#  Kelly Betting with Quantum Payoff: a continuous variable approach

Salvatore Tirone S. Tirone is with Scuola Normale Superiore, I-56126 Pisa,
Italy  Maddalena Ghio M. Ghio is with Scuola Normale Superiore, I-56126 Pisa,
Italy Giulia Livieri G. Livieri is with Scuola Normale Superiore, I-56126
Pisa, Italy  Vittorio Giovannetti V. Giovannetti is with NEST, Scuola Normale
Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy Stefano Marmi S.
Marmi is with Scuola Normale Superiore, I-56126 Pisa, Italy

###### Abstract

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 free-energy (ergotropy) 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. In particular, the evolution of the quantum memory
results in a stochastic trajectory 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.

###### Index Terms:

Quantum information, Gambling Theory, Quantum Communication, Bosonic Gaussian
Channels, Ergotropy, Squeezing, Kelly Criterion.

##  I Introduction

The problem of assessing the maximum growth of an optimal investment, or
equivalently of the maximization of the long-run interest rate, is one of the
central issues in quantitative finance. An especially simple but fundamental
example is provided by horse-race markets, i.e. markets with the property that
at every time step one of the assets pays off and all the other assets pay
nothing (the wealth invested is lost completely). Remarkably, horse-race
markets are very special cases of general markets, being, in a sense, the
extremal points of the distribution of asset returns [see, for instance 5,
23]. As such they have been extensively investigated: they possess many of the
usual attributes of financial markets but they also have the important
additional property that each bet has a well defined end point at which its
value becomes certain. This is rarely the case in finance, where asset values
depend on an intrinsically uncertain future [36]. In the 1950s, J. L. Kelly,
Jr. worked out a striking interpretation of the rate of transmission of
information over a noisy communication channel in the setting of gambling
theory [24]. In this context, the input symbols to the communication line were
interpreted as the outcomes of an uncertain event on which betting is
possible. Exploiting this construction, under “fair" odds and independent and
identically distributed (i.i.d.) horse-race markets assumptions (with no
transactions costs), Kelly proved the optimality of proportional betting
strategies; specifically, he proved that the asymptotic growth rate of the
cumulative wealth is maximal when the fraction of capital bet on each horse is
proportional to its true winning probability. Remarkably, over the years, both
theory and practice of the Kelly criterion in gambling and investment have
developed prolifically [27].
In this paper, we present a theory of classical betting with a quantum payoff
which investigates the behaviour of an economic agent whose initial capital is
represented by the value of a classical parameter encoded into the state of a
quantum memory that faces uncertainty under action of a stochastic
environment. In order to achieve this, we rely on the versatility of Bosonic
Gaussian channels (BGCs) to mimic the action of the random gains and losses
that affect the capital in a horse-race market.
BGCs play a fundamental role in quantum information theory [21, 35] where they
act as the proper counterparts of Gaussian channels of classical information
theory [see, for instance 22, 10, 17, and references therein]. Formally
speaking, they can be identified with the set of quantum evolutions which
preserve the “Gaussian character" of the transmitted signals and possess the
striking feature, heavily used in this paper, to be closed under composition
[32]. At the physical level BGCs describe the most relevant transformations an
optical pulse may experience when propagating in a noisy environment. In
particular, they provide a proper representation of attenuation and
amplification processes for continuous variable quantum systems [22, 17].
Differently from their classical counterparts, which can be always realized
without extra added noise effects, BGCs include vacuum fluctuation and display
a non-commutative character which has an intrinsic quantum mechanical origin.
As a result, the setting we introduce here is inherently different from
previously analyzed horse-market models [36, 20], giving rise to a non-trivial
generalization of classical betting procedures.
The connection between game theory and quantum information has a long history
dating back to the pioneering results presented in Refs. [28, 18, 13]. The
common denominator of these works is to grant players access to quantum
resources like entanglement or mere quantum coherence, that help them in
defining new strategies, thus enlarging the range of possible operations. Our
approach on the contrary relays on quantizing not the protocol itself but only
the resource that is used to encode the payoff of the game. Specifically, in
our model we identify the capital of our economic agent with the quantum
ergotropy functional ${\cal E}(\hat{\rho})$ [1] associated to the density
matrix $\hat{\rho}$ of a single Bosonic mode $A$ that is under the control of
the agent. Ergotropy plays an important role in many different contexts,
spanning from the characterization of optimal thermodynamical cycles [2, 6] to
energetic instabilities [31, 26], and to the study of quantum batteries models
[1, 4, 16]: it represents the maximum mean energy decrement obtainable when
forcing the quantum system $A$ to undergo through unitary evolutions induced
by cyclic external modulations of its Hamiltonian. Our choice of using the
ergotropy as a quantifier of the wealth of the agent is legitimated by the
fact that, in the Kelvin-Planck formulation of the second law of
thermodynamics, ${\cal E}(\hat{\rho})$ gauges the ultimate free-energy
associated with the state $\hat{\rho}$ of $A$, i.e. the maximum work that can
be extracted from it in the absence of heat dissipation. Moreover this choice
follows, at a quantum level, the long history of analogy making between
economics (and finance) and (stastical) thermodynamics, among recent papers on
the subject see, for instance, [33, 25], and references therein. Regarding the
uncertainty of the outcome of the horse race, we describe it with the random
selection of one among a collection of a finite set of one-mode BGCs operating
on $A$. Their action includes both an attenuation part – associated with the
partition of the capital determined by the strategy adopted by the gambler –
and an amplification part –associated instead with the odds offered by the
bookmaker. Each channel is therefore randomly selected and applied to $A$
according to the winning probability $p_{j}$ of the corresponding horse. The
whole procedure is iterated to reproduce the i.i.d. horse-race markets
framework, resulting into a stochastic trajectory for $A$ in which the state
of the system at the discrete time step $t$, is obtained by a concatenation of
$t$ elements of the selected BGCs set. The resulting payoff for the gambler is
therefore computed by evaluating the ergotropy functional on the final state
of the problem. Aim of our work is characterizing the probability distribution
of the resulting accumulated wealth and determining which, among all possible
choices of the input state $\hat{\rho}_{0}$ of $A$, provides the best
performance in terms of capital-gain for a given game strategy. It is worth
stressing that at the level of the mere internal energy of the system $A$,
irrespectively from the choice of the initial state $\hat{\rho}_{0}$, the
process (III.16) behaves essentially as its classical horse-market
counterpart, a part from a noisy contribution, where the BGCs are replaced by
standard betting operations. On the contrary, at the level of ergotropy, the
non-commutative nature of the involved quantum operation, combined with the
classical randomness of the bet, results in the injection of extra noise terms
which affect negatively the performance of the procedure, paving the way for a
non trivial optimization of the initial resource $\hat{\rho}_{0}$.
The rest of the paper is organized as follows: we start in Sec. II with a
brief review on BGCs. The Kelly model with quantum payoff is hence introduced
in Sec. III. In Sec. IV we study its statistical properties and present some
numerical analysis. Conclusions are presented in Sec. V. The manuscript also
contains an Appendix devoted to elucidate some technical aspects of the
problem.

##  II Preliminaries

###  II-A Gaussian states

Let $\mathcal{H}$ be a complex separable Hilbert space of infinite dimension
associated with one mode $A$ of the electromagnetic radiation (or equivalently
a 1-dimensional, quantum harmonic oscillator) of frequency $\nu$. The model is
fully described by the assignment of the canonical coordinate vector
$\hat{r}:=(\hat{q},\hat{p})^{T}$ and the quadratic Hamiltonian

$\hat{H}:=h\nu\;\frac{\hat{r}^{T}\cdot\hat{r}}{2}=h\nu\left(\frac{\hat{p}^{2}+\hat{q}^{2}}{2}\right)\;,$
(II.1)

expressed in terms of (rescaled) position and momentum operators $\hat{q}$ and
$\hat{p}$ which satisfy the (Bosonic) canonical commutation relation:
$[\hat{q},\hat{p}]=i$ ($h$ being the Planck constant). We let
$\mathcal{D}(\mathcal{H})$ be the set of the positive semidefinite operators
with trace $1$, and we call $\hat{\rho}$ a quantum state of $A$ if
$\hat{\rho}\in\mathcal{D}(\mathcal{H})$. For each element of such set we can
then associate a column vector of $\mathbb{R}^{2}$,

$m:=\text{Tr}[\hat{\rho}\;\hat{r}]\;,$ (II.2)

given by the expectation values of the canonical coordinates, and a $2\times
2$ covariance matrix $\sigma$ whose elements are

$\sigma_{ij}:=\text{Tr}[\hat{\rho}\\{(\hat{r}_{i}-m_{i}),(\hat{r}_{j}-m_{j})\\}],$
(II.3)

with $\\{\,,\\}$ denoting the anti-commutator operation, which describes
instead the second order moments of the coordinate distributions on
$\hat{\rho}$. Although no constraints hold on $m$, on $\sigma$ quantum
uncertainty relations impose the inequality $\sigma\geq\sigma_{2}$, with
$\sigma_{2}:=\tiny{\left[\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right]}$
being the second Pauli matrix. An exhaustive description of
$\mathcal{D}(\mathcal{H})$ is finally provided by the characteristic function
formalism which, given a density matrix $\hat{\rho}$ allows us to faithfully
represents it in terms of the following complex functional

$\phi(\hat{\rho};z):=\text{Tr}\left[\hat{\rho}\hat{V}(z)\right]\;,$ (II.4)

where $z:=(x,y)^{T}$ is a column vector of $\mathbb{R}^{2}$, and
$\hat{V}(z):=\exp[i\hat{r}^{T}\cdot z]$ is the Weyl operator of the system. In
this context, the density matrix $\hat{\rho}\in\mathcal{D}(\mathcal{H})$ is
thus said to be Gaussian if its associated characteristic function has a
Gaussian form, i.e. namely if we have

$\phi_{G}(\hat{\rho};z)=\exp[{-\tfrac{1}{4}z^{T}\sigma z+im^{T}\cdot z}]\;,$
(II.5)

with $m$ and $\sigma$ defined as in Eqs. (II.2) and (II.3) respectively [32].
At the physical level Gaussian states represent thermal Gibbs density matrices
of generic Hamiltonians which are (non-negative) quadratic forms of the
canonical vector $\hat{r}$. A convenient parametrization is given by the
following expression

$\hat{\rho}=\hat{V}(m)\hat{S}({\zeta})\hat{\rho}_{\beta}\hat{S}^{\dagger}(\zeta)\hat{V}^{\dagger}(m),$
(II.6)

where for $\beta\geq 0$, $\hat{\rho}_{\beta}:=\exp[-\beta\hat{H}]/Z_{\beta}$
defines a thermal state of the mode Hamiltonian (II.1) with inverse
temperature $\beta$ ($Z_{\beta}=\mbox{Tr}[\exp[-\beta\hat{H}]]$ being the
associated partition function) and where for $\zeta:=|\zeta|e^{i\varphi}$
complex we introduce the squeezing operator

$\displaystyle\hat{S}({\zeta}):=\exp\left[i\tfrac{|\zeta|}{2}\hat{r}^{T}R(\varphi)\hat{r}\right]\;,$
(II.7)

with $R(\varphi):=\cos(\varphi)\sigma_{1}-\sin(\varphi)\sigma_{3}$ being a
rotation matrix defined in terms of the first and third Pauli matrices
$\sigma_{1}:=\tiny{\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right]}$ and
$\sigma_{3}:=\tiny{\left[\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right]}$.
With this choice we get that the associated characteristic function of (II.6)
is in the form (II.5) with

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle(2{{{n}}}+1)[\cosh(2|\zeta|)\mathbb{I}$ (II.8)
$\displaystyle+\sinh(2|\zeta|)(\sin(\varphi)\sigma_{1}-\cos(\varphi)\sigma_{3})]\;,$

where $\mathbb{I}$ is the $2\times 2$ identity matrix and
${{{n}}}:=1/(e^{\beta h\nu}-1)$ is the average photon number of
$\hat{\rho}_{\beta}$. Special cases of Gaussian states are the coherent states
which often are referred to as semi-classical configurations of the electro-
magnetic field obtained by setting ${{{n}}}=\zeta=0$ in (II.6).

