# Analytical techniques in single and multi-parameter quantum estimation
theory: a focused review

Abdallah Slaoui<EMAIL_ADDRESS>LPHE-Modeling and Simulation,
Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco,  Centre
of Physics and Mathematics, CPM, Faculty of Sciences, Mohammed V University in
Rabat, Rabat, Morocco.  Lalla Btissam Drissi<EMAIL_ADDRESS>LPHE-
Modeling and Simulation, Faculty of Sciences, Mohammed V University in Rabat,
Rabat, Morocco,  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  El Hassan Saidi
<EMAIL_ADDRESS>LPHE-Modeling and Simulation, Faculty of Sciences, Mohammed
V University in Rabat, Rabat, Morocco,  Centre of Physics and Mathematics,
CPM, Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco.
Rachid Ahl Laamara<EMAIL_ADDRESS>LPHE-Modeling and Simulation,
Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco,  Centre
of Physics and Mathematics, CPM, Faculty of Sciences, Mohammed V University in
Rabat, Rabat, Morocco.

###### Abstract

As we enter the era of quantum technologies, quantum estimation theory
provides an operationally motivating framework for determining high precision
devices in modern technological applications. The aim of any estimation
process is to extract information from an unknown parameter embedded in a
physical system such as the estimation converges to the true value of the
parameter. According to the Cramér-Rao inequality in mathematical statistics,
the Fisher information in the case of single-parameter estimation procedures,
and the Fisher information matrix in the case of multi-parameter estimation,
are the key quantities representing the ultimate precision of the parameters
specifying a given statistical model. In quantum estimation strategies, it is
usually difficult to derive the analytical expressions of such quantities in a
given quantum state. This review provides comprehensive techniques on the
analytical calculation of the quantum Fisher information as well as the
quantum Fisher information matrix in various scenarios and via several
methods. Furthermore, it provides a mathematical transition from classical to
quantum estimation theory applied to many freedom quantum systems. To clarify
these results, we examine these developments using some examples. Other
challenges, including their links to quantum correlations and saturating the
quantum Cramér-Rao bound, are also addressed.

Keywords: Classical estimation theory, Quantum Cramér-Rao bounds, Quantum
information theory, Multi-parameter quantum estimation, Quantumness.

###### pacs:

03.65.Ta, 03.65.Yz, 03.67.Mn, 42.50.-p, 03.65.Ud

##  I Introduction

During the last two decades, quantum information has proven, both
theoretically [1] and experimentally [2], to have made several advances and to
provide the key tools for a modern technological revolution. These advances
are mainly due to the prospects of quantum communications [3, 4, 5] and
quantum computers [6, 7, 8, 9] that allow to storage, process and transform
information using quantum mechanical systems. Besides, several quantities of
interest in quantum information theory are not completely accessible via
observation and cannot be directly evaluated by quantum measurements [10, 11,
12]. One may quote for instance; a quantum phase [13], the purity of a quantum
state [14] or quantum correlations [15]. In all these cases, one must resort
to indirect measurement and deduce the value of the quantity of interest from
its inuence on a given source. The canonical way to deal with this problem is
to use the tools of quantum estimation theory [16, 17]. For example, during
the famouse LIGO experiment [18], the gravitational wave inserted a relative
phase into the light source which is not a directly measurable quantity,
however this phase was estimated through the interference pattern observed.

