Skip to main content
SpringerLink
Log in

Rényi and Tsallis formulations of separability conditions in finite dimensions

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Separability conditions for a bipartite quantum system of finite-dimensional subsystems are formulated in terms of Rényi and Tsallis entropies. Entropic uncertainty relations often lead to entanglement criteria. We propose new approach based on the convolution of discrete probability distributions. Measurements on a total system are constructed of local ones according to the convolution scheme. Separability conditions are derived on the base of uncertainty relations of the Maassen–Uffink type as well as majorization relations. On each of subsystems, we use a pair of sets of subnormalized vectors that form rank-one POVMs. We also obtain entropic separability conditions for local measurements with a special structure, such as mutually unbiased bases and symmetric informationally complete measurements. The relevance of the derived separability conditions is demonstrated with several examples.

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

Similar content being viewed by others

References

  1. Schrödinger, E.: Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften 23, 807–812, 823–828, 844–849 (1935)

  2. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Horodecki, M., Horodecki, P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206–4216 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  6. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Gühne, O., Lewenstein, M.: Entropic uncertainty relations and entanglement. Phys. Rev. A 70, 022316 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Giovannetti, V.: Separability conditions from entropic uncertainty relations. Phys. Rev. A 70, 012102 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  9. de Vicente, J.I., Sánchez-Ruiz, J.: Separability conditions from the Landau–Pollak uncertainty relation. Phys. Rev. A 71, 052325 (2005)

    Article  ADS  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, 010301(R) (2006)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  12. Huang, Y.: Entanglement criteria via concave-function uncertainty relations. Phys. Rev. A 82, 012335 (2010)

    Article  ADS  Google Scholar 

  13. Rastegin, A.E.: Separability conditions based on local fine-grained uncertainty relations. Quantum Inf. Process. 15, 2621–2638 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Huang, Y.: Entanglement detection: complexity and Shannon entropic criteria. IEEE Trans. Inf. Theor. 59, 6774–6778 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Heisenberg, W.: Über den anschaulichen inhalt der quanten theoretischen kinematik und mechanik. Zeitschrift für Physik 43, 172–198 (1927)

    Article  ADS  MATH  Google Scholar 

  16. Hall, M.J.W.: Universal geometric approach to uncertainty, entropy, and information. Phys. Rev. A 59, 2602–2615 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  17. Busch, P., Heinonen, T., Lahti, P.J.: Heisenberg’s uncertainty principle. Phys. Rep. 452, 155–176 (2007)

    Article  ADS  Google Scholar 

  18. Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  19. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  20. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  21. Hirschman, I.I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  22. Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Białynicki-Birula, I., Mycielski, J.: Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129–132 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  24. Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys. 12, 025009 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Białynicki-Birula, I., Rudnicki, Ł.: Entropic uncertainty relations in quantum physics. In: Sen, K.D. (ed.) Statistical Complexity, 1–34. Springer, Berlin (2011)

    Google Scholar 

  26. Coles, P.J., Berta, M., Tomamichel, M., Wehner, S.: Entropic uncertainty relations and their applications. Rev. Mod. Phys. 89, 015002 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  27. Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072–1074 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Ren, L.-H., Fan, H.: General fine-grained uncertainty relation and the second law of thermodynamics. Phys. Rev. A 90, 052110 (2014)

    Article  ADS  Google Scholar 

  29. Rastegin, A.E.: Fine-grained uncertainty relations for several quantum measurements. Quantum Inf. Process. 14, 783–800 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Partovi, M.H.: Majorization formulation of uncertainty in quantum mechanics. Phys. Rev. A 84, 052117 (2011)

    Article  ADS  Google Scholar 

  31. Puchała, Z., Rudnicki, Ł., Życzkowski, K.: Majorization entropic uncertainty relations. J. Phys. A: Math. Theor. 46, 272002 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Friedland, S., Gheorghiu, V., Gour, G.: Universal uncertainty relations. Phys. Rev. Lett. 111, 230401 (2013)

