

    Article  ADS  Google Scholar 

  15. Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)

    Article  ADS  Google Scholar 

  16. Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)

    Article  ADS  Google Scholar 

  17. Madhok, V., Datta, A.: Interpreting quantum discord through quantum state merging. Phys. Rev. A 83, 032323 (2011)

    Article  ADS  Google Scholar 

  18. Cavalcanti, D., Aolita, L., Boixo, S., Modi, K., Piani, M., Winter, A.: Operational interpretations of quantum discord. Phys. Rev. A 83, 032324 (2011)

    Article  ADS  Google Scholar 

  19. Dakić, B., Lipp, Y.O., Ma, X., et al.: Quantum discord as resource for remote state preparation. Nat. Phys. 8, 666 (2012)

    Article  Google Scholar 

  20. Li, B., Fei, S.M., Wang, Z.X., Fan, H.: Assisted state discrimination without entanglement. Phys. Rev. A 85, 022328 (2012)

    Article  ADS  Google Scholar 

  21. Luo, S.L.: Using measurement-induced disturbance to characterize correlations as classical or quantum. Phys. Rev. A 77, 022301 (2008)

    Article  ADS  Google Scholar 

  22. Modi, K., Paterek, T., Son, W., Vedral, V., Williamson, M.: Unified view of quantum and classical correlations. Phys. Rev. Lett. 104, 080501 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  23. Dakić, B., Vedral, V., Brukner, C.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)

    Article  ADS  Google Scholar 

  24. Luo, S.L.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)

    Article  ADS  Google Scholar 

  25. Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010)

    Article  ADS  Google Scholar 

  26. Huang, Y.C.: Quantum discord for two-qubit X states: analytical formula with very small worst-case error. Phys. Rev. A 88, 014302 (2013)

    Article  ADS  Google Scholar 

  27. Huang, Y.C.: Scaling of quantum discord in spin models. Phys. Rev. B 89, 054410 (2014)

    Article  ADS  Google Scholar 

  28. Huang, Y.C.: Computing quantum discord is NP-complete. New J. Phys. 16, 033027 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  29. Giorda, P., Paris, M.G.A.: Gaussian quantum discord. Phys. Rev. Lett. 105, 020503 (2010)

    Article  ADS  Google Scholar 

  30. Rulli, C.C., Sarandy, M.S.: Global quantum discord in multipartite systems. Phys. Rev. A 84, 042109 (2011)

    Article  ADS  Google Scholar 

  31. Ye, B.L., Liu, Y.M., Chen, J.L., Liu, X.S., Zhang, Z.J.: Analytic expressions of quantum correlations in qutrit Werner states. Quantum Inf. Process. 12, 2355 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Ye, B.L., Liu, Y.M., Liu, X.S., Zhang, Z.J.: Quantum correlations in a family of bipartite qubit–qutrit separable states. Chin. Phys. Lett. 30, 020302 (2013)

    Article  ADS  Google Scholar 

  33. Zhang, Z.J., Ye, B.L., Fei, S.M.: quant-ph/1206.0221.

  34. Zhang, Z.J.: quant-ph/1202.3640.

  35. Tang, H., Liu, Y.M., Chen, J., Ye, B., Zhang, Z.J.: Analytic expressions of discord and geometric discord in Werner derivatives. Quantum Inf. Process. 14, 1331 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  36. Wei, H.R., Ren, B.C., Deng, F.G.: Geometric measure of quantum discord for a two-parameter class of states in a qubit–qutrit system under various dissipative channels. Quantum Inf. Process. 12, 1109 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008)

    Article  ADS  Google Scholar 

  38. Werlang, T., Souza, S., Fanchini, F.F.: Robustness of quantum discord to sudden death. Phys. Rev. A 80, 024103 (2009)

    Article  ADS  Google Scholar 

  39. Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655 (2012)

    Article  ADS  Google Scholar 

  40. Aharon, B., Daniel, R.T.: Quantum discord, local operations, and Maxwells demons. Phys. Rev. A 81, 062103 (2010)

    Article  MathSciNet  Google Scholar 

  41. Auccaise, R., Céleri, L.C., Soares-Pinto, D.O., deAzevedo, E.R., Maziero, J., Souza, A.M., Bonagamba, T.J., Sarthour, R.S., Oliveira, I.S., Serra, R.M.: Environment-induced sudden transition in quantum discord dynamics. Phys. Rev. Lett. 107, 140403 (2011)

