# Rapid mixing implies exponential decay of correlations

@article{Kastoryano2013RapidMI, title={Rapid mixing implies exponential decay of correlations}, author={Michael J. Kastoryano and Jens Eisert}, journal={Journal of Mathematical Physics}, year={2013}, volume={54}, pages={102201-102201} }

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…

## 48 Citations

### The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions

- Mathematics
- 2020

Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the…

### Exponential decay of mutual information for Gibbs states of local Hamiltonians

- Computer ScienceQuantum
- 2022

This work shows that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespective of the temperature, and finds that the Gibbs states of the systems the authors consider are superexponentially close to saturating the data-processing inequality for the Belavkin-Staszewski relative entropy.

### Stability of Local Quantum Dissipative Systems

- Mathematics
- 2015

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…

### Random Lindblad dynamics

- Mathematics, PhysicsJournal of Physics A: Mathematical and Theoretical
- 2019

We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the long-time behavior…

### IBBS SAMPLERS : THE COMMUTING CASE

- Computer Science, Mathematics
- 2016

It is shown that for every one-dimensional lattice system, or for systems in lattices of any dimension at temperatures above a certain threshold, the Gibbs samplers of commuting Hamiltonians are always gapped, giving an efficient way of preparing the associated Gibbs states on a quantum computer.

### Quantum Gibbs Samplers: The Commuting Case

- Computer Science, Mathematics
- 2014

It is shown that for every one-dimensional lattice system, or for systems in lattices of any dimension at temperatures above a certain threshold, the Gibbs samplers of commuting Hamiltonians are always gapped, giving an efficient way of preparing the associated Gibbs states on a quantum computer.

### Lower bounds to the spectral gap of Davies generators

- Mathematics
- 2013

We construct lower bounds to the spectral gap of a family of Lindblad generators known as Davies maps. These maps describe the thermalization of quantum systems weakly coupled to a heat bath. The…

### Hypercontractivity of quasi-free quantum semigroups

- Mathematics
- 2014

Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic…

### Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing.

- PhysicsPhysical review letters
- 2021

The relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect is investigated, and it is shown that the longest relaxation time τ that is maximized over initial states and local observables is given by τ∼Δ^{-1}(1+L/ξ) with L being the system size.

### Relaxation times of dissipative many-body quantum systems.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2015

We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system…

## References

SHOWING 1-10 OF 55 REFERENCES

### Spectral Gap and Exponential Decay of Correlations

- Physics, Mathematics
- 2005

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide…

### Exponential Decay of Correlations Implies Area Law

- Computer Science
- 2015

It is proved that a finite correlation length implies an area law for the entanglement entropy of quantum states defined on a line, and it is shown that 1D quantum states with exponential decay of correlations have an efficient classical approximate description as a matrix product state of polynomial bond dimension.

### Mixing in time and space for lattice spin systems: A combinatorial view

- MathematicsRANDOM
- 2002

A sharp equivalence is proved between exponential decay with distance of spin correlations and “super-fast” mixing time of the Glauber dynamics of the Markov chain Monte Carlo algorithm.

### Stability of Local Quantum Dissipative Systems

- Mathematics
- 2015

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 χ2-divergence and mixing times of quantum Markov processes

- Mathematics, Physics
- 2010

We introduce quantum versions of the χ2-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach…

### Topological quantum order: Stability under local perturbations

- Mathematics
- 2010

We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of…

### Area laws for the entanglement entropy - a review

- Physics
- 2010

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay…

### Quantum logarithmic Sobolev inequalities and rapid mixing

- Mathematics
- 2013

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative Lp-spaces is reviewed and the relationship between quantum…

### An area law and sub-exponential algorithm for 1D systems

- Computer Science, Mathematics
- 2013

A new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result, and establishes a new, “random-walk like”, bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian.

### Topology by dissipation

- Physics
- 2013

Topological states of fermionic matter can be induced by means of a suitably engineered dissipative dynamics. Dissipation then does not occur as a perturbation, but rather as the main resource for…