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 $n{r}_n$ 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 ($n\to \infty$), 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.

Figure 1

Schematic representation of cumulative distribution functions for ${\mathit{P}}^n$ and ${\mathit{Q}}^n$, and the positions of $z_{\mathrm{C}}$ and $z_{\mathrm{T}}$ in different regimes.