Hierarchy of non-Markovianity and k -divisibility phase diagram of quantum processes in open systems

Department of Physics, National Chung Hsing University, Taichung 402, Taiwan(Dated: April 29, 2015)In recent years, much e ort has been devoted to the construction of a proper measure of quantumnon-Markovianity. However, those proposed measures are shown to be at variance with di erentsituations. In this work, we utilize the theory of k-positive maps to generalize a hierarchy of k-divisibility and develop a powerful tool, called k-divisibility phase diagram, which can provide afurther insight into the nature of quantum non-Markovianity. By exploring the phase diagram withseveral paradigms, we can explain the origin of the discrepancy between two frequently used measureand nd the condition under which the two measures coincide with each other.

[1]  P. Hänggi,et al.  Driven quantum tunneling , 1998 .

[2]  Jyrki Piilo,et al.  Measure for the degree of non-markovian behavior of quantum processes in open systems. , 2009, Physical review letters.

[3]  Chi-Kwong Li,et al.  A new criterion and a special class of k-positive maps , 2015 .

[4]  M. Paternostro,et al.  Geometrical characterization of non-Markovianity , 2013, 1302.6673.

[5]  Jyrki Piilo,et al.  Measure for the non-Markovianity of quantum processes , 2010, 1002.2583.

[6]  J Eisert,et al.  Assessing non-Markovian quantum dynamics. , 2007, Physical review letters.

[7]  Gert-Ludwig Ingold,et al.  Quantum Brownian motion: The functional integral approach , 1988 .

[8]  U. Weiss.  Quantum Dissipative Systems , 1993 .

[9]  Y. Tanimura,et al.  Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[10]  M. Paternostro,et al.  Competition between memory-keeping and memory-erasing decoherence channels , 2013, 1311.2045.

[11]  Shin-Liang Chen,et al.  Examining non-locality and quantum coherent dynamics induced by a common reservoir , 2013, Scientific Reports.

[12]  Hao-Sheng Zeng,et al.  Equivalence of the measures of non-Markovianity for open two-level systems , 2011, 1104.0070.

[13]  Mak,et al.  Path-integral Monte Carlo simulations without the sign problem: multilevel blocking approach for effective actions , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Susana F Huelga,et al.  Entanglement and non-markovianity of quantum evolutions. , 2009, Physical review letters.

[15]  J. Ross,et al.  Statistical reduction for strongly driven simple quantum systems , 1978 .

[16]  Y. Tanimura.  Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems , 2006 .

[17]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

[18]  T. Arimitsu,et al.  Expansion Formulas in Nonequilibrium Statistical Mechanics , 1980 .

[19]  S. Huelga,et al.  Quantum non-Markovianity: characterization, quantification and detection , 2014, Reports on progress in physics. Physical Society.

[20]  Li Li,et al.  Canonical form of master equations and characterization of non-Markovianity , 2010, 1009.0845.

[21]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[22]  A. Wallraff,et al.  Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate , 2014, Nature Communications.

[23]  Neill Lambert,et al.  Environmental dynamics, correlations, and the emergence of noncanonical equilibrium states in open quantum systems , 2013, 1311.0016.

[24]  R. Kubo,et al.  Time Evolution of a Quantum System in Contact with a Nearly Gaussian-Markoffian Noise Bath , 1989 .

[25]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[26]  D. Chruściński,et al.  Degree of non-Markovianity of quantum evolution. , 2013, Physical review letters.

[27]  J. Cirac,et al.  Dividing Quantum Channels , 2006, math-ph/0611057.

[28]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[29]  Michael J. W. Hall,et al.  Finding the Kraus decomposition from a master equation and vice versa , 2007, 0801.4100.

[30]  J. Onuchic,et al.  Effect of friction on electron transfer in biomolecules , 1985 .

[31]  S. Maniscalco,et al.  Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective , 2014, Scientific Reports.

[32]  Kedar S. Ranade,et al.  The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive Maps , 2007, Open Syst. Inf. Dyn..

[33]  R. Silbey,et al.  Local and nonlocal approximation for a simple quantum system , 2001 .

[34]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[35]  A. Leggett,et al.  Dynamics of the dissipative two-state system , 1987 .

[36]  S. Luo,et al.  Quantifying non-Markovianity via correlations , 2012 .

[37]  Karol Zyczkowski,et al.  Cones of positive maps and their duality relations , 2009, 0902.4877.

[38]  D. Chru'sci'nski,et al.  Measures of non-Markovianity: Divisibility versus backflow of information , 2011, 1102.4318.

[39]  John Goold,et al.  Non-Markovianity, Loschmidt echo, and criticality: a unified picture , 2012, 1202.2997.

[40]  Filip A. Wudarski,et al.  Non-Markovian random unitary qubit dynamics , 2012, 1212.2029.

[41]  S. Maniscalco,et al.  Comparative study of non-Markovianity measures in exactly solvable one- and two-qubit models , 2014, 1402.4975.