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.).

Figures (4)

• Figure 1: Universal combination of two processes in a process theory. For simplicity, the input and output wires of $f$ and $g$ are of the same type here. We can read the right-hand side as a set of five diagrams, the $\cup$ symbol indicates union of sets and each diagram in the union is thought of as a singleton set of diagrams. The light blue background indicates the type of the resulting process---a channel with two inputs and two outputs in the first case, a channel with a single input and single output in the next two and a 1-comb for the last two diagrams.
• Figure 2: The $f_W$-yield relative to $D$-image map of a channel $\phi$ given a function $f_W$ defined on states only. Note that the right hand side is equal to $-\infty$ if $D$ contains no states.
• Figure 3: A pictorial depiction of the optimal convex decompositions for each of the four monotones mentioned in this section: (a) resource weight $M_{\rm w}$, (b) resource robustness $M_{\rm rob}$, (c) free robustness $M_{\rm f.\,rob}$, and (d) resource non-convexity $M_{\rm nc}$. Grey disc represents the set $R$ of all resources, while the yellow "hourglass" witin represents the free resources among them. In order to illustrate each of the four optimal decompositions, we select a distinct resource (element of $X$), depicted by a green node. These demopositions are given by the three points along one of the line segments with an orange and purple portion. The value of each of the monotones for these; $M_{\rm w}(a), M_{\rm rob}(b), M_{\rm f.\,rob}(c)$, and $M_{\rm nc}(d)$; can be read off as the length of the respective orange segment divided by the total lenth of the orange and purple segments combined.
• Figure :

Theorems & Definitions (91)

• Example 1: translating measures of nonuniformity to measures of entanglement
• Example 2: translating measures of distinguishability to measures of asymmetry
• Example 3: resource theory of universally combinable processes
• Definition 4
• Lemma 5: compatibility of $\boxtimes$ and $\succeq$
• proof
• Example 6: universally combinable processes
• Remark 7: \ref{['def:resource theory']} excludes type-restricted theories
• Example 8: quantum resource theories
• Example 9: classical resource theories