Quantifying the unextendibility of entanglement

The unextendibility or monogamy of entangled states is a key property of quantum entanglement. Unlike conventional ways of expressing entanglement monogamy via entanglement measure inequalities, we introduce a state-dependent set of free states to quantify the unextendibility of a bipartite quantum state. First, we define a family of entanglement measures called \textit{unextendible entanglement}. Given a bipartite state $\rho_{AB}$, the key idea behind these measures is to minimize a divergence between $\rho_{AB}$ and any possible reduced state $\rho_{AB'}$ of an extension $\rho_{ABB'}$ of $\rho_{AB}$. These measures are intuitively motivated by the fact that the more that a bipartite state is entangled, the less that each of its individual systems can be entangled with a third party. Second, we show that the unextendible entanglement is an entanglement monotone under two-extendible operations, which include local operations and one-way classical communication as a special case. Unextendible entanglement has several other desirable properties, including normalization and faithfulness. As practical applications, we show that the unextendible entanglement provides efficiently computable benchmarks for the rate of exact secret key distillation and entanglement distillation and the overhead of probabilistic secret key or entanglement distillation.