Characterizing scalable measures of quantum resources

TL;DR

This paper introduces a method to analyze how quantum resource measures like entanglement scale with multiple copies, using self-similarity and recursive relations, enabling efficient estimation of resources in large quantum systems.

Contribution

It proposes a scalable framework for quantum resource measures based on self-similarity, allowing determination of resource quantities for large tensor powers without extensive computations.

Findings

One-shot distillable entanglement can be accurately estimated for large tensor powers.

Fibonacci polynomials characterize the functional form of certain resource measures.

Superactivation of non-additivity can occur in the context of scalable quantum resources.

Abstract

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 ${\cal E}(\rho^{\otimes N})$ is {\it scalable} if it can be described as a function of the variables $\{{\cal E}(\rho^{\otimes i_1}),\dots,{\cal E}(\rho^{\otimes i_q}); N\}$ for $N>i_j$, while, preserving the tensor-product structure. If analyticity is assumed, recursive relations can be derived for the Maclaurin series of ${\cal E}(\rho^{\otimes N})$, which enable us to determine its possible functional forms (in terms of the mentioned variables). In particular, we find that if ${\cal E}(\rho^{\otimes 2^n})$ depends only on ${\cal E}(\rho)$, ${\cal E}(\rho^{\otimes 2})$, 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 $\varrho$, 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 non-additivity may occur in this context.