Kelly Betting with Quantum Payoff: a continuous variable approach
Abstract
The main purpose of this study is to introduce a semiclassical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the freeenergy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all her/his initial resources into coherent state amplitude.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.11395
 Bibcode:
 2020arXiv200111395T
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory;
 Quantitative Finance  Mathematical Finance
 EPrint:
 18 pages, 8 figures