  We study the properties of mixed states obtained from eigenstates of
many-body lattice Hamiltonians after tracing out part of the lattice. Two
scenarios emerge for generic systems: (i) the diagonal entropy becomes
equivalent to the thermodynamic entropy when a few sites are traced out (weak
typicality); and (ii) the von Neumann (entanglement) entropy becomes equivalent
to the thermodynamic entropy when a large fraction of the lattice is traced out
(strong typicality). Remarkably, the results for few-body observables obtained
with the reduced, diagonal, and canonical density matrices are very similar to
each other, no matter which fraction of the lattice is traced out. Hence, for
all physical quantities studied here, the results in the diagonal ensemble
match the thermal predictions.