###  II-B One-mode Bosonic Gaussian channels

A BGC is a Completely Positive, Trace Preserving (CPTP) dynamical
transformation $\Phi$ [21, 35] defined by assigning a column vector $\lambda$
of $\mathbb{R}^{2}$, and two positive real matrices $X$ and $Y$ fulfilling the
constraint

$\displaystyle 2Y+X^{T}\sigma_{2}X\geq\sigma_{2}\;.$ (II.9)

When operating on a generic $\hat{\rho}\in\mathcal{D}(\mathcal{H})$, ${\Phi}$
sends it into a new density matrix ${\Phi}(\hat{\rho})$ whose characteristic
function is expressed by the formula

$\phi({\Phi}(\hat{\rho});z)=\phi(\hat{\rho};Xz)\exp\left[-\tfrac{1}{2}z^{T}Yz+i\lambda^{T}\cdot
z\right]\;.$ (II.10)

At the level of first and second moments, this mapping induces the
transformation

$\displaystyle m$ $\displaystyle\rightarrow$ $\displaystyle
m^{\prime}=X^{T}m+\lambda\;,$ $\displaystyle\sigma$ $\displaystyle\rightarrow$
$\displaystyle\sigma^{\prime}=X^{T}\sigma X+2Y\;,$ (II.11)

the couple $m$, $\sigma$ being associated with the input state $\hat{\rho}$,
and $m^{\prime}$, $\sigma^{\prime}$ being instead associated with the output
state ${\Phi}(\hat{\rho})$. These transformations inherit their name from the
notable fact that when applied to an arbitrary Gaussian input they will
produce Gaussian outputs. Most importantly for us, these processes are closed
under super-operator composition, meaning that given two BGC maps ${\Phi_{1}}$
and ${\Phi_{2}}$ identified respectively by the triples
$\lambda_{1},X_{1},Y_{1}$ and $\lambda_{2},X_{2},Y_{2}$, the transformation
${\Phi_{2}}\circ{\Phi_{1}}$ obtained by concatenating them in sequence is
still a BGC channel $\Phi_{3}$ characterized by the new triple

$\displaystyle\mu_{3}$ $\displaystyle:=$ $\displaystyle
X_{2}^{T}\lambda_{1}+\lambda_{2}\;,$ $\displaystyle X_{3}$ $\displaystyle:=$
$\displaystyle X_{1}X_{2}\;,$ $\displaystyle Y_{3}$ $\displaystyle:=$
$\displaystyle X_{2}^{T}Y_{1}X_{2}+Y_{2}\;.$ (II.12)

As already mentioned in the introduction, at physical level BGCs describe a
vast variety of processes. In what follows we shall focus on the special
subset of BGCs associated with amplification and attenuation events, possibly
accompanied by thermalization effects. These maps have $\lambda=0$ and $X$,
$Y$ proportional to $\mathbb{I}$. In view of the constraint (II.9) and
following a rather standard convention, we find it useful to express them in
terms of two non-negative constants $g,N$ as $X=g\;\mathbb{I}$,
$Y=\alpha\;\mathbb{I}$ with

$\displaystyle\alpha:=\left\\{\begin{array}[]{ll}|g^{2}-1|\left(N+\frac{1}{2}\right)&\mbox{for
$g\neq 1$}\;,\\\ N&\mbox{for $g=1$}\;,\end{array}\right.$ (II.15)

and use the notation $\Phi[g;{\alpha}]$ to represent the associated channel.
Notice that in this case Eq. (II.11) gets simplified as

$\displaystyle m$ $\displaystyle\rightarrow$ $\displaystyle
m^{\prime}=m\,g\;,$ (II.16) $\displaystyle\sigma$ $\displaystyle\rightarrow$
$\displaystyle\sigma^{\prime}=g^{2}\,\sigma+2\alpha\;\mathbb{I}\;,$ (II.17)

which allows us to interpret $g$ as a stretching parameter for the first
moments of the state, and $\alpha$ as an additive noise term. Furthermore from
(II.12) it trivially follows that the BGC subset formed by the maps
$\Phi[g;\alpha]$ is also closed under channel composition leading to the
following identity

$\displaystyle\Phi[{g_{2}};{\alpha_{2}}]\circ\Phi[{g_{1}};{\alpha_{1}}]=\Phi[{g_{3}};{\alpha_{3}}]$
(II.18)

with

$\displaystyle{g_{3}:=g_{2}g_{1}}\;,\qquad{\alpha_{3}:=g_{2}^{2}\alpha_{1}+\alpha_{2}}\;.$
(II.19)

For $g<1$, $\Phi[g;{\alpha}]$ defines a process where the mode $A$ interacts
dispersively with an external thermal Bosonic envinronment characterized by a
mean photon number $N$ (thermal attenuator channel). In particular for
$g=\eta\in[0,1[$ and $N=0$ (i.e. $\alpha=(1-\eta^{2})/2$) we get

$\displaystyle\Phi_{\eta}^{(\text{loss})}:=\Phi[\eta;{(1-\eta^{2})/2}]\;,$
(II.20)

which describes a pure loss of energy from the system, corresponding to the
evolution the mode $A$ experiences when passing through a medium (beam-
splitter) of transmissivity $\eta=g$. A special case of particular interest
for us is depicted in Fig. 1 where a collection of beam-splitters
(semitransparent mirrors [34]) are arranged to split an input optical signal
$\hat{\rho}$ into a different direction by properly mixing it with the vacuum.
In this construction, while the joint state emerging from the set up may
exhibit non trivial correlations, the local density matrices associated to the
various output ports are described as
$\Phi_{\eta}^{(\text{loss})}[\hat{\rho}]$ with $\eta$ being the corresponding
effective transmissivity.

Figure 1: Schematic of an optical splitter formed by a collection of beam-
splitters (blue elements) that induce a fractioning of the input state into a
collection of outputs which, locally, are associated with the action of purely
lossy channels (II.20). In the scheme $\tau_{j}$ are the transmissivities of
the individual beam-splitters, while $\eta_{j}$ are the resulting
transmissivities that emerge from their combinations which, by construction,
fulfil the normalization condition $\sum_{j}\eta_{j}^{2}=1$.

For $g>1$, $\Phi[g;{\alpha}]$ describes instead the interaction of $A$ and the
thermal bath in terms of an amplification event (thermal amplifier). In this
case for $g=k>1$ and $N=0$ (i.e. $\alpha=(k^{2}-1)/2$) the transformation

$\displaystyle\Phi_{k}^{(\text{amp})}:=\Phi[k;{(k^{2}-1)/2}]\;$ (II.21)

defines a purely amplifying process induced by the interaction between $A$ and
the vacuum mediated by a non-linear optical crystal [34]. Finally, we observe
that in the special case where $g=1$, the transformation

$\displaystyle\Phi_{1}^{(\text{add})}:=\Phi[1,{N}]\;,$ (II.22)

corresponds to the noise-additive channel which describes a random Gaussian
displacement in the phase space [22].

###  II-C Average energy and ergotropy

The mean energy of a generic state $\hat{\rho}$ of the mode $A$ is given by

$E(\hat{\rho}):=\text{Tr}[\hat{\rho}\hat{H}]=\frac{\text{Tr}(\sigma)}{4}+\frac{m^{2}}{2}\;,$
(II.23)

where $m^{2}=m^{T}\cdot m$ is the square norm of $m$, where in the second
identity we made explicit use of Eqs. (II.2) and (II.3), and where without
loss of generality we set $h\nu=1$. For a Gaussian state, exploiting the
parametrization introduced in Eq. (II.6), this can be cast in a formula that
highlights the squeezing, displacement, and thermal contributions to the final
result, i.e.

$\displaystyle E(\hat{\rho})=\frac{(2{{{n}}}+1)\cosh(2|\zeta|)+{m}^{2}}{2}\;.$
(II.24)

Equations (II.23) and (II.24) represent the total energy stored in
$\hat{\rho}$ which one could extract from the $A$, e.g. by putting it in
thermal contact with an external zero-temperature environment. Yet, as pointed
out in Ref. [3, 9], not all such energy will be nicely converted into useful
work: part of $E(\hat{\rho})$ will indeed necessarily emerge as dissipated
heat. The quantity that instead properly quantifies the maximum amount of
extractable work is given by the ergotropy. At variance with $E(\hat{\rho})$,
the latter is a non linear functional of $\hat{\rho}$ that we can compute as

$\mathcal{E}(\hat{\rho}):=E(\hat{\rho})-\min_{\hat{U}}E(\hat{U}\,\hat{\rho}\,\hat{U}^{\dagger}),$
(II.25)

where the minimization is performed over all unitary transformations
$\hat{U}$. For a one-mode Bosonic Gaussian state a closed expression in terms
of $\sigma$ and $m$ is known which we shall use in the following, i.e. [16]

$\displaystyle\mathcal{E}(\hat{\rho})$ $\displaystyle=$
$\displaystyle\frac{\text{Tr}(\sigma)}{4}+\frac{{m}^{2}}{2}-\frac{\sqrt{\text{Det}(\sigma)}}{2}$
(II.26) $\displaystyle=$
$\displaystyle\frac{(2{{{n}}}+1)(\cosh(2|\zeta|)-1)+{m}^{2}}{2}\;.$

This equation shows that the gap between $E(\hat{\rho})$ and
$\mathcal{E}(\hat{\rho})$ only depends upon the thermal component of
$\hat{\rho}$, i.e.

