Population Capacity and the Geometry of Thermomajorization

Abstract

We characterize finite-dimensional quasiclassical state conversion under thermal operations through population capacities: the maximum population that thermal operations can concentrate on each subset of energy levels. We show that these capacities form a normalized, nondecreasing submodular rank function whose base polytope coincides with the thermally reachable population region. This representation provides complete accessibility constraints and a greedy construction of states maximizing the expectation value of any system observable. For a prescribed target, exact conversion can be certified by the optimized responses of at most $n-1$ target-adapted binary work-extraction probes. For approximate conversion, we exploit the Lorenz-curve structure of population capacities to construct, in $O(n^2\log n)$ arithmetic and comparison operations, a reachable state that simultaneously minimizes every convex separable residual error at fixed positive comparison weights. This state also realizes a common first contact with the corresponding target-centered smoothing regions. Finally, the optimal trace-distance error equals the largest capacity advantage of the target over the initial state, attained at a target thermomajorization prefix and determined by the same optimized binary-probe responses. Thus population capacities provide a unified, constructive framework for thermal accessibility, observable optimization, conversion certification, and universal approximation.

arXiv commentimproved and extended version; 15 pages, 2 figures