Quantum horse betting, dependence measures, quantum R\'enyi divergences, and resource monotones: a four-way correspondence

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We introduce the operational tasks of quantum horse betting (QHB) games with a risk factor parametrised by a coefficient in the extended line of real numbers α ∈ R ∪ {+∞, −∞}. We do this by modifying and extending, to the quantum regime, the operational tasks of (classical) horse betting (CHB) games with risk. We prove that the operational tasks of CHB and QHB games are characterised by the Arimoto’s dependence measure of order α. Explicitly, the Arimoto’s measure quantifies the ratio between CHB games with and without side information. In QHB games on the other hand, and within the quantum resource theory of measurement informativeness, the Arimoto’s measure quantifies the ratio between QHB games being played with a fixed (potentially informative) measurement against the best uninformative measurement. Furthermore, we prove that QHB games recover, as limit cases, the operational tasks of: quantum state discrimination (QSD) (α → +∞) and quantum state exclusion (QSE) (α → −∞). Additionally, Arimoto’s measure recovers, as limit cases, the information-theoretic quantities of: accessible information (α → +∞) and excludible information (α → −∞) of quantum-classical channels. Inspired by these connections, we also introduce quantum R´enyi divergences for measurements, and derive a family of resource monotones for the quantum resource theory of measurement informativeness. This family of resource monotones recovers, as limit cases, the known measures of: generalised robustness of informativeness (α → +∞) and weight of informativeness (α → −∞). Altogether, these results therefore establish a four-way correspondence between: operational tasks, dependence measures, quantum R´enyi divergences, and resource monotones. This correspondence recovers the known four-way correspondences at the extremes α ∈ {+∞, −∞}, whilst generalising it to include the spectrum α ∈ R.
Original languageEnglish
JournalPhysical Review X
Publication statusIn preparation - 2020

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