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Interplay of Quantum Stochastic and Dynamical Maps to Discern Markovian and Non-Markovian Transitions

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It is known that the dynamical evolution of a system, from an initial tensor product state of system and environment, to any two later times, t1, t2 (t2 > t1), are both completely positive (CP) but in the intermediate times between t1 and t2 it need not be CP. This reveals the key to the Markov (if CP) and non Markov (if it is not CP) avataras of the intermediate dynamics. This is brought out here in terms of the quantum stochastic map A and the associated dynamical map B—without resorting to master equation approaches. We investigate these features with four examples which have entirely different physical origins: 1) A two qubit Werner state map with time dependent noise parameter; 2) Phenomenological model of a recent optical experiment (Nature Physics, 7, 931 (2011)) on the open system evolution of photon polarization; 3) Hamiltonian dynamics of a qubit coupled to a bath of N qubits; 4) Two qubit unitary dynamics of Jordan et al. (Phys. Rev. A 70, 052110 (2004) with initial product states of qubits. In all these models, it is shown that the positivity/negativity of the eigenvalues of intermediate time dynamical B map determines the Markov/non-Markov nature of the dynamics.

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A. Devi, A. Rajagopal, S. Shenoy and R. Rendell, "Interplay of Quantum Stochastic and Dynamical Maps to Discern Markovian and Non-Markovian Transitions,"

*Journal of Quantum Information Science*, Vol. 2 No. 3, 2012, pp. 47-54. doi: 10.4236/jqis.2012.23009.

[1] | H. P. Breuer and F. Petruccione, “The Theory of Open Quantum Systems,” Oxford University Press, Oxford, 2007. HHUdoi:10.1093/acprof:oso/9780199213900.001.0001U |

[2] | R. Alicki and K. Lendi, “Quantum Dynamical Semi-Groups and Applications,” Springer, Berlin, 1987. |

[3] | M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press, Cambridge, 2000. |

[4] | M. D. Choi, “Positive Linear Maps on C*-Algebras,” Canadian Journal of Mathematics, Vol. 24, No. 3, 1972, pp. 520-529. |

[5] | M. D. Choi, “Completely Positive Linear Maps on Complex Matrices,” Linear Algebra and Its Applications, Vol. 10, No. 3, 1975, pp. 285-290. HHUdoi:10.1016/0024-3795(75)90075-0U |

[6] | A. Jamiolkowski, “Linear Transformations Which Preserve Trace and Positive Semidefiniteness of Operators,” Reports on Mathematical Physics, Vol. 3, No. 4, 1972, pp. 275-278. HHUdoi:10.1016/0034-4877(72)90011-0U |

[7] | E. C. G. Sudarshan, P. Mathews and J. Rau, “StochasticDynamics of Quantum-Mechanical Systems,” Physical Review, Vol. 121, No. 3, 1961, pp. 920-924. |

[8] | T. F. Jordan and E. C. G. Sudarshan, “Dynamical Map-Pings of Density Operators in Quantum Mechanics,” Journal of Mathematical Physics, Vol. 2, No. 6, 1961, pp. 772-775. |

[9] | G. Lindblad, “On the Generators of Quantum Dynamical Semigroups,” Communications in Mathematical Physics, Vol. 48, No. 2, 1976, pp. 119-130. HHUdoi:10.1007/BF01608499U |

[10] | V. Gorini, A. Kossakowski and E. C. G. Sudarshan, “Completely Positive Dynamical Semigroups of N-level Systems,” Journal of Mathematical Physics, Vol. 17, No. 5, 1976, pp. 821-825. HHUdoi:10.1063/1.522979U |

[11] | M. M. Wolf, J. Eisert, T. S. Cubitt and J. I. Cirac, “Assessing Non-Markovian Quantum Dynamics,” Physical Review Letters, Vol. 101, No. 15, 2008, pp. 150402.1-150402.4. |

[12] | A. Rivas, S. F. Huelga and M. B. Plenio, “Entanglement and Non-Markovianity of Quantum Evolutions,” Physical eview Letters, Vol. 105, No. 5, 2010, pp. 050403.1-050403.4. |

[13] | T. F. Jordan, A. Shaji and E. C. G. Sudarshan, “Dynamics of Initially Entangled Open Quantum Systems,” Physical Review A, Vol. 70, No. 5, 2004, pp. 052110.1-052110.14. |

[14] | C. A. Rodríguez-Rosario and E. C. G. Sudarshan, “Non-Markovian Open Quantum Systems,” Bulletin of the American Physical Society, Vol. 53, No. 2, 2008. |

[15] | C. A. Rodríguez-Rosario, K. Modi, A. Kuah, A. Shaji and E. C. G. Sudarshan, “Completely Positive Maps and Classical Correlations,” Journal of Physics A, Vol. 41, No. 20, 2008, pp. 205301.1-205301.8. |

