Quantum logarithmic Sobolev inequalities and rapid mixing

Kastoryano, Michael J.; Temme, Kristan

doi:10.48550/arxiv.1207.3261

Cited by 8 publications

(19 citation statements)

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“…For a more comprehensive exposition, consult Ref. [19]. The mixing time of a quantum Markov process is the time it takes for the process to become close to the stationary state, starting from an arbitrary initial state.…”

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

“…A huge amount of effort has been invested in bounding the mixing time of classical Markov processes; especially in the setting of Markov chain Monte Carlo [35]. Recently, a set of functional tools have been developed for analyzing the quantum analogue of Markov chain mixing for one parameter semi-groups [20,19]. In particular, trace norm convergence of primitive semi-groups can be very well characterized in terms of two quantities: the inverse of the smallest eigenvalue of the stationary state σ −1 , and one of two exponential decay rates, the χ 2 constant λ s [20] or the Log-Sobolev constant α s [19].…”

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

“…Recently, a set of functional tools have been developed for analyzing the quantum analogue of Markov chain mixing for one parameter semi-groups [20,19]. In particular, trace norm convergence of primitive semi-groups can be very well characterized in terms of two quantities: the inverse of the smallest eigenvalue of the stationary state σ −1 , and one of two exponential decay rates, the χ 2 constant λ s [20] or the Log-Sobolev constant α s [19]. Each of these quantities has a convenient variational characterization.…”

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

“…( 9) and ( 10) are generalizations of similar expressions defined in Ref. [19], where only the s = 1/2 case was considered. Given that Γ s σ is only (completely)-positive for s = 1/2, certain results only hold for that case.…”

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

“…The functional methods developed in Refs. [19,20] distinguish between two fundamental convergence behaviors: an asymptotic convergence rate which decreases with the system size and goes to zero in the thermodynamic limit (non gapped), and an asymptotic rate which is lower bounded for all system sizes (gapped). We will focus on the second case, and point out that there are two subclasses in the analysis.…”

Section: Introductionmentioning

confidence: 99%

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Rapid mixing implies exponential decay of correlations

Kastoryano

Eisert

2013

Journal of Mathematical Physics

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We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the Log-Sobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information - a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped free-fermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrix-product operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.Comment: 25 pages, 2 figures, replaced by final versio

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Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

Section: Mixing Times Of Semi-groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rapid mixing implies exponential decay of correlations

Kastoryano

Eisert

2013

Journal of Mathematical Physics

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Hypercontractivity for Semigroups of Unital Qubit Channels

King

2014

Commun. Math. Phys.

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Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form e −t 1 H 1 ⊗ · · · ⊗ e −tnHn to be a contraction from L p to L q , where L p is the algebra of 2 n -dimensional matrices equipped with the normalized Schatten norm, and each generator H j is a self-adjoint positive semidefinite operator on the algebra of 2-dimensional matrices. As a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels.

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Stability of Local Quantum Dissipative Systems

Cubitt

Lucia

Michalakis

et al. 2015

Commun. Math. Phys.

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Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are potentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time which scales logarithmically with the system size. The proof relies on Lieb-Robinson bounds, which describe a finite group velocity for propagation of information in local systems. As a main example, we prove that classical Glauber dynamics is stable under local perturbations, including perturbations in the transition rates which may not preserve detailed balance.

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Quantum logarithmic Sobolev inequalities and rapid mixing

Cited by 8 publications

References 0 publications

Rapid mixing implies exponential decay of correlations

Rapid mixing implies exponential decay of correlations

Hypercontractivity for Semigroups of Unital Qubit Channels

Stability of Local Quantum Dissipative Systems

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