Is there a finite complete set of monotones in any quantum resource theory?

TL;DR

This paper investigates the limitations of finite resource monotones in quantum resource theories, showing that such theories cannot fully determine state transformations if they include resource-free pure states.

Contribution

It proves the non-existence of finite complete sets of monotones in resource theories with resource-free pure states and characterizes totally ordered theories where a single monotone suffices.

Findings

Finite monotones cannot fully determine state convertibility in certain resource theories.

Totally ordered theories are characterized by a single resource monotone.

Complete state transformation criteria are provided for single-qubit systems in totally ordered theories.

Abstract

Entanglement quantification aims to assess the value of quantum states for quantum information processing tasks. A closely related problem is state convertibility, asking whether two remote parties can convert a shared quantum state into another one without exchanging quantum particles. Here, we explore this connection for quantum entanglement and for general quantum resource theories. For any quantum resource theory which contains resource-free pure states, we show that there does not exist a finite set of resource monotones which completely determines all state transformations. We discuss how these limitations can be surpassed, if discontinuous or infinite sets of monotones are considered, or by using quantum catalysis. We also discuss the structure of theories which are described by a single resource monotone and show equivalence with totally ordered resource theories. These are theories where a free transformation exists for any pair of quantum states. We show that totally ordered theories allow for free transformations between all pure states. For single-qubit systems, we provide a full characterization of state transformations for any totally ordered resource theory.