$\displaystyle E(\hat{\rho})-\mathcal{E}(\hat{\rho})={{{n}}}+1/2\;,$ (II.27)

reaching its minimum value for pure (squeezed coherent) states (i.e.
${{{n}}}=0$) where it matches the vacuum energy level $1/2$. Notice also that
states $\hat{\rho}$ which are characterized by same ergotropy value form a
continuous 2D manifold in the phase space associated with the variables $m$,
$\zeta$, and ${{{n}}}$ – see Fig. 2.

Figure 2: Geometric representation of the iso-ergotropic 2D manifold
associated with gaussian states (II.6) with ${\cal E}(\hat{\rho})=50$ in the
phase space defined by the parameters $x:=m^{2}$, $y:=(\cosh(2|\zeta|)-1)$,
$z:=2{{{n}}}+1$. The green dot represents the pure coherent element of the set
($y=0$, $z=1$), while the red dot the pure squeezed element ($x=0$, $z=1$).
The orange line at the bottom represents the pure states of the manifold.

##  III The playing field

Motivated by the Kelly Criterion for optimal betting and the versatility of
the one-mode BGCs, we present here a theory of classical betting with a
quantum payoff. First, in Subsection III-A we briefly review the conventional
Kelly setting. Then in Subsection III-B, we introduce the repeated-quantum-
betting scheme as the iteration of single quantum bets, each bet being a
probabilistic transmission of the quantum state of a single Bosonic mode
through a one-mode BGC. We hence embed the protocol into a rigorous
mathematical framework. Not only this allows for a more compact representation
of the procedure over time, but also it enables to see the iterative scheme
from the higher perspective of random dynamical systems: the action of a
family of quantum transformations, corresponding to one-mode BGCs in the
procedure, over a phase space of (single Bosonic mode) quantum states. Finally
in Subsection IV we construct a parallel between the quantum and the classical
Kelly criterion.

###  III-A The Kelly Criterion for optimal betting

Imagine that one has $J$ horses competing in a race, each characterized by a
winning probability $\\{p_{j}\\}_{j=1,\ldots,J}$, $\sum_{j=1}^{J}p_{j}=1$.
Formally speaking the $p_{j}$’s define a distribution of a random variable $Z$
taking values in the alphabet $\left\\{1,\ldots,J\right\\}$ with probability
$\mathcal{P}(Z=j)=p_{j}$ (symbols $j$ of the alphabet representing the horses
whereas the variable $Z$ the winning horse). If a gambler invests one dollar
on horse $j$ then she/he receives $o_{j}$ dollars if horse $j$ wins and zero
dollars if horse $j$ loses: the game is then said to be under unfair odds
condition if $\sum_{j=1}^{J}\frac{1}{o_{j}}>1$; on the contrary, if
$\sum_{j=1}^{J}\frac{1}{o_{j}}=1$ or $\sum_{j=1}^{J}\frac{1}{o_{j}}<1$ we say
that the odds are fair  or super-fair respectively. In what follows we shall
always focus on these last two cases for which one can show that the best
gaming option for the gambler is always to distribute all of his/her wealth
across the horses. Accordingly we call $b_{j}$ the fraction of the gambler’s
wealth invested in horse $j$, $b_{j}\geq 0$ and $\sum_{j=1}^{J}b_{j}=1$. We
consider henceforth a sequence of $t$ repeated gambles on this race under
i.i.d. assumptions, i.e. requiring that neither the probabilities $p_{j}$s,
the odds $o_{j}$s, and the fractions $b_{j}$s vary from betting stage to
betting stage. The wealth of the gambler during such a sequence is a random
variable defined by the expression

$\displaystyle\bar{S}_{t}:=S_{j_{t}}\cdots S_{j_{2}}S_{j_{1}}\;,$ (III.1)

where for $\ell\in\\{1,\cdots,t\\}$, $j_{\ell}\in\left\\{1,\ldots,J\right\\}$
represents the horse that has won the $\ell$-th race delivering a payoff
$S_{j_{\ell}}=o_{j_{\ell}}b_{j_{\ell}}$, the corresponding joint probability
being

$\displaystyle\bar{p}_{t}:=p_{j_{t}}\cdots p_{j_{2}}p_{j_{1}}\;,$ (III.2)

(to help readability hereafter we adopt the compact convention of dropping
explicit reference of the functional dependence upon the indexes ${j_{t}}$,
$\cdots$, ${j_{2}}$, ${j_{1}}$ of quantities evaluated along a given
stochastic trajectory). In the asymptotic limit of large $t$, the value of
$\bar{S}_{t}$ can be directly linked to the doubling rate of the horse race,
i.e. the quantity

$\displaystyle
W(\mathbf{b},\mathbf{o},\mathbf{p}):=\sum_{j=1}^{J}p_{j}{\log_{2}}(o_{j}b_{j})\;,$
(III.3)

where $\mathbf{p}:=(p_{1},p_{2},\cdots,p_{J})$,
$\mathbf{o}:=(o_{1},o_{2},\cdots,o_{J})$ and
$\mathbf{b}:=(b_{1},b_{2},\cdots,b_{J})$ are the three vectors which, under
i.i.d. assumptions fully define the model: the first assigning the stochastic
character of the race, the second the selected odds of the broker, and the
last one the strategy the gambler adopts in placing his bets. More
specifically, the connection between $\bar{S}_{t}$ and
$W(\mathbf{b},\mathbf{o},\mathbf{p})$ is provided by the strong Law of Large
Numbers (LLN) which in the present case establishes that for sufficiently
large $t$ we have $\bar{S}_{t}\approx 2^{t\,W(\mathbf{b},\mathbf{p})}$ almost
surely, or more precisely that

${\Pr}\left[\lim_{t\rightarrow\infty}\frac{{\log_{2}}\bar{S}_{t}}{t}=W(\mathbf{b},\mathbf{o},\mathbf{p})\right]=1\;,$
(III.4)

see for instance [11], Theorem 6.6.1. Exploiting this result, one can now
identify the optimal gambling strategy that ensures the largest asymptotic
wealth increase, by selecting $\mathbf{b}$ which allows
$W(\mathbf{b},\mathbf{o},\mathbf{p})$ to reach its maximum value
$W^{*}(\mathbf{o},\mathbf{p})$ for fixed $\mathbf{o}$ and $\mathbf{p}$. Such
maximum is achieved by the proportional gambling scheme where
$\mathbf{b}=\mathbf{p}$ (Kelly criterion) so that

$W^{*}(\mathbf{o},\mathbf{p})=\sum_{j=1}^{J}p_{j}{\log_{2}}(o_{j}p_{j})=\sum_{j=1}^{J}p_{j}{\log_{2}}o_{j}-\mathcal{H}(\mathbf{p})\;,$
(III.5)

with $\mathcal{H}(\mathbf{p}):=\sum_{j=1}^{J}p_{j}\log_{2}p_{j}$ the Shannon
entropy of the Bernoulli distribution defined by the vector $\mathbf{p}$ (see
[11], Theorem 6.1.2).

###  III-B BGC betting model

As in the previous section we consider a horse-race model where at each step
of the gambling scheme $J$ horses compete, their probability of winning being
defined by a Bernoulli distribution $\\{p_{j}\\}_{j=1,\cdots,J}$. At variance
with the typical Kelly setting we assume that the capital the gambler invests
in the game is represented not by conventional money, but instead by the
ergotropy (II.25) he/she stores into a single Bosonic mode $A$ initially
prepared into the density matrix $\hat{\rho}_{0}$.

To place his/her bet, at each race the gambler divides his capital exploiting
an optical splitter analogous to one shown in Fig. 1 which distributes the
input mode $A$ into $J$ independent outputs characterized by effective
transmissivities $\eta_{j}\in]0,1[$, each associated with an individual horse
and fulfilling the normalization condition

$\displaystyle\sum_{j=1}^{J}\eta_{j}^{2}=1\;.$ (III.6)

This implies that the capital the gambler is placing on the horse $j$ is
locally stored into the density matrix obtained by applying the purely lossy
BGC mapping $\Phi_{\eta_{j}}^{(\text{loss})}$ of Eq. (II.20) to the entry
capital of the betting stage. The parameters $\eta_{j}$ clearly reflect the
gambler game strategy and play the role of the fractions $b_{j}$ of the
original Kelly’s model (more precisely, due to constraint (III.6) the exact
correspondence is between the $b_{j}$s and the squares $\eta_{j}^{2}$ of the
network transmissivities). In case the $j$-th horse win, the corresponding
output port will hence undergo to a coherent amplification event which pumps
up the energy level of the system via the gain parameter $k_{j}>1$ and is
returned to the gambler, the other sub-modes being instead absorbed away. At
the level of $A$ this results in a transformation described by the pure
amplifier BGC mapping $\Phi_{k_{j}}^{(\text{amp})}$ introduced in Eq. (II.21).
The parameters $k_{j}$ reflect the odds parameters $o_{j}$ of the classical
model. Again the exact correspondence is between the $o_{j}$s and the squares
$k^{2}_{j}$ of the amplification amplitudes which allows us to translate the
fair odds and super-fair odds conditions into the requirements

$\displaystyle\sum_{j=1}^{J}\frac{1}{k^{2}_{j}}$ $\displaystyle=$
$\displaystyle 1\;,\qquad(\mbox{fair odds})$ (III.7)
$\displaystyle\sum_{j=1}^{J}\frac{1}{k^{2}_{j}}$ $\displaystyle<$
$\displaystyle 1\;,\qquad(\mbox{super-fair odds})$ (III.8)

respectively, which we shall assume hereafter.

In summary, after each betting stage the gambler receives back the state of
his/here single mode resource quantum memory, transformed via the combined
action of a purely lossy map and a purely amplifier, i.e. the BGC map which
according to (II.18) writes as

$\Phi^{\text{}}_{{j}}:=\Phi[{{g_{j}}};{\alpha_{j}}]=\Phi_{{k_{j}}}^{(\text{amp})}\circ\Phi_{{\eta_{j}}}^{(\text{loss})},$
(III.9)

where now, for all $j\in\\{1,\cdots,J\\}$, the stretching and noise terms are
given by

$\displaystyle g_{j}$ $\displaystyle:=$ $\displaystyle k_{j}\eta_{j}\;,$
(III.10) $\displaystyle{\alpha_{j}}$ $\displaystyle:=$
$\displaystyle\frac{{k_{j}}^{2}(1-{\eta_{j}}^{2})+{k_{j}}^{2}-1}{2}\;.$
(III.11)

The composition property (II.18) also lends itself well to the framework of
repeated gambles described in Sec. III-A.

