On an inequality of Lin, Kim and Hsieh and Strong Subadditivity
We give an elementary proof of an inequality of Lin, Kim and Hsieh that implies strong subadditivity of the non Neumann entropy.
quant-ph↗
arXiv subjects
Michael P. Loss
Publications and source records attributed to Michael P. Loss.
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↗
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↗
Spectral Gaps for Reversible Markov Processes with Chaotic Invariant Measures: The Kac Process with Hard Sphere Collisions in Three Dimensions
We develop a method for producing estimates on the spectral gaps of reversible Markov jump processes with chaotic invariant measures, and we apply it to prove the Kac conjecture for hard sphere collision in three dimensions.
math-ph↗
Chaos, Ergodicity and Equilibria in a Quantum Kac Model
We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the $N$ particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.
math-ph↗