\section*{Appendix I - Flavor associated energies for a two level system}

The time evolution of a quantum system of well-defined flavor quantum numbers described by the state vectors $\nu^{e}$ and $\nu^{\mu}$ respectively related to electron and muon neutrinos is given by
\begin{equation}
\left(\begin{array}{l}\nu^e_{(t)} \\ \nu^{\mu}_{(t)} \end{array}\right) =
U \, \left(\begin{array}{ll} e^{-i\, E_{\mbox{\tiny $1$}} t} & 0 \\ 0  &  e^{-i\, E_{\mbox{\tiny $2$}} t} \end{array}\right)\,
\left(\begin{array}{l}\nu_1 \\ \nu_2 \end{array}\right) =
U \, \left(\begin{array}{ll} e^{-i\, E_{\mbox{\tiny $1$}} t} & 0 \\ 0  &  e^{-i\, E_{\mbox{\tiny $2$}} t} \end{array}\right)
\, U^{\dagger} \left(\begin{array}{l} \nu^e_{(0)} \\ \nu^{\mu}_{(0)}  \end{array}\right),
\label{101A}
\end{equation}
where $\nu_{\mbox{\tiny $1$}}$ and $\nu_{\mbox{\tiny $2$}}$ are the mass eigenstates with well-defined energies, $E_{s} = \sqrt{p^{\mbox{\tiny $2$}} + m^{\mbox{\tiny $2$}}_{s}}$, with $s = 1,\, 2$, and the matrix $U$ parameterizes the mixing relation as
\begin{equation}
\left(\begin{array}{l}\nu^e_{(0)} \\ \nu^{\mu}_{(0)} \end{array}\right) =
U \, \left(\begin{array}{l}\nu_1 \\ \nu_2 \end{array}\right) =
\left(\begin{array}{rr} \cos\bb{\theta} & \sin\bb{\theta} \\ -\sin\bb{\theta}  & \cos\bb{\theta} \end{array}\right)
\left(\begin{array}{l}\nu_1 \\ \nu_2 \end{array}\right),
\label{req00A}
\end{equation}
where $\theta$ is the mixing angle.
Since the Hamiltonian of the system in the mass eigenstate basis can be extracted from Eq.~(\ref{req00A}) as $H = Diag\{E_{\mbox{\tiny $1$}},\, E_{\mbox{\tiny $2$}}\}$, the flavor projection operators can be easily defined as
\begin{equation}
M^{e}_{(t)} = |\nu^{e}_{(t)} \rangle \langle \nu^{e}_{(t)}| =
\left[
\begin{array}{ll}
\cos^{\mbox{\tiny $2$}}\bb{\theta} & \sin\bb{\theta} \,\cos\bb{\theta} \,e^{-i\,\Delta E\, t}\\
\sin\bb{\theta}\, \cos\bb{\theta} \, e^{i\,\Delta E\, t} &  \sin^{\mbox{\tiny $2$}}\bb{\theta}
\end{array}
\right]
\label{req01A}
\end{equation}
and
\begin{equation}
M^{\mu}_{(t)} = |\nu^{\mu}_{(t)} \rangle \langle \nu^{\mu}_{(t)}|  =
\left[
\begin{array}{ll}
\sin^{\mbox{\tiny $2$}}\bb{\theta} & -\sin\bb{\theta} \,\cos\bb{\theta} \,e^{-i\,\Delta E\, t}\\
-\sin\bb{\theta}\, \cos\bb{\theta} \, e^{i\,\Delta E\, t} &  \cos^{\mbox{\tiny $2$}}\bb{\theta} \end{array}
\right]
\label{req01B}
\end{equation}
where $\Delta E = E_{\mbox{\tiny $1$}} - E_{\mbox{\tiny $2$}}$ and it can be verified that $M^{e}_{(t)} + M^{\mu}_{(t)}  = \mathbf{1}$.