Figure 3: Schematic of the quantum payoff betting protocol: Panel (a) single
betting stage; Panel (b) concatenation of multiple bettings. In the figures
the blue rectangles represents optical splitters analogous to the one depicted
in Fig. 1; the first red arrow represent the state of $A$ which encode the
invested capital whereas the second one is the one associated with the winning
stake. Black arrows represent the losing fractions of capital, while the
winning (green) one $\eta_{j}$ is amplified by the odd $k_{j}$; Panel (c)
schematic representation of the associated lumped-element model scheme
describing the propagation of e.m. modes inside a multi-layer material (blue
area): here the red crosses represent complete photon absorption, the the
green diamond stimulated emission process, and the yellow elements multi-modes
scattering events.

Specifically, assuming the initial state of $A$ to be described by the density
matrix $\hat{\rho}_{0}$, after the fist betting event, with probability
$p_{j_{1}}$, the gambler will get the state

$\displaystyle\hat{\rho}_{0}\longrightarrow\hat{\rho}_{1}:=\Phi^{\text{}}_{j_{1}}(\hat{\rho}_{0})\;,$
(III.12)

which he/she will use to place the second bet (notice that following the same
convention introduced in the previous section, the functional dependence of
$\hat{\rho}_{1}$ upon the index $j_{1}$ is left implicit). Accordingly, with
conditional probability $p_{j_{2}}$ the second horse race will induce the
following mapping on $A$

$\displaystyle\hat{\rho}_{1}\longrightarrow\hat{\rho}_{2}:=\Phi^{\text{}}_{j_{2}}(\hat{\rho}_{1})\;.$
(III.13)

At the level of input density matrix $\hat{\rho}_{0}$, Eq. (III.13)
corresponds to the stochastic mapping

$\displaystyle\hat{\rho}_{0}\longrightarrow\hat{\rho}_{2}=\bar{\Phi}_{2}(\hat{\rho}_{0})\;,$
(III.14)

where

$\displaystyle\bar{\Phi}_{2}:=\Phi^{\text{}}_{j_{2}}\circ\Phi^{\text{}}_{j_{1}}\;,$
(III.15)

is the composite BGC map obtained by merging together $\Phi^{\text{}}_{j_{2}}$
and $\Phi^{\text{}}_{j_{1}}$ and occurs with joint probability
$\bar{p}_{2}=p_{j_{2}}p_{j_{1}}$. More generally, indicating with
$\hat{\rho}_{j_{1},\cdots,j_{t}}$ the state of $A$ after $t$ steps along a
given stochastic realization of the horse race described by a winning string
of event $j_{1},j_{2},\cdots,j_{t}$, the transformation (III.13) gets replaced
by the mapping

$\displaystyle\hat{\rho}_{t-1}\longrightarrow\hat{\rho}_{t}:=\Phi^{\text{}}_{j_{t}}(\hat{\rho}_{t-1})\;,$
(III.16)

which occurs with probability $p_{j_{t}}$. Similarly the transformation
(III.14) becomes

$\displaystyle\hat{\rho}_{0}\longrightarrow\hat{\rho}_{t}:=\bar{\Phi}_{t}(\hat{\rho}_{0})\;,$
(III.17)

where now the $t$-fold mapping $\bar{\Phi}_{t}$ is given by

$\displaystyle\bar{\Phi}_{t}:=\Phi^{\text{}}_{j_{t}}\circ\cdots\circ\Phi^{\text{}}_{j_{2}}\circ\Phi^{\text{}}_{j_{1}}\;,$
(III.18)

and occurs with joint probability $\bar{p}_{t}$ defined as in Eq. (III.2) –
the whole process being visualized in Fig. 3. We notice that from Eq. (III.9)
and the general composition rule (II.18) it follows that

$\bar{\Phi}_{t}=\Phi[\bar{g}_{t};\bar{\alpha}_{t}]\;,$ (III.19)

where the parameter $\bar{g}_{t}$ is given by

$\displaystyle\bar{g}_{t}:=g_{j_{t}}\cdots g_{j_{2}}g_{j_{1}}\;,$ (III.20)

while $\bar{\alpha}_{t}$ is defined by the recursive formula

$\displaystyle\left\\{\begin{array}[]{l}\bar{\alpha}_{t=1}:={\alpha}_{j_{1}}\;,\\\
\\\
\bar{\alpha}_{t}=g_{j_{t}}^{2}\,\bar{\alpha}_{t-1}+\alpha_{j_{t}}\;,\end{array}\right.$
(III.24)

which, as shown in Lemma .1 of the Appendix, admits the following solution

$\bar{\alpha}_{t}=\bar{g}^{2}_{t}\sum_{\ell=1}^{t}\frac{\alpha_{j_{\ell}}}{\bar{g}_{\ell}^{2}}.$
(III.25)

Before proceeding any further it is worth stressing that the map
$\bar{\Phi}_{t}$ defined above describes the state of $A$ after $t$ steps
along a given trajectory of the betting process. Taking the weighted mean of
this super-operators with probabilities $\bar{p}_{t}$ gives instead the
transformation which defines the evolution of the system irrespectively from
the history of the betting process. Explicitly this is given by

$\bar{\Psi}_{t}:=\sum_{j_{1},\cdots,j_{t}}p_{j_{1}}\cdots
p_{j_{t}}{\Phi}_{j_{1}}\circ\cdots\circ{\Phi}_{j_{t}}=\underbrace{\Psi\circ\cdots\circ\Psi}_{\text{$t$-times}}\;,$
(III.26)

with $\Psi$ the CPTP map defined by the convex convolution

$\displaystyle\Psi:=\sum_{j=1}^{J}{p}_{j}{\Phi}_{j}\;,$ (III.27)

which does not necessarily correspond to a BGC. It is finally worth stressing
that the scheme provides an effective, lumped-element model description of
propagation of light modes inside a multi-layer material where, depending on
the layer, they get randomly amplified (as in a stimulated emission process)
or attenuated (as in an absorption process), the stochastic channel
$\bar{\Phi}_{t}$ describing the system evolution conditioned to the
realization of a specific history of events – see panel (c) of Fig. 3.

###  III-C Quantum payoff of the BGC model

From the above results it is now easy to express the energy associated with
the state of $A$ at a generic step $t$ along a given trajectory. Indeed from
(II.16) and (II.17) it follows that the first order expectation vector
$\bar{m}_{t}$ and the covariance matrix $\bar{\sigma}_{t}$ of $\hat{\rho}_{t}$
can be written as

$\displaystyle\bar{m}_{t}$ $\displaystyle=$ $\displaystyle\bar{g}_{t}\;m\;,$
(III.28) $\displaystyle\bar{\sigma}_{t}$ $\displaystyle=$
$\displaystyle\bar{g}_{t}^{2}\sigma+2\bar{\alpha}_{t}\;\mathbb{I}\;,$ (III.29)

where $m$ and $\sigma$ are their corresponding counterparts associated with
the input state $\hat{\rho}_{0}$. Therefore, the mean energy of the system is
now given by

$\bar{E}_{t}:=E(\hat{\rho}_{t})=\bar{g}_{t}^{2}E_{0}+\bar{\alpha}_{t}=\bar{g}_{t}^{2}(E_{0}+\bar{\gamma}_{t})\;,$
(III.30)

where we introduced the quantity

$\displaystyle\bar{\gamma}_{t}:=\frac{\bar{\alpha}_{t}}{\bar{g}_{t}^{2}}=\sum_{\ell=1}^{t}\frac{\alpha_{j_{\ell}}}{\bar{g}_{\ell}^{2}},$
(III.31)

and where $E_{0}:=E(\hat{\rho}_{0})$ is the mean energy of the input state.
Parametrizing the latter as in (II.24), Eq. (III.30) can then be casted in the
form

$\displaystyle\bar{E}_{t}=\bar{g}_{t}^{2}\left(\frac{2{{{n}}}+1}{2}\cosh(2|\zeta|)+\bar{\gamma}_{t}+\frac{m^{2}}{2}\right)\;.$
(III.32)

Similarly, from Eqs. (II.27), (III.28), and (III.29), we can express the
ergotropy $\bar{\mathcal{E}}_{t}:=\mathcal{E}(\hat{\rho}_{t})$ of
$\hat{\rho}_{t}$ as

$\displaystyle\bar{\mathcal{E}}_{t}$ $\displaystyle=$
$\displaystyle\bar{E}_{t}-\bar{n}_{t}-1/2\;,$ (III.33)

with $\bar{n}_{t}\geq 0$ being the thermal contribution to the total photon
number of the final state computed as

$\bar{n}_{t}:=\bar{g}_{t}^{2}\frac{\sqrt{(2{{{n}}}+1)^{2}+4\bar{\gamma}_{t}(2{{{n}}}+1)\cosh(2|\zeta|)+4\bar{\gamma}_{t}^{2}}}{2}-\frac{1}{2}\;.$
(III.34)

In our construction Eq. (III.33) represents the payoff of the gambler after
$t$ steps. By re-organizing the various contributions, it can be conveniently
casted into the form

$\displaystyle\bar{\mathcal{E}}_{t}$ $\displaystyle=$
$\displaystyle\bar{g}_{t}^{2}\bigg{(}{\mathcal{E}}_{0}-\bar{\Delta}_{t}\bigg{)}\;,$
(III.35)

where

$\displaystyle{\mathcal{E}}_{0}:={\mathcal{E}(\hat{\rho}_{0})}=\frac{(2{{{n}}}+1)(\cosh(2|\zeta|)-1)+{m}^{2}}{2}\;,$
(III.36)

is the initial value of the ergotropy, i.e. the initial capital invested by
the player. In the above expression

$\bar{\Delta}_{t}:=\frac{(2{{{n}}}+1)}{2}\Big{(}{\sqrt{1+4\bar{\Gamma}_{t}\cosh(2|\zeta|)+4\bar{\Gamma}_{t}^{2}}}-{(1+2\bar{\Gamma}_{t})}\Big{)}\;,$
(III.37)

is a non-negative quantity which, while not being directly linked to ${\cal
E}_{0}$, is an explicit functional of the selected input state
$\hat{\rho}_{0}$ (here
$\bar{\Gamma}_{t}:=\frac{\bar{\gamma}_{t}}{2{{{n}}}+1}$). This fact has
profound consequences as it implies that, differently from the original
Kelly’s scheme, in our setting the final payoff $\bar{\mathcal{E}}_{t}$ is not
just proportional to the initial invested capital ${\cal E}_{0}$, but also
depends in a non trivial way on the type of input state $\hat{\rho}_{0}$ that
was selected to “carry" the value of such capital. Accordingly in discussing
the performance of the betting scheme we are now facing an extra layer of
optimization where the gambler, beside selecting the transmissivities
$\eta_{j}$ and deciding the value of ${\cal E}_{0}$ he/she wants to invest in
the game, has also the possibility of deciding which one of the input states
$\hat{\rho}_{0}$ that belongs to the same iso-ergotropy manifold (see of Fig.
2) he/she wants to adopt: different choices will indeed result in different
$\bar{\Delta}_{t}$ and hence in different payoff values
$\bar{\mathcal{E}}_{t}$.

