Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication

Abstract

While there is general consensus on the definition of incompatible POVMs, moving up to the level of instruments one finds a much less clear situation, with mathematically different and logically independent definitions of incompatibility. Here we close this gap by introducing the notion of q-compatibility, which unifies different notions of POVMs, channels, and instruments incompatibility into one hierarchy of resource theories of communication between separated parties. The resource theories that we obtain are complete, in the sense that they contain complete families of free operations and monotones providing necessary and sufficient conditions for the existence of a transformation. Furthermore, our framework is fully operational, in the sense that free transformations are characterized explicitly, in terms of local operations aided by causally-constrained directed classical communication, and all monotones possess a game-theoretic interpretation making them experimentally measurable in principle. We are thus able to pinpoint exactly what each notion of incompatibility consists of, in terms of information-theoretic resources.

arXiv commentv2: accepted in Quantum, 25 pages, 9 figures; v1: 22 pages, 8 figures

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Cited by in corpus (14)

• Incompatibility as a resource for programmable quantum instruments
• A complete and operational resource theory of measurement sharpness
• Resource Marginal Problems
• Thermodynamic Approach to Quantifying Incompatible Instruments
• Measurement incompatibility is strictly stronger than disturbance
• Unveiling quantum steering by quantum-classical uncertainty complementarity
• Incompatibility of quantum instruments
• A Compendious Review of Majorization-Based Resource Theories: Quantum Information and Quantum Thermodynamics
• Joint-measurability and quantum communication with untrusted devices
• A Theory of Quantum Instruments
• Multi-object operational tasks for measurement incompatibility
• Resource theory of interactive quantum instruments
• Tight conic approximation of testing regions for quantum statistical models and measurements
• Comparing quantum incompatibility of device sets from an operational perspective