Thus the temporal evolution of a flavor eigenstate can be described by
\begin{equation}
|\nu^{e,\mu}_{(t)} \rangle = (M^{e}_{(0)} + M^{\mu}_{(0)}) |\nu^{e,\mu}_{(t)} \rangle =
 \langle \nu^{e}_{(0)} |\nu^{e,\mu}_{(t)} \rangle \, |\nu^{e}_{(0)} \rangle
+
\langle \nu^{\mu}_{(0)} |\nu^{e,\mu}_{(t)} \rangle \, |\nu^{\mu}_{(0)} \rangle,
\label{req01C}
\end{equation}
and the supposedly relevant measurable quantities, or {\em observables}, of the closed quantum system can be summarized by the the flavor-{\em averaged} energies,
\begin{eqnarray}
E^{e,\mu}_{(t)} &=&  \langle \nu^{e,\mu}_{(t)} |H| \nu^{e,\mu}_{(t)}\rangle,
\label{req02}
\end{eqnarray}
that result in time-independent quantities,
\begin{eqnarray}
E^{e}_{(t)} &=& E^{e}_{(0)} = \bar{E} + (1/2) \Delta E\, \cos\bb{2\theta},\nonumber\\
E^{\mu}_{(t)} &=& E^{\mu}_{(0)} = \bar{E} - (1/2) \Delta E\, \cos\bb{2\theta},
\label{req02A}
\end{eqnarray}
with $\bar{E} = (1/2)(E_{\mbox{\tiny $1$}} + E_{\mbox{\tiny $2$}})$, and by the time-oscillating flavor probabilities,
\begin{eqnarray}
\mathcal{P}_{\alpha\rightarrow\beta} \bb{t} &=& Tr\{M^{\beta}_{(0)}\, M^{\alpha}_{(t)}\}, ~~~~\alpha,\,\beta = e,\, \mu,
\label{req02BB}
\end{eqnarray}
that result in
\begin{eqnarray}
\mathcal{P}_{e\rightarrow e} \bb{t} &=& P_{\mu\rightarrow\mu} \bb{t} = |\langle \nu^{e}_{(\mbox{\tiny $0$})}|\nu^{e}_{(t)}\rangle|^{\mbox{\tiny $2$}} = |\langle \nu^{\mu}_{(\mbox{\tiny $0$})}|\nu^{\mu}_{(t)}\rangle|^{\mbox{\tiny $2$}}  = 1 - \sin^{\mbox{\tiny $2$}}\bb{2\theta} \,\sin^{\mbox{\tiny $2$}}\left(\frac{\Delta E}{2}\, t\right),
\label{req02B}
\end{eqnarray}
and
\begin{eqnarray}
\mathcal{P}_{e\rightarrow \mu}\bb{t} &=& P_{\mu \rightarrow e} \bb{t}= |\langle \nu^{\mu}_{(\mbox{\tiny $0$})}|\nu^{e}_{(t)}\rangle|^{\mbox{\tiny $2$}}  = |\langle \nu^{e}_{(\mbox{\tiny $0$})}|\nu^{\mu}_{(t)}\rangle|^{\mbox{\tiny $2$}} = \sin^{\mbox{\tiny $2$}}\bb{2\theta} \,\sin^{\mbox{\tiny $2$}}\left(\frac{\Delta E}{2}\, t\right),
\label{req02C}
\end{eqnarray}
that are interpreted as the probabilities of $e(\mu)$-flavor states produced at time $t_{\mbox{\tiny $0$}}$ be measured as $e(\mu)$-flavor states or be converted into $\mu (e)$-flavor states after a time interval $t - t_{\mbox{\tiny $0$}} \sim t - 0 \sim t$.

