  The main purpose of this study is to introduce a semi-classical 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 free-energy
(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.
