Monotones in General Resource Theories

Abstract

A central problem in the study of resource theories is to find functions that are nonincreasing under resource conversions — termed monotones — in order to quantify resourcefulness. Various constructions of monotones appear in many different concrete resource theories. How general are these constructions? What are the necessary conditions on a resource theory for a given construction to be applicable? To answer these questions, we introduce a broad scheme for constructing monotones. It involves finding an order-preserving map from the preorder of resources of interest to a distinct preorder for which nontrivial monotones are previously known or can be more easily constructed; these monotones are then pulled back through the map. In one of the two main classes we study, the preorder of resources is mapped to a preorder of sets of resources, where the order relation is set inclusion, such that monotones can be defined via maximizing or minimizing the value of a function within these sets. In the other class, the preorder of resources is mapped to a preorder of tuples of resources, and one pulls back monotones that measure the amount of distinguishability of the different elements of the tuple (hence its information content). Monotones based on contractions arise naturally in the latter class, and, more surprisingly, so do weight and robustness measures. In addition to capturing many standard monotone constructions, our scheme also suggests significant generalizations of these. In order to properly capture the breadth of applicability of our results, we present them within a novel abstract framework for resource theories in which the notion of composition is independent of the types of the resources involved (i.e., whether they are states, channels, combs, etc.).

Thermodynamic state convertibility is determined by qubit cooling and heating

“…γ S ; β) can be seen as a generalized monotone [83][84][85] that implies an operational order, namely that (ρ R , γ R ) can be used to cool/heat more qubits to the desired target inverse temperature β than (σ S , γ S ).…”

Every Quantum Helps: Operational Advantage of Quantum Resources beyond Convexity

“…In fact, classical randomness can be regarded as a resource [35,36], meaning that convexity itself may well be expensive in general. Hence, the current lack of techniques towards resources beyond convexity significantly impedes a unified understanding of fundamental advantages and limitations of quantum mechanical properties, which general QRTs [4,5,13,16,[21][22][23][24][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] aim to achieve.…”