The question of how quantities, like entanglement and coherence, depend on the number of copies of a given state $\rho$ is addressed. This is a hard problem, often involving optimizations over Hilbert spaces of large dimensions. Here, we propose a way to circumvent the direct evaluation of such quantities, provided that the employed measures satisfy a self-similarity property. We say that a quantity $\mathcal{E}\left({\rho }^{\otimes N}\right)$ is scalable if it can be described as a function of the variables $\{\mathcal{E}\left({\rho }^{\otimes i_1}\right),\cdots ,\mathcal{E}\left({\rho }^{\otimes i_q}\right);N\}$ for $N>i_j$, while preserving the tensor-product structure. If analyticity is assumed, recursive relations can be derived for the Maclaurin series of $\mathcal{E}\left({\rho }^{\otimes N}\right)$, which enable us to determine its possible functional forms (in terms of the mentioned variables). In particular, we find that if $\mathcal{E}\left({\rho }^{\otimes 2^n}\right)$ depends only on $\mathcal{E}\left(\rho \right), \mathcal{E}\left({\rho }^{\otimes 2}\right)$, and $n$, then it is completely determined by Fibonacci polynomials, to leading order. We show that the one-shot distillable (OSD) entanglement is well described as a scalable measure for several families of states. For a particular two-qutrit state $\varrho$, we determine the OSD entanglement for ${\varrho }^{\otimes 96}$ from smaller tensorings, with an accuracy of $97\%$ and no extra computational effort. Finally, we show that superactivation of nonadditivity may occur in this context.

• Received 22 October 2019
• Accepted 23 June 2020

DOI:https://doi.org/10.1103/PhysRevA.102.012413