Simplest non-additive measures of quantum resources

TL;DR

This paper investigates simple non-additive quantum resource measures that depend nonlinearly on a single-variable function of multiple copies, providing recurrence relations and exemplifying with the $ ext{l}_1$-norm of coherence.

Contribution

It introduces a class of nonlinear quantum resource measures characterized by a single variable, extending understanding beyond additive measures, with analytical tools for their analysis.

Findings

Recurrence relations for Maclaurin coefficients of resource measures.

Identification of the $ ext{l}_1$-norm of coherence as a nontrivial example.

Framework for analyzing nonlinear resource measures in quantum information.

Abstract

Given an arbitrary state $\rho$ and some figure of merit ${\cal E}(\rho)$, it is usually a hard problem to determine the value of ${\cal E}(\rho^{\otimes N})$. One noticeable exception is the case of additive measures, for which we simply have ${\cal E}(\rho^{\otimes N}) = Ne$, with $e\equiv {\cal E}(\rho)$. In this work we study measures that can be described by ${\cal E}(\rho^{\otimes N}) =E(e;N) \ne Ne$, 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.