Moderate deviation analysis of majorisation-based resource interconversion

We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate r_n at which n copies of a resource state ρ can be transformed into nr_n copies of another resource state σ, and the error level varepsilon_n of the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion r_n approaches its optimum in the asymptotic limit of unbounded resources $n→infty$, while the error ε_n vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.