  Physical transformations are described by linear maps that are completely
positive and trace preserving (CPTP). However, maps that are positive (P) but
not completely positive (CP) are instrumental to derive
separability/entanglement criteria. Moreover, the properties of such maps can
be linked to entanglement properties of the states they detect. Here, we extend
the results presented in [Phys. Rev A 93, 042335 (2016)], where sufficient
separability criteria for bipartite systems were derived. In particular, we
analyze the entanglement depth of an $N$-qubit system by proposing linear maps
that, when applied to any state, result in a bi-separable state for the
$1:(N-1)$ partitions, i.e., $(N-1)$-entanglement depth. Furthermore, we derive
criteria to detect arbitrary $(N-n)$-entanglement depth tailored to states in
close vicinity of the completely depolarized state (the normalized identity
matrix). We also provide separability (or $1$- entanglement depth) conditions
in the symmetric sector, including for diagonal states. Finally, we suggest how
similar map techniques can be used to derive sufficient conditions for a set of
expectation values to be compatible with separable states or
local-hidden-variable theories. We dedicate this paper to the memory of the
late Andrzej Kossakowski, our spiritual and intellectual mentor in the field of
linear maps.
