We consider an isolated macroscopic quantum system in a pure state ψt evolving unitarily in a separable Hilbert space H and take for granted that different macro states ν correspond to mutually orthogonal subspaces Hν⊂H. Let Pν be the projection to Hν. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most ψ0∈Hμ are such that for most t ≥ 0, ‖Pνψt‖2 is close to a t-and ψ0-independent value Mμν provided that Mμν is not too small. Here, “most” refers to the uniform distribution on the sphere S(Hμ). In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix ρ on H, GAP(ρ) is the most spread out distribution on S(H) with density matrix ρ. We show that also for GAP(ρ)-most ψ0∈H for most t ≥ 0, ‖Pνψt‖2 is close to a fixed value MρPν (which must not be too small). Moreover, we prove a generalization for certain operators B instead of Pν and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.
We consider an isolated macroscopic quantum system in a pure state ψt evolving unitarily in a separable Hilbert space H and take for granted that different macro states ν correspond to mutually orthogonal subspaces Hν⊂H. Let Pν be the projection to Hν. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most ψ0∈Hμ are such that for most t ≥ 0, ‖Pνψt‖2 is close to a t-and ψ0-independent value Mμν provided that Mμν is not too small. Here, “most” refers to the uniform distribution on the sphere S(Hμ). In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix ρ on H, GAP(ρ) is the most spread out distribution on S(H) with density matrix ρ. We show that also for GAP(ρ)-most ψ0∈H for most t ≥ 0, ‖Pνψt‖2 is close to a fixed value MρPν (which must not be too small). Moreover, we prove a generalization for certain operators B instead of Pν and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.
Abstract We generalize Lévy’s lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix ρ ρ on a separable Hilbert space H H , GAP(ρ) GAP ( ρ ) is the most spread-out probability measure on the unit sphere of H H that has density matrix ρ ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue ∥ρ∥ ‖ ρ ‖ of ρ ρ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for “most” pure states ψ ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a ψ ψ -independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states ψ ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state ψt ψ t is very close to a ψ ψ -independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for GAP(ρ) GAP ( ρ ) , provided the density matrix ρ ρ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.
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).
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.
We study how the grand-canonical density matrix arises in macroscopic quantum systems. ``Canonical typicality'' is the known statement that for a typical wave function Ψ from a micro-canonical energy shell of a quantum system S weakly coupled to a large but finite quantum system B, the reduced density matrix ρ^ΨS=trB∣Ψ⟩⟨Ψ∣ is approximately equal to the canonical density matrix ρ^can=Zcan−1exp(−βH^S). Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix ρ^gc=Zgc−1exp(−β(H^S−μ1N^1S−…−μrN^rS)) with N^iS the number operator for molecules of type i in the system S. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems S defined by a spatial region which particles may enter or leave. It includes statements about how ρ^gc arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace Hgmc⊂HS⊗HB (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical Ψ in Hgmc, as well as statements about the distribution of the (conditional) wave function ψS of S, which turns out to be a so-called GAP or Scrooge measure. That is, we discuss the foundation and justification of both the density matrix and the distribution of the wave function in the grand-canonical case. To this end (particularly for the chemical reactions), we also need to extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.
An open quantum system (i.e., one that interacts with its environment) is almost always entangled with its environment; it is therefore usually not attributed a wave function but only a reduced density matrix ρ. Nevertheless, there is a precise way of attributing to it a wave function ψ1, called its conditional wave function, which is a random wave function of the system whose probability distribution µ1 depends on the entangled wave function ψ ∈ H1 ⊗ H2 in the Hilbert space of system and environment together. We prove several universality (or typicality) results about µ1; they show that if the environment is sufficiently large then µ1 does not depend much on the details of ψ and is approximately given by one of the so-called GAP measures. Specifically, for most entangled states ψ with given reduced density matrix ρ1, µ1 is close to GAP (ρ1). We also show that, if the coupling between the system and the environment is weak, then for most entangled states ψ from a microcanonical subspace corresponding to energies in a narrow interval [E, E + δE] (and most bases of H2), µ1 is close to GAP (ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β = β(E). This provides the mathematical justification of the claim that GAP (ρβ) is the thermal equilibrium distribution of ψ1.
Abstract We say of an isolated macroscopic quantum system in a pure state ψ ψ that it is in macroscopic thermal equilibrium (MATE) if ψ ψ lies in or close to a suitable subspace Heq H eq of Hilbert space. It is known that every initial state ψ0 ψ 0 will eventually reach and stay there most of the time (“thermalize”) if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation HθfF H θ fF of the Hamiltonian H0fF H 0 fF of N≫1 N ≫ 1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of H0fF H 0 fF . Here, we first point out that also for degenerate Hamiltonians all ψ0 ψ 0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for H0fF H 0 fF . Inspired by the fact that there is one eigenbasis of H0fF H 0 fF for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given H0 H 0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of H0 H 0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, H=H0+λV H = H 0 + λ V with λ≪1 λ ≪ 1 , for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.