Alternatives of entanglement depth and metrological entanglement criteria

TL;DR

This paper develops a comprehensive framework for various entanglement depth measures and introduces stronger metrological criteria based on quantum Fisher information, enhancing the understanding of multipartite entanglement.

Contribution

It generalizes the theory of partial entanglement properties, introduces new physically meaningful properties, and formulates improved metrological bounds using quantum Fisher information.

Findings

Quantum Fisher information provides lower bounds on average entangled subsystem size.

Stronger entanglement criteria are formulated using convex quantities.

The framework includes new properties like squareability, toughness, and entropic measures.

Abstract

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 criteria with convex quantities for both cases, which are much stronger than the original ones. In particular, the quantum Fisher information puts a lower bound on the average size of entangled subsystems. We also argue that one-parameter partial entanglement properties, which carry entropic meaning, are more suitable for the purpose of defining metrological bounds.