    Article  ADS  Google Scholar 

  42. Streltsov, A., Kampermann, H., Bruß, D.: Behavior of quantum correlations under local noise. Phys. Rev. Lett. 107, 170502 (2011)

    Article  ADS  Google Scholar 

  43. Gessner, M., Breuer, H.: Detecting nonclassical system-environment correlations by local operations. Phys. Rev. Lett. 107, 180402 (2011)

    Article  ADS  Google Scholar 

  44. Francesco, C., Vittorio, G.: Creating quantum correlations through local nonunitary memoryless channels. Phys. Rev. A 85, 010102 (2012)

    Article  Google Scholar 

  45. Hu, X., Fan, H., Zhou, D.L., Liu, W.M.: Quantum correlating power of local quantum channels. Phys. Rev. A 87, 032340 (2013)

    Article  ADS  Google Scholar 

  46. Lanyon, B.P., Jurcevic, P., Hempel, C., Gessner, M., Vedral, V., Blatt, R., Roos, C.F.: Experimental generation of quantum discord via noisy processes. Phys. Rev. Lett. 111, 100504 (2013)

    Article  ADS  Google Scholar 

  47. Shi, M., Sun, C., Jiang, F., Yan, X., Du, J.: Optimal measurement for quantum discord of two-qubit states. Phys. Rev. A 85, 064104 (2012)

    Article  ADS  Google Scholar 

  48. Chitambar, E.: Quantum correlations in high-dimensional states of high symmetry. Phys. Rev. A 86, 032110 (2012)

    Article  ADS  Google Scholar 

  49. Zhou, T., Cui, J., Long, G.L.: Measure of nonclassical correlation in coherence-vector representation. Phys. Rev. A 84, 062105 (2011)

    Article  ADS  Google Scholar 

  50. Xie, C.M., Liu, Y.M., Li, G.F., Zhang, Z.J.: A note on quantum correlations in Werner states under two collective noises. Quantum Inf. Process. 13, 2713 (2014)

  51. Xie, C., Liu, Y., Xing, H., Chen, J., Zhang, Z.: Quantum correlation swapping. Quantum Inf. Process. (2014). doi:10.1007/s11128-014-0875-y

  52. Bronzan, J.B.: Parametrization of SU(3). Phys. Rev. D 38, 1994 (1988)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NNSFC) under Grant Nos. 11375011 and 11372122, the Natural Science Foundation of Anhui province under Grant No. 1408085MA12, and the 211 Project of Anhui University.

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Authors and Affiliations

  1. School of Physics and Material Science, Anhui University, Hefei, 230039, Anhui, China

    Guofeng Li, Haojie Tang, Xiaofeng Yin & Zhanjun Zhang

  2. Department of Physics, Shaoguan University, Shaoguan, 512005, China

    Yimin Liu

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  1. Guofeng Li
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Correspondence to Zhanjun Zhang.

Appendix

Appendix

Some partial derivatives:

