Enlarging the notion of additivity of resource quantifiers

Whenever a physical quantity becomes essential to the realization of useful tasks, it is desirable to define proper measures or monotones to quantify it. In quantum mechanics, coherence, entanglement, and Bell nonlocality are examples of such quantities. Given a quantum state $\varrho$ and a quantifier ${\cal E}(\varrho)$, both arbitrary, it is a hard task to determine ${\cal E}(\varrho^{\otimes N})$. However, if the figure of merit $\cal{E}$ turns out to be additive, we simply have ${\cal E}(\varrho^{\otimes N})=N e$, with $e={\cal E}(\varrho)$. In this work we generalize this useful notion through the inner product ${\cal E}(\varrho^{\otimes N}) = \vec{N}\cdot \vec{e}$, where $\vec{e}=({\cal E}(\varrho^{\otimes i_1}), {\cal E}(\varrho^{\otimes i_2}),\dots,{\cal E}(\varrho^{\otimes i_q}) )$ is a vector whose $q$ entries are the figure of merit under study calculated for some numbers of copies smaller than $N$ ($1 \le i_1<i_2<\dots <i_q<N$), where $\vec{N}=(N_{i_1}, N_{i_2}, \dots ,N_{i_q})$, is a string of numbers that depends only on $N$ and on the set of integers $\{ {i_j}\}$. We show that the one shot distillable entanglement of certain spherically symmetric states can be quantitatively approximated by such an augmented additivity.