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arXiv:2210.10018 · Journal of Statistical Physics

Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

Stefan Teufel, Roderich Tumulka, Cornelia Vogel

Published Feb 22, 2023• 12 citations
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Abstract

Abstract We consider a macroscopic quantum system with unitarily evolving pure state $$\psi _t\in \mathcal {H}$$ ψ t ∈ H and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $$\mathcal {H}_\nu $$ H ν (macro spaces) of $$\mathcal {H}$$ H . Let $$P_\nu $$ P ν denote the projection to $$\mathcal {H}_\nu $$ H ν . We prove two facts about the evolution of the superposition weights $$\Vert P_\nu \psi _t\Vert ^2$$ ‖ P ν ψ t ‖ 2 : First, given any $$T>0$$ T > 0 , for most initial states $$\psi _0$$ ψ 0 from any particular macro space $$\mathcal {H}_\mu $$ H μ (possibly far from thermal equilibrium), the curve $$t\mapsto \Vert P_\nu \psi _t\Vert ^2$$ t ↦ ‖ P ν ψ t ‖ 2 is approximately the same (i.e., nearly independent of $$\psi _0$$ ψ 0 ) on the time interval [0, T]. And second, for most $$\psi _0$$ ψ 0 from $$\mathcal {H}_\mu $$ H μ and most $$t\in [0,\infty )$$ t ∈ [ 0 , ∞ ) , $$\Vert P_\nu \psi _t\Vert ^2$$ ‖ P ν ψ t ‖ 2 is close to a value $$M_{\mu \nu }$$ M μ ν that is independent of both t and $$\psi _0$$ 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.

DOI
10.1007/s10955-023-03074-x

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Abstract We consider a macroscopic quantum system with unitarily evolving pure state ψt∈H\psi _t\in \mathcal {H}ψt​∈H ψ t ∈ H and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces Hν\mathcal {H}_\nu Hν​ H ν (macro spaces) of H\mathcal {H}H H . Let PνP_\nu Pν​ P ν denote the projection to Hν\mathcal {H}_\nu Hν​ H ν . We prove two facts about the evolution of the superposition weights ∥Pνψt∥2\Vert P_\nu \psi _t\Vert ^2∥Pν​ψt​∥2 ‖ P ν ψ t ‖ 2 : First, given any T>0T>0T>0 T > 0 , for most initial states ψ0\psi _0ψ0​ ψ 0 from any particular macro space Hμ\mathcal {H}_\mu Hμ​ H μ (possibly far from thermal equilibrium), the curve t↦∥Pνψt∥2t\mapsto \Vert P_\nu \psi _t\Vert ^2t↦∥Pν​ψt​∥2 t ↦ ‖ P ν ψ t ‖ 2 is approximately the same (i.e., nearly independent of ψ0\psi _0ψ0​ ψ 0 ) on the time interval [0, T]. And second, for most ψ0\psi _0ψ0​ ψ 0 from Hμ\mathcal {H}_\mu Hμ​ H μ and most t∈[0,∞)t\in [0,\infty )t∈[0,∞) t ∈ [ 0 , ∞ ) , ∥Pνψt∥2\Vert P_\nu \psi _t\Vert ^2∥Pν​ψt​∥2 ‖ P ν ψ t ‖ 2 is close to a value MμνM_{\mu \nu }Mμν​ M μ ν that is independent of both t and ψ0\psi _0ψ0​ 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\psi _tψt​ ψ t evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces Hν{\mathcal {H}}_\nu 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\psi _tψt​ ψ t looks like macroscopically, specifically on how much of ψt\psi _tψt​ ψ t lies in each Hν{\mathcal {H}}_\nu Hν​ H ν . Previous bounds concerned the absolute error for typical ψ0\psi _0ψ0​ 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\psi _0ψ0​ 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).
We consider a closed macroscopic quantum system in a pure state ψψψ evolving unitarily with Hamiltonian HHH in a finite-dimensional Hilbert space H\mathscr{H}H. Following von Neumann, we assume a decomposition of H\mathscr{H}H into mutually orthogonal subspaces Hν\mathscr{H}_νHν​, where each ννν represents a macro state, and Hν\mathscr{H}_νHν​ contains the pure states compatible with ννν. We define the quantum Boltzmann entropy for a macro state ννν as SqB(ν)=kBlog⁡dim⁡HνS_\mathrm{qB}(ν) = k_{\mathrm{B}} \log \dim \mathscr{H}_νSqB​(ν)=kB​logdimHν​, where kBk_\mathrm{B}kB​ is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that SqBS_\mathrm{qB}SqB​ predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of HHH and Hν\mathscr{H}_νHν​.
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 say of an isolated macroscopic quantum system in a pure state ψ\psi ψ ψ that it is in macroscopic thermal equilibrium (MATE) if ψ\psi ψ ψ lies in or close to a suitable subspace Heq\mathcal {H}_\textrm{eq}Heq​ H eq of Hilbert space. It is known that every initial state ψ0\psi _0ψ0​ 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 HθfFH_\theta ^\textrm{fF}HθfF​ H θ fF of the Hamiltonian H0fFH_0^\textrm{fF}H0fF​ H 0 fF of N≫1N\gg 1N≫1 N ≫ 1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of H0fFH_0^\textrm{fF}H0fF​ H 0 fF . Here, we first point out that also for degenerate Hamiltonians all ψ0\psi _0ψ0​ ψ 0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for H0fFH_0^\textrm{fF}H0fF​ H 0 fF . Inspired by the fact that there is one eigenbasis of H0fFH_0^\textrm{fF}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 H0H_0H0​ H 0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of H0H_0H0​ H 0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, H=H0+λVH=H_0+\lambda VH=H0​+λV H = H 0 + λ V with λ≪1\lambda \ll 1λ≪1 λ ≪ 1 , for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.
We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time $τ__\mathrm{B}:=h/(k_\mathrm{B}T)$. This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in the present setting. The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. According the present result, what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly. Mathematically our result is based on an interesting property of the maximum eigenvalue of the Hadamard product of a positive semi-definite matrix and a random projection matrix. The recent exact formula by Collins for the integral with respect to the Haar measure of the unitary group plays an essential role in our proof.
Abstract Typicality is a well-established and very general property of quantum many-body systems, referring to the phenomenon that the expectation values of any given observable are practically indistinguishable for the overwhelming majority of all pure states (normalized vectors) in a sufficiently high-dimensional Hilbert (sub-)space. Here, we provide very simple and general arguments that analogous typicality properties of pure states (phase space points) in classical many-body systems are still expected to hold true for macroscopic observables, but not any more for microscopic (few-body) observables.