$$\begin{aligned}&\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial \theta _n} \nonumber \\&\quad =\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial p_a^0} \left( \frac{\partial p_a^0}{\partial |\varGamma _{00}|^2} \frac{\partial |\varGamma _{00}|^2}{\partial \theta _n} +\frac{\partial p_a^0}{\partial |\varGamma _{10}|^2} \frac{\partial |\varGamma _{10}|^2}{\partial \theta _n} +\frac{\partial p_a^0}{\partial |\varGamma _{20}|^2} \frac{\partial |\varGamma _{20}|^2}{\partial \theta _n}\right) \nonumber \\&\quad \quad +\,\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial p_a^1} \left( \frac{\partial p_a^1}{\partial |\varGamma _{01}|^2} \frac{\partial |\varGamma _{01}|^2}{\partial \theta _n} +\frac{\partial p_a^1}{\partial |\varGamma _{11}|^2} \frac{\partial |\varGamma _{11}|^2}{\partial \theta _n} +\frac{\partial p_a^1}{\partial |\varGamma _{21}|^2} \frac{\partial |\varGamma _{21}|^2}{\partial \theta _n}\right) \nonumber \\&\quad \quad +\,\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial p_a^2} \left( \frac{\partial p_a^2}{\partial |\varGamma _{02}|^2} \frac{\partial |\varGamma _{02}|^2}{\partial \theta _n} \!+\!\frac{\partial p_a^2}{\partial |\varGamma _{12}|^2} \frac{\partial |\varGamma _{12}|^2}{\partial \theta _n} \!+\!\frac{\partial p_a^2}{\partial |\varGamma _{22}|^2} \frac{\partial |\varGamma _{22}|^2}{\partial \theta _n}\right) , \theta _n=\theta _1, \theta _2, \theta _3.\nonumber \\ \end{aligned}$$
(20)
$$\begin{aligned}&\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial \phi _4}\nonumber \\&\quad =\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial p_a^0} \left( \frac{\partial p_a^0}{\partial |\varGamma _{00}|^2} \frac{\partial |\varGamma _{00}|^2}{\partial \phi _4} +\frac{\partial p_a^0}{\partial |\varGamma _{10}|^2} \frac{\partial |\varGamma _{10}|^2}{\partial \phi _4} +\frac{\partial p_a^0}{\partial |\varGamma _{20}|^2} \frac{\partial |\varGamma _{20}|^2}{\partial \phi _4}\right) \nonumber \\&\quad \quad +\,\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial p_a^1} \left( \frac{\partial p_a^1}{\partial |\varGamma _{01}|^2} \frac{\partial |\varGamma _{01}|^2}{\partial \phi _4} +\frac{\partial p_a^1}{\partial |\varGamma _{11}|^2} \frac{\partial |\varGamma _{11}|^2}{\partial \phi _4} +\frac{\partial p_a^1}{\partial |\varGamma _{21}|^2} \frac{\partial |\varGamma _{21}|^2}{\partial \phi _4}\right) \nonumber \\&\quad \quad +\,\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial p_a^2} \left( \frac{\partial p_a^2}{\partial |\varGamma _{02}|^2} \frac{\partial |\varGamma _{02}|^2}{\partial \phi _4} +\frac{\partial p_a^2}{\partial |\varGamma _{12}|^2} \frac{\partial |\varGamma _{12}|^2}{\partial \phi _4} +\frac{\partial p_a^2}{\partial |\varGamma _{22}|^2} \frac{\partial |\varGamma _{22}|^2}{\partial \phi _4}\right) . \end{aligned}$$
(21)
$$\begin{aligned}&\frac{\partial |\varGamma _{00}|^2}{\partial \theta _1}=-\sin 2\theta _1\cos ^2\theta _2, \ \ \frac{\partial |\varGamma _{01}|^2}{\partial \theta _1}=\sin 2\theta _1,\ \ \frac{\partial |\varGamma _{02}|^2}{\partial \theta _1}=-\sin 2\theta _1\sin ^2\theta _2, \\&\frac{\partial |\varGamma _{10}|^2}{\partial \theta _1}=-\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4 +\sin 2\theta _1\cos ^2\theta _2\cos ^2\theta _3, \\&\frac{\partial |\varGamma _{11}|^2}{\partial \theta _1}=-\sin 2\theta _1\cos ^2\theta _3,\ \ \frac{\partial |\varGamma _{21}|^2}{\partial \theta _1}=-\sin 2\theta _1\sin ^2\theta _3, \\&\frac{\partial |\varGamma _{12}|^2}{\partial \theta _1}=\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4 +\sin 2\theta _1\sin ^2\theta _2\cos ^2\theta _3, \\&\frac{\partial |\varGamma _{20}|^2}{\partial \theta _1}=\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4 +\sin 2\theta _1\cos ^2\theta _2\sin ^2\theta _3, \\&\frac{\partial |\varGamma _{22}|^2}{\partial \theta _1}=-\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4 +\sin 2\theta _1\sin ^2\theta _2\sin ^2\theta _3.