# Exact nonequilibrium quantum observable statistics: A large-deviation
approach

Stefano Gherardini<EMAIL_ADDRESS>Department of Physics and
Astronomy & LENS, University of Florence, via G. Sansone 1, I-50019 Sesto
Fiorentino, Italy.  INFN Sezione di Firenze, via G. Sansone 1, I-50019 Sesto
Fiorentino, Italy.

###### Abstract

The exact statistics of an arbitrary quantum observable is analytically
obtained. Due to the probabilistic nature of a sequence of intermediate
measurements and stochastic fluctuations induced by the interaction with the
environment, the measurement outcomes at the end of the system’s evolution are
random variables. Here, we provide the exact large-deviation form of their
probability distribution, which is given by an exponentially decaying profile
in the number of measurements. The most probable distribution of the
measurement outcomes in a single realization of the system transformation is
then derived, thus achieving predictions beyond the expectation value. The
theoretical results are confirmed by numerical simulations of an
experimentally reproducible two-level system with stochastic Hamiltonian.

PACS numbers

05.30.-d, 03.65.Yz, 03.65.Ta

##  I Introduction

Observables of a quantum system out-of-equilibrium return random outcomes
fluctuating with a specific probability distribution BookJacobs . By following
the operational approach DaviesBook , a quantum stochastic process
BookPetruccione ; ReviewHuelga  can be envisaged as a nonequilibrium
transformation modeled as the composition of stochastic evolutions
characterized by semi-classical fluctuations of the system parameters
GherardiniErgodicity ; MuellerAnnalen ; RossiPRA2017  and intermediate quantum
measurements CampisiPRL2010 ; ThielPRL2018 . In the former case, especially in
quantum computing, stochastic fluctuations can arise from imperfections in the
physical realization of the quantum system or through the interaction with the
environment MishraNATCOMM2018 . On the other hand, repeated quantum
measurements could correspond to a process exchanging photons with the
environment HekkingPRL2013  or could be adopted to ensure the protection of
coherent evolutions of a quantum system by decoherence (quantum Zeno dynamics)
FacchiPRL2002 . Experimentally, together with strong coupling methods, such
dynamics have been realized in several physical setups such as solid-state
spins, superconducting qubits or ultracold atomic gases UhrigPRL2007 ;
LangeScience2010 ; PokharelPRL2018 ; SchaferNAT2014 ; SignolesNAT2014 ;
GherardiniErgodicity . They are also relevant in quantum metrology
PezzeReview2018  to probe the phase evolution of an atomic ensemble by means
of interleaved interrogations and feedback corrections KohlhaasPRX2015 ;
SchioppoNAT2017 . In this scenario, by observing only once the dynamics of the
system (single realization), the ensemble average of the measurement outcomes
does not provide complete information about the statistics of the measured
results. This becomes more evident when also one rare random event, i.e. a
stochastic fluctuation with very small probability, occurs within the system
dynamics Chantasri2013 ; WeberNature2014 ; Chantasri2015 . Such a concept is
at the heart of the Large Deviation (LD) theory Ellis1 ; Touchette1 , dealing
with the exponential decay of probabilities associated to large fluctuations
in stochastic classical Dembo1  and quantum systems Gallavotti2002 ;
Netocny2004 ; GarrahanPRL2010 ; LesanovskyPRL2013 ; PigeonPRA2015 ;
Collura2019 .

In this paper, we derive a closed-form expression for the outcomes’statistics
obtained by measuring a quantum system repeatedly monitored by an external
observer and whose dynamical evolution depends on random parameters. In
particular, we prove that for a sufficiently large number $m$ of intermediate
quantum measurements, the probability distribution of the last measurement
results obeys the so-called _LD principle_ Touchette1 ; GherardiniNJP ;
thesis_Stefano . This means that the behaviour of the measurement outcomes’
distribution is a decaying exponential in $m$, whose exponent is equal to the
relative Shannon entropy between the configurations of the stochastic system
dynamics. In other words, only a rigorous description of the occurrence
combinatorics of the parameters defining the stochastic evolution of the
system allows for the full characterization of the outcomes’ statistics, also
_beyond_ the Gaussian approximation given by the sole description of the
measurement apparatus.

##  II Model

Let us consider an arbitrary quantum system $\mathcal{S}$ within the Hilbert
space $\mathcal{H}$. We assume that $\mathcal{S}$ is initialized in the
quantum state with density matrix $\rho_{0}$ and that the Hamiltonian $H$ of
the system is time-independent. At the level of the single trajectory, we
assume that the random interaction between $\mathcal{S}$ and an environment
$\mathcal{E}$ gives rise to a sequence of stochastic dynamical evolutions,
separated by consecutive quantum projective measurements footnote1 , following
the postulates of quantum mechanics Sakurai1994 . Hereafter, the index $j$
denotes the dimension of $\mathcal{H}$, while the index $\alpha$ denotes the
instants composing the temporal sequence of measurements. More specifically,
we assume that the first $m-1$ measurements are performed on the quantum
observable $\mathcal{O}\equiv\sum_{j}o_{j}\Pi_{o_{j}}$, where $o_{j}$ are the
outcomes of $\mathcal{O}$ and $\\{\Pi_{o_{j}}\\}$ is the set of projectors
corresponding to the measured eigenvalues at time $t_{\alpha}$. The $m-$th
measurement, instead, is performed on the quantum observable
$\Theta\equiv\sum_{j}\theta_{j}\Pi_{\theta_{j}}$, whose outcomes $\theta_{j}$
are recorded by the observer. According to the postulate of quantum
measurement, the state $\rho_{\alpha}$ of $\mathcal{S}$ after the projective
measurement at $t_{\alpha}$ is identically equal to one of the projectors
defining the measurement observable. Then, between each projection event the
system undergoes a dynamics, that is governed by the Hamiltonian $H$ and
described by the completely-positive and trace-preserving quantum map
$\Phi(t_{\alpha},t_{0})[\rho_{0}]\equiv\Phi_{\alpha}[\rho_{0}]$
Caruso_RevModPhys_2014 . If we assume quite large number of intermediate
quantum measurements, the dynamics between measurements is described by a
unitary operator, so that $\Phi_{\alpha}[\rho_{\alpha-1}]$ is simply given by
the super-operator $\mathcal{U}_{\alpha}[\rho_{\alpha-1}]\equiv
U_{\alpha}\rho_{\alpha-1}U_{\alpha}^{\dagger}$, where
$U_{\alpha}\equiv\exp(-iH\tau_{\alpha})$, $\hbar$ is set to unity and
$\tau_{\alpha}\equiv t_{\alpha}-t_{\alpha-1}$. We also assume that, in
accordance with the recently introduced stochastic quantum Zeno phenomena
GherardiniNJP ; GherardiniErgodicity ; MuellerAnnalen , there exists for each
propagator $U_{\alpha}$ at least one dynamical parameter $\lambda$, that is a
fluctuating variable. For example, one could consider as in
Gherardini_heat_stat ; MukherjeePRE2018  that each $\lambda_{\alpha}$ is equal
to the time interval $\tau_{\alpha}$, with $\tau_{\alpha}$ random. Moreover,
the $\lambda$’s between the measurements are taken as constant and have a
different random value only after the occurrence of a new measurement,
according to the probability density function $p(\lambda)$. Here,
$\vec{\lambda}\equiv(\lambda_{1},\ldots,\lambda_{m})$ denotes the sequence of
dynamical parameters $\lambda_{\alpha}$ in a single realization of the system
transformation. The fluctuating dynamical parameters $\lambda_{\alpha}$, then,
are taken as independent and identically distributed (i.i.d.) random variables
sampled from $p(\lambda)$, since the environment is a-priori unknown.