###  III-D Input state optimization

Let start observing that from Eq. (III.35) it follows that if the input state
of the system $\hat{\rho}_{0}$ is a purely Gibbs thermal state of the system
Hamiltonian (i.e. if $m=\zeta=0$) then the capital gain is strictly null at
all time steps, i.e. $\bar{\mathcal{E}}_{t}=0$ for all $t$: this implies that
in order to have at a chance of getting some positive payoff, the player needs
to invest at least a non-zero capital (a very reasonable contraint). Notice
also that if we restrict to a semiclassical case in which $\hat{\rho}_{0}$ is
described by a noisy coherent state (i.e. for $\zeta=0$), we get
$\bar{\Delta}_{t}=0$ and hence

$\displaystyle\bar{\mathcal{E}}_{t}=\bar{g}_{t}^{2}\frac{m^{2}}{2}=\bar{g}_{t}^{2}{\mathcal{E}}_{0}\;,$
(III.38)

implying that under this assumption the capital evolves as the classical case
of a geometric game. Departures from Eq. (III.38) can instead be observed when
squeezing is present in the initial state of the system paving the way to the
optimization problem mentioned at the end of the previous paragraph, i.e.
deciding what is the best choice of $\hat{\rho}_{0}$ to be employed in the
betting scheme for assigned values of $\eta_{j}$ and ${\cal E}_{0}$.

To tackle this issue we find it useful to define two different figures of
merit, i.e. the renomalized ergotropy functional $\bar{\mu}_{t}$ which, as
$\bar{\mathcal{E}}_{t}$ gauges the absolute payoff, and the output ergotropy
ratio $\bar{r}_{t}$, aimed instead to characterize the amount of resources
wasted in the process. The first quantity is simply obtained by rescaling the
payoff capital $\bar{\mathcal{E}}_{t}$ by the term
$\bar{g}^{2}_{t}\mathcal{E}_{0}$ in order to contrast the potential divergent
behaviour of the former, i.e.

$\displaystyle\bar{\mu}_{t}:=\frac{\bar{\mathcal{E}}_{t}}{\bar{g}^{2}_{t}\mathcal{E}_{0}}=1-\frac{\bar{\Delta}_{t}}{\mathcal{E}_{0}}\;.$
(III.39)

For fixed choices of $\eta_{j}$ and ${\cal E}_{0}$, $\bar{\mu}_{t}$ exhibits
the same dependence upon $\bar{\mathcal{E}}_{t}$ upon the input state
$\hat{\rho}_{0}$, with the main advantage of being bounded from above by 1 (on
the contrary, as we already mentioned, $\bar{\mathcal{E}}_{t}$ can explode).
The second quantity instead is defined as the ratio between the ergotropy and
the mean energy of the state $\hat{\rho}_{t}$, i.e.

$\displaystyle\bar{r}_{t}$ $\displaystyle:=$
$\displaystyle\frac{\bar{\mathcal{E}}_{t}}{\bar{E}_{t}}=\frac{\bar{\mathcal{E}}_{t}}{\bar{\mathcal{E}}_{t}+\bar{{{n}}}_{t}+1/2}$
$\displaystyle=$ $\displaystyle
r_{0}+(1-r_{0})\left(1-\frac{\sqrt{1+4\bar{\Gamma}_{t}\cosh(2|\zeta|)+4\bar{\Gamma}_{t}^{2}}}{1+2(1-r_{0})\bar{\Gamma}_{t}}\right)\;,$

where

$\displaystyle r_{0}$ $\displaystyle:=$
$\displaystyle\frac{\mathcal{E}_{0}}{E_{0}}=\frac{\mathcal{E}_{0}}{\mathcal{E}_{0}+{{{n}}}+1/2}\;,$
(III.41)

is the value of the ratio computed on the input state $\hat{\rho}_{0}$. Notice
that by construction $\bar{r}_{t}$ and ${r}_{0}$ are bounded quantities that
cannot exceed the value 1. Furthermore irrespectively from the parameters of
the model and from the outcome of the stochastic process, $\bar{r}_{t}$ cannot
be larger than its initial value $r_{0}$: indeed an upper bound for
$\bar{r}_{t}$ can be obtained by setting equal to zero the squeezing parameter
of the initial state (i.e. $\zeta=0$) which leads to

$\displaystyle\bar{r}_{t}\leq\frac{r_{0}}{1+2(1-r_{0})\bar{\Gamma}_{t}}\leq
r_{0}\;,$ (III.42)

the rightmost inequality following as a consequence of the positivity of
$\bar{\Gamma}_{t}$. Equation (III.42) implies that, even though the gambler
has a chance of increasing its wealth (i.e the value of the ergotropy of the
mode $A$), in the quantum model we are considering here this always occurs at
the cost of an inevitable waste of resources.

It goes without mentioning that, as in the case of the payoff capital
$\bar{\cal E}_{t}$, the figures of merit introduced above are stochastic
quantities which strongly depend on the random sequence of betting events.
Interestingly enough however a mere analysis of the functional dependence of
$\bar{\mu}_{t}$, $\bar{r}_{t}$ upon the input data of the problem, allows us
to draw some important conclusions even without carrying over a full
statistical analysis of the process (a task which, as we shall see in the next
section, is rather complex). For this purpose we first notice that from Eq.
(III.38) it follows that setting $\zeta=0$, i.e. using input states
$\hat{\rho}_{0}$ which have no squeezing, is the proper choice to bust
$\bar{\mu}_{t}$ to its upper bound 1. This choice also ensures the saturation
of the first inequality in Eq. (III.42), leading us to an output ergotropy
ratio that expressed in terms of ${\cal E}_{0}$ can be written as

$\displaystyle\bar{r}_{t}=\frac{{\cal E}_{0}}{{\cal
E}_{0}+\bar{\gamma}_{t}+{{{n}}}+1/2}\leq\frac{{\cal E}_{0}}{{\cal
E}_{0}+\bar{\gamma}_{t}+1/2}\;,$ (III.43)

the last inequality being saturated by setting ${{{n}}}=0$. Accordingly we can
conclude that, for fixed values of ${\cal E}_{0}$ and game strategy
$\eta_{j}$, the best choice for the input density matrix $\hat{\rho}_{0}$ is
the pure coherent input state of the selected iso-ergotropy manifold (i.e. the
green dot of Fig. 2) since, irrespectively from the specific outcomes of the
stochastic process that drives the system dynamics, this will ensure the
saturation of the functional upper bounds for both the two figures of merit we
have introduced.

##  IV Statistical analysis

In Kelly’s language each bet in our gambling problem is identified with the
drawing of a Bernoulli random variable and the overall gain/loss of each bet
is identified with the parameters $g_{j}$ and ${\alpha_{j}}$. In particular,
in the new playing field, the analogous of the gambler’s wealth after $t$
races, $\bar{S}_{t}$ of Sec. III-A, is the quantity
$\bar{\mathcal{E}}_{t}/\mathcal{E}_{0}$ of Eq. (III.33). As discussed in the
previous section the latter is computed through the action of the BGC
$\bar{\Phi}_{t}$ of Eq. (III.19) which determines the evolution of the quantum
mode $A$ via Eq. (III.17) and which is fully characterized by the parameters
$\bar{g}_{t}$ and $\bar{\alpha}_{t}$. As anticipated in Eq. (III.38), in the
absence of input squeezing (i.e. $\zeta=0$) the first of these two quantities,
i.e. $\bar{g}_{t}$, represents the amplification of the input signal in the
model, the gambler’s wealth being directly proportional to $\bar{g}^{2}_{t}$.
The presence of squeezing in the initial state, makes the connection between
$\bar{g}_{t}$ and the payoff more complex (see Eq. (III.33)) still a direct
comparison between Eq. (III.20) and (III.1) makes it explicit that
$\bar{g}_{t}$ shares the same statistical properties of the function
$\bar{S}_{t}$ of the classical setting. In particular also in this case we can
use the strong LLN to replace Eq. (III.4) with

${\Pr}\left[\lim_{t\rightarrow\infty}\frac{{\log_{2}}\bar{g}_{t}}{t}=G(\boldsymbol{\eta},\mathbf{k},\mathbf{p})\right]=1\;,$
(IV.1)

where now

$\displaystyle
G(\boldsymbol{\eta},\mathbf{k},\mathbf{p}):=\sum_{j=1}^{J}p_{j}{\log_{2}}(g_{j})=\sum_{j=1}^{J}p_{j}{\log_{2}}(k_{j}\eta_{j})\;,$
(IV.2)

substitute the doubling rate $W(\mathbf{b},\mathbf{o},\mathbf{p})$ of Eq.
(III.3) – here $\boldsymbol{\eta}:=(\eta_{1},\cdots,\eta_{J})$,
$\mathbf{k}:=(k_{1},\cdots,k_{J})$ and $\mathbf{p}:=(p_{1},\cdots,p_{J})$.
Accordingly we can claim that almost surely $\bar{g}_{t}\approx
2^{tG(\boldsymbol{\eta},\mathbf{k},\mathbf{p})}$ for $t$ sufficiently large.
Furthermore simple algebra allows one to show that the optimal energy
splitting strategy which yields the maximum doubling rate increase

$\displaystyle
G^{*}(\mathbf{k},\mathbf{p}):=\max_{\boldsymbol{\eta}}G(\boldsymbol{\eta},\mathbf{k},\mathbf{p})\;,$
(IV.3)

is obtained by setting

$\displaystyle\eta^{2}_{j}=p_{j}\;,$ (IV.4)

which represents precisely the Kelly criterion for the optimal choice of the
betting strategy.

Let us now turn our attention to the second stochastic quantity which defines
the model, i.e. the parameter $\bar{\alpha}_{t}$. This term measures the noise
generated in the process due to the random application of quantum BGC, and it
appears thanks to the non-commutative nature of our quantum model. From a
close inspection of Eq. (III.24) we observe that $\bar{\alpha}_{t}$ is
generated by the iteration of a family of random Lipschitz maps [8], precisely
affine random maps with random slope and random intercept. For instance in the
case where the total number of horses in the race is $J=2$ with $p_{1}=p$,
$p_{2}=1-p$, this can be made explicit by starting with $\bar{\alpha}_{0}=0$
and writing $\bar{\alpha}_{t}=f(\bar{\alpha}_{t-1})$ with

$f(x):=\begin{cases}g_{1}^{2}x+\alpha_{1}\quad\text{with probability}\quad
p\\\ g_{2}^{2}x+\alpha_{2}\quad\text{with probability}\quad 1-p\;.\end{cases}$
(IV.5)

Depending on the value of $G^{*}(\mathbf{k},\mathbf{p})$ we can distinguish
different scenarios. The one that is best characterized at mathematical level
is when the model is contracting on average [30], i.e. when
$G^{*}(\mathbf{k},\mathbf{p})<0$, which unfortunately is the less interesting
case for our model as it represents a betting scheme where the player
asymptotically looses all its capital. Under this condition there exists a
unique stationary invariant probability measure for the process which can be
singular or absolutely continuous [12]. For instance, referring to the
parametrization introduced in Eq. (IV.5), when $p=\frac{1}{2}$ in the case
$\frac{1}{2}<g_{1}^{2}=g_{2}^{2}<1$ if $\alpha_{1}\not=\alpha_{2}$ then, up to
an affine map, the invariant measure of the system is a Bernoulli convolution
[19] which is known to be singular when the inverse of the Lipschitz constant
is a Pisot number. When $g_{1}^{2}=g_{2}^{2}=\frac{1}{2}$ instead, the measure
is the Lebesgue measure whereas if $g_{1}^{2}=g_{2}^{2}<\frac{1}{2}$ it is
uniform measure on a Cantor set or, equivalently, the Hausdorff measure of
dimension equal to the dimension of the Cantor set. Generally speaking in the
contractive-on-average case the fundamental fact which one can exploit is that
the random family of maps contracts the Wasserstein metric on compactly
supported probability measures. If this assumption is not met then the
situation is considerably more complex: orbits can be dense on the real half-
line [29] and one can investigate to what extent their distribution is uniform
[7].