Now let us suppose that the density matrix of a composite quantum system of two neutrino flavor states is given by
\begin{equation}
\hat{\rho}\bb{t} \equiv \hat{\rho} = w_e \, M^{e}_{(t)} + w_{\mu} \, M^{\mu}_{(t)},
\label{req05}
\end{equation}
with $w_{e} + w_{\mu} = 1$.
One easily finds that the re-defined probabilities of measuring the electron and muon flavor eigenstates at time $t$ are given by
\begin{eqnarray}
P^{e}_{(t)}   = \mbox{Tr}\{M^{e}_{(0)}\, \hat{\rho}\} &=& w_e \, \mbox{Tr}\{M^{e}_{(0)}\,M^{e}_{(t)}\} + w_{\mu} \, \mbox{Tr}\{M^{e}_{(0)}\,M^{\mu}_{(t)}\} = \nonumber\\
                        &=& w_e \, \mathcal{P}_{e\rightarrow e}\bb{t} + w_{\mu} \, \mathcal{P}_{\mu\rightarrow e}\bb{t},\\
P^{\mu}_{(t)} = \mbox{Tr}\{M^{\mu}_{(0)}\, \hat{\rho}\} &=& w_e \, \mbox{Tr}\{M^{\mu}_{(0)}\,M^{e}_{(t)}\} + w_{\mu} \, \mbox{Tr}\{M^{\mu}_{(0)}\,M^{\mu}_{(t)}\} = \nonumber\\
                        &=& w_e \, \mathcal{P}_{e\rightarrow \mu}\bb{t} + w_{\mu} \, \mathcal{P}_{\mu\rightarrow \mu}\bb{t},
\label{req06}
\end{eqnarray}
where we have used the results from Eqs.~(\ref{req02B}-\ref{req02C}).
One also easily notices that
\begin{eqnarray}
P^{e}_{(t)} +  P^{\mu}_{(t)} &=&
w_{e} (\mathcal{P}_{e\rightarrow e}\bb{t} + \mathcal{P}_{e\rightarrow \mu}\bb{t})
+
w_\mu (\mathcal{P}_{\mu\rightarrow e}\bb{t} + \mathcal{P}_{\mu\rightarrow \mu}\bb{t})
= w_{e} + w_\mu = 1
\label{req06B}
\end{eqnarray}
and that the properties of a statistical mixture are immediate.
It leads to a reinterpretation of the energy related to each flavor quantum number.

The standard {\em total} averaged energy for a composite quantum system is defined through the density matrix as
\begin{eqnarray}
E_{(t)} = \mbox{Tr}\{H \, \hat{\rho}\} &=& w_{e} \, \mbox{Tr}\{H \, M^{e}_{(t)}\} + w_{\mu} \, \mbox{Tr}\{H \, M^{\mu}_{(t)}\}\nonumber\\
                            &=&   w_{e} \, E^{e}_{(t)} + w_{\mu} \, E^{\mu}_{(t)},
\label{req07}
\end{eqnarray}
from which one can notice the explicit dependence on the flavor-{\em averaged} energies, $E^{e,\mu}_{(t)}$, recovered from Eq.~(\ref{req02}).
In this context $E^{e}_{(t)}$ and $E^{\mu}_{(t)}$ are respectively decoupled from the statistical weights $w_{\mu}$ and $w_{e}$.
It just ratifies our previous arguments that such flavor energies are noway correlated with the flavor probabilities from Eq~(\ref{req07}), $P^{e}_{(t)}$ and $P^{\mu}_{(t)}$ since both of them depend simultaneously on both statistical weights, $w_{\mu}$ and $w_{e}$.
Thus the arguments that assert the ambiguity and the insufficiency in defining the flavor eigenstate averaged energies through $E^{e,\mu}_{(t)}$ are maintained.

After simple mathematical manipulations involving the definitions from Eq.~(\ref{req01B}) and the probabilities from Eq.~(\ref{req06}), one easily finds that
\begin{eqnarray}
M^{\mu}_{(0)} \, \hat{\rho}\, M^{\mu}_{(0)} &=& (w_e \, \mathcal{P}_{e\rightarrow \mu} \bb{t}+ w_{\mu} \, \mathcal{P}_{\mu\rightarrow \mu}\bb{t}) M^{\mu}_{(0)} = P^{\mu}_{(t)}\, M^{\mu}_{(0)},\nonumber\\
M^{e}_{(0)} \, \hat{\rho}\, M^{e}_{(0)}     &=& (w_e \, \mathcal{P}_{e\rightarrow e}\bb{t} + w_{\mu} \, \mathcal{P}_{\mu\rightarrow e}\bb{t}) M^{e}_{(0)} = P^{e}_{(t)}\, M^{e}_{(0)},
\label{req08}
\end{eqnarray}
Observing the cyclic properties of the trace, the flavor-{\em weighted} energies can be defined as
\begin{eqnarray}
\epsilon^{e,\mu}_{(t)} = \mbox{Tr}\{M^{e,\mu}_{(0)}\,H\, M^{e,\mu}_{(0)} \, \hat{\rho} \} &=& \mbox{Tr}\{H\, M^{e,\mu}_{(0)} \, \hat{\rho}\, M^{e,\mu}_{(0)}\}
= P^{e,\mu}_{(t)}\mbox{Tr}\,\{M^{e,\mu}_{(0)}\,H\} =
P^{e,\mu}_{(t)}\, E^{e,\mu}_{(0)},
\label{req09}
\end{eqnarray}
which can be promptly compared with the previous definition through the relation
\begin{equation}
\frac{|\epsilon^{e,\mu}_{(t)} - w_{e,\mu} E^{e,\mu}_{(0)}|}{E^{e,\mu}_{(0)}} = |w_{e} - w_{\mu}| \sin^{\mbox{\tiny $2$}}\bb{2\theta}
\, \sin^{\mbox{\tiny $2$}}\left(\frac{\Delta E}{2}\,t\right).
\label{req10A}
\end{equation}

