Adiabaticity in open quantum systems

We provide a rigorous generalization of the quantum adiabatic theorem for open systems described by a Markovian master equation with time-dependent Liouvillian $\mathcal{L}(t)$. We focus on the finite system case relevant for adiabatic quantum computing and quantum annealing. Adiabaticity is defined in terms of closeness to the instantaneous steady state. While the general result is conceptually similar to the closed system case, there are important differences. Namely, a system initialized in the zero-eigenvalue eigenspace of $\mathcal{L}(t)$ will remain in this eigenspace with a deviation that is inversely proportional to the total evolution time $T$. In the case of a finite number of level crossings the scaling becomes $T^{-\eta}$ with an exponent $\eta$ that we relate to the rate of the gap closing. For master equations that describe relaxation to thermal equilibrium, we show that the evolution time $T$ should be long compared to the corresponding minimum inverse gap squared of $\mathcal{L}(t)$. Our results are illustrated with several examples.

[1]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[2]  Daniel A. Lidar,et al.  Adiabatic approximation with exponential accuracy for many-body systems and quantum computation , 2008, 0808.2697.

[3]  Daniel A. Lidar,et al.  Distance bounds on quantum dynamics , 2008, 0803.4268.

[4]  Germany,et al.  Quantum states and phases in driven open quantum systems with cold atoms , 2008, 0803.1482.

[5]  R. Werner,et al.  The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation , 2006, IEEE Transactions on Information Theory.

[6]  M. Ruskai,et al.  Bounds for the adiabatic approximation with applications to quantum computation , 2006, quant-ph/0603175.

[7]  Walid K. Abou Salem,et al.  On the Quasi-Static Evolution of Nonequilibrium Steady States , 2006, math-ph/0601046.

[8]  Peter Hänggi,et al.  Gauging a quantum heat bath with dissipative Landau-Zener transitions. , 2006, Physical review letters.

[9]  Peter D. Jarvis,et al.  Reports on Mathematical Physics , 2006 .

[10]  Jean,et al.  Henri Poincare,为科学服务的一生 , 2006 .

[11]  Johan Åberg,et al.  Adiabatic approximation for weakly open systems , 2005 .

[12]  J. Butcher Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods , 2005 .

[13]  Daniel A. Lidar,et al.  Adiabatic approximation in open quantum systems , 2004, quant-ph/0404147.

[14]  H. Breuer Genuine quantum trajectories for non-Markovian processes , 2004, quant-ph/0403117.

[15]  R. Xu,et al.  Theory of open quantum systems , 2002 .

[16]  G. Hagedorn,et al.  Elementary Exponential Error Estimates for the Adiabatic Approximation , 2002 .

[17]  K. Lendi,et al.  Virtues and Limitations of Markovian Master Equations with a Time-Dependent Generator , 2000 .

[18]  A. Elgart,et al.  Adiabatic Theorem without a Gap Condition , 1998, math/9803153.

[19]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[20]  A. Auerbach Interacting electrons and quantum magnetism , 1994 .

[21]  G. Nenciu,et al.  Linear adiabatic theory. Exponential estimates , 1993 .

[22]  Dénes Petz,et al.  The Bogoliubov inner product in quantum statistics , 1993 .

[23]  J. Lambert Numerical Methods for Ordinary Differential Equations , 1991 .

[24]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[25]  Ruedi Seiler,et al.  Adiabatic theorems and applications to the quantum hall effect , 1987 .

[26]  Herbert Spohn,et al.  Open quantum systems with time-dependent Hamiltonians and their linear response , 1978 .

[27]  V. Gorini,et al.  Quantum detailed balance and KMS condition , 1978 .

[28]  R. Alicki On the detailed balance condition for non-hamiltonian systems , 1976 .

[29]  Rudolf Haag,et al.  On the equilibrium states in quantum statistical mechanics , 1967 .

[30]  Tosio Kato Perturbation theory for linear operators , 1966 .

[31]  L. Garrido,et al.  Degree of approximate validity of the adiabatic invariance in quantum mechanics , 1962 .

[32]  Tosio Kato On the Adiabatic Theorem of Quantum Mechanics , 1950 .

[33]  V. Fock,et al.  Beweis des Adiabatensatzes , 1928 .

[34]  October I Physical Review Letters , 2022 .