The situation becomes even more problematic in the case which is more
interesting for us, i.e. for maps that are expanding-on-average, i.e. when
$G^{*}(\mathbf{k},\mathbf{p})>0$. Not much attention has been devoted in the
study of this scenario: as a matter of fact, the only reference we could find
[14] investigates only the orbits of the associated topological dynamical
systems obtained by suitably renormalizing the $n$–th iterates. We contribute
to the effort by noticing that in these cases due to the fact that
$\bar{g}_{t}$ is almost surely a divergent quantity (see e.g. (IV.1)), it is
convenient to focus on the renormalized version $\bar{\gamma}_{t}$ of
$\bar{\alpha}_{t}$ introduced in Eq. (III.31), which determines the stochastic
evolution of indicators $\bar{r}_{t}$ and $\bar{\mu}_{t}$ defined in Sec. III.
It turns out that $\bar{\gamma}_{t}$ is rather regular and for finite $t$, its
first and second moments can be computed (see Lemma .2 of the Appendix).
Furthermore it is possible to show that the random parameter
$\bar{\gamma}_{t}$ converges almost surely whenever $\bar{g}_{t}^{2}$ diverges
exponentially almost surely: this is a consequence of the fact that
$\bar{\gamma}_{t}$ is $t$-th partial sum of the infinite series with $t$-th
term $\alpha_{j_{t}}/\bar{g}^{2}_{t}$ that always exists due to the non-
negativity of the terms, and it is guaranteed to be finite when the
denominator can be upper bounded by an exponentially diverging term, as it
gives a trivial upper bound by a convergent geometric series (see the Theorem
.1 of the Appendix for a more detailed derivation of this result). We also
observe that, since both $\bar{\mu}_{t}$ and $\bar{r}_{t}$ are positive
quantities bounded from above, it makes sense to determine the asymptotic
values $r$ and $\mu$ as the (point-wise) limit of the distribution
$\bar{r}_{t}$ for $t\rightarrow\infty$:

$\mu:=\lim_{t\rightarrow\infty}\bar{\mu}_{t}\qquad
r:=\lim_{t\rightarrow\infty}\bar{r}_{t}.$ (IV.6)

Despite this nice property, determining the effective distribution of $\mu$
and $r$ (or those of $\bar{\mu}_{t}$ and $\bar{r}_{t}$ for $t$ finite),
remains however a rather complex task and at present we are not able to make
general claims. To compensate for this, we do however present a numerical
analysis that allows us to at least get some insights on the problem.

Figure 4: Time evolution of the empirical distributions of $\bar{r}_{t}$ of
Eq. (III-D) and of the (renormalized) ergotropy of the model $\bar{\mu}_{t}$
of Eq. (III.39), for a two horses game ($J=2$) with probability vector
$\mathbf{p}=(0.7,0.3)$, under super-fair odds assumptions (III.8) defined by
the vector $\mathbf{k}=(\sqrt{3},\sqrt{3})$, and pure squeezed input state of
initial ergotropy value ${\cal E}_{0}=25$ ($m^{2}=0$, $\cosh(2|\zeta|)=51$,
${{{n}}}=0$). Panels (a) and (b): empirical distributions of $\bar{r}_{t}$ and
$\bar{\mu}_{t}$ obtained under optimal betting conditions (IV.4) corresponding
to $\boldsymbol{\eta}=(\sqrt{0.7},\sqrt{0.3})$; Panels (c) and (d): empirical
distributions of $\bar{r}_{t}$ and $\bar{\mu}_{t}$ under sub-optimal betting
strategy with $\boldsymbol{\eta}=(\sqrt{0.3},\sqrt{0.7})$. All the data were
obtained over samples of $10.000$ simulations. The red lines in the plots
represent the empirical average of the sample; the cyan dots are the mean
field approximation of the mean $\tilde{r}_{t}$ defined in Eq. (IV.7); the
insets present instead the asymptotic histograms of the associated values of
$\bar{r}_{t}$ and $\bar{\mu}_{t}$ sampled at $t=100$ (panels (a) and (b)), and
$t=150$ (panels (c) and (d)).

###  IV-A Numerics

In this section we simulate numerically the evolution of $\bar{\mu}_{t}$ and
$\bar{r}_{t}$ over a sufficiently long time horizon to infer their asymptotic
behaviour. In Figs. 4–8 the temporal evolution of the empirical distributions
of these quantities have been reconstructed by sampling $10.000$ simulations
for each temporal step and for various choices of the parameters and in the
two-horse case (i.e. $J=2$). In the plots we also report the temporal
evolutions of the average values of $\bar{r}_{t}$ and $\bar{\mu}_{t}$
extracted from the sampling. It goes without mentioning that due to the
nonlinear dependence of these functionals upon the state of the system, they
do not coincide with the corresponding values computed on the average density
matrix of the system obtained by applying the average map (III.26) to
$\hat{\rho}_{0}$. In all the figures we also exhibit an educated guess for the
average value of $\bar{r}_{t}$ obtained by naively replacing the
$\bar{\Gamma}_{t}$ terms that appears in Eq. (III-D) with its average value
$\mathbb{E}[\bar{\Gamma}_{t}]$ computed as in the Appendix. This quantity can
be thought of as a mean field approximation of our random variable and it is
written as

$\tilde{r}_{t}:=1-\frac{\sqrt{1+4\mathbb{E}\left[\bar{\Gamma}_{t}\right]\cosh(2|\zeta|)+4\mathbb{E}\left[\bar{\Gamma}_{t}\right]^{2}}}{\cosh(2|\zeta|)+2\mathbb{E}\left[\bar{\Gamma}_{t}\right]+F},$
(IV.7)

with

$\displaystyle\mathbb{E}\left[\bar{\Gamma}_{t}\right]=\frac{1-\mathbb{E}\left[1/g^{2}\right]^{t}}{1-\mathbb{E}\left[1/g^{2}\right]}\;\frac{\mathbb{E}\left[{\alpha}/{g^{2}}\right]}{2{{{n}}}+1}\;,$
(IV.8)

converging to the asymptotic value
$\frac{1}{1-\mathbb{E}\left[1/g^{2}\right]}\;\frac{\mathbb{E}\left[{\alpha}/{g^{2}}\right]}{2{{{n}}}+1}$.

Figure 5: Time evolution of the empirical distributions of $\bar{r}_{t}$ of
Eq. (III-D) for a two horses game ($J=2$), under super-fair odds assumptions
(III.8) defined by the vector $\mathbf{k}=(\sqrt{3},\sqrt{3})$, and pure
coherent input state of ergotropy value ${\cal E}_{0}=25$ ($m^{2}=50$,
$\zeta=0$, ${{{n}}}=0$). Panels (a) and (b): empirical distributions of
$\bar{r}_{t}$ under optimal betting conditions (IV.4) obtained by setting
$\mathbf{p}=(0.7,0.3)$, $\boldsymbol{\eta}=(\sqrt{0.7},\sqrt{0.3})$ (for panel
(a)), and $\mathbf{p}=(0.6,0.4)$, $\boldsymbol{\eta}=(\sqrt{0.6},\sqrt{0.4})$
(for panel (b)). Panels (c) and (d): empirical distributions of $\bar{r}_{t}$
under sub-optimal betting strategies for the same probability setting of (a)
(i.e. $\mathbf{p}=(0.7,0.3)$) but assuming
$\boldsymbol{\eta}=(\sqrt{0.6},\sqrt{0.4})$ (panel (c)) and
$\boldsymbol{\eta}=(\sqrt{0.7},\sqrt{0.3})$ (panel (d)). As in Fig. 4, the
data were obtained over samples of $10.000$ simulations. The red lines in the
plots represent the empirical average of the sample while the cyan dots are
the mean field approximation $\tilde{r}_{t}$ of Eq. (IV.7). The insets show
the asymptotic histograms of the associated values of $\bar{r}_{t}$ sampled at
$t=100$ (panel (a) and (b)), and $t=150$ (panels (c) and (d)). It should be
stressed in all cases one has $\bar{\mu}_{t}=1$ for all $t$ due to Eq.
(III.38).  Figure 6: Values of the asymptotic averages for $r$ and $\mu$ of
Eq. (IV.6) for different types of input state $\hat{\rho}_{0}$ and for
different values of input ergotropy $\mathcal{E}_{0}$ ranging from $50$ to
$50.000$, logarithmically spaced. The red circles represent the squeezed input
state where $\mathcal{E}_{0}=\frac{\cosh(2|\zeta|)-1}{2}$, while the green
circles show the coherent input state with $\mathcal{E}_{0}=m^{2}/2$. The
orange circles show the result for mixed input state where
$m^{2}=3\mathcal{E}_{0}/4$, finally the blue circles represent the input state
where $m^{2}=7\mathcal{E}_{0}/8$. All of the aforementioned states have
${{{n}}}=0$. The cyan lines correspond to the first two isoergotropic curves.
The quantities $r$ and $\mu$ have been obtained as empirical averages over a
sample of $10.000$ simulations for a two horses game $J=2$, with super-fair
odds conditions $\mathbf{k}=(\sqrt{3},\sqrt{3})$ (III.8) and optimal betting
strategy (IV.4) with $\mathbf{p}=(0.7,0.3)$ and
$\boldsymbol{\eta}=(\sqrt{0.7},\sqrt{0.3})$. The plots makes it explicit that
for fixed value of the initial capital ${\cal E}_{0}$, optimal perfomances in
terms of $\mu$ and $r$ are always attained by using coherent input states.

