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Convex geometry of Markovian Lindblad dynamics and witnessing non-Markovianity

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Abstract

We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad-type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn–Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. Furthermore, we have devised another method of detection NM of various qubit channels, by using projective measurements in the Bell basis. This can be done without the knowledge of what specific kind of channel we are dealing with. This gives us a huge operational advantage for NM detection. We further investigate the geometric structure of the Markovian Choi states to find that they do not form a polytope. This presents a platform to consider nonlinear improvement of non-Markovianity witnesses.

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References

  1. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Lecture notes in Physics. (Springer-Verlag, Berlin Heidelberg (2007)

    MATH  Google Scholar 

  2. Breuer, H. P., Petruccione, F.: The theory of open quantum systems ( Oxford University Press, address Great Clarendon Street) (2002)

  3. Rivas, A., Huelga, S. F., Plenio, M. B.: Quantum non-Markovianity: characterization, quantification and detection, Vol. 77 ( 2014) p. 094001

  4. Breuer, H.-P., Laine, E.-M., Piilo, J., Vacchini, B.: Colloquium, Vol. 88 ( American Physical Society, 2016) p. 021002

  5. Wolf, M. M., Eisert, J., Cubitt, T. S., Cirac, J. I.: Assessing Non-Markovian Quantum Dynamics, Vol. 101 ( American Physical Society, 2008) p. 150402

  6. de Vega, I., Alonso, D.: Dynamics of non-Markovian open quantum systems, Vol. 89 ( American Physical Society, 2017) p. 015001

  7. Laine, E.-M., Piilo, J., Breuer, H.-P.: Measure for the non-Markovianity of quantum processes, Vol. 81 ( American Physical Society, 2010) p. 062115

  8. Rivas, A., Huelga, S. F., Plenio, M. B.: Entanglement and Non-Markovianity of Quantum Evolutions, Vol. 105 ( American Physical Society, 2010) p. 050403

  9. Bellomo, B., Lo Franco, R., Compagno, G.: Non-Markovian Effects on the Dynamics of Entanglement, Vol. 99 ( American Physical Society, 2007) p. 160502

  10. Dijkstra, A. G., Tanimura, Y.: Non-Markovian Entanglement Dynamics in the Presence of System-Bath Coherence, Vol. 104 ( American Physical Society, 2010) p. 250401

  11. Bhattacharya, S., Misra, A., Mukhopadhyay, C., Pati, A. K.: Exact master equation for a spin interacting with a spin bath: Non-Markovianity and negative entropy production rate, Vol. 95 ( American Physical Society, 2017) p. 012122

  12. Mukhopadhyay, C., Bhattacharya, S., Misra, A., Pati, A. K.: Dynamics and thermodynamics of a central spin immersed in a spin bath, Vol. 96 ( American Physical Society, 2017) p. 052125

  13. Awasthi, N., Bhattacharya, S., Sen(De), A., Sen, U.: Universal quantum uncertainty relations between nonergodicity and loss of information, Vol. 97 ( American Physical Society, 2018) p. 032103

  14. Lu, X.-M., Wang, X., Sun, C. P.: Quantum Fisher information flow and non-Markovian processes of open systems, Vol. 82 ( American Physical Society, 2010) p. 042103

  15. Rajagopal, A. K., Usha Devi, A. R., Rendell, R. W.: Kraus representation of quantum evolution and fidelity as manifestations of Markovian and non-Markovian forms, Vol. 82 ( American Physical Society, 2010) p. 042107

  16. Luo, S., Fu, S., Song, H.: Quantifying non-Markovianity via correlations, Vol. 86 ( American Physical Society, 2012) p. 044101

  17. Jiang, M., Luo, S.: Comparing quantum Markovianities: Distinguishability versus correlations, Vol. 88 ( American Physical Society, 2013) p. 034101

  18. Lorenzo, S., Plastina, F., Paternostro, M.: Geometrical characterization of non-Markovianity, Vol. 88 ( American Physical Society, 2013) p. 020102

