Alternatives of entanglement depth and metrological entanglement criteria

We work out the general theory of one-parameter families of partial entanglement properties and the resulting entanglement depth-like quantities. Special cases of these are the depth of partitionability, the depth of producibility (or simply entanglement depth) and the depth of stretchability, which are based on one-parameter families of partial entanglement properties known earlier. We also construct some further physically meaningful properties, for instance the squareability, the toughness, the degree of freedom, and also several ones of entropic motivation. Metrological multipartite entanglement criteria with the quantum Fisher information fit naturally into this framework. Here we formulate these for the depth of squareability, which therefore turns out to be the natural choice, leading to stronger bounds than the usual entanglement depth. Namely, the quantum Fisher information turns out to provide a lower bound not only on the maximal size of entangled subsystems, but also on the average size of entangled subsystems for a random choice of elementary subsystems. We also formulate convex criteria for both cases, which are much stronger than the original ones. This means that the aforementioned bounds hold also for the average in every decomposition of the quantum state. We also argue for that one-parameter partial entanglement properties bearing entropic meaning are more suitable for the purpose of defining metrological bounds.