Linear maps as sufficient criteria for entanglement depth and compatibility in many-body systems

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.