No-go theorem for norm-based quantumness-certification with linear functionals

TL;DR

This paper introduces a convex resource-theoretic framework to quantify optical quantumness using linear functionals, revealing fundamental limitations on universal quantumness measures without optimization.

Contribution

It develops a novel framework based on norms of linear functionals and proves a no-go theorem showing the impossibility of universal quantumness measures without optimization.

Findings

Framework allows direct quantification of optical quantumness

No universal measure exists without optimization

Validated with Gaussian and non-Gaussian states

Abstract

Despite several approaches proposed to operationally characterize quantum states of light-those that cannot be sampled with a positive distribution over classical states-most existing formulations suffer from limited practicality or rely on convex optimization procedures that are computationally demanding. In this work, we develop a general convex resource-theoretic framework to quantify optical quantumness directly from the norms of linear functionals of quantum states, thereby avoiding any optimization. We further establish a no-go theorem demonstrating that no universal measure of quantumness can exist in the absence of optimization. Finally, we substantiate our theoretical result through explicit examples involving both Gaussian and non-Gaussian states.