The stochastic nature of the measurement outcomes $\theta_{m}$’s at time
$t_{m}$ lies in the specific values assumed by $\vec{o}$ and $\vec{\lambda}$.
Thus, being the dynamics of $\mathcal{S}$ stochastic, the single realization
of the system density matrix $\rho_{m,\vec{o},\vec{\lambda}}$ at the end of
its evolution is a fluctuating variable, i.e., given the sequences $\vec{o}$
and $\vec{\lambda}$, it is mapped into

$\rho_{m,\vec{o},\vec{\lambda}}=\frac{\mathcal{P}_{\theta_{m}}\mathcal{U}_{m}\mathcal{P}_{o_{m-1}}\mathcal{U}_{m-1}\cdots\mathcal{P}_{o_{1}}\mathcal{U}_{1}[\rho_{0}]}{p_{\theta_{m}}(\vec{o},\vec{\lambda})},$
(1)

where
$\mathcal{P}_{\mu_{\alpha}}[(\cdot)]\equiv\Pi_{\mu_{\alpha}}(\cdot)\Pi_{\mu_{\alpha}}$
is the measurement super-operator acting on $(\cdot)$ at $t_{\alpha}$, with
$\mu\in\\{o,\theta\\}$. Consequently

$p_{\theta_{m}}(\vec{o},\vec{\lambda})\equiv\textrm{Tr}\left[\mathcal{P}_{\theta_{m}}\mathcal{U}_{m}\mathcal{P}_{o_{m-1}}\mathcal{U}_{m-1}\cdots\mathcal{P}_{o_{1}}\mathcal{U}_{1}[\rho_{0}]\right],$
(2)

i.e.
$p_{\theta_{m}}(\vec{o},\vec{\lambda})=\textrm{Tr}[\Pi_{\theta_{m}}\rho_{m,\vec{o},\vec{\lambda}}]$,
denotes the conditional probability to obtain the outcome $\theta_{m}$ from
the measurement of $\Theta$ given the specific realization of the sequences
$\vec{o}$ and $\vec{\lambda}$.

##  III Quantum observable statistics

Figure 1: Statistics of $\Theta$’s outcomes. By repeating several times the
stochastic evolution of the system and measuring each of the outcomes
$\theta_{m,j}$ at the final time instant $t_{m}$, an ensemble of conditional
probabilities $p_{\theta_{m,j}}$ is obtained, where each of them is computed
after the single realization of the system dynamics. Thus, just by counting
the occurrence relative frequencies of the $p_{\theta_{m,j}}$’s, one can
derive the corresponding probability distributions (blue dashed lines). If the
number of realizations is relatively small, such distributions do not obey the
Gaussian approximation. This means that one has to distinguish between two
different statistics for $\theta$’s: One (green solid line) linking the
ensemble averages $\langle p_{\theta_{m,j}}(\vec{o},\vec{\lambda})\rangle$
(red dots) of the conditional probabilities for each measurement outcome, and
the other (orange dotted line) – also called _most probable distribution_ –
connecting all the realizations of $p_{\theta_{m,j}}$ in correspondence to the
maximum value of the conditional probability distributions (blue dots).

In a single realization of the system transformation,
$\rho_{m,\vec{o},\vec{\lambda}}$ and $\theta_{m}$ depend on $\vec{o}$ and
$\vec{\lambda}$ and are random quantities. Thus, the following question
naturally emerges: Which is the _best_ description for the statistics of the
measurement results $\theta_{m,j}$ from the observation of $\mathcal{S}$ at
$t_{m}$? Three possible answers, characterized by an increasing degree of
prediction accuracy, can be provided. First, one could describe the
probabilistic expected result from the measured outcomes by using the
expectation value $\textrm{Tr}[\Theta\rho_{m}]$, with

$\rho_{m}\equiv\langle\rho_{m,\vec{o},\vec{\lambda}}\rangle=\sum_{\vec{o}}\int_{\vec{\lambda}}d^{m}\vec{\lambda}\,p(\vec{\lambda})\rho_{m,\vec{o},\vec{\lambda}}$
(3)

and $p(\vec{\lambda})$ denoting the occurrence joint probability of the
dynamical parameters $\lambda_{\alpha}$. It is worth noting that
$\textrm{Tr}[\Theta\rho_{m}]$ is equal to the ensemble average of all possible
measurement outcomes. Second, the probability distribution of $\Theta$’s
outcomes (green line in Fig. 1) can be introduced:

$\textrm{Prob}(\theta)=\sum_{j}\delta(\theta-\theta_{m,j})\overline{p}_{\theta_{m,j}}\,,$
(4)

where $\overline{p}_{\theta_{m,j}}\equiv\langle
p_{\theta_{m,j}}(\vec{o},\vec{\lambda})\rangle=\textrm{Tr}[\Pi_{\theta_{m,j}}\rho_{m}]$
and $\delta(\cdot)$ is the Kronecker delta. Eq. (4) defines the statistics of
$\Theta$, while $\overline{p}_{\theta_{m,j}}$ (red dots in Fig. 1) is the
probability to obtain on average the $j-$th outcome $\theta_{m,j}$. Otherwise,
the third option that we are here proposing is to adopt the probability
distribution of the conditional probabilities
$p_{\theta_{m,j}}(\vec{o},\vec{\lambda})$, i.e.
$\textrm{Prob}(p_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$ (in Fig. 1, the blue
dashed lines for each outcome $\theta_{m,j}$), defined over all the possible
realizations of the sequences $\vec{o}$ and $\vec{\lambda}$. Only in this way,
by deriving $\textrm{Prob}(p_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$, one will
be able to get the most probable statistics (orange dotted line) for the
quantum observable $\Theta$ connecting the blue dots in Fig. 1.

We start by observing that the $p_{\theta_{m,j}}(\vec{o},\vec{\lambda})$’s can
be written as the product of the probabilities
$|\langle\pi_{\mu_{\alpha-1}}|U_{\alpha}(\lambda_{\alpha})|\pi_{\mu_{\alpha}}\rangle|^{2}$
– also called dynamical transition probabilities – that the quantum state
moves from $|\pi_{\mu_{\alpha-1}}\rangle$ to $|\pi_{\mu_{\alpha}}\rangle$ via
the propagator $U_{\alpha}(\lambda_{\alpha})$ (see the Appendix for more
details). Specifically, $|\pi_{\mu_{\alpha}}\rangle$’s are the eigenvectors
that define the measurement projectors $\Pi_{\mu_{\alpha}}$, with $\mu$ equal
to $\theta$ or $o$, depending on whether the observable $\Theta$ or
$\mathcal{O}$ is measured. Thus, one has that

$\displaystyle\displaystyle{p_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\prod_{\alpha=1}^{m}|\langle\pi_{\mu_{\alpha-1}}|U_{\alpha}(\lambda_{\alpha})|\pi_{\mu_{\alpha}}\rangle|^{2},\,\,\,\text{i.e.,}}$
(5)
$\displaystyle\displaystyle{\overline{p}_{\theta_{m,j}}=\sum_{\vec{o}}\prod_{\alpha=1}^{m}\int_{\lambda_{\alpha}}d\lambda_{\alpha}p(\lambda_{\alpha})q_{\alpha}(\mu_{\alpha-1},\mu_{\alpha},\lambda_{\alpha}),}$
(6)

where
$q_{\alpha}\equiv|\langle\pi_{\mu_{\alpha-1}}|U_{\alpha}(\lambda_{\alpha})|\pi_{\mu_{\alpha}}\rangle|^{2}$,
$\pi_{\mu_{0}}\equiv\pi_{0}$ (with $\rho_{0}=|\pi_{0}\rangle\langle\pi_{0}|$),
$\mu_{\alpha}\equiv o_{\alpha}$ for $\alpha=1,\ldots,m-1$ and
$\pi_{\mu_{m}}\equiv\pi_{\theta_{m,j}}$.

##  IV Large deviation formalism

We now derive the main result of this paper, namely a closed-form expression
of the $p_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ distributions. Notice that if
we just characterize the quantum observable $\Theta$ at $t_{m}$ and at the
same time the hypotheses of the central limit theorem are respected, then the
statistics of its outcomes is well represented by a Gaussian probability
distribution. In fact, with this assumption, outliers in the outcomes
statistics are simply classified as the result of a non-modeled experimental
noise on the measurement apparatus. Thus, by increasing the number of
realizations, the occurrence probability of the outliers naturally decreases
and the Gaussian distribution well fits the data. However, this evidence is no
longer valid if we assume that the statistics of $\theta_{m,j}$’s is given by
an arbitrary stochastic transformation governing the dynamics of the system.
In such a case, the configuration space defined by the occurrence of random
events during its evolution becomes exponentially larger, but allows for the
description of the outliers’statistics with occurrence probability greater
than zero and not satisfying the Gaussian approximations. Therefore, we expect
to increase our prediction power on the outcomes’ distribution by
statistically characterizing the trajectories of the system before the
measurement of $\Theta$. This amounts to computing the statistics of the
dynamical transition probabilities
$q_{\alpha}(\mu_{\alpha-1},\mu_{\alpha},\lambda_{\alpha})$ as defined in Eq.
(5). Briefly, the procedure to derive the exact LD form of
$\textrm{Prob}(p_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$ is to take the
logarithm of the conditional probabilities
$p_{\theta_{m,j}}(\vec{o},\vec{\lambda})$, i.e.
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})\equiv\ln
p_{\theta_{j,m}}(\vec{o},\vec{\lambda})=\sum_{\alpha=1}^{m}\ln
q_{\alpha}(\mu_{\alpha-1},\mu_{\alpha},\lambda_{\alpha})$, compute its
distribution and then apply the contraction principle from LD theory
Touchette1 .

