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arXiv:2404.14067 · Physical review. B./Physical review. B

Quantum master equation for many-body systems based on the Lieb-Robinson bound

Koki Shiraishi, Masaya Nakagawa, Takashi Mori, Masahito Ueda

Published May 12, 2025• 11 citations
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Abstract

The authors derive here a local quantum master equation for many-body open quantum systems on the basis of the Lieb-Robinson bound, which places a fundamental upper bound on the propagation speed of correlations of observables. This approach clarifies the conditions under which local quantum master equations are valid and provides a microscopic justification for their use in strongly interacting systems. Numerical simulations on a one-dimensional fermionic chain demonstrate the accuracy of the derived equation.

DOI
10.1103/physrevb.111.184311

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The authors derive here a local quantum master equation for many-body open quantum systems on the basis of the Lieb-Robinson bound, which places a fundamental upper bound on the propagation speed of correlations of observables. This approach clarifies the conditions under which local quantum master equations are valid and provides a microscopic justification for their use in strongly interacting systems. Numerical simulations on a one-dimensional fermionic chain demonstrate the accuracy of the derived equation.
The Lieb-Robinson bound (LRB) states that the range and strength of interactions between the constituents of a complex many-body system impose upper limits to how fast the signal can propagate. It manifests in a light conelike growth of correlation function connecting two distant subsystems. Here, we employ the techniques of quantum information to demonstrate that the LRB can be determined from local measurements performed on a single qubit that is connected to a many-body system. This formulation provides an operational recipe for estimating the LRB in complex systems, replacing the measurement of the correlation function with simple single-particle manipulations. We demonstrate the potency of this approach by deriving the upper limit to the speed of signal propagation in the XY spin chain.
The dynamics of quantum systems strongly depends on the local structure of the Hamiltonian. For short-range interacting systems, the well-known Lieb–Robinson bound defines the effective light cone with an exponentially small error with respect to the spatial distance, whereas we can obtain only polynomially small errors for distance in long-range interacting systems. In this paper, we derive a qualitatively new bound for quantum dynamics by considering how many spins can correlate with each other after time evolution. Our bound characterizes the number of spins which support the many-body entanglement with exponentially small errors and is valid for a large class of Hamiltonians including long-range interacting systems. To demonstrate the advantage of our approach in quantum many-body systems, we apply our bound to prove several fundamental properties which have not been derived from the Lieb–Robinson bound.
We consider open many-body systems governed by a time-dependent quantum master equation with short-range interactions. With a generalized Lieb-Robinson bound, we show that the evolution in this very generic framework is quasilocal; i.e., the evolution of observables can be approximated by implementing the dynamics only in a vicinity of the observables' support. The precision increases exponentially with the diameter of the considered subsystem. Hence, time evolution can be simulated on classical computers with a cost that is independent of the system size. Providing error bounds for Trotter decompositions, we conclude that the simulation on a quantum computer is additionally efficient in time. For experiments and simulations in the Schrödinger picture, our result can be used to rigorously bound finite-size effects.
This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
The Lieb-Robinson correlation function captures propagation of quantum correlations in a many-body system. We calculate the value of the leading order of the correlation function, not its bound, for a system of interacting qubits at early times. The general analytical result is compared to numerical calculations and is applied to regular qubit lattices in one, two, and three dimensions. The Lieb-Robinson velocity and the approximately exponential leading edge of correlations emerge in the limit of large arrays.
We study equilibration of an isolated quantum system by mapping it onto a network of classical oscillators in Hilbert space. By choosing a suitable basis for this mapping, the degree of locality of the quantum system reflects in the sparseness of the network. We derive a Lieb-Robinson bound on the speed of propagation across the classical network, which allows us to estimate the timescale at which the quantum system equilibrates. The bound contains a parameter that quantifies the degree of locality of the Hamiltonian and the observable. Locality was disregarded in earlier studies of equilibration times, and it is believed to be a key ingredient for making contact with the majority of physically realistic models. The more local the Hamiltonian and observables, the longer the equilibration timescale predicted by the bound.