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

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.

6+3 pages, close to the published version

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• Complete Characterization of Entanglement Embezzlement
• Coherence and imaginarity of quantum states
• Teleportation with Embezzling Catalysts
• Communication with Quantum Catalysts
• Thermodynamic state convertibility is determined by qubit cooling and heating
• Complete characterisation of state conversions by work extraction
• State convertibility under genuinely incoherent operations
• Insufficiency of Pure-State Ensembles in Characterizing Transformations of Entangled States under LOCC
• Quantum geometric tensor determines the pure-state i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group
• Macroscopicity and observational deficit in states, operations, and correlations
• Incoherent Operations Enable State Transformations Impossible under Dephasing-covariant Incoherent Operations
• A finite sufficient set of conditions for catalytic majorization