All bets are off

Itamar Pitowsky

We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the other measurements is returned to the agent. We show how the rules of rational betting imply all the interesting features of quantum probability, even in such finite gambles. These include the uncertainty principle and the violation of Bell's inequality among others. Quantum gambles are closely related to quantum logic and provide a new semantics to it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.

Jennifer Maier

Maite Arcos, Renato Renner, Jonathan Oppenheim

Betting games provide a natural setting to capture how information yields strategic advantage. The Kelly criterion for betting, long a cornerstone of portfolio theory and information theory, admits an interpretation in the limit of infinitely many repeated bets. We extend Kelly's seminal result into the single-shot and finite-betting regimes, recasting it as a resource theory of adversarial information. This allows one to quantify what it means for the gambler to have more information than the odds-maker. Given a target rate of return, after a finite number of bets, we compute the optimal strategy which maximises the probability of successfully reaching the target, revealing a risk-reward trade-off characterised by a hierarchy of Rényi divergences between the true distribution and the odds. The optimal strategies in the one-shot regime coincide with strategies maximizing expected utility, and minimising hypothesis testing errors, thereby bridging economic and information-theoretic viewpoints. We then generalize this framework to a distributed side-information game, in which multiple players observe correlated signals about an unknown state. Recasting gambling as an adversarial resource theory provides a unifying lens that connects economic and information-theoretic perspectives, and allows for generalisation to the quantum domain, where quantum side-information and entanglement play analogous roles.

Andrés F. Ducuara, Paul Skrzypczyk

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 $\alpha$ determining the risk aversion of the gambler. In limiting cases, we show that QSB recovers the tasks of quantum state discrimination when $\alpha \rightarrow \infty$, and quantum state exclusion when $\alpha \rightarrow -\infty$. 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.