Lieb–Robinson bounds for open quantum systems with long-ranged interactions

Ryan Sweke, Jens Eisert, Michael Kästner

Published Jan 1, 2019• 17 citations

Abstract

We state and prove four types of Lieb–Robinson bounds valid for many-body open quantum systems with power law decaying interactions undergoing out of equilibrium dynamics. We also provide an introductory and self-contained discussion of the setting and tools necessary to prove these results. The results found here apply to physical systems in which both long-ranged interactions and dissipation are present, as commonly encountered in certain quantum simulators, such as Rydberg systems or Coulomb crystals formed by ions.

DOI: 10.17169/refubium-26857

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Lieb–Robinson bounds for open quantum systems with long-ranged interactions

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Published Jan 1, 2019• 17 citations

Publisher

Abstract

DOI: 10.17169/refubium-26857

Related papers

Lieb–Robinson bounds for open quantum systems with long-ranged interactions

Ryan Sweke, Jens Eisert, Michael Kästner

Abstract We state and prove four types of Lieb–Robinson bounds valid for many-body open quantum systems with power law decaying interactions undergoing out of equilibrium dynamics. We also provide an introductory and self-contained discussion of the setting and tools necessary to prove these results. The results found here apply to physical systems in which both long-ranged interactions and dissipation are present, as commonly encountered in certain quantum simulators, such as Rydberg systems or Coulomb crystals formed by ions.Abstract We state and prove four types of Lieb–Robinson bounds valid for many-body open quantum systems with power law decaying interactions undergoing out of equilibrium dynamics. We also provide an introductory and self-contained discussion of the setting and tools necessary to prove these results. The results found here apply to physical systems in which both long-ranged interactions and dissipation are present, as commonly encountered in certain quantum simulators, such as Rydberg systems or Coulomb crystals formed by ions.

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Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In $D$ spatial dimensions, interactions decaying as a power-law $r^{-α}$ with $α> 2 D+1$ exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold $2D +1$ is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form $t \, r^{-α}$ for any $α>0$ , and this scaling is sharp. In particular, the linear light cone occurs at $α= 1$ in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In $D$ spatial dimensions, interactions decaying as a power-law $r^{-α}$ with $α> 2 D+1$ exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold $2D +1$ is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form $t \, r^{-α}$ for any $α>0$, and this scaling is sharp. In particular, the linear light cone occurs at $α= 1$ in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.

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Andrew Y. Guo, Minh C. Tran, Andrew M. Childs +2 more

Strongly long-range interacting quantum systems---those with interactions decaying as a power law $1/{r}^{\ensuremath{\alpha}}$ in the distance $r$ on a $D$ -dimensional lattice for $\ensuremath{\alpha}\ensuremath{\le}D$---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum-information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. In a step towards rectifying this problem, we prove two Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for $\ensuremath{\alpha}\ensuremath{\le}D/2$. Our second bound pertains to generic long-range interacting spin Hamiltonians and gives a tight lower bound for the signaling time to extensive subsets of the system for all $\ensuremath{\alpha}<D$. This many-site signaling time lower bounds the scrambling time in strongly long-range interacting systems.Strongly long-range interacting quantum systems---those with interactions decaying as a power law $1/{r}^{\ensuremath{\alpha}}$ in the distance $r$ on a $D$-dimensional lattice for $\ensuremath{\alpha}\ensuremath{\le}D$---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum-information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. In a step towards rectifying this problem, we prove two Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for $\ensuremath{\alpha}\ensuremath{\le}D/2$. Our second bound pertains to generic long-range interacting spin Hamiltonians and gives a tight lower bound for the signaling time to extensive subsets of the system for all $\ensuremath{\alpha}<D$. This many-site signaling time lower bounds the scrambling time in strongly long-range interacting systems.

arXiv:1906.02662Physical Review AJul 8, 202049 citations

In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary

k

-body interactions, so long as they decay with a power law greater than

kd

, where

d

is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to

R

, acting on any one site, decays as a power law $1/{R}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>d$. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to $1/{r}^{\ensuremath{\alpha}}$. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.

Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In

D

spatial dimensions, interactions decaying as a power-law

r^{-α}

with

α> 2 D+1

exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold

2D +1

is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form

t \, r^{-α}

for any

α>0

, and this scaling is sharp. In particular, the linear light cone occurs at

α= 1

in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.

Strongly long-range interacting quantum systems---those with interactions decaying as a power law $1/{r}^{\ensuremath{\alpha}}$ in the distance

r

on a

D

-dimensional lattice for $\ensuremath{\alpha}\ensuremath{\le}D$---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum-information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. In a step towards rectifying this problem, we prove two Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for $\ensuremath{\alpha}\ensuremath{\le}D/2$. Our second bound pertains to generic long-range interacting spin Hamiltonians and gives a tight lower bound for the signaling time to extensive subsets of the system for all $\ensuremath{\alpha}<D$. This many-site signaling time lower bounds the scrambling time in strongly long-range interacting systems.