Flood of multipartite Rains entanglement

Quantifying how much useful entanglement can be extracted from quantum states is a central task in quantum information theory. In bipartite systems, the Rains relative entropy provides the tightest efficiently computable upper bound on distillable entanglement via semidefinite programming. However, generalizing bipartite entanglement measures to multipartite systems is non-trivial because multipartite entanglement exhibits unique phenomena such as the activation of genuine multipartite entanglement (GME) and inequivalent classes of entanglement (such as GHZ and W states). This paper addresses the lack of unique multipartite generalizations by introducing a family of measures based on the Rains relative entropy.

The authors define four multipartite entanglement measures—the Rains, monsoon, hurricane, and squall entanglement—as generalizations of the bipartite Rains relative entropy using generalized divergences. They establish an explicit analytical ordering among them, proving that the Rains entanglement is the smallest. To analyze their operational significance, the paper investigates their behavior under local operations and classical communication (LOCC) and positive partial transpose (PPT) operations. Additionally, the authors formulate the multipartite max-Rains entanglement as a semidefinite program (SDP), derive its dual program, and provide analytical expressions for the Rains entanglement of GHZ and W states.

The paper proves that the squall and hurricane entanglement are tensor-product and tensor-power subadditive, respectively, while the monsoon, hurricane, and squall entanglement are monotone under PPT operations. Furthermore, the Rains entanglement satisfies selective PPT monotonicity, making it useful for bounding distillation rates. By studying pure-state entanglement conversion, the authors establish single-letter upper bounds on the one-shot and asymptotic rates at which an arbitrary state can be converted to a fixed pure state. The tightest one-shot bound is given by the Rains entanglement, while asymptotic distillation rates are bounded using the hurricane and squall entanglement, leading directly to bounds on GHZ- and W-distillable entanglement.