Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values

Schmid, Jochen

doi:10.1142/9789814618144_0031

Cited by 11 publications

(11 citation statements)

References 29 publications

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“…and (b) is the holonomy after a closed adiabatic deformation unitary? Regarding the first question, the adiabatic theorem has indeed been generalized to Lindblad master equations [70,72,[99][100][101][102] and all orders of corrections to adiabatic evolution have been derived (see, e.g., Ref. [72], Theorem 6).…”

Section: Earlier Workmentioning

confidence: 99%

Geometry and Response of Lindbladians

et al. 2016

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Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to study several important features of the unique state and subspace cases. We quantify how subspaces retain information about initial states and show how to use Lindbladians to simulate any quantum channels. We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected subspaces. We derive a Kubo formula governing linear response of subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zerofrequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either Hamiltonian or jump-operator perturbations or corrections to adiabatic evolution, is different from the conventional Lindbladian dissipative gap and, in certain cases, is equivalent to the excitation gap of a related Hamiltonian. DOI: 10.1103/PhysRevX.6.041031 Subject Areas: Condensed Matter Physics, Quantum Information, Statistical Physics I. MOTIVATION AND OUTLINEConsider coupling a quantum mechanical system to a Markovian reservoir which evolves initial states of the system into multiple nonequilibrium (i.e., nonthermal) asymptotic states in the limit of infinite time. After tracing out the degrees of freedom of the reservoir, the time evolution of the system is governed by a Lindbladian L [1,2] (see also ), and its various asymptotic states ρ ∞ are elements of an asymptotic subspace AsðHÞ-a subspace of OpðHÞ, the space of operators on the system Hilbert space H. The asymptotic subspace attracts all initial states ρ in ∈ OpðHÞ, is free from the decoherence effects of L, and any remaining time evolution within AsðHÞ is exclusively unitary. If AsðHÞ has no time evolution, all ρ ∞ are stationary or steady. This work provides a thorough investigation into the response and geometrical properties of the various asymptotic subspaces.On one hand, AsðHÞ that support quantum information [7][8][9][10] are promising candidates for storing, preserving, and manipulating such information, particularly when their states can be engineered to possess favorable features (e.g., topological protection [11][12][13])....

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Section: Earlier Workmentioning

confidence: 99%

Geometry and Response of Lindbladians

et al. 2016

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“…However, most adiabatic results (for example [6,9,27,40] in continuous time, or [17,39] in discrete time) apply to unitary dynamics only, whereas here the L k,T are not unitary. On the other hand, adiabatic results for non unitary dynamics (see [1,7,26,37]) are proven only in continuous time, whereas we work here in discrete time. We will therefore need to derive a specific form of adiabatic theorem for products of slowly changing CPTP maps.…”

Section: The Discrete Non Unitary Adiabatic Theoremmentioning

confidence: 99%

“…We note that adiabatic approximations for unitary evolution operators generated by slowly varying time dependent self-adjoint generators with gaps in their spectrum can be found in [8,27,32] and, without gap assumptions, in [6,40]. Extensions to non-unitary semigroups of contractions with or without gap condition can be found in [1,7,26,37]. On the other hand, discrete time adiabatic theorems have been proven in [17,39] for unitary groups only.…”

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confidence: 98%

Landauer’s Principle in Repeated Interaction Systems

Hanson

Joye

Pautrat

et al. 2016

Commun. Math. Phys.

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We study Landauer's Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system S in contact with a structured environment E made of a chain of independent quantum probes; S interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer's lower bound, which relates the energy variation of the environment E to a decrease of entropy of the system S during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment E displaying small variations of order T −1 between the successive probes encountered by S, after n ≃ T interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of S in this regime, in order to tackle the adiabatic limit of Landauer's bound. We find that saturation of Landauer's bound is related to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauer's bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.

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“…Section 4.1 contains a qualitative adiabatic theorem which, in simplified form, can be formulated as follows (with I := [0, 1]). See [61]. If A(t) : D ⊂ X → X for every t ∈ I generates a contraction semigroup, if λ(t) for every t ∈ I is an eigenvalue of A(t) such that λ(t) + δe iϑ(t) ∈ ρ(A(t)) for every δ ∈ (0, δ 0 ], and if P (t) is weakly associated with A(t) and λ(t) for almost every t ∈ I and of finite rank and the reduced resolvent estimate…”

Section: Introductionmentioning

confidence: 99%

Adiabatic theorems for general linear operators with time-independent domains

Schmid

2019

Rev. Math. Phys.

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We establish adiabatic theorems with and without spectral gap condition for general -typically dissipative -linear operators A(t) : D(A(t)) ⊂ X → X with timeindependent domains D(A(t)) = D in some Banach space X. Compared to the previously known adiabatic theorems -especially those without spectral gap condition -we do not require the considered spectral values λ(t) of A(t) to be (weakly) semisimple. We also impose only fairly weak regularity conditions. Applications are given to slowly time-varying open quantum systems and to adiabatic switching processes. Subject classification (2010) and key words: 34E15, 34G10, 35Q41, 47D06, 81Q12, 81S22 Adiabatic theorems for general linear operators, dissipative operators, time-independent domains, non-semisimple spectral values, spectral gap, open quantum systems, adiabatic switching arXiv:1804.11213v2 [math-ph] 17 Sep 2018Some typical spectral situations are illustrated below for the special case of skew-adjoint operators A(t): the spectrum σ(A(t)) is plotted on the vertical axis iR against the horizontal t-axis and the red line represents the considered spectral values λ(t). In the first two figures, we have a spectral gap which is uniform in the first and non-uniform in the second picture. And the third figure depicts a situation without spectral gap.What adiabatic theory is interested in is how certain distinguished solutions to (1.2) behave in the singular limit where the slowness parameter ε tends to 0. In more specific terms, the basic goal of adiabatic theory can be described -for skew-adjoint and then for general operators -as follows. In the special case of skew-adjoint operators A(t), one wants to show that for small values of ε and every t the solution operator U ε (t, 0)with Z 0 (t) being the generator of the semigroup on S p (h) defined by e Z 0 (t)τ (ρ) := e −iH(t)τ ρ e iH(t)τ , where H(t) : D(H(t)) ⊂ h → h is a self-adjoint operator and B j (t) for every j in the arbitrary index set J is a bounded opertor in h such that j∈J B j (t)B j (t) * = j∈J B j (t) * B j (t) < ∞ (1.14)

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Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values

Cited by 11 publications

References 29 publications

Geometry and Response of Lindbladians

Geometry and Response of Lindbladians

Landauer’s Principle in Repeated Interaction Systems

Adiabatic theorems for general linear operators with time-independent domains

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