  The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap
between quantum mechanical and statistical mechanical descriptions of isolated
quantum systems. Here, we define unbiased measures for how well the ETH works
in various regimes, by mapping general interacting quantum systems on regular
lattices onto a single particle living on a high-dimensional graph. By
numerically analyzing deviations from ETH behavior in the non-integrable Ising
model, we propose a quantity that we call the $n$-$weight$ to democratically
characterize the average deviations for all operators residing on a given
number of sites, irrespective of their spatial structure. It appears to have a
simple scaling form, that we conjecture to hold true for all non-integrable
systems. A closely related quantity, that we term the $n$-$distinguishability$,
tells us how well two states can be distinguished if only $n$-site operators
are measured. Along the way, we discover that complicated operators on average
are worse than simple ones at distinguishing between neighboring eigenstates,
contrary to the naive intuition created by the usual statements of the ETH that
few-body (many-body) operators acquire the same (different) expectation values
in nearby eigenstates at finite energy density. Finally, we sketch heuristic
arguments that the ETH originates from the limited ability of simple operators
to distinguish between quantum states of a system, especially when the states
are subject to constraints such as roughly fixed energy with respect to a local
Hamiltonian.
