arXiv subjects

Cornelia Vogel

Publications and source records attributed to Cornelia Vogel.

8 recordsLinked to original sources

arXiv2026

Invariant measures of randomized quantum trajectories

Quantum trajectories are Markov chains modeling quantum systems subjected to repeated indirect measurements. Their stationary regime depends on what observables are measured on the probes used to indirectly measure the system. In this article we explore the properties of quantum trajectories when the choice of probe observable is randomized. The randomization induces some regularization of the quantum trajectories. We show that non-singular randomization ensures that quantum trajectories purify and therefore accept a unique invariant probability measure. We furthermore study the regularity of that invariant measure. In that endeavour, we introduce a new notion of ergodicity for quantum channels, which we call multiplicative primitivity. It is a priory stronger than primitivity but weaker than positivity improving. Finally, we compute some invariant measures for canonical quantum channels and explore the limits of our assumptions with several examples.

math-ph↗

arXiv2026

Grand-Canonical 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 $\Psi$ 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 $\hat{\rho}^S_\Psi=\mathrm{tr}^B |\Psi\rangle\langle \Psi|$ is approximately equal to the canonical density matrix $\hat{\rho}_\mathrm{can}=Z^{-1}_\mathrm{can} \exp(-\beta \hat{H}^S)$. Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix $\hat{\rho}_\mathrm{gc}=Z^{-1}_\mathrm{gc} \exp(-\beta(\hat{H}^S-\mu_1 \hat{N}_{1}^S-\ldots-\mu_r\hat{N}_{r}^S))$ with $\hat{N}_{i}^S$ 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 $\hat{\rho}_\mathrm{gc}$ arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace $\mathscr{H}_\mathrm{gmc} \subset \mathscr{H}^S \otimes \mathscr{H}^B$ (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical $\Psi$ in $\mathscr{H}_\mathrm{gmc}$, as well as statements about the distribution of the (conditional) wave function $\psi^S$ 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.

quant-ph↗

arXiv2025

Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model

We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices [arXiv:2403.10359] and an intricate analysis of the deterministic approximation.

math-ph↗

arXiv2024

Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles

We consider an isolated macroscopic quantum system in a pure state $\psi_t$ evolving unitarily in a separable Hilbert space $\mathcal{H}$ and take for granted that different macro states $\nu$ correspond to mutually orthogonal subspaces $\mathcal{H}_\nu\subset\mathcal{H}$. Let $P_\nu$ be the projection to $\mathcal{H}_\nu$. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most $\psi_0\in\mathcal{H}_\mu$ are such that for most $t\geq 0$, $\|P_\nu\psi_t\|^2$ is close to a $t$- and $\psi_0$-independent value $M_{\mu\nu}$ provided that $M_{\mu\nu}$ is not too small. Here, ``most'' refers to the uniform distribution on the sphere $\mathbb{S}(\mathcal{H}_\mu)$. 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 $\rho$ on $\mathcal{H}$, $\mathrm{GAP}(\rho)$ is the most spread out distribution on $\mathbb{S}(\mathcal{H})$ with density matrix $\rho$. We show that also for $\mathrm{GAP}(\rho)$-most $\psi_0\in\mathcal{H}$ for most $t\geq 0$, $\|P_\nu\psi_t\|^2$ is close to a fixed value $M_{\rho P_\nu}$ (which must not be too small). Moreover, we prove a generalization for certain operators $B$ instead of $P_\nu$ 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.

math-ph↗

arXiv2024

Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

We say of an isolated macroscopic quantum system in a pure state $\psi$ that it is in macroscopic thermal equilibrium (MATE) if $\psi$ lies in or close to a suitable subspace $\mathcal{H}_{eq}$ of Hilbert space. It is known that every initial state $\psi_0$ 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 $H_\theta^{fF}$ of the Hamiltonian $H_0^{fF}$ of $N\gg 1$ free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $H_0^{fF}$. Here, we first point out that also for degenerate Hamiltonians all $\psi_0$ thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for $H_0^{fF}$. Inspired by the fact that there is one eigenbasis of $H_0^{fF}$ 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 $H_0$ that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of $H_0$ lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, $H=H_0+\lambda V$ with $\lambda\ll 1$, for most perturbations $V$ the perturbed Hamiltonian $H$ satisfies ETH and all states thermalize.

cond-mat.stat-mech↗

arXiv2023

Canonical Typicality For Other Ensembles Than Micro-Canonical

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

math-ph↗

arXiv2023

Typical Macroscopic Long-Time Behavior for Random Hamiltonians

We consider a closed macroscopic quantum system in a pure state $\psi_t$ evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces $\mathcal{H}_\nu$ (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of $\psi_t$ looks like macroscopically, specifically on how much of $\psi_t$ lies in each $\mathcal{H}_\nu$. Previous bounds concerned the \emph{absolute} error for typical $\psi_0$ and/or $t$ and are valid for arbitrary Hamiltonians $H$; now, we provide bounds on the \emph{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 $\psi_0$ 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.

math-ph↗

arXiv2023

Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

We consider a macroscopic quantum system with unitarily evolving pure state $ψ_t\in \mathcal{H}$ and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $\mathcal{H}_ν$ (macro spaces) of $\mathcal{H}$. Let $P_ν$ denote the projection to $\mathcal{H}_ν$. We prove two facts about the evolution of the superposition weights $\|P_νψ_t\|^2$: First, given any $T>0$, for most initial states $ψ_0$ from any particular macro space $\mathcal{H}_μ$ (possibly far from thermal equilibrium), the curve $t\mapsto \|P_νψ_t\|^2$ is approximately the same (i.e., nearly independent of $ψ_0$) on the time interval $[0,T]$. And second, for most $ψ_0$ from $\mathcal{H}_μ$ and most $t\in[0,\infty)$, $\|P_νψ_t\|^2$ is close to a value $M_{μν}$ that is independent of both $t$ and $ψ_0$. 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.

quant-ph↗