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  • Open Access

Characterization of Quantum Betting Tasks in Terms of Arimoto Mutual Information

Andrés F. Ducuara1,2,3,4,* and Paul Skrzypczyk3,†

  • 1Quantum Engineering Centre for Doctoral Training, University of Bristol, Bristol BS8 1FD, United Kingdom
  • 2Quantum Engineering Technology Labs, University of Bristol, Bristol BS8 1FD, United Kingdom
  • 3H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
  • 4Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1FD, United Kingdom

  • *andres.ducuara@bristol.ac.uk
  • †paul.skrzypczyk@bristol.ac.uk
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PRX Quantum 3, 020366 – Published 30 June, 2022

DOI: https://doi.org/10.1103/PRXQuantum.3.020366

Abstract

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 α-mutual information, with the order α 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 α→∞, and quantum state (channel) exclusion when α→−∞. 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.

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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.

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PRX Quantum 3, 020366– Published 30 June, 2022
  • Received 19 July 2021
  • Revised 14 December 2021
  • Accepted 26 May 2022
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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