Let us now enter into the details of our numerical analysis. In Fig. 4,
focusing on the same betting scenario, we compare the results obtained when
adopting the optimal (Kelly) betting strategy (IV.4) (panels (a) and (b)) with
those of a sub-optimal one (panels (c) and (d)), for a pure squeezed input
state, under super-fair odds conditions. As clear from the plots the Kelly
strategy leads to a beneficial increment both in terms of the mean values of
$\bar{r}_{t}$ and $\bar{\mu}_{t}$ (attaining higher values in the (a) and (b)
panels), as well as in terms of their spreads (more concentrated toward higher
values in the (a) and (b) panels). Similar findings are confirmed in Fig. 5
were instead we focus on the case of a pure coherent state (in this case we
only present $\bar{r}_{t}$, since one always get $\bar{\mu}_{t}=1$ for all $t$
due to Eq. (III.38)). A direct comparison between the panels (a) of the Figs.
4 and 5, also show that under the same betting conditions, and input ergotropy
resources, the coherent inputs provide better performances than pure squeezed
states in agreement with the results of Sec. III-D. This fact is also
confirmed by the data of Fig. 6 which present a numerical estimation of the
asymptotic averaged values of $\mu$ and $r$ of Eq. (IV.6) for various choices
of pure (displaced-squeezed) input state while maintaining the same (optimal)
betting strategy. The detrimental influence of thermal photons in the input
state is instead addressed in Fig. 7 where the temporal distribution of
$\bar{r}_{t}$ is studied under optimal betting strategies and the same game
conditions discussed in the panels (a) of Figs. 4 and 5: despite exhibiting
the same input ergotropy one notices that the range and the mean value
exhibited by $\bar{r}_{t}$ get contracted with respect to both the coherent
and squeezed case.

As a general remark we point out that the shape that we observe for the
asymptotic distributions of $\bar{r}_{t}$ and $\bar{\mu}_{t}$ are a direct
consequence of the underlying binomial process which generates a random walk
on $[0,1]$ with steps whose length and direction at time $t+1$ depend on the
position at time $t$ [15]. We highlight also that when choosing
$\boldsymbol{\eta}$ as the optimal Kelly-betting strategy (IV.4) and $\bf k$
to be a vector of super-fair odds (III.8), then we have finiteness of the
first moment of $\bar{\gamma}_{t}$ – see panels (a) and (b) of Figs. 4–6. The
same stability condition has been enforced also for all the other
configurations we have considered, a part from the case reported in Fig. 8:
here choosing $\boldsymbol{\eta}$ as the optimal Kelly-betting strategy (IV.4)
and setting $\bf k$ to be a vector of fair odds (III.7), while all the moments
of $\bar{r}_{t}$ remain finite (because it has support contained in the
interval $[0,1]$), one has
$\lim_{t\rightarrow\infty}{\mathbb{E}[\bar{\gamma}}_{t}]=\infty$ (see Lemma
(.2) in appendix) which tends to compromise the convergency of the simulations
as evident from the reported data and the matching with the mean field
estimation (IV.7).

Figure 7: Time evolution of the empirical distribution of $\bar{r}_{t}$ of Eq.
(III-D) for a two horses game ($J=2$), under super-fair odds assumptions
(III.8) defined by the vector $\mathbf{k}=(\sqrt{3},\sqrt{3})$, and thermal
displaced (zero squeezing) density matrix ($m^{2}=50$, $\zeta=0$,
${{{n}}}=10$) with ergotropy value ${\cal E}_{0}=25$. Here we assumed optimal
betting strategy (IV.4) obtained by setting $\mathbf{p}=(0.7,0.3)$ and
$\boldsymbol{\eta}=(\sqrt{0.7},\sqrt{0.3})$. As in Fig. 4 and 5, the data were
obtained over samples of $10.000$ simulations. The red lines in the plots
represent the empirical average of the sample while the cyan dots are the mean
field approximation $\tilde{r}_{t}$ of Eq. (IV.7). The insets show the
asymptotic histograms of the associated values of $\bar{r}_{t}$ sampled at
$t=100$.  Figure 8: Time evolution of the empirical distribution of
$\bar{r}_{t}$ of Eq. (III-D) over a sample of $10.000$ simulations ($J=2$).
The red line is the empirical average of the sample. The cyan dots represent
$\tilde{r}_{t}$, as we see in this case due to the irregular moments of
$\bar{\gamma}_{t}$, not only is the convergence slower, but $\tilde{r}_{t}$ is
also completely different from the sampled mean. The inset figure is the
asymptotic histogram, sampled a $t=250$. Here we assumed fair odds condition
(III.7) $\mathbf{k}=(\sqrt{2},\sqrt{2})$, and betting strategy which is
slightly sub-optimal setting $\mathbf{p}=(0.7,0.3)$ and
$\boldsymbol{\eta}=(\sqrt{0.71},\sqrt{0.29})$. The input is a coherent state
with $m^{2}=50$, $\zeta=0$, ${{{n}}}=0$. Notice that the convergence rate is
slower, with respect to the previous plots, due to the increase of the
characteristic time of the process that is proportional to $1/G$.

##  V Conclusions and future perspectives

Betting (or gambling) is a practical tool for studying decision-making in face
of classical uncertainty. The optimal betting strategy has been in the
mathematical literature since the 1950s. Known as the classical Kelly
criterion for optimal betting, it holds that under “fair" odds and i.i.d.
repeated horse-races you should bet a fraction of capital on each horse that
is proportional to the true winning probabilities of the latter in order to
maximize the asymptotic growth rate of the cumulative wealth.
In this paper, we presented a semi-classical model describing betting
scenarios where the payoff of the gambler is encoded into the internal degrees
of freedom of a quantum memory element. We provided a translation of the
classical Kelly framework into the new semi-classical playing field.
Specifically, the invested capital was associated with the ergotropy of a
single mode of the electromagnetic radiation; the losses and winning events
with the attenuation and amplification processes of the just-mentioned single-
mode; instead, the (random) evolution of the capital, represented by the
evolution of the quantum memory, is characterized within the theoretical
setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for
optimal betting, we defined the asymptotic doubling rate of the model and
identified the optimal gambling strategy for fixed odds and winning
probabilities. We carried out an accurate statistical and numerical analysis
to determine the performance of the model. We found that if the input capital
state belongs to the set of Gaussian density matrices then the best option for
the gambler is to devote all her/his initial resources into coherent state
amplitude.
A first natural problem to address is the generalization of our constructions
to multimode scenarios. This would pave the way for the study of non-classical
interference effects in betting games where the gambler can distribute her/his
capital on parallel independent horse-races. In particular, this happens quite
often in reality [36, 27] and the comparison across different racetrack
markets is fundamental in analysing their informational efficiency. Moreover,
since the late nineties, platforms allowing short as well as long bets have
been introduced (betting exchanges). This allows bettors to wager against
over-priced horses just as hedge fund managers can short financial contracts.
Finding an adequate quantum setting for betting exchanges, including possibly
general financial markets, would considerably extend the set of applications
of our framework.

V. G. acknowledges support by MIUR via PRIN 2017 (Progetto di Ricerca di
Interesse Nazionale): project QUSHIP (2017SRNBRK). The authors acknowledge the
financial support of UniCredit Bank R$\&$D group through the _Dynamics and
Information Research Institute_ at the Scuola Normale Superiore.

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######  Lemma .1 (Recurrence relation for $\bar{\alpha}_{t}$).

The recurrence relation (III.24) admits the solution

$\bar{\alpha}_{t}=\bar{g}^{2}_{t}\sum_{\ell=1}^{t}\frac{\alpha_{j_{\ell}}}{\bar{g}_{\ell}^{2}}.$
(.1)

###### Proof.

The parameters $\bar{\alpha}_{t}$ and $\bar{g}_{t}$ define the action of the
BGC (III.19). By the equation (III.24) we have:

$\bar{\alpha}_{t+1}=g_{j_{t+1}}^{2}\bar{\alpha}_{t}+\alpha_{j_{t+1}}.$ (.2)

Now using $\bar{\gamma}_{t}$ as defined in (III.20), we can find an explicit
formula for $\bar{\alpha}_{t}$:

$\bar{\gamma}_{t+1}-\bar{\gamma}_{t}=\frac{\bar{\alpha}_{t+1}}{\bar{g}_{t+1}^{2}}-\frac{\bar{\alpha}_{t}}{\bar{g}_{t}^{2}}=\frac{\alpha_{j_{t+1}}}{\bar{g}_{t+1}^{2}}.$

So we can solve for $\bar{\gamma}_{t}$, integrating the finite difference:

$\sum_{\ell=1}^{t-1}(\bar{\gamma}_{\ell+1}-\bar{\gamma}_{\ell})=\bar{\gamma}_{t}-\bar{\gamma}_{1}=\sum_{\ell=1}^{t-1}\frac{\alpha_{j_{\ell+1}}}{\overline{g}_{\ell+1}^{2}},$
(.3)

since we have that $\bar{\gamma}_{1}=\frac{\alpha_{j_{1}}}{\bar{g}^{2}_{1}}$
we obtain:

$\bar{\gamma}_{t}=\sum_{\ell=1}^{t}\frac{\alpha_{j_{\ell}}}{\bar{g}_{\ell}^{2}},$
(.4)

and recalling the definition III.31 of $\bar{\gamma}_{t}$ we obtain the
thesis. ∎

We conclude this part by deriving some properties of $\bar{\gamma}_{t}$.

######  Lemma .2 (Moments of $\bar{\gamma}_{t}$).

If we call

$\mathbb{E}\left[\frac{1}{g^{2}}\right]:=\sum_{j=1}^{J}\frac{p_{j}}{g_{j}^{2}}\;,\qquad\mathbb{E}\left[\frac{1}{g^{4}}\right]:=\sum_{j=1}^{J}\frac{p_{j}}{g_{j}^{4}}\;,$
$\mathbb{E}\left[\frac{\alpha}{g^{2}}\right]:=\sum_{j=1}^{J}p_{j}\frac{\alpha_{j}}{g_{j}^{2}}\;,\qquad\mathbb{E}\left[\frac{\alpha^{2}}{g^{4}}\right]:=\sum_{j=1}^{J}p_{j}\frac{\alpha_{j}^{2}}{g_{j}^{4}},$

then

$\mathbb{E}\left[\bar{\gamma}_{t}\right]=\mathbb{E}\left[\frac{\alpha}{g^{2}}\right]\frac{1-\mathbb{E}\left[1/g^{2}\right]^{t}}{1-\mathbb{E}\left[1/g^{2}\right]}\;.$
(.5)

and

$\begin{split}&\mathbb{E}\left[\bar{\gamma}_{t}^{2}\right]=\mathbb{E}\left[\frac{\alpha^{2}}{g^{4}}\right]\frac{1-\mathbb{E}\left[1/g^{4}\right]^{t}}{1-\mathbb{E}\left[1/g^{4}\right]}-\frac{1}{\mathbb{E}[1/g^{2}]-\mathbb{E}\left[1/g^{4}\right]}\\\
&\times\left(\frac{1-\mathbb{E}\left[1/g^{2}\right]^{t}}{1-\mathbb{E}\left[1/g^{2}\right]}-\frac{1-\mathbb{E}\left[1/g^{4}\right]^{t}}{1-\mathbb{E}\left[1/g^{4}\right]}\right)\;.\end{split}$
(.6)

###### Proof.

The expected value of $\bar{\gamma}_{t}$ is:

$\displaystyle\mathbb{E}[\bar{\gamma}_{t}]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{\bar{\alpha}_{t}}{\bar{g}^{2}_{t}}\right]=\sum_{\ell=1}^{t}\mathbb{E}\Bigg{[}\frac{\alpha}{g^{2}}\Bigg{]}\mathbb{E}\Bigg{[}\frac{1}{\bar{g}_{\ell-1}^{2}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{\alpha}{g^{2}}\right]\sum_{\ell=1}^{t}\mathbb{E}\left[\frac{1}{g^{2}}\right]^{\ell-1}=\mathbb{E}\left[\frac{\alpha}{g^{2}}\right]\frac{1-\mathbb{E}\left[1/g^{2}\right]^{t}}{1-\mathbb{E}\left[1/g^{2}\right]},$

where we used the fact that the different bets are independent.