For the sake of clarity, let us here consider that the measurement bases of
$\mathcal{O}$ and $\Theta$, with $[\mathcal{O},\Theta]\neq 0$, belong to a set
of finite dimension, i.e. that each measurement eigenvector
$|\pi_{\mu_{\alpha}}\rangle$ admits only a finite number $d_{\pi}$ of elements
$|\pi^{(j)}\rangle$. Then, the distribution of
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ is obtained as following. By
following a common procedure in LD theory, the terms of
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ are grouped as a function of the
number of times each dynamical transition probability $q_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)\equiv|\langle\pi^{(j_{{\rm B}})}|U(\lambda)|\pi^{(j_{{\rm
A}})}\rangle|^{2}$ occurs, where the superscripts ${\rm B}$ and ${\rm A}$
stand, respectively, for “Before” and “After” the evolution via the propagator
$U(\lambda)$. In this way, $l_{\theta_{m,j}}$ is recast in the following sum
of i.i.d. random variables:

$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\sum_{j_{{\rm B}},j_{{\rm
A}}=1}^{d_{\pi}}\int_{\lambda}n_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\ln
q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)d\lambda,$ (7)

with $n_{j_{{\rm B}},j_{{\rm A}}}(\lambda)$ denoting the relative frequencies
for the occurrence of the $q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)$’s, usually
different from the corresponding probability value $p_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)$. The deviation between them vanishes only by attaining a
complete statistics for each admissible value of $q_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)$. This latter condition can be in principle realized, but likely
a quite long system dynamical evolution could be required.

The probability distribution of a sum of $n$ i.i.d. random terms can be always
written as an exponential, linearly decaying in $n$ with $n$ large. In
particular, regarding $\textrm{Prob}(l_{\theta_{m,j}})$, it is given by an
exponential distribution decaying in the number $m$ of projective measurements
(see Appendix for more details), i.e.

$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))\asymp\exp(-m\,I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)).$
(8)

In Eq. (8), the function $I(l_{\theta_{m,j}}/m)$, also called the _rate
function_ associated to the probability distribution
$\textrm{Prob}(l_{\theta_{m,j}})$, equals to

$I(l_{\theta_{m,j}}/m)\equiv\sum_{j_{{\rm B}},j_{{\rm
A}}=1}^{d_{\pi}}\int_{\lambda}f_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)\ln\left(\frac{f_{j_{{\rm B}},j_{{\rm A}}}(\lambda)}{p_{j_{{\rm
B}},j_{{\rm A}}}(\lambda)}\right)d\lambda,$ (9)

where $f_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\equiv n_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)/m$ for each set $(j_{{\rm B}},j_{{\rm A}},\lambda)$ of system
parameters. The rate function $I(l_{\theta_{m,j}}/m)$ is the _Kullback-Leibler
distance_ (or relative entropy) between the set $\\{f_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)\\}$ of _scaled_ relative frequencies and the set of
probabilities $\\{p_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\\}$, and, thus, has the
properties to be positive and convex. Eq. (8) is valid in the limit of $m$
large and indicates a _unique nonequilibrium weighted partition_ of the system
configuration space (see the Appendix for the proof). For a small value of
$m$, indeed, the distribution $\textrm{Prob}(l_{\theta_{m,j}})$ cannot be
uniquely determined. The latter can be considered as a universal property of
any dynamical evolutions given by the composition of quantum maps and
projections. The alternative LD expression of Eq. (8), providing the formal
definition of $I(l_{\theta_{m,j}}/m)$, is

$\lim_{m\rightarrow\infty}-\frac{1}{m}\ln\,\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))=I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m),$
(10)

where the number $m$ of measurements is assumed ideally infinite. If Eq. (10),
also called the _large-deviation approximation_ , holds, it means that the
dominant behaviour of $\textrm{Prob}(l_{\theta_{m,j}})$ is _convergent_ and
identically equal to a decaying exponential in $m$.

As last, through the contraction principle footnote4 , the distribution
$\textrm{Prob}(p_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$ is obtained:

$\textrm{Prob}(p_{\theta_{m,j}}(\vec{o},\vec{\lambda}))=\int\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))\delta(l_{\theta_{m,j}}-\ln
p_{\theta_{m,j}})d\,l_{\theta_{m,j}}.$ (11)

Then, by applying the saddle point method BookWong1989 , one has that

$\textrm{Prob}(p_{\theta_{m,j}}(\vec{o},\vec{\lambda}))\asymp\exp(-m\,J(p_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)),$
(12)

where

$J(p_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)\equiv\min_{l_{\theta_{j}}:\,l_{\theta_{j}}=\,\ln
p_{\theta_{j}}}I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m).$ (13)

The result is that a quantum system exhibiting stochastic evolutions (e.g. due
to random interactions with the environment) and repeatedly monitored by an
observer tends to reach in probability a unique configuration defined by
specific probability distributions of its characteristic parameters.

###  IV.1 Most probable distribution

The expression of the most probable distribution of $\Theta$’s outcomes is
here discussed. In general, from the knowledge of the rate function $I(\xi/m)$
associated to $\textrm{Prob}(\xi)$, we can then compute the most probable
value $\xi^{\star}$, representing the best prediction for the random variable
$\xi$ in a single realization of the system dynamics. Specifically, the most
probable value of the log-conditional-probability
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$, i.e., $l^{\star}_{\theta_{m,j}}$,
is obtained by evaluating the value at which the rate function
$I(l_{\theta_{m,j}}/m)$ is minimized as a function of $l_{\theta_{m,j}}$. As
proved in the Appendix, the closed-form expression of
$l^{\star}_{\theta_{m,j}}$ is

$l^{\star}_{\theta_{m,j}}=m\sum_{j_{{\rm B}},j_{{\rm
A}}=1}^{d_{\pi}}\int_{\lambda}p_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\ln
q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\,d\lambda,$ (14)

corresponding to the statement that $n_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)=m\,p_{j_{{\rm B}},j_{{\rm A}}}(\lambda)$ for every $(j_{{\rm
B}},j_{{\rm A}},\lambda)$. This means that for large $m$ the most probable
trajectories of the system dynamics are those allowing for the equality
between the scaled relative frequencies $f_{j_{{\rm B}},j_{{\rm A}}}(\lambda)$
and the occurrence probabilities $p_{j_{{\rm B}},j_{{\rm A}}}(\lambda)$. Only
this condition minimizes the relative Shannon entropy between the
configurations induced by the stochastic dynamics of the system. Once more, it
is worth observing the importance of imposing $m\rightarrow\infty$; indeed,
the value of the scaled relative frequencies $f_{j_{{\rm B}},j_{{\rm A}}}$ can
get closer to that of the probabilities $p_{j_{{\rm B}},j_{{\rm A}}}$ only if
the number of configurations generated by the stochasticity within the
dynamics is as large as possible. Finally, to derive the most probable value
of the conditional probabilities $p^{\star}_{\theta_{m,j}}$, we use again the
contraction principle from LD theory, but this time on the functional relation
between $l_{\theta_{m,j}}$ and $p_{\theta_{m,j}}$, i.e.
$p_{\theta_{m,j}}=e^{l_{\theta_{m,j}}}$:

$p^{\star}_{\theta_{m,j}}=\exp\left(m\sum_{j_{{\rm B}},j_{{\rm
A}}=1}^{d_{\pi}}\int_{\lambda}p_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\ln
q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\,d\lambda\right).$ (15)

##  V Discussion

Having proved that the nonequilibrium statistics of an arbitrary observable
obeys the LD principle for a sufficiently large number of intermediate
projection events, the answer to the original question is given by the
distribution of $\Theta$’s outcomes with occurrence conditional probabilities
equal to $p^{\star}_{\theta_{m,j}}$. Such a distribution is denoted as the
_most probable distribution_ (see e.g. Fig. 2 inset (a)):

$\textrm{Prob}^{\star}(\theta)\equiv\frac{1}{\mathcal{N}}\sum_{j}\delta(\theta-\theta_{m,j})p^{\star}_{\theta_{m,j}},$
(16)

which has to be normalized by the factor $\mathcal{N}$ so as to ensure that
$\int\textrm{Prob}^{\star}(\theta)d\theta=1$. Accordingly, the predictions
about the result of single-shot measurements from $\Theta$ according to the
most probable distribution $\textrm{Prob}^{\star}(\theta)$ are expected to be
more accurate than the ones that we would obtain by directly computing the
expectation value of the measurement outcomes. This is because _large
fluctuations_ within the evolution of the system are now properly weighted and
thus correctly included in the outcome distributions. In this regard, also
quantum noise sensing techniques DegenReview2017  could be used, so as to
improve the a-priori information on $p(\lambda)$.