[16] | K. Modi and E. C. G. Sudarshan, “Role of Preparation in Quantum Process Tomography,” Physical Review A, Vol. 81, No. 5, 2010, pp. 052119.1-052119.10. |

[17] | A. R. Usha Devi, A. K. Rajagopal and Sudha, “Open-System Quantum Dynamics with Correlated Initial States, Not Completely Positive Maps, and Non-Markovianity,” Physical Review A, Vol. 83, No. 2, 2011, pp. 022109.1-022109.8. |

[18] | H.-P. Breuer, “Exact Quantum Jump Approach to Open Systems in Bosonic and Spin Baths,” Physical Review A, Vol. 69, No. 2, 2004, pp. 022115.1-022115.8. |

[19] | H. P. Breuer, “Genuine Quantum Trajectories for Non-Markovian Processes,” Physical Review A, Vol. 70, No. 1, 2004, pp. 012106.1-012106.12. |

[20] | S. Daffer, K. Wódkiewicz, J. D. Cresser and K. McIver, “Depolarizing Channel as a Completely Positive Map with Memory,” Physical Review A, Vol. 70, No. 1, 2004, pp. 010304.1-010304.4. |

[21] | H.-P. Breuer and B. Vacchini, “Quantum Semi-Markov Processes,” Physical Review Letters, Vol. 101, No. 14, 2008, pp. 140402.1-140402.4. |

[22] | H.-P. Breuer and B. Vacchini, “Structure of Completely Positive Quantum Master Equations with Memory Kernel,” Physical Review E, Vol. 79, No. 4, 2009, pp. 041147.1-041147.12. |

[23] | A. Kossakowski and R. Rebolledo, “On Non-Markovian Time Evolution in Open Quantum Systems,” Open Systems and Information Dynamics, Vol. 14, No. 3, 2007, pp. 265-274. HHUdoi:10.1007/s11080-007-9051-5U |

[24] | A. Kossakowski and R. Rebolledo, “On Completely Positive Non-Markovian Evolution of a d-Level System,” Open Systems and Information Dynamics, Vol. 15, No. 2, 2008, pp. 135-141. |

[25] | A. Kossakowski and R. Rebolledo, “On the Structure of Generators for Non-Markovian Master Equations,” Open Systems and Information Dynamics, Vol. 16, No. 2-3, 2009, pp. 259-268. |

[26] | D. Chru?ciński and A. Kossakowski, “Non-Markovian Quantum Dynamics: Local versus Nonlocal,” Physical Review Letters, Vol. 104, No. 7, 2010, pp. 070406.1-070406.4. |

[27] | A. K. Rajagopal, A. R. Usha Devi and R. W. Rendell, “Kraus Representation of Quantum Evolution and Fidelity as Manifestations of Markovian and Non-Markovian Forms,” Physical Review A, Vol. 82, No. 4, 2010, pp. 042107.1-042107.7. |

[28] | H.-P. Breuer, E. M. Laine and J. Piilo, “Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems,” Physical Review Letters, Vol. 103, No. 21, 2009, pp. 210401.1-210401.4. |

[29] | E. M. Laine, J. Piilo and H.-P. Breuer, “Measure for the Non-Markovianity of Quantum Processes,” Physical Review A, Vol. 81, No. 6, 2010, pp. 062115.1-062115.8. |

[30] | P. Haikka, J. D. Cresser and S. Maniscalco, “Comparing Different Non-Markovianity Measures in a Driven Qubit System,” Physical Review A, Vol. 83, No. 1, 2011, pp. 012112.1-012112.5. |

[31] | D. Chru?ciński, A. Kossakowski and A. Rivas, “Measures of Non-Markovianity: Divisibility versus Backflow of Information,” Physical Review A, Vol. 83, No. 5, 2011, pp. 052128.1-052128.6. |

[32] | S. C. Hou, X. X. Yi, S. X. Yu and C. H. Oh, “Alternative Non-Markovianity Measure by Divisibility of Dynamical Maps,” Physical Review A, Vol. 83, No. 6, 2011, pp. 062115.1-062115.6 |

[33] | K. G. H. Vollbrecht and R. F. Werner, “Entanglement Measures under Symmetry,” Physical Review A, Vol. 64, No. 6, 2001, pp. 062307.1-062307.15 |

[34] | B.-L. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M Laine, H.-P. Breuer and J. Piilo, “Experimental Control of the Transition from Markovian to Non-Markovian Dynamics of Open Quantum Systems,” Nature Physics, Vol. 7, No. 12, 2011, pp. 931-934. HUdoi:10.1038/nphys2085U |

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