31 problems

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…

Convex-set-ordering conjecture. Mean entropy is decreasing under this operation, in the sense that

Uniform-temperature LPPL conjecture. For the systems considered, the LPPL bound actually holds uniformly in temperature.

Uniqueness conjecture. The state $\omega_{\mathbb{Z}^D}^\beta$ obtained in this way is the only thermodynamic-limit $\beta$-KMS state.

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…

Non-integrability conjecture. Any $\hat{V}$ that makes the total Hamiltonian $\hat{H}_L+\lambda\hat{V}$ non-integrable will lead to thermalization of the time-evolved state for suf…

Consider a quantum lattice system in spatial dimension $D$, with a local perturbation $V_{i_0}$ and the quasi-locality estimates described in the paper. In the general bound, the r…

Let $H^\ast_L$ be the entanglement Hamiltonian of a contiguous region $L$ in a quantum lattice system, and let $\tilde{H}^\ast_{L,R}$ be an approximation whose effective interactio…

Van Kampen's conjecture. The entropy of the Universe is not affected by measurements, in the sense that the mean entropy is conserved under measurement.

Hugenholtz's conjecture. Under the selection rule used in the weak-coupling expansion, only two-point correlations survive at higher orders in $lambda$.

Two-peak mixing conjecture. The average density matrix is

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

Eigenstate Thermalization Hypothesis. The diagonal expectation values satisfy

Consider models in the same universality class as interacting massless Dirac fermions, with density-density correlations having an interaction-dependent amplitude and mass-mass cor…

Non-commutative central limit theorem. The limit exists for all $omega$ in a full-measure set $tilde{\Omega}\subset\Omega$, is independent of $omega$ in this set, and uniquely defi…

For an MBL system with $N$ degrees of freedom, assign its many-body eigenstates bijectively to binary strings $|i_{b1_1}i_{b1_2}cdots i_{b1_N}\rangle$, with $i_b1=\pm1$, and let…

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…

The conserved quantities are those fixed by the kinetic equation, and the parameters are the parameters of the corresponding Fermi–Dirac state, including $\beta$ and the additional…

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 $H$ be the random-field Ising-chain Hamiltonian on $L$ sites, with eigenfunctions $\Psi$, and let $S^{(3)}_{L/2}$ be the Pauli matrix at the central site. At infinite temperatu…

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…

Reservoir-energy conjecture. If the range of reservoir energy goes from $- fty$ to $+ fty$, then it is not possible to satisfy the sufficient conditions derived for the existence o…

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…