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A family of separability criteria and lower bounds of concurrence

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Abstract

Background and Aim

The problem on detecting the entanglement of a bipartite state is significant in quantum information theory.

Methods

In this article, we apply the Ky Fan norm to the revised realignment matrix of a bipartite state.

Results

We consider a family of separable criteria for bipartite states and present when the density matrix corresponds to a state is real, the criterion is equivalent to the enhanced realignment criterion. Moreover, we present analytical lower bounds of concurrence and the convex-roof extended negativity for arbitrary dimensional systems.

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Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. Plenio, M.B., Virmani, S.S.: An introduction to entanglement theory. In: Quantum Information and Coherence, pp. 173–209. Springer (2014)

  3. Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67(6), 661 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Bennett, C.H., Brassard, G., Crépeau, 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(13), 1895 (1993)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. Bennett, C.H., Wiesner, S.J.: Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69(20), 2881 (1992)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77(8), 1413 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Gurvits, L.: Classical deterministic complexity of Edmonds’ problem and quantum entanglement. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 10–19 (2003)

  8. Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4(3), 219–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, K., Wu, L.-A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 6, 66 (2022)

    Google Scholar 

  10. Gühne, O., Mechler, M., Tóth, G., Adam, P.: Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion. Phys. Rev. A 74(1), 010301 (2006)

    Article  ADS  Google Scholar 

  11. De Vicente, J.I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. 6, 66 (2022)

    Google Scholar 

  12. Zhang, C.-J., Zhang, Y.-S., Zhang, S., Guo, G.-C.: Entanglement detection beyond the computable cross-norm or realignment criterion. Phys. Rev. A 77(6), 060301 (2008)

    Article  ADS  Google Scholar 

  13. Shen, S.-Q., Wang, M.-Y., Li, M., Fei, S.-M.: Separability criteria based on the realignment of density matrices and reduced density matrices. Phys. Rev. A 92(4), 042332 (2015)

    Article  ADS  Google Scholar 

  14. Shang, J., Asadian, A., Zhu, H., Gühne, O.: Enhanced entanglement criterion via symmetric informationally complete measurements. Phys. Rev. A 98(2), 022309 (2018)

    Article  ADS  Google Scholar 

  15. Sarbicki, G., Scala, G., Chruściński, D.: Family of multipartite separability criteria based on a correlation tensor. Phys. Rev. A 101(1), 012341 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  16. Sarbicki, G., Scala, G., Chruściński, D.: Enhanced realignment criterion vs linear entanglement witnesses. J. Phys. A Math. Theor. 53(45), 455302 (2020)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Jivulescu, M.A., Lancien, C., Nechita, I.: Multipartite entanglement detection via projective tensor norms. Annales Henri Poincaré 23(11), 3791–3838 (2022)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  18. Yan, X., Liu, Y.-C., Shang, J.: Operational detection of entanglement via quantum designs. Annalen der Physik 534(5), 2100594 (2022)

    Article  MathSciNet  ADS  Google Scholar 

  19. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245 (1998)

    Article  MATH  ADS  Google Scholar 

  20. Vedral, V., Plenio, M.B.: Entanglement measures and purification procedures. Phys. Rev. A 57(3), 1619 (1998)

    Article  ADS  Google Scholar 

  21. Vidal, G., Tarrach, R.: Robustness of entanglement. Phys. Rev. A 59(1), 141 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  22. Terhal, B.M., Horodecki, P.: Schmidt number for density matrices. Phys. Rev. A 61(4), 040301 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  23. Vidal, G.: Entanglement monotones. J. Mod. Opt. 47(2–3), 355–376 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  24. Wei, T.-C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68(4), 042307 (2003)

    Article  ADS  Google Scholar 

  25. Christandl, M., Winter, A.: “Squashed entanglement": an additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  26. Lee, S., Chi, D.P., Oh, S.D., Kim, J.: Convex-roof extended negativity as an entanglement measure for bipartite quantum systems. Phys. Rev. A 68(6), 062304 (2003)

    Article  ADS  Google Scholar 

  27. Huang, Y.: Computing quantum discord is np-complete. New J. Phys. 16(3), 033027 (2014)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. Chen, K., Albeverio, S., Fei, S.-M.: Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett. 95(4), 040504 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  29. Brandao, F.G.: Quantifying entanglement with witness operators. Phys. Rev. A 72(2), 022310 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  30. de Vicente, J.I.: Lower bounds on concurrence and separability conditions. Phys. Rev. A 75(5), 052320 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  31. Chen, Z.-H., Ma, Z.-H., Gühne, O., Severini, S.: Estimating entanglement monotones with a generalization of the Wootters formula. Phys. Rev. Lett. 109(20), 200503 (2012)

    Article  ADS  Google Scholar 

  32. Li, M., Wang, Z., Wang, J., Shen, S., Fei, S.-M.: Improved lower bounds of concurrence and convex-roof extended negativity based on Bloch representations. Quantum Inf. Process. 19(4), 1–11 (2020)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  33. Bhatia, R.: Matrix Analysis, vol. 169. Springer (2013)

  34. Bruß, D., Peres, A.: Construction of quantum states with bound entanglement. Phys. Rev. A 61(3), 030301 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  35. Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232(5), 333–339 (1997)

    Article  MathSciNet  MATH  ADS  Google Scholar 

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Authors and Affiliations

  1. College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, 100029, China

    Xian Shi & Yashuai Sun

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  1. Xian Shi
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  2. Yashuai Sun
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Correspondence to Xian Shi.

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Shi, X., Sun, Y. A family of separability criteria and lower bounds of concurrence. Quantum Inf Process 22, 131 (2023). https://doi.org/10.1007/s11128-023-03875-9

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