    Article  ADS  Google Scholar 

  33. Rudnicki, Ł., Puchała, Z., Życzkowski, K.: Strong majorization entropic uncertainty relations. Phys. Rev. A 89, 052115 (2014)

    Article  ADS  MATH  Google Scholar 

  34. Rudnicki, Ł.: Majorization approach to entropic uncertainty relations for coarse-grained observables. Phys. Rev. A 91, 032123 (2015)

    Article  ADS  Google Scholar 

  35. Rastegin, A.E., Życzkowski, K.: Majorization entropic uncertainty relations for quantum operations. J. Phys. A: Math. Theor. 49, 355301 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  37. Watrous J.: Theory of Quantum Information, a draft of book. University of Waterloo, Waterloo (2017) http://www.cs.uwaterloo.ca/~watrous/TQI/

  38. Rényi, A.: On measures of entropy and information. In: Neyman, J. (ed.) Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 547–561. University of California Press, Berkeley (1961)

  39. Jizba, P., Arimitsu, T.: The world according to Rényi: thermodynamics of multifractal systems. Ann. Phys. 312, 17–59 (2004)

    Article  ADS  MATH  Google Scholar 

  40. Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  41. Ben-Bassat, M., Raviv, J.: Rényi’s entropy and error probability. IEEE Trans. Inf. Theory 24, 324–331 (1978)

    Article  MATH  Google Scholar 

  42. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  44. Durt, T., Englert, B.-G., Bengtsson, I., Życzkowski, K.: On mutually unbiased bases. Int. J. Quantum Inf. 8, 535–640 (2010)

    Article  MATH  Google Scholar 

  45. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  46. Klappenecker, A., Rötteler, M.: Constructions of mutually unbiased bases. Finite Fields and Applications. Lecture Notes in Computer Science, vol. 2948, pp. 137–144. Springer, Berlin (2004)

  47. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Appleby, D.M.: Symmetric informationally complete-positive operator valued measures and the extended Clifford group. J. Math. Phys. 46, 052107 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Scott, A.J., Grassl, M.: Symmetric informationally complete positive-operator-valued measures: a new computer study. J. Math. Phys. 51, 042203 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Appleby, D.M., Dang, H.B., Fuchs, C.A.: Symmetric informationally-complete quantum states as analogues to orthonormal bases and minimum-uncertainty states. arXiv:0707.2071 [quant-ph] (2007)

  51. Ruskai, M.B.: Some connections between frames, mutually unbiased bases, and POVM’s in quantum information theory. Acta Appl. Math. 108, 709–719 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kalev, A., Gour, G.: Mutually unbiased measurements in finite dimensions. New J. Phys. 16, 053038 (2014)

    Article  ADS  Google Scholar 

  53. Gour, G., Kalev, A.: Construction of all general symmetric informationally complete measurements. J. Phys. A: Math. Theor. 47, 335302 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Riesz, M.: Sur les maxima des forms bilinéaires et sur les fonctionnelles linéaires. Acta Math. 49, 465–497 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  55. Rastegin, A.E.: Entropic uncertainty relations for extremal unravelings of super-operators. J. Phys. A: Math. Theor. 44, 095303 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Hall, M.J.W.: Quantum information and correlation bounds. Phys. Rev. A 55, 100–113 (1997)

    Article  ADS  MATH  Google Scholar 

  57. Rastegin, A.E.: Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies. Eur. Phys. J. D 67, 269 (2013)

    Article  ADS  Google Scholar 

  58. Rastegin, A.E.: On uncertainty relations and entanglement detection with mutually unbiased measurements. Open Sys. Inf. Dyn. 22, 1550005 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Wu, S., Yu, S., Mølmer, K.: Entropic uncertainty relation for mutually unbiased bases. Phys. Rev. A 79, 022104 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  60. Spengler, C., Huber, M., Brierley, S., Adaktylos, T., Hiesmayr, B.C.: Entanglement detection via mutually unbiased bases. Phys. Rev. A 86, 022311 (2012)