As described above, flavor-{\em weighted} energies establish a unique correspondence between flavor eigenstate energies and the statistical definitions of probabilities, $P^{e,\mu}_{(t)}$.
It can not be identified through the definition of flavor-{\em averaged} energies.

\section*{Appendix II - The von-Neumann entropy and quantum measurements}

The connection to the quantum measurement theory arises from assuming the von-Neumann entropy as an important entropy functional defined in terms of the density matrix by
\begin{equation}
S\bb{\hat{\rho}} = - \mbox{Tr}\{\hat{\rho}\bb{t}\, \ln\bb{\hat{\rho}\bb{t}}\},
\label{eq26Z}
\end{equation}
where we have set the multiplicative Boltzmann constant, $k_{B}$, equal to unity.
The entropy $S\bb{\hat{\rho}}$ quantifies the departure of a composite quantum system from a pure state by measuring its time-evolved degree of mixture.
The quantum measurements can also induce modifications on the the von-Neumann entropy of the system.
The entropy changes due to a {\em non-selective} measurement scheme described by {\em operations} parameterized by the projection operators $M^{\alpha}_{(0)}$ are given by \cite{Breuer}
\begin{equation}
\Delta S = S\bb{\hat{\rho}^{\prime}} - S\bb{\hat{\rho}} \geq 0,
\label{eq26A}
\end{equation}
where $\alpha$ is the relevant quantum number and $\hat{\rho}^{\prime} = \sum_{\alpha} {P^{\alpha}_{(t)} \hat{\rho}_{\alpha}}$ so that
\begin{equation}
S\bb{\hat{\rho}^{\prime}} =
S\left(\sum_{\alpha} {P^{\alpha}_{(t)} \hat{\rho}_{\alpha}}\right).
\label{eq26B}
\end{equation}
Since $\Delta S \geq 0$, the {\em non-selective} quantum measurement never decreases the von-Neumann entropy.
To quantify the relation between {\em selective} and {\em non-selective} levels of measurement, the mixing entropy described by
\begin{equation}
\delta S =
S\left(\sum_{\alpha} {P^{\alpha}_{(t)} \hat{\rho}_{\alpha}}\right) - \sum_{\alpha} {P^{\alpha}_{(t)} S\left(\hat{\rho}_{\alpha}\right)}.
\label{eq26C}
\end{equation}
gives the difference between the entropy of a system projected by a {\em non-selective} quantum measurement, $S\left(\sum_{\alpha} {P^{\alpha}_{(t)} \hat{\rho}_{\alpha}}\right)$, and the average of the entropies of the sub-ensembles $\hat{\rho}_{\alpha}$, described by $M^{\alpha}_{(0)}$.
For the {\em selective} measurement scheme, with $M^{\alpha}_{(0)}$ denoting the creation of a single-flavor (pure) state, the mixing entropy is reduced to
\begin{equation}
\delta S = S\left(\sum_{\alpha} {P^{\alpha}_{(t)} \hat{\rho}_{\alpha}}\right),
\label{eq26D}
\end{equation}
since $S\left(\hat{\rho}_{\alpha}\right) = 0$.
The {\em non-selective} measurement described by observables like flavor-{\em weighted} energies always modifies the von-Neumann entropy by $\Delta S$ (c. f. Eq.~(\ref{eq26A})).
In the particular case of an ensemble described as a pure state, the {\em selective} measurement of flavor-{\em averaged} energies which are not expressed in terms of flavor conversion probabilities does not modify the von-Neumann entropy.
From the theoretical perspective, the von-Neumann entropy is therefore an auxiliary variable in distinguishing the measurement procedure and in classifying the measurement interventions as {\em selective} and {\em non-selective} ones, which however, as noticed above, are not complementary concepts.

All the above defined entropies satisfy some set of inequalities \cite{Breuer} which have been extensively used in different forms in the framework of quantum information theory and quantum entanglement.

