Noether’s theorem for dissipative quantum dynamical semi-groups

Gough, John E.; Raţiu, Tudor S.; Smolyanov, O. G.

doi:10.1063/1.4907985

Cited by 19 publications

(39 citation statements)

References 7 publications

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“…For Δ ¼ 0, such J are conserved quantities, so a natural question is whether they always commute with the Hamiltonian and the jump operators. It turns out that they do not always commute [50,78], and so various generalizations of Noether's theorem have to be considered [70,128]. Using the following analysis, we can say that J's always commute with both the Hamiltonian and jump operators of L when there is no decaying subspace (P ¼ I).…”

Section: B Nonsteady Asymptotic Subspacesmentioning

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: B Nonsteady Asymptotic Subspacesmentioning

confidence: 99%

Geometry and Response of Lindbladians

et al. 2016

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“…This is a one-way Noether-type theorem linking conserved quantities to symmetries (see Refs. [8,67] or Ref. [68], Ch.…”

Section: Limited Noether-type Theoremmentioning

confidence: 99%

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

Albert

2019

Quantum

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This work derives an analytical formula for the asymptotic state-the quantum state resulting from an infinite number of applications of a general quantum channel on some initial state. For channels admitting multiple fixed or rotating points, conserved quantities-the left fixed/rotating points of the channeldetermine the dependence of the asymptotic state on the initial state. The formula stems from a Noether-like theorem stating that, for any channel admitting a full-rank fixed point, conserved quantities commute with that channel's Kraus operators up to a phase. The formula is applied to adiabatic transport of the fixed-point space of channels, revealing cases where the dissipative/spectral gap can close during any segment of the adiabatic path. The formula is also applied to calculate expectation values of noninjective matrix product states (MPS) in the thermodynamic limit, revealing that those expectation values can also be calculated using an MPS with reduced bond dimension and a modified boundary.

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“…Finally, our framework has a number of interesting generalisations connected with other ongoing mathematical work on quantum stochastic evolutions. In particular, when the stationary state manifold is nontrivial (non-ergodic case), one can discuss conserved quantities and adiabatic transport [3,26,1]. From the more technical point of view, our manifold of dynamical parameters actually has a natural Lie group structure [17]; reformulation of our results in this more structured framework could be useful especially for applications to control theory.…”

Section: System Inputmentioning

confidence: 99%

Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics

Guţǎ

Kiukas

2017

Journal of Mathematical Physics

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This paper deals with the problem of identifying and estimating dynamical parameters of continuous-time Markovian quantum open systems, in the input-output formalism. First, we characterise the space of identifiable parameters for ergodic dynamics, assuming full access to the output state for arbitrarily long times, and show that the equivalence classes of undistinguishable parameters are orbits of a Lie group acting on the space of dynamical parameters. Second, we define an information geometric structure on this space, including a principal bundle given by the action of the group, as well as a compatible connection, and a Riemannian metric based on the quantum Fisher information of the output. We compute the metric explicitly in terms of the Markov covariance of certain ?fluctuation operators? and relate it to the horizontal bundle of the connection. Third, we show that the system-output and reduced output state satisfy local asymptotic normality, i.e., they can be approximated by a Gaussian model consisting of coherent states of a multimode continuous variables system constructed from the Markov covariance ?data.? We illustrate the result by working out the details of the information geometry of a physically relevant two-level systemauthorsversionPeer reviewe

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Noether’s theorem for dissipative quantum dynamical semi-groups

Cited by 19 publications

References 7 publications

Geometry and Response of Lindbladians

Geometry and Response of Lindbladians

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics

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