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

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References in corpus (4)

• Measuring Quantum Coherence with Entanglement
• Observable measure of quantum coherence in finite dimensional systems
• Distilling entanglement from arbitrary resources
• Quantifying Bell non-locality with the trace distance

Cited by in corpus (2)

• Simplest non-additive measures of quantum resources
• Enlarging the notion of additivity of resource quantifiers