Universal convex and monotone quantification relative to the attainable state set

Develop a universal quantitative method for quantifying relative resources with respect to the experimentally accessible or preparable state set \(\mathcal{E}\) that guarantees both convexity and monotonicity under free operations.

Background

The paper introduces $\mathcal{E}$ as the set of quantum states realizable under specified technological, operational, or economic constraints, and seeks to evaluate states relative to this practically accessible supply environment rather than solely relative to the conventional free-state set. A valid relative resource measure is expected to satisfy standard resource-theoretic properties, particularly convexity and monotonicity under free operations.

The caption of Figure 1 explicitly identifies the unresolved issue that no universal quantitative method is known to guarantee both properties when resources are quantified relative to an arbitrary accessible set $\mathcal{E}$. The paper addresses this difficulty by constructing extended reference sets, such as the free-operation reachable set and the max-performance-equivalence set, but the stated universal-method problem remains the motivating open issue.