Moderate deviation analysis of majorization-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 majorization 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 nontrivial trade-off between the rate r(n) at which n copies of a resource state rho can be transformed into nr(n), copies of another resource state sigma, and the error level epsilon(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 -> infinity), while the error epsilon(n) vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behavior 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.