Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2306.07975

Fundamental connections between utility theories of wealth and information theory

Andrés F. Ducuara, Paul Skrzypczyk

Published May 22, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2306.07975

Related papers

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

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.

•Jun 24, 2021•0 citations

Multivariate Rényi divergences characterise betting games with multiple lotteries

Andrés F. Ducuara, Erkka Haapasalo, Ryo Takakura

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 $D_{\underlineα}(\vec{P}_X)$ of probability distributions $\vec{P}_X =(p^{(0)}_X,\dots,p^{(d)}_X)$ and real-valued orders $\underlineα = (α_0, \dots, α_d)$ quantifies the economic-theoretic value that a rational agent assigns to $d$ lotteries with odds $o^{(k)}_X \propto (p_X^{(k)})^{-1}$ ($k=1,\dots,d$) on a random event described by $p^{(0)}_X$. 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 $w^{\mathrm{ICE}}_{\underline{R}}=\exp[D_{\underlineα}(\vec{P}_X)]$, where $\underline{R}=(R_1,\dots,R_d)$ is a risk-aversion vector with $R_k = 1+α_k/α_0$ 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.

quant-pharXiv:2601.17850•Jan 27, 2026

A resource theory of gambling

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.

arXiv:2510.08418•Oct 9, 2025•0 citations

Characterization of Quantum Betting Tasks in Terms of Arimoto Mutual Information

Andrés F. Ducuara, Paul Skrzypczyk

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.

arXiv:2106.12711•PRX Quantum•Jun 30, 2022•11 citations

Rényi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations

Alexander Clyde McKinlay

We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.

arXiv:2106.10415•Jun 19, 2021•0 citations

Chain Rules for Renyi Information Combining

Christoph Hirche, Guan, Xinyue, Marco Tomamichel

Bounds on information combining are a fundamental tool in coding theory, in particular when analyzing polar codes and belief propagation. They usually bound the evolution of random variables with respect to their Shannon entropy. In recent work this approach was generalized to Renyi $α$-entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of $α=2$ we give the first optimal information combining bounds with quantum side information.

arXiv:2305.02589•May 4, 2023•0 citations
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 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 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.
We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.
Bounds on information combining are a fundamental tool in coding theory, in particular when analyzing polar codes and belief propagation. They usually bound the evolution of random variables with respect to their Shannon entropy. In recent work this approach was generalized to Renyi ααα-entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of α=2α=2α=2 we give the first optimal information combining bounds with quantum side information.