  Ergodicity in quantum many-body systems is - despite its fundamental
importance - still an open problem. Many-body localization provides a general
framework for quantum ergodicity, and may therefore offer important insights.
However, the characterization of many-body localization through simple
observables is a difficult task. In this article, we introduce a measure for
distances in Hilbert space for spin-1/2 systems that can be interpreted as a
generalization of the Anderson localization length to the many-body Hilbert
space. We show that this many-body localization length is equivalent to a
simple local observable in real space, which can be measured in experiments of
superconducting qubits, polar molecules, Rydberg atoms, and trapped ions. Using
the many-body localization length and a necessary criterion for ergodicity that
it provides, we study many-body localization and quantum ergodicity in
power-law-interacting Ising models subject to disorder in the transverse field.
Based on the nonequilibrium dynamical renormalization group, numerically exact
diagonalization, and an analysis of the statistics of resonances we find a
many-body localized phase at infinite temperature for small power-law
exponents. Within the applicability of these methods, we find no indications of
a delocalization transition.
