Exponential Stability of Subspaces for Quantum Stochastic Master Equations

Benoist, Tristan; Pellegrini, Clément; Ticozzi, Francesco

doi:10.1007/s00023-017-0556-3

Cited by 23 publications

(23 citation statements)

References 55 publications

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“…Statements similar to Proposition 2 exist for fixed points of quantum channels [79,88], and their extension to rotating points will be a subject of future work. These results may also be useful in determining properties of asymptotic algebras of observables [168,169] and properties of quantum jump trajectories when the Lindbladian is "unraveled" [170,171].…”

Section: -21mentioning

confidence: 97%

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: -21mentioning

confidence: 97%

Geometry and Response of Lindbladians

et al. 2016

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“…Then (56) can be concluded from (63), (65) and (66). 2 It is noted that Tr(P Rρθ t ) in (56) serves as a linear Lyapunov function candidate for the subspace H S = C |ψ 0 (See Theorem 1.2 in ( [Benoist et al (2017)])). Thus Theorem 3.3 can be treated as an input-to-state stability result for the exponential quantum projection filter.…”

Section: Practical Stability Of the Quantum Projection Filtermentioning

confidence: 99%

An exponential quantum projection filter for open quantum systems

Gao

Petersen

2019

Automatica

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“…Without this assumption, we would have to take into account the almost sure Lyapunov exponent corresponding to the escape from the transient part. See [3] for a precise account of these ideas.…”

Section: Lyapunov Exponentsmentioning

confidence: 99%

Invariant measure for quantum trajectories

Benoist

Fraas

Pautrat

et al. 2018

Probab. Theory Relat. Fields

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We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices. It is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations.

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Exponential Stability of Subspaces for Quantum Stochastic Master Equations

Cited by 23 publications

References 55 publications

Geometry and Response of Lindbladians

Geometry and Response of Lindbladians

An exponential quantum projection filter for open quantum systems

Invariant measure for quantum trajectories

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