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Eric A. Carlen

Publications and source records attributed to Eric A. Carlen.

At least 19 recordsLinked to original sources
arXiv2026

The structure of gauge invariant Gaussian quantum operations on finite Fermion systems

Let ${\mathcal H}_1$ be a finite dimensional complex Hilbert space. Let $ψ\mapsto Z(ψ)$ be a canonical anti-commutation relations (CAR) field over ${\mathcal H}_1$ acting irreducibly on a Hilbert space ${\mathord{\mathscr K}}$. The $*$-algebra ${\mathscr A}_{{\mathcal H}_1}$ generated by the $Z(ψ)$, $ψ\in {\mathcal H}_1$, is simply all operators on ${\mathscr K}$. However, the CAR field endows ${\mathscr A}_{{\mathcal H}_1}$ with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' $ψ\mapsto e^{it}ψ$. The fixed point algebra of the gauge group, ${\mathscr G}_{{\mathcal H}_1}$, is a sub-algebra of ${\mathscr A}_{{\mathcal H}_1}$ studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, ${\mathfrak S}_{GIG}$. Our focus is on semigroups $\{e^{t{\mathscr L}}\}_{t\geq 0}$ of quantum operations on ${\mathscr A}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Each $e^{t{\mathscr L}}$ is one-to-one, and our first main result is a structure theorem for such quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. We apply this to study semigroups of quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs $(G,A)$ where $G$ is a contraction semigroup generator on ${\mathcal H}_1$, and $0 \leq A \leq -G -G^*$. We then show that each of these semigroups has a natural extension to the full CAR algebra ${\mathscr A}_{{\mathcal H}_1}$. Further results are obtained under further assumptions on the pair $(G,A)$.

math.FA↗
arXiv2026

Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior

We study composite open quantum systems with a finite-dimensional state space ${\mathcal H}_A\otimes {\mathcal H}_B$ governed by a Lindblad equation $ρ'(t) = {\mathcal L}_γρ(t)$ where ${\mathcal L}_γρ= -i[H,ρ] + γ{\mathcal D} ρ$, and ${\mathcal D}$ is a dissipator ${\mathcal D}_A\otimes I$ acting non-trivially only on part $A$ of the system, which can be thought of as the boundary, and $γ$ is a parameter. It is known that the dynamics simplifies for large $γ$: after a time of order $γ^{-1}$, $ρ(t)$ is well approximated for times small compared to $γ^2$ by $π_A\otimes R(t)$ where $π_A$ is a steady state of ${\mathcal D}_A$, and $R(t)$ is a solution of $\frac{\rm d}{{\rm d}t}R(t) = {\mathcal L}_{P,γ}R(t)$ where ${\mathcal L}_{P,γ} R := -i[H_P,R] + γ^{-1} {\mathcal D}_P R$ with $H_P$ being a Hamiltonian on ${\mathcal H}_B$ and ${\mathcal D}_P$ being a Lindblad generator over ${\mathcal H}_B$. We prove this assuming only that ${\mathcal D}_A$ is ergodic and gapped. In order to better control the long time behavior, and study the steady states $\barρ_γ$, we introduce a third Lindblad generator ${\mathcal D}_P^\sharp$ that does not involve $γ$, but still closely related to ${\mathcal L}_γ$. We show that if ${\mathcal D}_P^\sharp$ is ergodic and gapped, then so is ${\mathcal L}_γ$ for all large $γ$, and if $\barρ_γ$ denotes the unique steady state for ${\mathcal L}_γ$, then $\lim_{γ\to\infty}\barρ_γ= π_A\otimes \bar R$ where $\bar R$ is the unique steady state for ${\mathcal D}_P^\sharp$. We show that there is a convergent expansion $\barρ_γ= π_A\otimes\bar R +γ^{-1} \sum_{k=0}^\infty γ^{-k} \bar n_k$ where, defining $\bar n_{-1} := π_A\otimes\bar R$, ${\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$.

quant-ph↗
arXiv2026

The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system.

math.DS↗
arXiv2025

Spectral Gap for the Stochastic Exchange Model

We prove a spectral gap inequality for the stochastic exchange model studied by Gaspard and Gilbert and by Grigo, Khanin and Szász in connection with understanding heat conduction in a deterministic billiards model. The bound on the spectral gap that we prove is uniform in the number of particles, as had been conjectured. We adapt techniques that were originally developed to prove spectral gap bounds for the Kac model with hard sphere collisions, which, like the stochastic exchange model, has degenerate jump rates.

math.PR↗
arXiv2024

Stationary states of boundary driven quantum systems: some exact results

We study finite-dimensional open quantum systems whose density matrix evolves via a Lindbladian, $\dotρ=-i[H,ρ]+{\mathcal D}ρ$. Here $H$ is the Hamiltonian of the isolated system and ${\mathcal D}$ is the dissipator. We consider the case where the system consists of two parts, the "boundary'' $A$ and the ``bulk'' $B$, and ${\mathcal D}$ acts only on $A$, so ${\mathcal D}={\mathcal D}_A\otimes{\mathcal I}_B$, where ${\mathcal D}_A$ acts only on part $A$, while ${\mathcal I}_B$ is the identity superoperator on part $B$. Let ${\mathcal D}_A$ be ergodic, so ${\mathcal D}_A\hatρ_A=0$ only for one unique density matrix $\hatρ_A$. We show that any stationary density matrix $\barρ$ on the full system which commutes with $H$ must be of the product form $\barρ=\hatρ_A\otimesρ_B$ for some $ρ_B$. This rules out finding any ${\mathcal D}_A$ that has the Gibbs measure $ρ_β\sim e^{-βH}$ as a stationary state with $β\neq 0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\barρ$ for systems with interactions between $A$ and $B$. Related results for non-ergodic cases are also discussed.