  19. Dhar, H. S., Bera, M. N., Adesso, G.: Characterizing non-Markovianity via quantum interferometric power, Vol. 91 ( American Physical Society, 2015) p. 032115

  20. Bylicka, B., Chruściński, D., Maniscalco, S.: Non-Markovianity as a Resource for Quantum Technologies (2013) arXiv:1301.2585 [quant-ph]

  21. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement, Vol. 81 ( American Physical Society, 2009) pp. 865–942

  22. Bhattacharya, S., Bhattacharya, B., Majumdar, A. S.: Resource theory of non-Markovianity: A thermodynamic perspective (2018) arXiv:1803.06881 [quant-ph]

  23. Horodecki, M., Horodecki, P., Horodecki, R.: Separability Mix. States: Necessar. Suffic. Cond. 223, 1–8 (1996)

    Google Scholar 

  24. Bourennane, M., Eibl, M., Kurtsiefer, C., Gaertner, S., Weinfurter, H., Gühne, O., Hyllus, P., Bruß, D., Lewenstein, M., Sanpera, A.: Experimental Detection of Multipartite Entanglement using Witness Operators, Vol. 92 (American Physical Society, 2004) p. 087902

  25. Terhal, B.M.: Bell Inequal. Separability Criteri. 271, 319–326 (2000)

    Google Scholar 

  26. Lewenstein, M., Kraus, B., Cirac, J. I., Horodecki, P.: Optimization of entanglement witnesses, Vol. 62 (American Physical Society, 2000) p. 052310

  27. Gühne, O., Hyllus, P., Bruß, D., Ekert, A., Lewenstein, M., Macchiavello, C., Sanpera, A.: Detection of entanglement with few local measurements, Vol. 66 (American Physical Society, 2002) p. 062305

  28. Guhne, O., Toth, G.: Entanglement Detect. 474, 1–75 (2009)

    Google Scholar 

  29. Chruściński, D., Sarbicki, G.: Entanglement witnesses: construction, analysis and classification, Vol. 47 (2014) p. 483001

  30. Gühne, O., Lu, C.-Y., Gao, W.-B., Pan, J.-W.: Toolbox for entanglement detection and fidelity estimation, Vol. 76 (American Physical Society, 2007) p. 030305

  31. Zhang, C.-J., Zhang, Y.-S., Zhang, S., Guo, G.-C.: Entanglement detection beyond the computable cross-norm or realignment criterion, Vol. 77 (American Physical Society, 2008) p. 060301

  32. Chruściński, D., Macchiavello, C., Maniscalco, S.: Detecting Non-Markovianity of Quantum Evolution via Spectra of Dynamical Maps, Vol. 118 (American Physical Society, 2017) p. 080404

  33. Macchiavello, C., Rossi, M.: Quantum channel detection, Vol. 88 (American Physical Society, 2013) p. 042335

  34. Orieux, A., Sansoni, L., Persechino, M., Mataloni, P., Rossi, M., Macchiavello, C.: Experimental Detection of Quantum Channels, Vol. 111 (American Physical Society, 2013) p. 220501

  35. Kołodyński, J., Rana, S., Streltsov, A.: Entanglement negativity as a universal non-Markovianity witness, Vol. 101 (American Physical Society, 2020) p. 020303

  36. Jamiolkowski, A.: Linear Trans. Preserve Trace Positive Semidefinit. Oper. 3, 275–278 (1972)

    Google Scholar 

  37. Choi, M.-D.: Completely Positive Linear Maps Complex Matrices 10, 285–290 (1975)

    Google Scholar 

  38. Wolf, M.M., Cirac, J.I.: Div. Quantum Channels 279, 147–168 (2008)

    Google Scholar 

  39. Berk, G. D., Garner, A. J. P., Yadin, B., Modi, K., Pollock, F. A.: Resource theories of multi-time processes: A window into quantum non-Markovianity ( 2019) p. , arXiv:1907.07003 [quant-ph]

