13 problems

Let $\rho_\beta$ be a reduced density matrix on a subsystem, and consider the cluster-expansion method applied to the logarithm of reduced density matrices in order to compute cond…

Let $\rho_\beta$ be an arbitrary quantum Gibbs state on a system partitioned as $\Lambda=A\sqcup B\sqcup C$, and let $d_{A,C}$ denote the distance between $A$ and $C$. Write…

Eigenstate Thermalization Hypothesis. The local statistics of the off-diagonal matrix elements satisfy

Eigenstate Thermalization Hypothesis. The diagonal expectation values satisfy

Let $\rho$ be a Gibbs state of a short-ranged Hamiltonian defined on a $D$-dimensional spin lattice. Let $A$, $B$, and $C$ be three regions, with $B$ shielding $A$ from $C$, and le…

Let $A$ be a few-body observable, and let $|E_k\rangle$ be an eigenvector of the Hamiltonian of a large interacting many-body system with energy $E_k$. Eigenstate thermalisation hy…

Consider a many-particle system placed in a heat bath and represent its equilibrium state as a Hilbert-space vector. The authors assume that the many-particle system and its dynami…

Let $\Gamma$ be the finite graph under consideration, let $\mathcal M_{\Gamma}$ be its parameter space, and let…

Let $C$ and $u$ be the parameters appearing in relation (6.24'), and let $Ls$ denote the function used there. For $n=1$ and $u<0$, assume … Relation (6.24') is the inequality est…

Let $Z_q(\beta,h)=\sum_{n=1}^{\infty}e^{-\beta E_n(h)}$ be the quantum statistical sum for the Schrödinger system, where $h>0$ is Planck's constant, $\beta>0$, and $N$ is the dimen…

Let $Z_q(\beta,h)$ and $Z_c(\beta)$ be the quantum and classical statistical sums, and let $E_q(\beta,h)$ and $E_c(\beta)$ be the corresponding quantum and classical mean energies.…

Quantum-classical mean-energy inequality. The inequality

Let $Z_q(\beta,h)=\sum_{n=1}^{\infty}e^{-\beta E_n(h)}$ be the quantum statistical sum, where $E_n(h)$ are the energy eigenvalues, and let … be the classical statistical sum for th…