Unified Framework for Calculating Convex Roof Resource Measures

TL;DR

This paper introduces a unified, efficient computational framework for convex roof quantum resource measures, reformulating the problem as an optimization over a Stiefel manifold, applicable to various quantum resources.

Contribution

The authors develop a novel gradient-based optimization method reformulating convex roof calculations as Stiefel manifold problems, outperforming existing SDP-based approaches.

Findings

Superior computational efficiency over SDP methods

Effective application to entanglement, coherence, and magic states

Framework extendable to other convex roof quantities

Abstract

Quantum resource theories (QRTs) provide a comprehensive and practical framework for the analysis of diverse quantum phenomena. A fundamental task within QRTs is the quantification of resources inherent in a given quantum state. In this letter, we introduce a unified computational framework for a class of widely utilized quantum resource measures, derived from convex roof extensions. We establish that the computation of these convex roof resource measures can be reformulated as an optimization problem over a Stiefel manifold, which can be further unconstrained through polar projection. Compared to existing methods employing semi-definite programming (SDP), gradient-based techniques or seesaw strategy, our approach not only demonstrates superior computational efficiency but also maintains applicability across various scenarios within a streamlined workflow. We substantiate the efficacy of our method by applying it to several key quantum resources, including entanglement, coherence, and magic states. Moreover, our methodology can be readily extended to other convex roof quantities beyond the domain of resource theories, suggesting broad applicability in the realm of quantum information theory.