Regarding the second moment of $\bar{\gamma}_{t}$, we split it into two terms
as:

$\displaystyle\mathbb{E}\left[\bar{\gamma}_{t}^{2}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\sum_{\ell=1}^{t}\frac{\alpha_{j_{\ell}}}{g^{2}_{j_{\ell}}}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{\ell=1}^{t}\sum_{m=1}^{t}\frac{\alpha_{j_{\ell}}}{\bar{g}_{\ell}^{2}}\frac{\alpha_{j_{m}}}{\bar{g}^{2}_{m}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{\alpha^{2}}{g^{4}}\right]\frac{1-\mathbb{E}\left[1/g^{4}\right]^{t}}{1-\mathbb{E}\left[1/g^{4}\right]}+2\mathbb{E}\left[\sum_{m<\ell}\frac{\alpha_{j_{\ell}}}{\bar{g}^{2}_{\ell}}\frac{\alpha_{j_{m}}}{\bar{g}^{2}_{m}}\right]$
$\displaystyle:=$ $\displaystyle(I)+(II)\;,$

where the second term can also be written as:

$\displaystyle(II)$ $\displaystyle=$ $\displaystyle
2\sum_{\ell=2}^{t}\sum_{m=1}^{\ell-1}\mathbb{E}\left[\alpha_{j_{\ell}}\alpha_{j_{m}}\left(\prod_{i=m+1}^{\ell}g_{j_{i}}^{2}\prod_{s=1}^{m}g_{j_{s}}^{4}\right)^{-1}\right]$
(.9) $\displaystyle=$ $\displaystyle
2\mathbb{E}\left[\frac{\alpha}{g^{4}}\right]\mathbb{E}\left[\frac{\alpha}{g^{2}}\right]\sum_{\ell=2}^{t}\sum_{m=1}^{\ell-1}\mathbb{E}\left[\frac{1}{g^{2}}\right]^{\ell-m-1}\mathbb{E}\left[\frac{1}{g^{4}}\right]^{m-1}$
$\displaystyle=$ $\displaystyle
2\mathbb{E}\left[\frac{\alpha_{1}}{g^{4}}\right]\mathbb{E}\left[\frac{\alpha_{1}}{g^{2}}\right]\sum_{\ell=2}^{t}\mathbb{E}\left[\frac{1}{g^{2}}\right]^{\ell-2}\sum_{m=1}^{\ell-1}\left(\frac{\mathbb{E}\left[1/g^{4}\right]}{\mathbb{E}\left[1/g^{2}\right]}\right)^{m-1}$
$\displaystyle=$
$\displaystyle\frac{1}{1-\mathbb{E}\left[1/g^{4}\right]/\mathbb{E}\left[1/g^{2}\right]}$
$\displaystyle\times\sum_{\ell=2}^{t}\mathbb{E}\left[\frac{1}{g^{2}}\right]^{\ell-2}\left(1-\left(\frac{\mathbb{E}\left[1/g^{4}\right]}{\mathbb{E}\left[1/g^{2}\right]}\right)^{\ell-1}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{\alpha^{2}}{g^{4}}\right]\frac{1-\mathbb{E}\left[1/g^{4}\right]^{t}}{1-\mathbb{E}\left[1/g^{4}\right]}-\frac{1}{\mathbb{E}[1/g^{2}]-\mathbb{E}\left[1/g^{4}\right]}$
$\displaystyle\times\left(\frac{1-\mathbb{E}\left[1/g^{2}\right]^{t}}{1-\mathbb{E}\left[1/g^{2}\right]}-\frac{1-\mathbb{E}\left[1/g^{4}\right]^{t}}{1-\mathbb{E}\left[1/g^{4}\right]}\right)\;.$

∎

######  Corollary .1.

Under the assumptions that
$\mathbb{E}\left[\frac{1}{g^{2}}\right],\>\mathbb{E}\left[\frac{1}{g^{4}}\right]<1$
and
$\mathbb{E}\left[\frac{1}{g^{2}}\right]-\mathbb{E}\left[\frac{1}{g^{4}}\right]\neq
0$ it results that:

$\displaystyle\mathbb{E}\left[\lim_{t\rightarrow\infty}\bar{\gamma}_{t}\right]=\mathbb{E}\left[\frac{\alpha}{g^{2}}\right]\frac{1}{1-\mathbb{E}\left[1/g^{2}\right]}\;,$
$\displaystyle\mathbb{E}\left[\lim_{t\rightarrow\infty}\bar{\gamma}_{t}^{2}\right]=\mathbb{E}\left[\frac{\alpha^{2}}{g^{4}}\right]\frac{1}{1-\mathbb{E}\left[1/g^{4}\right]}-\frac{1}{\mathbb{E}[1/g^{2}]-\mathbb{E}\left[1/g^{4}\right]}$
$\displaystyle\times\left(\frac{1}{1-\mathbb{E}\left[1/g^{2}\right]}-\frac{1}{1-\mathbb{E}\left[1/g^{4}\right]}\right)\;.$
(.10)

###### Proof.

By definition $\forall$$c\in\mathbb{R}^{+}$ $\\{\bar{\gamma}_{t}^{c}\\}$ is a
positive and monotonically increasing sequence, so for the monotone
convergence theorem we have that:

$\lim_{t\rightarrow\infty}\mathbb{E}\left[\bar{\gamma}_{t}^{c}\right]=\mathbb{E}\left[\lim_{t\rightarrow\infty}\bar{\gamma}_{t}^{c}\right].$

If we take the limit of the formulas in lemma (.2) we obtain the thesis ∎

######  Theorem .1.

If $\bar{g}_{t}^{2}$ diverges exponentially almost surely then
$\bar{\gamma}_{t}$ converges almost surely.

###### Proof.

We just need to prove that the sequence
$\\{\bar{\gamma}_{t}\\}_{t\in\mathbb{N}}$ is a Cauchy sequence almost
everywhere in our probability space. To formally embed the statement in a
rigorous mathematical framework, we should introduce a probability space
$(\Omega,\Sigma,\mathcal{P})$ where $\mathcal{P}$ the Bernoulli distribution
on $\left\\{1,\ldots,J\right\\}$,
$\Omega:=\left\\{1,\ldots,J\right\\}^{\mathbb{N}}$ and $\Sigma$ to be the
$\sigma$-algebra generated by the cylinder sets. Then taking $\omega\in\Omega$
we identify $\bar{g}_{t}(\omega)$ with the define quantity (III.20 ) computed
on the first $t$ elements of $\omega$, and define
$\bar{G}_{t}(\omega):=\frac{{\log_{2}}\bar{g}_{t}(\omega)}{t}$. In this
language Eq. (IV.1) formally translates into

$\Pr[\omega\in\Omega:\forall\varepsilon>0\>\exists\,\bar{t}\;s.t.\;\forall\,t\geq\bar{t}\,\>|\bar{G}_{t}(\omega)-G|<\varepsilon]=1,$
(.11)

where $G$ is a short hand notation for
$G(\boldsymbol{\eta},\mathbf{k},\mathbf{p})$. Accordingly given

$\tilde{\Omega}:=\\{\omega\in\Omega:\forall\varepsilon>0\>\exists\,\bar{t}\;s.t.\;\forall\,t\geq\bar{t}\,\>|\bar{G}_{t}(\omega)-G|<\varepsilon\\}\;,$
(.12)

it follows that

$\displaystyle\Pr(\tilde{\Omega}):=\Pr(\omega\in\tilde{\Omega})=1\;.$ (.13)

Now, let $\omega\in\tilde{\Omega}$. Fix $\varepsilon\in(0,\frac{G}{2})$,
$\delta>0$ and let $\bar{t}(\omega,\varepsilon)$ as in Eq.(.12). Then for all
$t\geq\bar{t}(\omega,\varepsilon)$

$2^{t(G-\varepsilon)}\leq\bar{g}_{t}(\omega)\leq 2^{t(G+\varepsilon)}.$ (.14)

Let $\overline{m}>\bar{t}(\omega,\varepsilon)$, $p,q>\overline{m}$ with $p<q$.
For a chosen $\omega$

$\displaystyle\bar{\gamma}_{q}(\omega)-\bar{\gamma}_{p}(\omega)$
$\displaystyle=$
$\displaystyle\sum_{\ell=p}^{q}\frac{\alpha_{j_{\ell}}(\omega)}{g_{\ell}^{2}(\omega)}\frac{1}{\bar{g}_{\ell-1}^{2}(\omega)}$
(.15) $\displaystyle\leq$ $\displaystyle
K\sum_{\ell=p}^{q}2^{-2(\ell-1)(G-\varepsilon)}$ $\displaystyle\leq$
$\displaystyle K^{\prime}2^{-(\overline{m}-1)G},$

for some constants $K,K^{\prime}>0$ – we are using the convention of
indicating with $\bar{\gamma}_{q}(\omega)$ the quantity (III.31) associated
with the first $q$ elements of $\omega$. For a sufficiently large
$\overline{m}$ we have
$K^{\prime}2^{-(\overline{m}-1)G}<\delta\;\forall\delta>0$. Hence we proved
that

$\displaystyle\Pr(\omega\in\Omega:$
$\displaystyle\forall\delta>0\>\exists\,\overline{m}\;s.t.\;$
$\displaystyle\forall\,p,q\geq\overline{m}\;,|\bar{\gamma}_{p}(\omega)-\bar{\gamma}_{q}(\omega)|<\delta)=1,$

or equivalently that $\\{\bar{\gamma}_{t}\\}_{t\in\mathbb{N}}$ is a Cauchy
sequence almost everywhere. ∎