  We introduce a method for extracting meaningful entanglement measures of
tensor network states in general dimensions. Current methods require the
explicit reconstruction of the density matrix, which is highly demanding, or
the contraction of replicas, which requires an effort exponential in the number
of replicas and which is costly in terms of memory. In contrast, our method
requires the stochastic sampling of matrix elements of the classically
represented reduced states with respect to random states drawn from simple
product probability measures constituting frames. Even though not corresponding
to physical operations, such matrix elements are straightforward to calculate
for tensor network states, and their moments provide the R\'enyi entropies and
negativities as well as their symmetry-resolved components. We test our method
on the one-dimensional critical XX chain and the two-dimensional toric code in
a checkerboard geometry. Although the cost is exponential in the subsystem
size, it is sufficiently moderate so that - in contrast with other approaches -
accurate results can be obtained on a personal computer for relatively large
subsystem sizes.
