Moderate deviation analysis of majorization-based resource interconversion
Christopher T. Chubb, Marco Tomamichel, Kamil Korzekwa · Physical Review A · 2019
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 $\ensuremath{\rho}$ can be transformed into $n{r}_{n}$ copies of another resource state $\ensuremath{\sigma}$, and the error level ${\ensuremath{\epsilon}}_{n}$ 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 ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), while the error ${\ensuremath{\epsilon}}_{n}$ 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.