Comparability of multipartite entanglement

Due to its numerous important applications[1–5] in the area of quantum information and computation, entanglement has been extensively studied in the past decade. The concept of considering entanglement as a new type of useful resource has been widely accepted. Efforts have been made to quantitatively and qualitatively measure this kind of resource in both bipartite and multipartite cases[6–8]. In the bipartite case, a unique measure has been found[6] to quantify entanglement asymptotically, however, it is a much more complicated problem for multipartite entanglements even in the asymptotic manner[7, 9–12]. Entanglement measure is in fact an approach of quantifying entanglement that relies closely on the reversibility of entanglement transformation. We assign a same scalar number to two different entangled states if they can convert interchangeably to each other by some predefined physical transformation. Because of the reversibility, we can simply do quantification by comparing the entanglement under consideration with some standard entanglements, for instance, the maximally entangled states or minimal reversible entanglement generating set (MREGS)[7]. However, the structure of entanglement is so complicated that it is difficult to define proper entanglement measure based on this interconvertible scheme except for the bipartite asymptotic case. Various approaches can be adopted to cope with this difficulty. One common approach is to use multicomponent measures. For example, in the bipartite case, Nielsen’s criteria of exact LOCC transformation[13] indicates that the vector of Schmidt coefficients, instead of any scalar, is a proper entanglement measure when exact transformations are considered. The disadvantage of this approach is that there exist incomparable states due to lack of total order on multicomponent measure[11, 12]. Another approach that we are concerned with is to measure entanglement relatively. In this approach, the requirement of reversibility is discarded and there are no so called standard entanglements. Instead, we focus on how many target entangled states can be obtained asymptotically, taking both initial state and target state as parameters. Specifically, define the entanglement exchange rate between |ψ⟩ and |φ⟩ as[17]