Simplest non-additive measures of quantum resources

Abstract

Given an arbitrary state $\rho$ and some figure of merit $\mathcal{E}(\rho )$, it is usually a hard problem to determine the value of $\mathcal{E}({\rho }^{\otimes N})$. One noticeable exception is the case of additive measures, for which we simply have $\mathcal{E}({\rho }^{\otimes N})=Ne$, with $e\equiv \mathcal{E}(\rho )$. In this work we study measures that can be described by $\mathcal{E}({\rho }^{\otimes N})=E(e;N)\ne Ne$, that is, measures for which the amount of resources of $N$ copies is still determined by the single real variable $e$, but in a nonlinear way. If, in addition, the measures are analytic around $e=0$, recurrence relations can be found for the Maclaurin coefficients of $E$ for larger $N$. As an example, we show that the ${\ell }_1$-norm of coherence is a nontrivial case of such a behavior.

References in corpus (3)

• Measuring Quantum Coherence with Entanglement
• Distilling entanglement from arbitrary resources
• Flag Additivity in Quantum Resource Theories