  40. Milz, S., Kim, M. S., Pollock, F. A., Modi, K.: Completely Positive Divisibility Does Not Mean Markovianity, Vol. 123 ( American Physical Society, 2019) p. 040401

  41. Taranto, P., Pollock, F. A., Milz, S., Tomamichel, M., Modi, K.: Quantum Markov Order, Vol. 122 ( American Physical Society, 2019) p. 140401

  42. Lindblad, G.: On the Generators of Quantum Dynamical Semigroups 48, 119–130 (1976)

    Google Scholar 

  43. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely Positive Dynam Semigroups N-level Syst. 17, 821–825 (1976)

    Google Scholar 

  44. Hall, M. J. W., Cresser, J. D., Li, L., Andersson, E.: Canonical form of master equations and characterization of non-Markovianity, Vol. 89 ( American Physical Society, 2014) p. 042120

  45. Cohen, J.E., Friedland, S., Kato, T., Kelly, F.P.: Eigenvalue Inequalities Products Matrix Exponentials 45, 55–95 (1982)

    Google Scholar 

  46. Rockafellar, R.: Convex Analysis (Princeton University Press, address 41 William Street, 1997)

  47. Schlosshauer, M.: Quantum decoherence, Vol. 831 ( 2019) pp. 1–57, quantum decoherence

  48. Pittenger, A.O., Rubin, M.H.: Convexity Separability Problem Quantum Mech. Density Matrices 346, 47–71 (2002)

    Google Scholar 

  49. Gühne, O., Lütkenhaus, N.: Nonlinear Entanglement Witnesses, Vol. 96 ( American Physical Society, 2006) p. 170502

  50. Arrazola, J. M., Gittsovich, O., Lütkenhaus, N.: Accessible nonlinear entanglement witnesses, Vol. 85 ( American Physical Society, 2012) p. 062327

  51. Gühne, O., Lütkenhaus, N.: Nonlinear entanglement witnesses, covariance matrices and the geometry of separable states, Vol. 67 ( 2007) p. 012004

  52. Ioannou, L. M., Travaglione, B. C.: Quantum separability and entanglement detection via entanglement-witness search and global optimization, Vol. 73 ( American Physical Society, 2006) p. 052314

  53. Nemhauser, G. L., Wolsey, L.: Integer and Combinatorial Optimization ( John Wiley and Sons, address Chichester, 2014)

  54. Li, L., Hall, M. J., Wiseman, H. M.: Concepts of quantum non-Markovianity: A hierarchy, Vol. 759 ( 2018) pp. 1 – 51, concepts of quantum non-Markovianity: A hierarchy

  55. Chruściński, D., Kossakowski, A., Rivas, A.: Measures of non-Markovianity: Divisibility versus backflow of information, Vol. 83 ( American Physical Society, 2011) p. 052128

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Acknowledgements

Authors thank Manik Banik of SNBNCBS, Kolkata, for illuminating discussion. SB thanks SERB, DST, Government of India, for financial support. BB thanks DST INSPIRE programme for financial support.

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Authors and Affiliations

  1. S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata, 700 098, India

    Bihalan Bhattacharya & Samyadeb Bhattacharya

  2. Centre for Quantum Optical Technologies IRAU, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097, Warsaw, Poland

    Samyadeb Bhattacharya

  3. Center for Security, Theory and Algorithmic Research, International Institute of Information Technology-Hyderabad, Gachibowli, Telangana, 500 032, Hyderabad, India

    Samyadeb Bhattacharya

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  1. Bihalan Bhattacharya
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  2. Samyadeb Bhattacharya
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Correspondence to Samyadeb Bhattacharya.

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Bhattacharya, B., Bhattacharya, S. Convex geometry of Markovian Lindblad dynamics and witnessing non-Markovianity. Quantum Inf Process 20, 253 (2021). https://doi.org/10.1007/s11128-021-03177-y

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