Dynamical typicality refers to the property that two pure states, which initially exhibit (almost) the same expectation value for some given observable A, are very likely to exhibit also very similar expectation values when evolving in time according to the pertinent Schrödinger equation. We unify and generalize a variety of previous findings of this type for sufficiently high-dimensional quantum mechanical model systems. Particular emphasis is put on the necessary and sufficient conditions, which the initial expectation value and the spectrum of A have to fulfill.
We show that typicality holds for a class of nonequilibrium systems, i.e., nonequilibrium steady states (NESSs): almost all the pure states properly sampled from a certain Hilbert space well represent a NESS and characterize its intrinsic thermal nature. We clarify the relevant Hilbert space from which the pure states are to be sampled, and construct practically all the typical pure NESSs. The scattering approach leads us to the natural extension of the typicality for equilibrium systems. Each pure NESS correctly yields the expectation values of observables given by the standard ensemble approach. It means that we can calculate the expectation values in a NESS with only a single pure NESS. We provide an explicit construction of the typical pure NESS for a model with two reservoirs, and see that it correctly reproduces the Landauer-type formula for the current flowing steadily between the reservoirs.
Characterizing the time evolution of generic quantum many-body systems is a fundamental challenge, as representing the exact state requires exponentially scaling computational resources. While hydrodynamics and statistical mechanics successfully simplify this task by predicting the expectation values of local observables, these macroscopic frameworks provide no information about nonlinear characteristics of the quantum state. In this work, we demonstrate that for systems exhibiting a timescale separation, with dynamics governed by the transport of conserved charges, this lost information can be systematically recovered. By constraining the maximum-entropy Scrooge ensemble solely by the system's slow modes, we accurately reconstruct complex, nonlinear quantum properties of the global time-evolved state, including the half-chain entanglement entropy and the participation entropy in the computational basis. Our results generalize the paradigm of canonical typicality, revealing that a hydrodynamically bottlenecked system behaves as a typical pure state within a dynamically restricted submanifold of the Hilbert space - a phenomenon we term global restricted typicality.
We provide a summary of both seminal and recent results on typical entanglement. By ‘typical’ values of entanglement, we refer here to values of entanglement quantifiers that (given a reasonable measure on the manifold of states) appear with arbitrarily high probability for quantum systems of sufficiently high dimensionality. We shall focus on pure states and work within the Haar measure framework for discrete quantum variables, where we report on results concerning the average von Neumann and linear entropies as well as arguments implying the typicality of such values in the asymptotic limit. We then proceed to discuss the generation of typical quantum states with random circuitry. Different phases of entanglement, and the connection between typical entanglement and thermodynamics are discussed. We also cover approaches to measures on the non-compact set of Gaussian states of continuous variable quantum systems.
Abstract We consider a macroscopic quantum system with unitarily evolving pure state
ψt∈H ψ t ∈ H and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces
Hν H ν (macro spaces) of
H H . Let
Pν P ν denote the projection to
Hν H ν . We prove two facts about the evolution of the superposition weights
∥Pνψt∥2 ‖ P ν ψ t ‖ 2 : First, given any
T>0 T > 0 , for most initial states
ψ0 ψ 0 from any particular macro space
Hμ H μ (possibly far from thermal equilibrium), the curve
t↦∥Pνψt∥2 t ↦ ‖ P ν ψ t ‖ 2 is approximately the same (i.e., nearly independent of
ψ0 ψ 0 ) on the time interval [0, T]. And second, for most
ψ0 ψ 0 from
Hμ H μ and most
t∈[0,∞) t ∈ [ 0 , ∞ ) ,
∥Pνψt∥2 ‖ P ν ψ t ‖ 2 is close to a value
Mμν M μ ν that is independent of both t and
ψ0 ψ 0 . The first is an instance of the phenomenon of dynamical typicality observed by Bartsch, Gemmer, and Reimann, and the second modifies, extends, and in a way simplifies the concept, introduced by von Neumann, now known as normal typicality.
Abstract We consider a closed macroscopic quantum system in a pure state
ψt ψ t evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces
Hν H ν (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of
ψt ψ t looks like macroscopically, specifically on how much of
ψt ψ t lies in each
Hν H ν . Previous bounds concerned the absolute error for typical
ψ0 ψ 0 and/or t and are valid for arbitrary Hamiltonians H; now, we provide bounds on the relative error, which means much tighter bounds, with probability close to 1 by modeling H as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of H are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of
ψ0 ψ 0 from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin (Geom Funct Anal 26:1716–1776, 2016).