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

Andrés F. Ducuara, Paul Skrzypczyk

Published Jun 24, 2021• 0 citations
OpenAlex

Abstract

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.

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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\alphaα determining the risk aversion of the gambler. In limiting cases, we show that QSB recovers the tasks of quantum state discrimination when α→∞\alpha \rightarrow \inftyα→∞, and quantum state exclusion when α→−∞\alpha \rightarrow -\inftyα→−∞. 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 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)(q,r)(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.
We consider the quantum resource theory of measurement informativeness and introduce a weight-based quantifier of informativeness. We show that this quantifier has operational significance from the perspective of quantum state exclusion, by showing that it precisely captures the advantage a measurement provides in minimising the error in this game. We furthermore introduce information theoretic quantities related to exclusion, in particular the notion of excludible information of a quantum channel, and show that for the case of quantum-to-classical channels it is determined precisely by the weight of informativeness. This establishes a three-way correspondence which sits in parallel to the recently discovered correspondence in quantum resource theories between robustness-based quantifiers, discrimination games, and accessible information. We conjecture that the new correspondence between a weight-based quantifier and an exclusion-based task found in this work is a generic correspondence that holds in the context of quantum resource theories.
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‾(P⃗X)D_{\underlineα}(\vec{P}_X)Dα​​(PX​) of probability distributions P⃗X=(pX(0),…,pX(d))\vec{P}_X =(p^{(0)}_X,\dots,p^{(d)}_X)PX​=(pX(0)​,…,pX(d)​) and real-valued orders α‾=(α0,…,αd)\underlineα = (α_0, \dots, α_d)α​=(α0​,…,αd​) quantifies the economic-theoretic value that a rational agent assigns to ddd lotteries with odds oX(k)∝(pX(k))−1o^{(k)}_X \propto (p_X^{(k)})^{-1}oX(k)​∝(pX(k)​)−1 (k=1,…,dk=1,\dots,dk=1,…,d) on a random event described by pX(0)p^{(0)}_XpX(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 wR‾ICE=exp⁡[Dα‾(P⃗X)]w^{\mathrm{ICE}}_{\underline{R}}=\exp[D_{\underlineα}(\vec{P}_X)]wR​ICE​=exp[Dα​​(PX​)], where R‾=(R1,…,Rd)\underline{R}=(R_1,\dots,R_d)R​=(R1​,…,Rd​) is a risk-aversion vector with Rk=1+αk/α0R_k = 1+α_k/α_0Rk​=1+αk​/α0​ being the risk-aversion parameter for lottery kkk. 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.
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.
Why does quantum relative entropy occupy such a privileged role among measures of distinguishability? We show that a single operational principle determines it. If one pair of states can be distinguished at least as successfully as another for every prior, when both are assisted by the same auxiliary pair, acting as a catalyst, a distinguishability measure should assign it no smaller value. Remarkably, for a normalized measure, this requirement singles out Umegaki relative entropy under mild classical continuity and expansibility on uniform distributions: adding impossible outcomes leaves their associated entropy unchanged. Qubit catalysts already suffice. No additivity or quantum continuity is assumed. We also prove that, without auxiliary assistance, respecting binary discrimination dominance uniquely determines the quantum extension of every regular classical divergence, including the Hirche--Tomamichel extensions of classical fff-divergences. For additive classical divergences, these extensions respect classical catalysis; allowing noncommuting qubit catalysts is what singles out Umegaki relative entropy.