Understand
We introduce operational quantum tasks based on betting with risk-aversion -- or quantum betting tasks for short -- inspired by standard quantum state discrimination and classical horse betting with risk-aversion and side information.
- In particular, we introduce the operational tasks of quantum state betting (QSB), noisy quantum state betting (nQSB), and quantum channel betting (QCB) played by gamblers with different risk tendencies.
- We prove that the advantage that informative measurements (non-constant channels) provide in QSB (nQSB) is exactly characterised by Arimoto's $\alpha$-mutual information, with the order $\alpha$ determining the risk aversion of the gambler.
- More generally, we show that Arimoto-type information-theoretic quantities characterise the advantage that resourceful objects offer at playing quantum betting tasks when compared to resourceless objects, for general quantum resource theories (QRTs) of measurements, channels, states, and state-measurement pairs, with arbitrary resources.