\\&\frac{\partial |\varGamma _{00}|^2}{\partial \theta _2}=-\frac{\partial |\varGamma _{02}|^2}{\partial \theta _2}=-\cos ^2\theta _1\sin 2\theta _2,\ \ \frac{\partial |\varGamma _{01}|^2}{\partial \theta _2}=\frac{\partial |\varGamma _{11}|^2}{\partial \theta _2}=\frac{\partial |\varGamma _{21}|^2}{\partial \theta _2}=0, \\&\frac{\partial |\varGamma _{10}|^2}{\partial \theta _2}=\sin 2\theta _2\sin ^2\theta _3-\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4 -\sin ^2\theta _1\sin 2\theta _2\cos ^2\theta _3, \\&\frac{\partial |\varGamma _{12}|^2}{\partial \theta _2}=-\sin 2\theta _2\sin ^2\theta _3+\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4 +\sin ^2\theta _1\sin 2\theta _2\cos ^2\theta _3, \\&\frac{\partial |\varGamma _{20}|^2}{\partial \theta _2}=\sin 2\theta _2\cos ^2\theta _3+\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4 -\sin ^2\theta _1\sin 2\theta _2\sin ^2\theta _3, \\&\frac{\partial |\varGamma _{22}|^2}{\partial \theta _2}=-\sin 2\theta _2\cos ^2\theta _3-\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4 +\sin ^2\theta _1\sin 2\theta _2\sin ^2\theta _3.\\&\frac{\partial |\varGamma _{00}|^2}{\partial \theta _3}=\frac{\partial |\varGamma _{01}|^2}{\partial \theta _3}=\frac{\partial |\varGamma _{02}|^2}{\partial \theta _3}=0,\ \ \frac{\partial |\varGamma _{11}|^2}{\partial \theta _3}=-\frac{\partial |\varGamma _{21}|^2}{\partial \theta _3}=-\cos ^2\theta _1\sin 2\theta _3, \\&\frac{\partial |\varGamma _{10}|^2}{\partial \theta _3}=\sin ^2\theta _2\sin 2\theta _3-\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4 -\sin ^2\theta _1\cos ^2\theta _2\sin 2\theta _3, \\&\frac{\partial |\varGamma _{12}|^2}{\partial \theta _3}=\cos ^2\theta _2\sin 2\theta _3+\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4 -\sin ^2\theta _1\sin ^2\theta _2\sin 2\theta _3, \\&\frac{\partial |\varGamma _{20}|^2}{\partial \theta _3}=-\sin ^2\theta _2\sin 2\theta _3+\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4 +\sin ^2\theta _1\cos ^2\theta _2\sin 2\theta _3, \\&\frac{\partial |\varGamma _{22}|^2}{\partial \theta _3}=-\cos ^2\theta _2\sin 2\theta _3-\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4 +\sin ^2\theta _1\sin ^2\theta _2\sin 2\theta _3. \end{aligned}$$
$$\begin{aligned}&\frac{\partial |\varGamma _{00}|^2}{\partial \phi _4}=\frac{\partial |\varGamma _{01}|^2}{\partial \phi _4}=\frac{\partial |\varGamma _{02}|^2}{\partial \phi _4}=\frac{\partial |\varGamma _{11}|^2}{\partial \phi _4}=\frac{\partial |\varGamma _{21}|^2}{\partial \phi _4}=0,\nonumber \\&\frac{\partial |\varGamma _{10}|^2}{\partial \phi _4}=\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4,\nonumber \\&\frac{\partial |\varGamma _{12}|^2}{\partial \phi _4}=-\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4, \end{aligned}$$
(22)
$$\begin{aligned}&\frac{\partial |\varGamma _{20}|^2}{\partial \phi _4}=-\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4,\\&\frac{\partial |\varGamma _{22}|^2}{\partial \phi _4}=\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4. \end{aligned}$$

Obviously, for any \(m,n \in \{0,1,2\}\), \(\frac{\partial |\varGamma _{mn}|^2}{\partial \theta _1}=\frac{\partial |\varGamma _{mn}|^2}{\partial \theta _2}= \frac{\partial |\varGamma _{mn}|^2}{\partial \theta _3}=\frac{\partial |\varGamma _{mn}|^2}{\partial \phi _4}=0\) hold if \(\theta _1=\theta _2=\theta _3=\phi _4=0\). Using them, it is easy to verify that \(\frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial \theta _1}= \frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial \theta _2}= \frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial \theta _3}= \frac{\partial S(\varrho _{ab}|\{\hat{E}^n_a\})}{\partial \phi _4}=0\). This means that \((\theta _1,\theta _2,\theta _3, \phi _4)=(0,0,0,0)\) is an extreme point of function \(S(\varrho _{ab}|\{\hat{E}^n_a\})\).

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Li, G., Liu, Y., Tang, H. et al. Analytic expression of quantum correlations in qutrit Werner states undergoing local and nonlocal unitary operations. Quantum Inf Process 14, 559–572 (2015). https://doi.org/10.1007/s11128-014-0888-6

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