Unconditional robustness of multipartite entanglement of superposition

The article delves into the intricate world of quantum entanglement, particularly focusing on genuine multipartite entanglement (GME) and its behavior under superpositions with product states. It begins by establishing the foundational role of the superposition principle in quantum entanglement and introduces the concept of unconditional robustness in entanglement, where certain entangled states remain entangled regardless of the superposition coefficients or the choice of product states. The study then extends this concept to multipartite systems, which lack universal Schmidt decompositions, making the problem more complex. The authors introduce the minimal and maximal Schmidt ranks of the first order for multipartite pure states, providing criteria to identify genuinely multipartite entangled states that are unconditionally robust in GME and inseparability. The article also defines the Schmidt rank of the second order to clarify whether a superposition of a genuinely multipartite entangled pure state and a set of fully product states results in a triple separable state. Through a series of detailed examples and theorems, the authors illustrate the conditions under which multipartite entanglement persists or vanishes under superpositions, offering insights into the minimal number of superposed product states required to achieve separability. The conclusions highlight the resistance of GME to mixtures with separable states and suggest potential generalizations for future research.