 also consider the problem of $\mathsf{p}$-estimation,
which we have shown to allow for a global efficient estimator within the frequentist approach.
As the binary outcome probability $\mathsf{p}$ does not constitute a circular parameter, we may directly
minimise the $\overline{\textrm{MSE}}$ \eref{eq:AvMSE}, $\left\langle \Delta^{2}\tilde{\mathsf{p}}_{N}^\t{\tiny MMSE}\right\rangle$, 
for which the MMSE estimator \eref{eq:MMSE_Est}
is always optimal. Thus, assuming a uniform prior distribution $p(\mathsf{p})\!=\!1$ for $\mathsf{p}\!\in\![0,1]$,
we calculate the posterior PDF defined with help of \eqnref{eq:PostPDF}
after substituting for the binomial PDF \eref{eq:PDF_BinPhi}, $p^{N}_\mathsf{p}\!(k)\!\equiv\!p^{N}\!(k|\mathsf{p})$, to obtain
$p^{N}\!(\mathsf{p}|k)\!=\!(N+1)\,p^{N}\!(k|\mathsf{p})$.
As a result, we may compute the MMSE estimator defined as the mean of $p^{N}\!(\mathsf{p}|k)$:
\begin{equation}
\tilde{\mathsf{p}}_{N}^\t{\tiny MMSE}(k)=\int_{0}^{1}\!\! \t{d}\mathsf{p}\; \mathsf{p}\; p^N\!(\mathsf{p}|k)=\frac{k+1}{N+2},
\label{eq:MMSE_BinP}
\end{equation}
and the corresponding minimal $\overline{\textrm{MSE}}$ \eref{eq:MinAvMSE}:
\begin{equation}
\left\langle \Delta^{2}\tilde{\mathsf{p}}_{N}^\t{\tiny MMSE}\right\rangle \;=\;\int_{0}^{1}\!\! d\mathsf{p}\; p(\mathsf{p})\sum_{k=0}^{N}p^{N}\!(k|\mathsf{p})\,\left(\frac{k+1}{N+2}-\mathsf{p}\right)^{2}\;=\;\frac{1}{6\left(N+2\right)}.
\label{eq:AvMSE_BinP}
\end{equation}
Again, in accordance with discussions of \secref{sub:AvMSE}, the (optimal Bayesian) 
MMSE estimator \eref{eq:MMSE_BinP} converges with $N$ to the global unbiased (and thus the ML) estimator
$\tilde{\mathsf{p}}_{N}^\t{ML}(k)\!=\! k/N$ derived within 
the frequentist approach.
On the other hand, as the corresponding CRB \eref{eq:CRBp} 
is $\mathsf{p}$-dependent, \eqnref{eq:MinAvMSE_Jensen} relating local
and global precision measures assures the minimal $\overline{\textrm{MSE}}$ \eref{eq:AvMSE_BinP} 
to converge to the average version of the CRB \eref{eq:CRBp}, so that
\begin{equation}
\left\langle \Delta^{2}\tilde{\mathsf{p}}_{N}^\t{\tiny MMSE}\right\rangle\;\overset{N\rightarrow\infty}{=}\;\left\langle \left.\Delta^{2}\tilde{\mathsf{p}}_{N}^\t{\tiny ML}\right|_{\mathsf{p}}\right\rangle_{p(\mathsf{p})}\;=\;\int_{0}^{1}\!\! d\mathsf{p}\,\frac{\mathsf{p}(\mathsf{p}-1)}{N}=\frac{1}{6N},
\end{equation}
what is consistent with \eqnref{eq:AvMSE_BinP} 
also yielding 
$\left\langle \Delta^{2}\tilde{\mathsf{p}}_{N}^\t{\tiny MMSE}\right\rangle\!\overset{N\rightarrow\infty}{=}\!1/(6N)$.

