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 [34], 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 biseparable 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 the 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.

物理变换由完全正向且保留迹线 (CPTP) 的线性映射描述。然而，正 (P) 但不完全正 (CP) 的图有助于推导可分离性/纠缠标准。此外，此类地图的属性可以与它们检测到的状态的纠缠属性相关联。在这里，我们扩展了 [34] 中提出的结果，其中导出了二分系统的充分可分离性标准。特别是，我们分析了一个纠缠深度$否$-qubit 系统通过提出线性映射，当应用于任何状态时，会导致双分离状态$\mathrm{1个}:(否-\mathrm{1个})$分区，即$(否-\mathrm{1个})$-纠缠深度。此外，我们推导出标准来检测任意$(否-n)$- 纠缠深度适合完全去极化状态（归一化单位矩阵）附近的状态。我们还提供可分离性（或$\mathrm{1个}$-纠缠深度）对称扇区中的条件，包括对角线状态。最后，我们建议如何使用相似的映射技术来导出一组期望值与可分离状态或局部隐藏变量理论兼容的充分条件。我们将本文献给已故的 Andrzej Kossakowski，他是我们在线性地图领域的精神和知识导师。