In conclusion, being the average $\overline{p}_{\theta_{m,j}}$ explicitly
equal to

$\langle p_{\theta_{m,j}}(\vec{o},\vec{\lambda})\rangle=\left(\sum_{j_{{\rm
B}},j_{{\rm A}}=1}^{d_{\pi}}\int_{\lambda}p_{j_{{\rm B}},j_{{\rm
A}}}(\lambda)q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)d\lambda\right)^{m},$ (17)

one can observe a deviation between the most probable values
$p^{\star}_{\theta_{m,j}}$ and the probabilities $\overline{p}_{\theta_{m,j}}$
to measure on average the outcome $\theta_{m,j}$ at the final time instant
$t_{m}$. Specifically, by applying Jensen’s inequality footnote2  to
$p^{\star}_{\theta_{m,j}}$ and $\overline{p}_{\theta_{m,j}}$, the inequality
$p^{\star}_{\theta_{m,j}}\leq\overline{p}_{\theta_{m,j}}$ is obtained. The
latter well justifies the introduction of the normalization factor
$\mathcal{N}$ in Eq. (16). In fact, the normalization of
$\textrm{Prob}^{\star}(\theta)$ has to be ensured, and the values of the
$p^{\star}_{\theta_{m,j}}$’s are thus corrected by dividing for
$\mathcal{N}\equiv\sum_{j}p^{\star}_{\theta_{m,j}}\leq 1$, so that
$p^{\star}_{\theta_{m,j}}/\mathcal{N}=1-(\sum_{k,\,k\neq
j}p^{\star}_{\theta_{m,k}})/\mathcal{N}$,
$\forall\,j=1,\ldots,\dim(\mathcal{H})$.

##  VI Experimental proposal

Figure 2: Experimental proposal: Exact quantum observable statistics of a 2LS
with stochastic Hamiltonian. (a) Theoretical most probable values
$p_{\theta_{1}}^{\star}$ and $p_{\theta_{2}}^{\star}$ as a function of $m$
compared with different values of the probability $p_{\theta_{1}}$, which has
been numerically obtained from single realizations of the system
transformation. (b) Time-behaviour of system populations $\rho^{(11)}$ and
$\rho^{(22)}$ by numerically solving the stochastic evolution of the 2LS for
one realization of $\vec{o}$ and $\vec{\varphi}$. (c) Plot of
$\ln(p_{\theta_{1}})/m$ from data of inset (a). (d) Histogram of
$p_{\theta_{1}}$, with $m=14\cdot 10^{3}$, over $20\cdot 10^{3}$ repetitions
of the system dynamics.

Let us consider a two-level system (2LS) with Hamiltonian
$H=\frac{1}{2}\omega(\cos\varphi\sigma_{z}-\sin\varphi\sigma_{x})$, with
$\omega=11$ (in natural units s.t. $\hbar=1$) and $\sigma_{x},\sigma_{z}$
Pauli matrices. This system, realized e.g. within a Bose-Einstein condensate
of Rubidium atoms confined by a magnetic micro-trap as in GherardiniErgodicity
, is intrinsically stochastic due to random fluctuations of the angle
$\varphi$. They can be produced for example by a random coupling between the
system and the laser source that coherently drives it. In particular, we
assume sampling of $\varphi$ from the Rayleigh probability distribution
$p(\varphi)=(\varphi/\sigma^{2})\exp(-\varphi^{2}/(2\sigma^{2}))$ with scale
parameter $\sigma=1$. Then, the 2LS, starting from the ground $|0\rangle$, is
monitored by a sequence of $m$ repeated projective measurements. They are
separated by the constant time interval $\tau=0.05$ (in natural units) and
defined by the projectors $\Pi_{1}\equiv|0\rangle\langle 0|$ and
$\Pi_{2}\equiv|1\rangle\langle 1|$. After each dynamical evolution with whole
duration $t_{m}=m\tau$, the quantum observable

$\Theta\equiv\sum_{j=1}^{2}\theta_{j}\Pi_{\theta_{j}}=\rho^{(11)}\Pi_{1}+\rho^{(22)}\Pi_{2}$
(18)

is measured, where the measurement outcomes $\theta_{1}\equiv\rho^{(11)}$ and
$\theta_{2}\equiv\rho^{(22)}=1-\rho^{(11)}$ identify the percentage of system
population respectively in the ground and excited states, $|0\rangle$ and
$|1\rangle$ respectively. Notice that, being $\mathcal{S}$ a 2LS, we analyze
the fluctuations of only the conditional probability
$p_{\theta_{1}}(\vec{o},\vec{\varphi})\equiv{\rm
Tr}[\Pi_{\theta_{1}}\rho_{m,\vec{o},\vec{\varphi}}]$.

In Fig. 2-(a) we show the most probable values of $p_{\theta_{1}}$, i.e.
$p_{\theta_{1}}^{\star}$ (solid red curve), which has been computed by
plotting the analytical expression of Eq. (15) as a function of $m$. The
following procedure has been followed. First, we have numerically computed the
probabilities $p_{\theta_{1}}$ (blue points in Fig. 2-(a)) to measure the
measurement outcome $\theta_{1}$ after each realization of the system
stochastic transformation (see inset (b)). Then, we have derived the quantity
$\ln(p_{\theta_{1}})/m$, by distinguishing between values of $p_{\theta_{1}}$
smaller (dotted blue curve – inset (c)) and greater (dashed red curve – inset
(c)) than $1/2$. As it can be observed, only the dotted blue curve is
practically constant for $m\geq 700$ and equal to $X\simeq-4.22\cdot 10^{-6}$.
In particular, from the theoretical findings, we can state that for $m$ large
$\ln(p_{\theta_{1}}^{\star})\asymp mX$ (solid red curve – inset (a)) with

$\displaystyle\displaystyle{X\equiv\sum_{j_{\rm B},j_{\rm A}}\int p_{j_{\rm
B},j_{\rm A}}(\varphi)\ln(q_{j_{\rm B},j_{\rm
A}}(\varphi))d\varphi},\,\,\,\text{i.e.,}$ (19)
$\displaystyle\displaystyle{X=\frac{1}{N_{\rm meas}}\int d\varphi
p(\varphi)\sum_{k,j=0}^{1}\ln|\langle k|U(\varphi)|j\rangle|^{2}}$ (20)

for this specific example, where $U(\varphi)\equiv\exp(-iH(\varphi)\tau)$ and
$N_{\rm meas}$ encodes the occurrence probabilities of the intermediate
quantum measurement projectors. As a further test, we have numerically
obtained the probability distribution of $p_{\theta_{1}}$ (inset (d)) with
$m=14\cdot 10^{3}$, by repeating $20\cdot 10^{3}$ times the dynamics of the
system, and computed the value of $p_{\theta_{1}}^{\rm peak}$ at which ${\rm
Prob}(p_{\theta_{1}})$ admits the peak; we measured $p_{\theta_{1}}^{\rm
peak}=0.0736$ and an average value $\overline{p}_{\theta_{1}}=0.5016$. This
value should equal to the most probable value $p_{\theta_{1}}^{\star}$ and
this has been effectively verified by the numerics, since from the solid red
curve in Fig. 2-(a) we find for $m=14\cdot 10^{3}$ the value
$p_{\theta_{1}}^{\star}=0.0731$.

##  VII Conclusions

These results combine out-of-equilibrium quantum systems with statistical
mechanics methods: They are expected to be a tool to defeat noise in a quantum
system, since they provide predictions of the stochastic evolution of the
system after the single realization.

The author gratefully acknowledges F. Caruso and M. Artoni for a detailed
reading of the paper, A. Trombettoni and M. Fattori for several useful
comments, and Shamik Gupta for insightful discussions on LD theory, in
particular on the derivation of the measurement outcomes’probability
distribution. S.G. was financially supported from the Fondazione CR Firenze
through the project Q-BIOSCAN.

