Moderate deviation analysis of majorisation-based resource interconversion

TL;DR

This paper analyzes the trade-off between interconversion rate and error in resource theories based on majorisation, deriving optimal moderate deviation results that reveal resonance phenomena allowing near-perfect finite-size transformations.

Contribution

It provides the first detailed moderate deviation analysis of resource interconversion under majorisation, revealing resonance effects that enable near-perfect finite-size transformations.

Findings

Optimal trade-off between rate and error in moderate deviation regime

Resonance behavior allows negligible error at asymptotic rates

Finite-size interconversions can be nearly perfect under certain conditions

Abstract

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 $\rho$ can be transformed into $nr_n$ copies of another resource state $\sigma$, 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 ($n\to\infty$), while the error $\epsilon_n$ 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.