    Article  ADS  Google Scholar 

  61. Chen, B., Ma, T., Fei, S.-M.: Entanglement detection using mutually unbiased measurements. Phys. Rev. A 89, 064302 (2014)

    Article  ADS  Google Scholar 

  62. Rastegin, A.E.: Notes on general SIC-POVMs. Phys. Scr. 89, 085101 (2014)

    Article  ADS  Google Scholar 

  63. Chen, B., Ma, T., Fei, S.-M.: General SIC measurement-based entanglement detection. Quantum Inf. Process. 14, 2281–2290 (2015)

    Article  ADS  MATH  Google Scholar 

  64. Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)

    Article  ADS  MATH  Google Scholar 

  65. Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A 58, 883–892 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  66. Das, S., Chanda, T., Lewenstein, M., Sanpera, A., Sen(De), A., Sen, U.: The separability versus entanglement problem. arXiv:1701.02187 [quant-ph] (2017)

  67. Walborn, S.P., Taketani, B.G., Salles, A., Toscano, F., de Matos Filho, R.L.: Entropic entanglement criteria for continuous variables. Phys. Rev. Lett. 103, 160505 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  68. Saboia, A., Toscano, F., Walborn, S.P.: Family of continuous-variable entanglement criteria using general entropy functions. Phys. Rev. A 83, 032307 (2011)

    Article  ADS  Google Scholar 

  69. Rastegin, A.E.: Rényi formulation of entanglement criteria for continuous variables. Phys. Rev. A 95, 042334 (2017)

    Article  ADS  Google Scholar 

  70. Białynicki-Birula, I.: Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A 74, 052101 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  71. Rastegin, R.E.: Rényi formulation of the entropic uncertainty principle for POVMs. J. Phys. A: Math. Theor. 43, 155302 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  72. Coles, P.J., Colbeck, R., Yu, L., Zwolak, M.: Uncertainty relations from simple entropic properties. Phys. Rev. Lett. 108, 210405 (2012)

    Article  ADS  Google Scholar 

  73. Linden, N., Popescu, S.: On multi-particle entanglement. Fortschr. Phys. 46, 567–578 (1998)

    Article  MathSciNet  Google Scholar 

  74. Bengtsson, I., Życzkowski, K.: A brief introduction to multipartite entanglement. arXiv:1612.07747 [quant-ph] (2016)

  75. Tóth, G., Gühne, O.: Detection of multipartite entanglement with two-body correlations. Appl. Phys. B 82, 237–241 (2006)

    Article  ADS  Google Scholar 

  76. Huang, Y., Qiu, D.W.: Concurrence vectors of multipartite states based on coefficient matrices. Quantum Inf. Process. 11, 235–254 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  77. Spengler, C., Huber, M., Gabriel, A., Hiesmayr, B.C.: Examining the dimensionality of genuine multipartite entanglement. Quantum Inf. Process. 12, 269–278 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  78. Zhao, C., Yang, G., Hung, W.N.N., Li, X.: A multipartite entanglement measure based on coefficient matrices. Quantum Inf. Process. 14, 2861–2881 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. Pittenger, A.O., Rubin, M.N.: Note on separability of the Werner states in arbitrary dimensions. Opt. Commun. 179, 447–449 (2000)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, Irkutsk State University, Gagarin Bv. 20, Irkutsk, Russia, 664003

    Alexey E. Rastegin

Authors
  1. Alexey E. Rastegin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexey E. Rastegin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rastegin, A.E. Rényi and Tsallis formulations of separability conditions in finite dimensions. Quantum Inf Process 16, 293 (2017). https://doi.org/10.1007/s11128-017-1746-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1746-0

Keywords

Search

Navigation