  In a recent work [A. Aloy et al. arXiv:1807:06027 (2018)] we have considered
the characterization of entanglement depth, from a device-independent
perspective, in a quantum many-body system. We have shown that the inequalities
introduced in [J. Tura et al. Science 344 1256 (2014)] can be used to obtain
device-independent witnesses of entanglement depth and that they enjoy two key
properties that allow to compute their $k$-producibility bounds more
efficiently for larger system sizes, as well as yielding
experimentally-friendlier device-independent witnesses of entanglement depth:
they involve at most two-body correlators and they are permutationally
invariant. While the main aim of our previous work was to illustrate the main
ideas and applicability of the method, here we outline the details and
complement its findings with detailed analysis and further case studies.
Specifically, we consider the problem of finding the $k$-producible bounds of
such DIWEDs under different assumptions. Not surprisingly, with the weakest
assumptions, we can compute $k$-producible bounds only for relatively small
number of parties; however we can still learn interesting features from these
solutions that motivate the search on larger systems under the assumption that
these features persist. This allows us to tackle the case where the system size
eventually reaches the thermodynamic limit.