## Appendix

### I. Measurement conditional probabilities

Let us express the measurement projectors $\Pi_{\mu_{\alpha}}$ applied at time
instants $t_{\alpha}$, $\alpha=1,\ldots,m$, as a function of their
eigenvectors $|\pi_{\mu_{\alpha}}\rangle$, i.e.
$\Pi_{\mu_{\alpha}}\equiv|\pi_{\mu_{\alpha}}\rangle\langle\pi_{\mu_{\alpha}}|$,
with $\mu\in\\{\theta,o\\}$. Thus, by substituting the definition of the
measurement super-operators $\mathcal{P}_{\theta_{m}}$ and
$\mathcal{P}_{o_{\alpha}}$ into the equation

$p_{\theta_{m}}(\vec{o},\vec{\lambda})\equiv\textrm{Tr}\left[\mathcal{P}_{\theta_{m}}\mathcal{U}_{m}\mathcal{P}_{o_{m-1}}\mathcal{U}_{m-1}\cdots\mathcal{P}_{o_{1}}\mathcal{U}_{1}[\rho_{0}]\right],$
(21)

the conditional probability $p_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ can be
written as the product of the transition probabilities (random terms) that the
quantum state moves from $|\pi_{\mu_{\alpha-1}}\rangle$ to
$|\pi_{\mu_{\alpha}}\rangle$ via the unitary operator
$U_{\alpha}(\lambda_{\alpha})$:

$\displaystyle
p_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\textrm{Tr}[\Pi_{\theta_{m,j}}U_{m}\Pi_{o_{m}-1}U_{m-1}\cdots\Pi_{o_{1}}U_{1}\rho_{0}U_{1}^{\dagger}\Pi_{o_{1}}\cdots
U_{m-1}^{\dagger}\Pi_{o_{m-1}}U_{m}^{\dagger}\Pi_{\theta_{m,j}}]$
$\displaystyle=\textrm{Tr}\left[|\pi_{\theta_{m,j}}\rangle\langle\pi_{\theta_{m,j}}|U_{m}|\pi_{o_{m-1}}\rangle\langle\pi_{o_{m-1}}|U_{m-1}\cdots|\pi_{o_{1}}\rangle\langle\pi_{o_{1}}|U_{1}\rho_{0}U_{1}^{\dagger}|\pi_{o_{1}}\rangle\langle\pi_{o_{1}}|\cdots
U_{m-1}^{\dagger}|\pi_{o_{m-1}}\rangle\langle\pi_{o_{m-1}}|U_{m}^{\dagger}|\pi_{\theta_{m,j}}\rangle\langle\pi_{\theta_{m,j}}|\right]$
$\displaystyle=\langle\pi_{o_{1}}|U_{1}\rho_{0}U_{1}^{\dagger}|\pi_{o_{1}}\rangle\cdot\prod_{k=2}^{m-1}|\langle\pi_{o_{k-1}}|U_{k}|\pi_{o_{k}}\rangle|^{2}\cdot|\langle\pi_{o_{m-1}}|U_{m}|\pi_{\theta_{m,j}}\rangle|^{2}.$
(22)

$\rho_{0}$ is the initial density matrix of the open quantum system
$\mathcal{S}$ before the transformation induced by random interactions with
the environment $\mathcal{E}$ and the monitoring by an observer. Moreover, we
also assume that $\rho_{0}$ is defined by the pure state $|\pi_{0}\rangle$,
i.e., $\rho_{0}\equiv|\pi_{0}\rangle\langle\pi_{0}|$. For example this
assumption is verified when the measurement outcomes at the final time $t_{m}$
are obtained by a two-time measurement scheme CampisiPRL2010 . In such a case,
indeed, $\rho_{0}$ is the density matrix of $\mathcal{S}$ after the first
measurement of the scheme, and thus it is described only by a pure state.
Therefore, under this further hypothesis, one has that
$\langle\pi_{o_{1}}|U_{1}\rho_{0}U_{1}^{\dagger}|\pi_{0_{1}}\rangle=|\langle\pi_{0}|U_{1}|\pi_{0_{1}}\rangle|^{2}$,
so that

$p_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\prod_{\alpha=1}^{m}|\langle\pi_{\mu_{\alpha-1}}|U_{\alpha}(\lambda_{\alpha})|\pi_{\mu_{\alpha}}\rangle|^{2},$
(23)

with $\pi_{\mu_{0}}\equiv\pi_{0}$, $\mu_{\alpha}\equiv o_{\alpha}$ for
$\alpha=1,\ldots,m-1$ and $\pi_{\mu_{m}}\equiv\pi_{\theta_{m,j}}$.

### II. Derivation of $\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$
via LD theory

The log-probability $l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ is the logarithm
of the conditional probability $p_{\theta_{m,j}}(\vec{o},\vec{\lambda})$
defining the probability to obtain the $j$-th measurement outcome
$\theta_{m,j}$ at the time instant $t_{m}$, conditioned to the specific
realizations of the sequences $\vec{o}$ and $\vec{\lambda}$. More formally,

$\displaystyle l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ $\displaystyle\equiv$
$\displaystyle\ln
p_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\sum_{\alpha=1}^{m}\ln
q_{\alpha}(\mu_{\alpha-1},\mu_{\alpha},\lambda_{\alpha})$ (24)
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{m}\ln\left(|\langle\pi_{\mu_{\alpha-1}}|U_{\alpha}(\lambda_{\alpha})|\pi_{\mu_{\alpha}}\rangle|^{2}\right).$

For the sake of clarity, let us assume the following two hypotheses: (i)
$p(\lambda)$ is assumed to be a $d_{\lambda}-$dimensional Bernoulli
distribution, with the result that at each time instant $t_{\alpha}$ the
parameter $\lambda_{\alpha}$ takes on $d_{\lambda}$ possible values
$\lambda^{(1)},\ldots,\lambda^{(d_{\lambda})}$ with corresponding
probabilities $p_{\lambda}^{(1)},\ldots,p_{\lambda}^{(d_{\lambda})}$ so that
$\sum_{i=1}^{d_{\lambda}}p_{\lambda}^{(i)}=1$. The index $i$ denotes the
values that can be assumed by $\lambda$. (ii) The measurement bases of
$\Theta$ and $\mathcal{O}$, given by the set of eigenvectors
$\\{|\pi_{\mu_{\alpha}}\rangle\\}$, belong to a set of finite dimension and
this means that each ket $|\pi_{\mu_{\alpha}}\rangle$ admits only a finite
number $d_{\pi}$ of elements $|\pi^{(j)}\rangle$ with probability $p_{\pi}$.
Notice that the index for the dimensionality of the measurement basis
configurations is $j$, as well as that for the dimension of $\mathcal{H}$. By
following a common procedure in LD theory, we group the terms of
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ as a function of the number of times
(relative frequencies) each dynamical transition probability $q_{j_{{\rm
B}},i,j_{{\rm A}}}\equiv|\langle\pi^{(j_{{\rm
B}})}|U(\lambda^{(i)})|\pi^{(j_{{\rm A}})}\rangle|^{2}$ occurs during the
transformation of the system. We denote with $n_{j_{{\rm B}},i,j_{{\rm A}}}$
the relative frequencies for the occurrence of the $q_{j_{{\rm B}},i,j_{{\rm
A}}}$’s. The corresponding occurrence probabilities, instead, are denoted as
$p_{j_{{\rm B}},i,j_{{\rm A}}}$. Over many realizations of the system
transformation, one can reasonably assume that $p_{\pi}$ is sampled by a
uniform distribution, such that $p_{\pi}=1/d_{\pi}$ and $p_{j_{{\rm
B}},i,j_{{\rm A}}}\equiv p_{\pi}^{(j_{{\rm
B}})}p_{\lambda}^{(i)}p_{\pi}^{(j_{{\rm A}})}=p_{\lambda}^{(i)}/d_{\pi}^{2}$.
The latter procedure directly leads to the analytical expression of
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$, i.e.,

$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\sum_{j_{{\rm
B}}=1}^{d_{\pi}}\sum_{i=1}^{d_{\lambda}}\sum_{j_{{\rm
A}}=1}^{d_{\pi}}n_{j_{{\rm B}},i,j_{{\rm A}}}\ln q_{j_{{\rm B}},i,j_{{\rm
A}}}.$ (25)

Eq. (25) shows us that $l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ can be
written as the sum of the i.i.d. dynamical transition probabilities
$q_{j_{{\rm B}},i,j_{{\rm A}}}$, weighted by the relative frequencies
$n_{j_{{\rm B}},i,j_{{\rm A}}}$ corresponding to the occurrence statistics of
the $q_{j_{{\rm B}},i,j_{{\rm A}}}$’s. Notice that the relative frequencies
$n_{j_{{\rm B}},i,j_{{\rm A}}}$ are usually different from the corresponding
probability values $p_{j_{{\rm B}},i,j_{{\rm A}}}$. Then, by taking Eq. (25),
the distribution probability of $l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ is
given by

