Abstract
The question of how quantities, like entanglement and coherence, depend on the number of copies of a given state ρ 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 E(ρ^⊗ N) is {\it scalable} if it can be described as a function of the variables { E(ρ^⊗ i₁),…, E(ρ^⊗ i_q); 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 E(ρ^⊗ N), which enable us to determine its possible functional forms (in terms of the mentioned variables). In particular, we find that if E(ρ^⊗ 2ⁿ) depends only on E(ρ), E(ρ^⊗ 2), 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 ϱ, we determine the OSD entanglement for ϱ^⊗ 96 from smaller tensorings, with an accuracy of 97 % and no extra computational effort. Finally, we show that superactivation of non-additivity may occur in this context.