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Annales Henri Poincaré

Canonical Typicality for Other Ensembles than Micro-canonical

Stefan Teufel, Roderich Tumulka, Cornelia Vogel

Published Jul 2, 2024• 7 citations
Publisher

Abstract

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 $$\rho $$ ρ on a separable Hilbert space $${\mathcal {H}}$$ H , $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) is the most spread-out probability measure on the unit sphere of $${\mathcal {H}}$$ H that has density matrix $$\rho $$ ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $$\Vert \rho \Vert $$ ‖ ρ ‖ of $$\rho $$ 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 $$\psi $$ ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $$\psi $$ ψ -independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states $$\psi $$ ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $$\psi _t$$ ψ t is very close to a $$\psi $$ 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 $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) , provided the density matrix $$\rho $$ 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.

DOI
10.1007/s00023-024-01466-7

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One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic ($N \rightarrow \infty$) limit the density matrix emerges for small subsystems from almost all pure states of large quantum many-body systems. This notion of canonical is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. We show that quantum chaos plays a crucial role in the emergence of typicality for large but finite quantum systems. We demonstrate that the degree of typicality can be quantitatively controlled and tuned by the degree of quantum chaoticity present in the many-body system.

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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\rho ρ ρ on a separable Hilbert space H{\mathcal {H}}H H , GAP(ρ){\textrm{GAP}}(\rho )GAP(ρ) GAP ( ρ ) is the most spread-out probability measure on the unit sphere of H{\mathcal {H}}H H that has density matrix ρ\rho ρ ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue ∥ρ∥\Vert \rho \Vert ∥ρ∥ ‖ ρ ‖ of ρ\rho 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\psi ψ ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a ψ\psi ψ ψ -independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states ψ\psi ψ ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state ψt\psi _tψt​ ψ t is very close to a ψ\psi 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){\textrm{GAP}}(\rho )GAP(ρ) GAP ( ρ ) , provided the density matrix ρ\rho 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.
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 study how the grand-canonical density matrix arises in macroscopic quantum systems. ``Canonical typicality'' is the known statement that for a typical wave function ΨΨΨ from a micro-canonical energy shell of a quantum system SSS weakly coupled to a large but finite quantum system BBB, the reduced density matrix ρ^ΨS=trB∣Ψ⟩⟨Ψ∣\hatρ^S_Ψ=\mathrm{tr}^B |Ψ\rangle\langle Ψ|ρ^​ΨS​=trB∣Ψ⟩⟨Ψ∣ is approximately equal to the canonical density matrix ρ^can=Zcan−1exp⁡(−βH^S)\hatρ_\mathrm{can}=Z^{-1}_\mathrm{can} \exp(-β\hat{H}^S)ρ^​can​=Zcan−1​exp(−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))\hatρ_\mathrm{gc}=Z^{-1}_\mathrm{gc} \exp(-β(\hat{H}^S-μ_1 \hat{N}_{1}^S-\ldots-μ_r\hat{N}_{r}^S))ρ^​gc​=Zgc−1​exp(−β(H^S−μ1​N^1S​−…−μr​N^rS​)) with N^iS\hat{N}_{i}^SN^iS​ the number operator for molecules of type iii in the system SSS. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems SSS defined by a spatial region which particles may enter or leave. It includes statements about how ρ^gc\hatρ_\mathrm{gc}ρ^​gc​ arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace Hgmc⊂HS⊗HB\mathscr{H}_\mathrm{gmc} \subset \mathscr{H}^S \otimes \mathscr{H}^BHgmc​⊂HS⊗HB (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical ΨΨΨ in Hgmc\mathscr{H}_\mathrm{gmc}Hgmc​, as well as statements about the distribution of the (conditional) wave function ψSψ^SψS of SSS, 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.
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).
It is well known that a system weakly coupled to a heat bath is described by the canonical ensemble when the composite S + B is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schrödinger in 1952.
We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is “normal”: it evolves in such a way that |ψt〉〈ψt | is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞N \rightarrow \inftyN→∞) limit the density matrix emerges for small subsystems from almost all pure states of large quantum many-body systems. This notion of canonical is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. We show that quantum chaos plays a crucial role in the emergence of typicality for large but finite quantum systems. We demonstrate that the degree of typicality can be quantitatively controlled and tuned by the degree of quantum chaoticity present in the many-body system.