Framework for resource quantification in infinite-dimensional general probabilistic theories

Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. Here we introduce methods for the quantification of resources in general probabilistic theories (GPTs), focusing in particular on the technical issues associated with infinite-dimensional state spaces. We define a universal resource quantifier based on the robustness measure, and show it to admit a direct operational meaning: in any GPT, it quantifies the advantage that a given resource state enables in channel discrimination tasks over all resourceless states. We show that the robustness acts as a faithful and strongly monotonic measure in any resource theory described by a convex and closed set of free states, and can be computed through a convex conic optimization problem. Specializing to continuous-variable quantum mechanics, we obtain additional bounds and relations, allowing an efficient computation of the measure and comparison with other monotones. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. In particular, we establish exact expressions for various classes of states, including Fock states and squeezed states in the resource theory of nonclassicality and general pure states in the resource theory of entanglement, as well as tight bounds applicable in general cases.

Figure 1

Evaluating bounds for the robustness of nonclassicality of the Schrödinger cat states (a) $|{\alpha }_{+}\rangle$ and (b) $|{\alpha }_{-}\rangle$. The plots show the various analytical and numerical bounds: (a) The lower bound obtained by choosing $|{\alpha }_{+}\rangle$ [Eq. (107)] and $|1_{+}\rangle$ [Eq. (108)] in Lemma t14, as well as the one obtained by numerically optimizing over the choice of $|{\gamma }_{+}\rangle$ [Eq. (112)]. (b) The upper bounds obtained by choosing ${\sigma }_1$ [Eq. (110)] or the optimized state ${\sigma }_x$ in Lemma t13, and the lower bounds obtained by choosing $|{\alpha }_{-}\rangle$ [Eq. (107)], $|1\rangle$ [Eq. (109)], or the optimized state $|{\gamma }_{-}\rangle$ in Lemma t14. The quantities plotted are dimensionless.

Figure 2

Robustness of nonclasicality [Eq. (77)] and non-Gaussianity [Eq. (145)] for the Fock states $|n\rangle$. The quantities plotted are dimensionless.