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The main purpose of this study is to introduce a semi-classical modeldescribing betting scenarios in which, at variance with conventionalapproaches, the payoff of the gambler is encoded into the internal degrees offreedom of a quantum memory element. In our scheme, we assume that the investedcapital is explicitly associated with the quantum analog of the free-energy(i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of asingle mode of the electromagnetic radiation which, depending on the outcome ofthe betting, experiences attenuation or amplification processes which modellosses and winning events. The resulting stochastic evolution of the quantummemory resembles the dynamics of random lasing which we characterize within thetheoretical setting of Bosonic Gaussian channels. As in the classical KellyCriterion for optimal betting, we define the asymptotic doubling rate of themodel and identify the optimal gambling strategy for fixed odds andprobabilities of winning. The performance of the model are hence studied as afunction of the input capital state under the assumption that the latterbelongs to the set of Gaussian density matrices (i.e. displaced, squeezedthermal Gibbs states) revealing that the best option for the gambler is todevote all her/his initial resources into coherent state amplitude.

Kelly Betting with Quantum Payoff: a continuous variable approach