Using the framework of quantum resource theories, inspired by the concepts of betting and risk aversion, a broad and continuous family of four-way correspondences between important information-theoretic quantities and operational tasks is established.
We introduce the operational tasks of quantum state betting (QSB) played by gamblers with different risk tendencies. We do this by adapting to the quantum regime the task of horse betting with risk-aversion and side information. We prove that the advantage that informative measurements provide in QSB is exactly characterised by Arimoto's dependence measure, with the order α determining the risk aversion of the gambler. In limiting cases, we show that QSB recovers the tasks of quantum state discrimination when α→∞, and quantum state exclusion when α→−∞. In these respective limiting cases, Arimoto's dependence measure becomes equal to the accessible information and excludible information of the associated quantum-classical channels. Inspired by these connections, we introduce new quantum Renyi divergences for measurements, and derive a family of resource monotones for the quantum resource theory of measurement informativeness. This family of resource monotones recovers in the same limiting cases as above, the generalised robustness and the weight of informativeness. Altogether, these results establish a broad and continuous family of four-way correspondences between operational tasks, dependence measures, quantum Renyi divergences, and resource monotones, that can be seen to generalise two limiting correspondences that were recently discovered.
We provide an operational interpretation of the multivariate Rényi divergence in terms of economic-theoretic tasks based on betting, risk aversion, and multiple lotteries. We show that the multivariate Rényi divergence Dα(PX) of probability distributions PX=(pX(0),…,pX(d)) and real-valued orders α=(α0,…,αd) quantifies the economic-theoretic value that a rational agent assigns to d lotteries with odds oX(k)∝(pX(k))−1 (k=1,…,d) on a random event described by pX(0). In particular, when the odds are fair and the rational agent maximises over all betting strategies, the economic-theoretic value (the isoelastic certainty equivalent) that the agent assigns to the lotteries is exactly given by wRICE=exp[Dα(PX)], where R=(R1,…,Rd) is a risk-aversion vector with Rk=1+αk/α0 being the risk-aversion parameter for lottery k. Furthermore, we introduce a new conditional multivariate Rényi divergence that characterises a generalised scenario where the agent uses side information. We prove that this new quantity satisfies a data processing inequality which can be interpreted as the increment in the economic-theoretic value provided by side information; crucially, such a data processing inequality is a consequence of the agent's economic-theoretically consistent risk-averse attitude towards every lottery and vice versa. Finally, we apply these results to the resource theory of informative measurements in general probabilistic theories (GPTs). By establishing quantitative connections between information theory, physics, and economics, our framework provides a novel operational foundation for quantum state betting games with multiple lotteries in the realm of quantum resource theories.
We establish fundamental connections between utility theories of wealth from the economic sciences and information-theoretic quantities. In particular, we introduce operational tasks based on betting where both gambler and bookmaker have access to side information, or betting tasks with double side information for short. In order to characterise these operational tasks we introduce new conditional Rényi divergences, and explore some of their properties. Furthermore, we introduce an utility theory of wealth ratios, and operationally interpret there the two-parameter (q,r) generalised mutual information measure recently introduced by V. M. Ilić and I. V. Djordjević; it quantifies the advantage provided by side information in betting tasks for utility theories of wealth ratios. Moreover, we show that the Ilić-Djordjević conditional entropy satisfies a type of generalised chain rule, which generalises that of Arimoto-Rényi. Finally, we address the implications of these results on the quantum resource theories of informative measurements and non-constant channels. Altogether, these results further help strengthening the bridge between the theory of expected utility from the economic sciences and Shannon's theory of information.