$\displaystyle\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))=\frac{m!}{\displaystyle{\prod_{j_{{\rm
B}},i,j_{{\rm A}}}n_{j_{{\rm B}},i,j_{{\rm A}}}!}}\prod_{j_{{\rm B}},i,j_{{\rm
A}}}(p_{j_{{\rm B}},i,j_{{\rm A}}})^{n_{j_{{\rm B}},i,j_{{\rm A}}}}$
$\displaystyle\times\delta\left(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})-\sum_{j_{{\rm
B}},i,j_{{\rm A}}}n_{j_{{\rm B}},i,j_{{\rm A}}}\ln q_{j_{{\rm B}},i,j_{{\rm
A}}}\right),$ (26)

where $\delta(\cdot)$ denotes the Kronecker delta. Let us observe that in Eq.
(II. Derivation of $\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$
via LD theory), to simplify the notation, we have used the symbols
$\displaystyle{\sum_{j_{{\rm B}},i,j_{{\rm A}}}}$ and
$\displaystyle{\prod_{j_{{\rm B}},i,j_{{\rm A}}}}$ to denote respectively
$\displaystyle{\sum_{j_{{\rm
B}}=1}^{d_{\pi}}\sum_{i=1}^{d_{\lambda}}\sum_{j_{{\rm A}}=1}^{d_{\pi}}}$ and
$\displaystyle{\prod_{j_{{\rm
B}}=1}^{d_{\pi}}\prod_{i=1}^{d_{\lambda}}\prod_{j_{{\rm A}}=1}^{d_{\pi}}}$.
Then, by imposing in Eq. (II. Derivation of
$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$ via LD theory) the
condition given by the Kronecker delta, one has that

$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))=\frac{m!}{\displaystyle{\prod_{j_{{\rm
B}},i,j_{{\rm A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}!}}\prod_{j_{{\rm
B}},i,j_{{\rm A}}}(p_{j_{{\rm B}},i,j_{{\rm A}}})^{\widehat{n}_{j_{{\rm
B}},i,j_{{\rm A}}}},$ (27)

where the relative frequencies $\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}$’s
have to satisfy the following constraints:

$\begin{cases}\displaystyle{\sum_{j_{{\rm B}},i,j_{{\rm A}}}n_{j_{{\rm
B}},i,j_{{\rm A}}}=m}\\\
\displaystyle{l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\sum_{j_{{\rm
B}},i,j_{{\rm A}}}n_{j_{{\rm B}},i,j_{{\rm A}}}\ln q_{j_{{\rm B}},i,j_{{\rm
A}}}}.\end{cases}$ (28)

By combining together the constraints (28), we obtain a unique constraint
equation for $l_{\theta_{m,j}}$, i.e.,

$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=m\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm
A}}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}\gamma_{j_{{\rm B}},i,j_{{\rm
A}}},$ (29)

where

$\gamma_{j_{{\rm B}},i,j_{{\rm A}}}\equiv\frac{\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}}{\ln q_{j_{{\rm B}},i,j_{{\rm A}}}}.$ (30)

It is worth observing that in deriving the constraint (29) we have chosen
$(d_{\pi},d_{\lambda},d_{\pi})$ as the _reference triplet_ of the
configuration space that defines the stochastic trajectory of the system
dynamics in a single realization. This choice is arbitrary and represents a
degree of freedom of the procedure. However, that is not surprising because
the number of constraints of formula (28) is smaller than the number of
relative frequencies $n_{j_{{\rm B}},i,j_{{\rm A}}}$, so that the values of
$\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}$ that satisfy Eq. (27) are generally
not uniquely determined. This means that in order to obtain a unique
analytical expression of
$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$, we need to answer
the following questions: Which are the _unique_ values of the relative
frequencies $\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}$ obeying the constraint
equation (29)? By generalizing the results in GherardiniNJP , we can prove
that _there exists a unique value for the $\widehat{n}_{j_{{\rm B}},i,j_{{\rm
A}}}$’s, under the hypothesis of a sufficiently large number $m$ of
intermediate projective measurements_. To see this, let us consider the
product $\widehat{n}_{a,b,c}\gamma_{a,b,c}$ with generic indices $(a,b,c)$:

$\displaystyle\displaystyle{\widehat{n}_{a,b,c}\gamma_{a,b,c}=m\left(\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m\right)}$
$\displaystyle-\displaystyle{\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm
A}};\,\,(j_{{\rm B}},i,j_{{\rm A}})\neq(a,b,c)}}\widehat{n}_{j_{{\rm
B}},i,j_{{\rm A}}}\gamma_{j_{{\rm B}},i,j_{{\rm A}}}},$ (31)

i.e.,

$\displaystyle\displaystyle{\frac{\widehat{n}_{a,b,c}}{m}=\frac{\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m}{\gamma_{a,b,c}}}$
(32) $\displaystyle-\displaystyle{\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm
A}};\,\,j_{{\rm B}}\neq a,i\neq b,j_{{\rm A}}\neq c}}\widehat{n}_{j_{{\rm
B}},i,j_{{\rm A}}}\frac{\widehat{n}_{j_{{\rm B}},i,j_{{\rm
A}}}}{m}\frac{\gamma_{j_{{\rm B}},i,j_{{\rm A}}}}{\gamma_{a,b,c}}}$ .

Being $\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}$’s relative frequencies, it
still holds that $\lim_{m\rightarrow\infty}\frac{\widehat{n}_{j_{{\rm
B}},i,j_{{\rm A}}}}{m}=0$, for each triplet $(j_{{\rm B}},i,j_{{\rm A}})$
except $(a,b,c)$, with the result that

$\displaystyle\displaystyle{\frac{\widehat{n}_{a,b,c}}{m}\asymp\frac{\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m}{\gamma_{a,b,c}}}$
(33) $\displaystyle=\displaystyle{\frac{1}{N_{\rm sf}}\left(\frac{\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m}{\gamma_{a,b,c}}\right)}$
,

with $N_{\rm sf}$ denoting the corresponding _scaling factor_. Thus, this
means that for _large_ $m$

$\widehat{n}_{a,b,c}\gamma_{a,b,c}\asymp\frac{m\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})}{N_{\rm
sf}}={\rm constant},$ (34)

for each triplet $(a,b,c)$. Eq. (34) is a quite important result, since it
denotes the existence of a _unique nonequilibrium weighted partition_ of the
system configuration space, once we have fixed the reference triplet
$(d_{\pi},d_{\lambda},d_{\pi})$. Such a property is thus the key point for the
derivation of $\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$. In
particular, by summing together the terms $\widehat{n}_{a,b,c}\gamma_{a,b,c}$
over all the possible values that can be assumed by $(a,b,c)$ except for the
triplet $(d_{\pi},d_{\lambda},d_{\pi})$, we find that

$\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm A}}}}\widehat{n}_{j_{{\rm
B}},i,j_{{\rm A}}}\gamma_{j_{{\rm B}},i,j_{{\rm A}}}\asymp\frac{(d_{\rm
tot}-1)}{N_{\rm sf}}\left(m\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})\right),$
(35)

where $d_{\rm tot}\equiv 2d_{\pi}+d_{\lambda}$ is the dimension of the
statistical ensemble defining the stochastic transformation of the system from
$t_{0}$ to $t_{m}$. Therefore, by comparing Eqs. (29) and (35), one can state
that $N_{\rm sf}=d_{\rm tot}-1$, so that for $m$ sufficiently large

$\widehat{n}_{a,b,c}\asymp\frac{m\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})}{(d_{\rm
tot}-1)\gamma_{a,b,c}}.$ (36)

Once we have obtained a closed solution for the value of the relative
frequencies obeying the constraints (28), we are able to validate the
exponential approximation given by the _large-deviation principle_ for the
probability distribution of $l_{\theta_{m,j}}$ in the thermodynamic limit of
$m\rightarrow\infty$. To practically derive the LD form of
$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$, just take Eq. (27)
and apply the Stirling approximation on $\ln(m!)$ and
$\ln(\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}!)$, which is valid again in the
limit of large $m$:

$\displaystyle\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$
$\displaystyle=$ $\displaystyle\exp\left(\ln(m!)-\sum_{j_{{\rm B}},i,j_{{\rm
A}}}\ln(\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}!)+\sum_{j_{{\rm B}},i,j_{{\rm
A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}\ln p_{j_{{\rm B}},i,j_{{\rm
A}}}\right)$ (37) $\displaystyle\asymp$ $\displaystyle\exp\left(m\ln
m-m-\sum_{j_{{\rm B}},i,j_{{\rm A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm
A}}}\ln\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}+\sum_{j_{{\rm B}},i,j_{{\rm
A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}+\sum_{j_{{\rm B}},i,j_{{\rm
A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}\ln p_{j_{{\rm B}},i,j_{{\rm
A}}}\right)$ $\displaystyle=$ $\displaystyle\exp\left(m\ln m-\sum_{j_{{\rm
B}},i,j_{{\rm A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm
A}}}\ln\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}+\sum_{j_{{\rm B}},i,j_{{\rm
A}}}\widehat{n}_{j_{{\rm B}},i,j_{{\rm A}}}\ln p_{j_{{\rm B}},i,j_{{\rm
A}}}\right).$

