Characterization of Quantum Betting Tasks in Terms of Arimoto Mutual Information

We introduce operational quantum tasks based on betting with risk aversion—or quantum betting tasks for short—inspired by standard quantum state discrimination and classical horse betting with risk aversion and side information. In particular, we introduce the operational tasks of quantum state betting (QSB), noisy quantum state betting (NQSB), and quantum-channel betting (QCB) played by gamblers with different risk tendencies. We prove that the advantage that informative measurements (nonconstant channels) provide in QSB (NQSB) is exactly characterized by Arimoto’s $\alpha$-mutual information, with the order $\alpha$ determining the risk aversion of the gambler. More generally, we show that Arimoto-type information-theoretic quantities characterize the advantage that resourceful objects offer at playing quantum betting tasks when compared to resourceless objects, for general quantum resource theories (QRTs) of measurements, channels, states, and state-measurement pairs, with arbitrary resources. In limiting cases, we show that QSB (QCB) recovers the known tasks of quantum state (channel) discrimination when $\alpha \to \mathrm{\infty }$, and quantum state (channel) exclusion when $\alpha \to -\mathrm{\infty }$. Inspired by these connections, we also introduce new quantum Rényi divergences for measurements, and derive a new family of resource monotones for the QRT of measurement informativeness. This family of resource monotones recovers in the same limiting cases as above, the generalized robustness and the weight of informativeness. Altogether, these results establish a broad and continuous family of four-way correspondences between operational tasks, mutual information measures, quantum Rényi divergences, and resource monotones, that can be seen to generalize two limiting correspondences that were recently discovered for the QRT of measurement informativeness.

Popular Summary

It is common to present quantum information tasks in terms of games. When games involve rewards or penalties, how a player chooses to play will depend upon their attitude to risk. However, to date, this aspect of games has not been studied in quantum information science. Here, we instigate this research direction, by introducing quantum betting games–which involve rewards and penalties –and which can be viewed as generalizations of discrimination and exclusion tasks. This allows us to incorporate the concept of risk-aversion–widely studied in many fields, ranging from the economic sciences to neuroscience, into quantum information science. Our main result is to show that, within the context of quantum resource theories, the advantage that a risk-averse player gains when having access to a resource for playing a betting game is exactly characterised by an information theoretic quantity based upon Renyi entropies, known as the Arimoto mutual information. In particular, in our setting we show that the Renyi parameter gains the operational significance of quantifying the risk attitude of the player.

Figure 1

Isoelastic utility function $u_R(w)$ (18) as a function of positive wealth ($1\le w\le 3$) for players with different risk tendencies (different values of $R$). The risk parameter $R$ quantifies different types of risk tendencies: (i) $R<0$ risk-seeking players (convex), (ii) $R=0$ risk-neutral players (linear), and (iii) $R>0$ risk-averse players (concave). Risk aversion for positive wealth then increases from $-\mathrm{\infty }$ to $\mathrm{\infty }$.

Figure 2

Operational tasks based on betting and risk aversion. Quantum state betting, quantum-subchannel betting, quantum-channel betting, quantum state discrimination and exclusion, quantum-channel discrimination and exclusion. $A\to B$ means that the task $A$ is more general than $B$.

Figure 3

Possible scenarios for quantum state discrimination and quantum state exclusion games being played by gamblers with different risk tendencies: risk averse, risk seeking, or risk neutral, with the risk being parametrized as $R(\alpha )=1/\alpha$. Result 1 establishes that Arimoto’s mutual information quantifies the shaded region for $\alpha \in \overline{\mathbb{R}}$, meaning that it characterizes risk-averse gamblers playing either QSD ($\alpha \ge 0$) and QSE games ($\alpha <0$). The left-bottom corner $(\alpha \to -\mathrm{\infty })$ and the top-right corner $(\alpha \to \mathrm{\infty })$ represent a risk-neutral gambler $R=0$ playing either standard exclusion or discrimination games, respectively. This means that standard QSD games can be understood as a risk-neutral gambler playing QSD games with risk. Similarly, standard QSE games can be understood as a risk-neutral gambler playing QSE games with risk. The middle point at $\alpha \to 0$ represents the transition between a maximally risk-averse gambler playing QSD games and a maximally risk-averse gambler playing QSE games.

Figure 4

A four-way correspondence for the QRT of measurement informativeness. The correspondence is parametrized by the Rényi parameter $\alpha \in \mathbb{R}\cup \{\mathrm{\infty },-\mathrm{\infty }\}$. The outer rectangle represents $\alpha =\mathrm{\infty }$, the inner rectangle represents $\alpha =-\mathrm{\infty }$, and the shaded region in between represents the values $\alpha \in \mathbb{R}$. This four-way correspondence links operational tasks, dependence measures, Rényi divergences, and resource monotones. The operational task is quantum state betting played by a gambler with risk aversion $R(\alpha )=1/\alpha$. This task generalizes quantum state discrimination (recovered when $\alpha \to \mathrm{\infty }$), and quantum state exclusion (recovered when $\alpha \to -\mathrm{\infty }$). $I_{\alpha }(p_{GX}^{(\mathbb{M},\mathcal{E})})$ is Arimoto’s dependence measure, from which we recover the accessible information $I_{\mathrm{\infty }}^{\mathrm{acc}}({\mathrm{\Lambda }}_{\mathbb{M}})$ when $\alpha \to \mathrm{\infty }$, and the excludible information $I_{-\mathrm{\infty }}^{\mathrm{exc}}({\mathrm{\Lambda }}_{\mathbb{M}})$ when $\alpha \to -\mathrm{\infty }$, with ${\mathrm{\Lambda }}_{\mathbb{M}}$ the measure-prepare channel of the measurement $\mathbb{M}$. We introduce $D_{\alpha }^{\mathcal{S}}(\mathbb{M}||\mathbb{N})$, the quantum Rényi divergence of two measurements $\mathbb{M}$ and $\mathbb{N}$ for a given set of states $\mathcal{S}=\{{\rho }_x\}$. We also introduce $M_{\alpha }(\mathbb{M})$, a new family of resource monotones, which generalize the robustness of informativeness $\mathrm{R}(\mathbb{M})$ (when $\alpha \to \mathrm{\infty })$ and the weight of informativeness $\mathrm{W}(\mathbb{M})$ (when $\alpha \to -\mathrm{\infty }$). The outer rectangle was uncovered in Ref. [12], whilst the inner rectangle was first uncovered in Ref. [30]. The main set of results of this paper is to fill the shaded region, and connect these two correspondences for all $\alpha \in \mathbb{R}$.

Figure 5

Hierarchical relationship between the Rényi divergence $D_{\alpha }(\cdot ||\cdot )$, conditional Rényi divergences $D_{\alpha }^V(\cdot ||\cdot |\cdot )$, mutual informations $I_{\alpha }^V(X;G)$, and the Rényi channel capacity $C_{\alpha }(p_{G|X})$ with $\alpha \ge 1$, and $V\in \{S,C,\mathrm{BLP}\}$ a label specifying the measures of Sibson [40], Csiszár [42], and Bleuler-Lapidoth-Pfister [38]. The mutual information associated to the BLP conditional Rényi divergence was independently derived by Lapidoth-Pfister [43] and Tomamichel-Hayashi [44]. In this work we address the capacities generated by Sibson and Arimoto.