quant-ph↗
arXiv2023

Connecting the Deep Quench Obstacle Problem with Surface Diffusion via their Steady States

In modeling phase transitions, it is useful to be able to connect diffuse interface descriptions of the dynamics with corresponding limiting sharp interface motions. In the case of the deep quench obstacle problem (DQOP) and surface diffusion (SD), while a formal connection was demonstrated many years ago, rigorous proof of the connection has yet to be established. In the present note, we show how information regarding the steady states for both these motions can provide insight into the dynamic connection, and we outline tools that should enable further progress. For simplicity, we take both motions to be defined on a planar disk.

math-ph↗
arXiv2023

Spectrum for some Quantum Markov semigroups describing $N$-particle systems evolving under a binary collision mechanism

We compute the spectrum for a class of quantum Markov semigroups describing systems of $N$ particle interacting through a binary collision mechanism. These quantum Markov semgroups are associated to a novel kind of quantum random walk on graphs, with the graph structure arising naturally in the quantization of the classical Kac model, and we show that the spectrum of the generator of the quantum Markov semigroup is closely related to the spectrum of the Laplacian on the corresponding graph. For the direct analog of the original classical Kac model, we determine the exact spectral gap for the quantum generator. We also give a new and simple method for studying the spectrum of certain graph Laplacians.

math.CO↗
arXiv2022

Monotonicity versions of Epstein's Concavity Theorem and related inequalities

Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.

math.FA↗
arXiv2022

Characterizing Schwarz maps by tracial inequalities

Let $ϕ$ be a linear map from the $n\times n$ matrices ${\mathcal M}_n$ to the $m\times m$ matrices ${\mathcal M}_m$. It is known that $ϕ$ is $2$-positive if and only if for all $K\in {\mathcal M}_n$ and all strictly positive $X\in {\mathcal M}_n$, $ϕ(K^*X^{-1}K) \geq ϕ(K)^*ϕ(X)^{-1}ϕ(K)$. This inequality is not generally true if $ϕ$ is merely a Schwarz map. We show that the corresponding tracial inequality ${\rm Tr}[ϕ(K^*X^{-1}K)] \geq {\rm Tr}[ϕ(K)^*ϕ(X)^{-1}ϕ(K)]$ holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.

math-ph↗
arXiv2022

A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality

The Golden-Thompson trace inequality which states that $Tr\, e^{H+K} \leq Tr\, e^H e^K$ has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here we make this G-T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, $H=Δ$ or $H= -\sqrt{-Δ+m}$ and $K=$ potential, $Tr\, e^{H+(1-u)K}e^{uK}$ is a monotone increasing function of the parameter $u$ for $0\leq u \leq 1$. Our proof utilizes an inequality of Ando, Hiai and Okubo (AHO): $Tr\, X^sY^tX^{1-s}Y^{1-t} \leq Tr\, XY$ for positive operators X,Y and for $\tfrac{1}{2} \leq s,\,t \leq 1 $ and $s+t \leq \tfrac{3}{2}$. The obvious conjecture that this inequality should hold up to $s+t\leq 1$, was proved false by Plevnik. We give a different proof of AHO and also give more counterexamples in the $\tfrac{3}{2}, 1$ range. More importantly we show that the inequality conjectured in AHO does indeed hold in this range if $X,Y$ have a certain positivity property -- one which does hold for quantum mechanical operators, thus enabling us to prove our G-T monotonicity theorem.

math-ph↗
arXiv2021

On a two-species cross attraction system in higher dimensions

We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d \geq 3$. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of free-energy solutions, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a set of critical values: there is a critical value of a parameter pair in the system of equations for which there is a global-in-time energy solution and there exist blowing-up free-energy solutions under a criticality condition is violated for the parameter pair.

math.AP↗
arXiv2021

Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction

In a recent paper we studied an equation (called the "simple equation") introduced by one of us in 1963 for an approximate correlation function associated to the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $ρ$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $ρ(e)$ for the density as a function of the energy $e$. We had conjectured that $ρ(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(ρ)$. We had also conjectured that $ρe(ρ)$ is convex as a function of $ρ$. We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities $ρ$. Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities.

math-ph↗
arXiv2021

On the convolution inequality $f \geq f\star f$

We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space where $f\star f$ denotes the convolution of $f$ with itself. We show that all such functions $f$ are non-negative, which is not the case for the same inequality in $L^p$ for any $1 < p \leqslant 2$, for which the convolution is defined. We also show that all integrable solutions $f$ satisfy $\int f(x){\rm d}x \leqslant \tfrac12$. Moreover, if $\int f(x){\rm d}x = \tfrac12$, then $f$ must decay fairly slowly: $\int |x| f(x){\rm d}x = \infty$, and this is sharp since for all $r< 1$, there are solutions with $\int f(x){\rm d}x = \tfrac12$ and $\int |x|^r f(x){\rm d}x <\infty$. However, if $\int f(x){\rm d}x = : a < \tfrac12$, the decay at infinity can be much more rapid: we show that for all $a<\tfrac12$, there are solutions such that for some $ε>0$, $\int e^{ε|x|}f(x){\rm d}x < \infty$.

math.FA↗