Then, by substituting the expression of $\widehat{n}_{j_{{\rm B}},i,j_{{\rm
A}}}$’s given by Eq. (36), after straightforward calculations one finds that

$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))\asymp\exp(-m\,I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)),$
(38)

where

$I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)=\sum_{j_{{\rm
B}}=1}^{d_{\pi}}\sum_{i=1}^{d_{\lambda}}\sum_{j_{{\rm
A}}=1}^{d_{\pi}}f_{j_{{\rm B}},i,j_{{\rm A}}}\ln\left(\frac{f_{j_{{\rm
B}},i,j_{{\rm A}}}}{p_{j_{{\rm B}},i,j_{{\rm A}}}}\right)$ (39)

is the rate function associated to the probability distribution
$\textrm{Prob}(l_{\theta_{m,j}}(\vec{o},\vec{\lambda}))$. In particular, in
Eq. (39),

$f_{j_{{\rm B}},i,j_{{\rm A}}}\equiv\frac{\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m}{(d_{\rm
tot}-1)\gamma_{j_{{\rm B}},i,j_{{\rm A}}}}$ (40)

for each triplet $(j_{{\rm B}},i,j_{{\rm
A}})\neq(d_{\pi},d_{\lambda},d_{\pi})$, while

$f_{d_{\pi},d_{\lambda},d_{\pi}}\equiv 1-\widetilde{\sum_{j_{{\rm
B}},i,j_{{\rm A}}}}f_{j_{{\rm B}},i,j_{{\rm A}}}.$ (41)

As a final remark, given the generic triplet $(a,b,c)$, it is worth noting
that $f_{a,b,c}$ and the relative frequency $\widehat{n}_{a,b,c}$ are simply
related by the following equation:

$f_{a,b,c}=\frac{\widehat{n}_{a,b,c}}{m}\,.$ (42)

### III. Most probable distribution $l^{\star}_{\theta_{m,j}}$

Here, the most probable value $l^{\star}_{\theta_{m,j}}$ of the log-
conditional probability $l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ is derived.
The latter is obtained by evaluating the value at which the rate function
$I(l_{\theta_{m,j}}/m)$ of Eq. (39) is minimized as a function of
$l_{\theta_{m,j}}$ in the thermodynamic limit of $m\rightarrow\infty$.
$I(l_{\theta_{m,j}}/m)$ is a positive and convex function – see Appendix II –
and $l_{\theta_{m,j}}$ is given by a convex sum of the dynamical transition
probabilities $q_{j_{{\rm B}},i,j_{{\rm A}}}$. Thus, sufficient condition for
its minimization is that the identities

$\left.\frac{\partial
I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)}{\partial\ln q_{j_{{\rm
B}},i,j_{{\rm
A}}}}\right|_{l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=l^{\star}_{\theta_{m,j}}}=\,\,0,$
(43)

(computed in correspondence to
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=l^{\star}_{\theta_{m,j}}$) are all
simultaneously verified for _each_ triplet $(j_{{\rm B}},i,j_{{\rm A}})$. If
we perform the derivative of $I(l_{\theta_{m,j}}/m)$ with respect to $\ln
q_{j_{{\rm B}},i,j_{{\rm A}}}$, then we find that the identities (43) for the
triplets $(j_{{\rm B}},i,j_{{\rm A}})$ apart from
$(d_{\pi},d_{\lambda},d_{\pi})$ can be equivalently written by means of an
unique equation, i.e.,

$p_{d_{\pi},d_{\lambda},d_{\pi}}f_{j_{{\rm B}},i,j_{{\rm A}}}=p_{j_{{\rm
B}},i,j_{{\rm A}}}\left(1-\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm
A}}}}f_{j_{{\rm B}},i,j_{{\rm A}}}\right).$ (44)

By summing both sides over $(j_{{\rm B}},i,j_{{\rm A}})$, we get

$p_{d_{\pi},d_{\lambda},d_{\pi}}\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm
A}}}}f_{j_{{\rm B}},i,j_{{\rm A}}}=\left(1-\widetilde{\sum_{j_{{\rm
B}},i,j_{{\rm A}}}}f_{j_{{\rm B}},i,j_{{\rm
A}}}\right)\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm A}}}}p_{j_{{\rm
B}},i,j_{{\rm A}}},$ (45)

which, by using $\displaystyle{\sum_{j_{{\rm B}},i,j_{{\rm A}}}p_{j_{{\rm
B}},i,j_{{\rm A}}}}=1$, gives

$\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm A}}}}f_{j_{{\rm B}},i,j_{{\rm
A}}}=\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm A}}}}p_{j_{{\rm B}},i,j_{{\rm
A}}}.$ (46)

It is worth observing that Eq. (46) represents the condition for the
minimization of the rate function
$I(l_{\theta_{m,j}}(\vec{o},\vec{\lambda})/m)$ with respect to
$l_{\theta_{m,j}}$. This means that the most probable trajectories of the
system dynamics are those allowing for the equality between the summations of
the relative frequencies $f_{j_{{\rm B}},i,j_{{\rm A}}}$ and the occurrence
probabilities $p_{j_{{\rm B}},i,j_{{\rm A}}}$, respectively. Therefore, this
also implies that in general the same value of $l^{\star}_{\theta_{m,j}}$ can
be obtained by more than one trajectory within the configuration space of the
system, each of them corresponding to a different realization of the
stochastic dynamics of the system.

By combining Eq. (46) with the expression of
$l_{\theta_{m,j}}(\vec{o},\vec{\lambda})=\sum_{j_{{\rm B}},j_{{\rm
A}}=1}^{d_{\pi}}\int_{\lambda}n_{j_{{\rm B}},j_{{\rm A}}}(\lambda)\ln
q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)d\lambda$, using Eqs. (30), (40) and
substituting $l_{\theta_{m,j}}(\vec{o},\vec{\lambda})$ with
$l^{\star}_{\theta_{m,j}}$, one has that

$\displaystyle\displaystyle{\left(\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}-\frac{l^{\star}_{\theta_{j,m}}}{m}\right)=(d_{\rm
tot}-1)}$ $\displaystyle\times\displaystyle{\left(1-\widetilde{\sum_{j_{{\rm
B}},i,j_{{\rm A}}}}p_{j_{{\rm B}},i,j_{{\rm A}}}\right)\frac{p_{j_{{\rm
B}},i,j_{{\rm A}}}}{p_{d_{\pi},d_{\lambda},d_{\pi}}}\gamma_{j_{{\rm
B}},i,j_{{\rm A}}}}$ (47)

for each triplet $(j_{{\rm B}},i,j_{{\rm
A}})\neq(d_{\pi},d_{\lambda},d_{\pi})$. Then, if we extend Eq. (46) by
assuming that $f_{j_{{\rm B}},i,j_{{\rm A}}}=p_{j_{{\rm B}},i,j_{{\rm A}}}$
$\forall\,(j_{{\rm B}},i,j_{{\rm A}})$, then

$\frac{p_{j_{{\rm B}},i,j_{{\rm A}}}\gamma_{j_{{\rm B}},i,j_{{\rm
A}}}}{p_{d_{\pi},d_{\lambda},d_{\pi}}}=\frac{f_{j_{{\rm B}},i,j_{{\rm
A}}}\gamma_{j_{{\rm B}},i,j_{{\rm A}}}}{p_{d_{\pi},d_{\lambda},d_{\pi}}}={\rm
constant}.$ (48)

Since Eqs. (III. Most probable distribution $l^{\star}_{\theta_{m,j}}$) have
to be verified for each triplet of the system configuration space, this means
that Eqs. (III. Most probable distribution $l^{\star}_{\theta_{m,j}}$) are
identically equivalent to the relation

$\displaystyle\displaystyle{\frac{l^{\star}_{\theta_{m,j}}}{m}=\ln
q_{d_{\pi},d_{\lambda},d_{\pi}}}$
$\displaystyle-\displaystyle{\left(1-\widetilde{\sum_{j_{{\rm B}},i,j_{{\rm
A}}}}p_{j_{{\rm B}},i,j_{{\rm A}}}\right)\left(\widetilde{\sum_{j_{{\rm
B}},i,j_{{\rm A}}}}\frac{p_{j_{{\rm B}},i,j_{{\rm
A}}}}{p_{d_{\pi},d_{\lambda},d_{\pi}}}\gamma_{j_{{\rm B}},i,j_{{\rm
A}}}\right)},$

finally providing us the analytical expression of the most probable
distribution $l^{\star}_{\theta_{m,j}}$ as given by Eq. (14).