Lastly, let us emphasise once more that due to $p_{\varphi/\mathsf{p}}(\mathbf{x})\!=\!\prod_{i=1}^{N}p_{\varphi/\mathsf{p}}(x_{i})$
the asymptotic $N$ limit corresponds to the regime of the infinitely sized, \emph{independently} distributed data.
From the frequentist perspective, this means that the locality of estimation is always guaranteed when $N\!\rightarrow\!\infty$,
whereas within the Bayesian framework one is thus able to invoke the asymptotic connection \eref{eq:MinAvMSE_Jensen} 
with the local results and relate the corresponding estimators and precision measures.
In the case of the Mach-Zehnder interferometer of \figref{fig:MZinter_cl}, 
such condition is fulfilled owing to the assumption that the photons employed are uncorrelated between 
one another and individually measured at the end. As a result, each of them constitutes an \emph{independent 
statistical object} and the infinite sampling regime may be attained just by increasing the photon number. 
Importantly, such an assumption is not valid when considering general quantum systems, 
for which we would like to investigate the positive impact of correlations in 
between the constituent particles. Thus, in the quantum setting we must not only account for the quantum nature 
of the process, but also slightly modify the overall estimation scheme, so that the regime of independently distributed 
data can be clearly identified.





\section{Quantum estimation theory} 
\label{sec:QEst}


\subsection{The parameter estimation problem in the quantum setting}
\label{sub:QEstProb}

In the quantum setting, we consider a general estimation scenario 
\citep{Giovannetti2004,Giovannetti2006} depicted in \figref{fig:Ph_Est_Scheme},
which for compatibility with the classical estimation problem of \secref{sub:ClEstProb} is only constrained so 
that the estimated parameter $\varphi$ is encoded \emph{independently} onto each of the $N$ particles 
contained in the initial, \emph{input state} of the system:~$\rho_{\t{in}}^N$ (see \secref{sub:QSboundtates} 
for introduction to the density-matrix description). As a result, $\varphi$ may in general represent 
any \emph{latent} variable discussed in \secref{sec:qmet_latpars} parametrising 
the \emph{single-particle} evolution. Yet,
before considering a general channel estimation scenario, 
we restrict ourselves in \figref{fig:Ph_Est_Scheme} to the 
\emph{phase estimation problem}, in which the parameter specifies the 
``angle" of a \emph{unitary} rotation%
\footnote{%
Following the notation of \chapref{chap:q_sys}, we do not alter the fount when labelling the \emph{operators} acting on quantum wave-functions --
e.g.~$U$, $\mathbb{I}$ and $K$  respectively representing unitary, identity and Kraus operators, 
whereas we utilise the calligraphic fount to denote \emph{super-operators} acting on density matrices --
e.g.~$\mathcal{U}[\bullet]\!=\!U\bullet U^\dagger$, $\mathcal{I}[\bullet]\!=\!\bullet$ and 
$\mathcal{D}[\bullet]\!=\!\sum_i K_i\bullet K_i^\dagger$
now corresponding respectively to unitary, identity super-operators and a general quantum channel.}:
$\mathcal{U}_\varphi[\bullet]\!=\!U_\varphi \bullet U_\varphi^\dagger$ with $U_\varphi\!=\!\mathrm{e}^{-\mathrm{i} \hat{H}\varphi}$, 
generated by some single-particle Hamiltonian $\hat H$.
Such a choice allows us to clearly explain the mechanism behind the quantum enhancement 
of attainable precision, but the frameworks described in this section apply also
to non-unitary estimation tasks, e.g.~see the decoherence-strength estimation scenarios discussed in \chapref{sec:loc_dec_est}. 
Nevertheless, the scheme of \figref{fig:Ph_Est_Scheme}
importantly accounts for the noise present in the setup, which we
represent by a general quantum channel $\mathcal{D}$ (see \secref{sub:Evo_QCh}) modelling 
the decoherence that may affect the particles in an uncorrelated, (\textbf{i}),  or correlated, (\textbf{ii}), manner.
One should note that such a picture is an oversimplification for quantum 
systems in which the parameter-encoding (here unitary) part of the evolution cannot 
be separated from the noise, i.e.~when they do not
commute with one another \citep{Chaves2013}.
In the case of uncorrelated noise and the frequentist approach, 
however, we show later in \chapref{chap:local_noisy_est}
that such an issue becomes irrelevant after 
utilising the language of $\varphi$-parametrised quantum channels