We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).

arXiv:2401.15263Physical review. EMay 30, 20255 citations

Quantum theory of irreversibility

A. Pérez-Madrid

arXiv:cond-mat/0510433Physica A Statistical Mechanics and its ApplicationsDec 26, 20063 citations

Lieb-Robinson bounds for irreversible dynamics via adaptive Dyson expansion

Carla Rubiliani, Marius Wesle, Tom Wessel

In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form $[A,τ_{s,t} \, B]$ in terms of self-avoiding paths of sets, where $A$ and $B$ are local operators and $τ_{s,t}$ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form $[A,τ_{s,t} \, B]$ in terms of self-avoiding paths of sets, where $A$ and $B$ are local operators and $τ_{s,t}$ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.

math-phCross-listedarXiv:2610.12731Oct 12, 20260 citations

New quantum rules for dissipative systems

Christophe George, Françoise Hénin, F. Mayné +1 more

It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« lessIt is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less

Hadronic J.; (United States)Jun 1, 19784 citations

[A,τ_{s,t} \, B]

in terms of self-avoiding paths of sets, where

A

and

B

are local operators and

τ_{s,t}

is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.