## References

  * (1) K. Jacobs. Quantum Measurement Theory and its Applications (Cambridge University Press, 2014).
  * (2) E.B. Davies, and J.T. Lewis. An operational approach to quantum probability, Commun. Math. Phys. 17, 239 (1970).
  * (3) H. Breuer, and F. Petruccione. The Theory of Open Quantum Systems (Oxford University Press, 2003).
  * (4) A. Rivas, and S. Huelga. Open Quantum Systems: An Introduction (Spriger Verlag, 2012).
  * (5) S. Gherardini, C. Lovecchio, M.M. Müller, P. Lombardi, F. Caruso, and F.S. Cataliotti. Ergodicity in randomly perturbed quantum systems, Quantum Sci. Technol. 2(1), 015007 (2017).
  * (6) M.M. Müller, S. Gherardini, and F. Caruso. Quantum Zeno dynamics through stochastic protocols, Annalen der Physik 529(9), 1600206 (2017).
  * (7) M.A.C. Rossi, C. Foti, A. Cuccoli, J. Trapani, P. Verrucchi, and M.G.A. Paris. Effective description of the short-time dynamics in open quantum systems, Phys. Rev. A 96, 032116 (2017).
  * (8) M. Campisi, P. Talkner, and P. Hänggi. Fluctuation Theorems for Continuously Monitored Quantum Fluxes, Phys. Rev. Lett. 105, 140601 (2010).
  * (9) F. Thiel, E. Barkai, and D.A. Kessler. First Detected Arrival of a Quantum Walker on an Infinite Line, Phys. Rev. Lett. 120, 040502 (2018).
  * (10) A. Mishra, T. Albash, and D.A. Lidar. Finite temperature quantum annealing solving exponentially small gap problem with nonmonotonic success probability, Nat. Comm. 9, 2917 (2018).
  * (11) F.W.J. Hekking and J.P. Pekola. Quantum Jump Approach for Work and Dissipation in a Two-Level System, Phys. Rev. Lett. 111, 093602 (2013).
  * (12) P. Facchi and S. Pascazio. Quantum Zeno Subspaces, Phys. Rev. Lett. 89, 080401 (2002).
  * (13) G.S. Uhrig. Keeping a Quantum Bit Alive by Optimized pi-Pulse Sequences, Phys. Rev. Lett. 98, 100504 (2007).
  * (14) G. de Lange, Z.H. Wang, D. Riste, V.V. Dobrovitski, R. Hanson. Universal Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath, Science 330, 60–63 (2010).
  * (15) B. Pokharel, N. Anand, B. Fortman, and D.A. Lidar. Demonstration of Fidelity Improvement Using Dynamical Decoupling with Superconducting Qubits, Phys. Rev. Lett. 121, 220502 (2018).
  * (16) F. Schäfer, I. Herrera, S. Cherukattil, C. Lovecchio, F.S. Cataliotti, F. Caruso, and A. Smerzi. Experimental realization of quantum Zeno dynamics, Nat. Commun. 5, 4194 (2014).
  * (17) A. Signoles, A. Facon, D. Grosso, I. Dotsenko, S. Haroche, J.-M. Raimond, M. Brune, and S. Gleyzes. Confined quantum Zeno dynamics of a watched atomic arrow, Nat. Phys. 10, 715–719 (2014).
  * (18) L. Pezzè, A. Smerzi, M.K. Oberthaler, R. Schmied, and P. Treutlein. Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys 90, 035005 (2018).
  * (19) R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer. Phase Locking a Clock Oscillator to a Coherent Atomic Ensemble, Phys. Rev. X 5, 021011 (2015).
  * (20) M. Schioppo, R.C. Brown, W.F. McGrew, N. Hinkley, R.J. Fasano, K. Beloy, T.H. Yoon, G. Milani, D. Nicolodi, J.A. Sherman, N.B. Phillips, C.W. Oates, and A.D. Ludlow. Ultrastable optical clock with two cold-atom ensembles, Nat. Phot. 11, 48-52 (2017).
  * (21) A. Chantasri, J. Dressel, and A.N. Jordan. Action principle for continuous quantum measurement, Phys. Rev. A 88, 042110 (2013).
  * (22) S.J. Weber, A. Chantasri, J. Dressel, A.N. Jordan, K.W. Murch, and I. Siddiqi. Mapping the optimal route between two quantum states, Nature 511, 570–573 (2014).
  * (23) A. Chantasri, and A.N. Jordan. Stochastic path-integral formalism for continuous quantum measurement, Phys. Rev. A 92, 032125 (2015).
  * (24) R. Ellis. Entropy, Large Deviations, and Statistical Mechanics (Springer New York, 2006).
  * (25) H. Touchette. The large deviation approach to statistical mechanics, Phys. Rep. 478, 1 (2009).
  * (26) A. Dembo, and O. Zeitouni. Large Deviations Techniques and Applications (Springer Berlin, 2010).
  * (27) G. Gallavotti, J.L. Lebowitz, and V. Mastropietro. Large deviation in rarefied quantum gases, J. Stat. Phys. 108 831 (2002).
  * (28) K. Netočný, and F. Redig. Large deviation for quantum spin systems, J. Stat. Phys. 117 521 (2004).
  * (29) J.P. Garrahan, and I. Lesanovsky. Thermodynamics of quantum jump trajectories, Phys. Rev. Lett. 104 160601 (2010).
  * (30) I. Lesanovsky, M. van Horssen, M. Guţă, and J.P. Garrahan. Characterization of dynamical phase transitions in quantum jump trajectories beyond the properties of the stationary state, Phys. Rev. Lett. 110 150401 (2013).
  * (31) S. Pigeon, L. Fusco, A. Xuereb, G. De Chiara, and M. Paternostro. Thermodynamics of trajectories of a quantum harmonic oscillator coupled to N baths, Phys. Rev. A 92, 013844 (2015).
  * (32) M. Collura, and F.H.L. Essler. How order melts after quantum quenches, Eprint arXiv:1901.04402 (2019).
  * (33) S. Gherardini, S. Gupta, F.S. Cataliotti, A. Smerzi, F. Caruso, and S. Ruffo. Stochastic Quantum Zeno by Large Deviation Theory, New J. Phys. 18(1), 013048 (2016).
  * (34) S. Gherardini. Noise as a resource. PhD Thesis, University of Florence (Italy). Eprint arXiv:1805.01800 (2018).
  * (35) The introduction of semi-classical fluctuations during the evolution of the system is responsible for the modeling of the effects on its reduced dynamics given by tracing out the environmental degrees of freedom RossiPRA2017 . Instead, the local couplings between the system and the environment are described by a sequence of projective measurements thesis_Stefano .
  * (36) J.J. Sakurai. Modern Quantum Mechanics (2nd edition, 1994).
  * (37) F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini. Quantum channels and memory effects, Rev. Mod. Phys. 86, 1203 (2014).
  * (38) S. Gherardini, L. Buffoni, M.M. Mueller, F. Caruso, M. Campisi, A. Trombettoni, and S. Ruffo. Nonequilibrium quantum-heat statistics under stochastic projective measurements, Phys. Rev. E 98(3), 032108 (2018).
  * (39) B. Mukherjee, K. Sengupta, and S.N. Majumdar. Quantum dynamics with stochastic reset, Phys. Rev. B 98, 104309 (2018).
  * (40) The contraction principle allows us to put in relation the probability distributions in LD form of two distinct quantities, one as a function of the other, by means of a continuous function.
  * (41) R. Wong. Asymptotic Approximations of Integrals: Computer Science and Scientific Computing. Acad. Press (1989).
  * (42) C.L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  * (43) In probability theory, the Jensen’s inequality can be simply formulated in the following way: Given the random variable $X$ and the _convex_ function $g(X)$, then $g(\langle X\rangle)$ is smaller or at most equal to $\langle g(X)\rangle$, i.e. $g(\langle X\rangle)\leq\langle g(X)\rangle$. In our case, $X=q_{j_{{\rm B}},j_{{\rm A}}}(\lambda)$ and $g$ is an exponentially decreasing function; thus the Jensen’s inequality is just given by $\langle\exp(q)\rangle\geq\exp(\langle q\rangle)$, where $\langle f(q)\rangle\equiv\sum_{j_{{\rm B}},j_{{\rm A}}}\int_{\lambda}p_{j_{{\rm B}},j_{{\rm A}}}(\lambda)f(q_{j_{{\rm B}},j_{{\rm A}}}(\lambda))d\lambda$.