Quantum estimation theory, as the mathematical language of quantum metrology
[19, 20, 21], is of critical importance in the development of high-precision
devices in several technological areas. Its main objective is to realize high
precision measurements by estimating the unknown parameters that specify a
given quantum system using quantum effects. In this sense, it aims at
developing new methods to improve the accuracy limit of physical parameters
beyond classical metrological methods [22, 23, 24, 25]. In short, it is
interested in achieving the highest possible accuracy in various parameter
estimation tasks, and in finding measurement schemes that achieve this
accuracy. Originally, metrology focused on measurements made using classical
systems, such as mechanical systems described by classical physics or optical
systems modeled by classical wave optics [26]. We must therefore ask how can
we deduce the limits of accuracy of parameter estimation, as well as methods
that can improve accuracy in quantum systems? Is there a fundamental limit? In
the classical scheme, the first answers appeared around the 1940s with the
seminal paper of Rao [27] and Cramer [28], who independently found a lower
bound on the variance of an arbitrary estimator. These results were extended
to the multiparametric strategy by Darmois [29]. This limit, generally called
the Cramér-Rao bound, is closely linked to the Fisher information, introduced
by Fisher in the 1920s [30]. The Fisher information thus plays a central role
in estimation theory. Its maximization on all possible quantum measures
defines the quantum Fisher information and provides a quantum lower bound to
the Cramér-Rao bound [31, 32].

The quantum estimation theory is the mathematical framework for approaching
the optimization problem of a quantum measurement. It applies to situations
where we need to predict the value of a parameter by performing a repeated
series of measurements on identical preparations of the system and then
processing the data to estimate the value of the unknown parameter [19]. To
achieve all these tasks, we would need quantum phenomena as resources, such as
quantum entanglement [33, 34], quantum discord [35, 36] and quantum coherence
[94, 38], to improve the sensitivity of a system. At the same time, quantum
systems place intrinsic restrictions on the sensitivity that can be achieved,
for example through Heisenberg uncertainty relations, which state that
complementary variables cannot be measured simultaneously with unlimited
precision. To determine how quantum properties enhance but also restrict the
sensitivity of the system, we model its dynamics and focus on a key quantity
which is the quantum Fisher information.

According to the quantum Cramér-Rao bound, a higher accuracy is obtained for
small variances, which correspond to the largest values of the quantum Fisher
information. Thus, the primary aim of all quantum metrology protocols is to
reach the smallest value of the variance. However, recent studies have
revealed broad connections between quantum Fisher information and other
aspects of quantum mechanics, including characterization of quantum
correlations [39], quantum phase transition [40], entanglement control [41],
quantum speed limit [42] and quantum thermodynamics [43]. These links indicate
that it is more than a concept in quantum metrology, but rather a fundamental
quantity in quantum mechanics. Besides, it is well known that the achievable
phase estimation accuracy, in the classical optical interferometry theory, is
bounded by the shot noise. Then the phase uncertainty can only reach the
accuracy scale $1/\sqrt{N}$, where $N$ stands for the average photon number of
the light field. This phase uncertainty limit is called the standard quantum
limit [16]. Utilizing the non-classical states of light, which in principle
contains a non-zero amount of correlations, the quantum metrology theory
allows to beat the shot noise and the accuracy of the phase shift estimation
in an optical interferometer. The use of NOON states for example, which are
maximally entangled states, can lead to a scaling of the precision $1/N$. This
bound being called the Heisenberg limit [44]. Several reasons explain why
quantum systems are able to outperform classical devices, and perhaps most
importantly, the classical measurement boundary can be exceeded using quantum
systems.

Quantum entanglement underlies many fundamental quantum tasks [45, 46] and is
often considered synonymous with quantum correlations in early studies,
although it is now recognized that the notion of quantum correlations is much
broader in scope, and that entanglement is a particular, though most
important, type of quantum correlation, i.e., quantum entanglement can be
identified as nonlocal quantum correlations [47]. The study of entanglement
explicitly dates back to the seminal work of Einstein, Podolsky and Rosen
[33], and of Schrödinger [48, 49] as early as the 1930s. Today, entanglement
is considered a key resource of quantum information and is often linked to
quantum non-locality [50, 51]. In 2002, after studying the correlation between
the device and the system during a measurement, Ollivier and Zurek [35]
realized that separable states, as defined by Werner [47], can still have some
correlation in the sense that they can be perturbed by local measurements.
This novel measure of quantum correlations beyond entanglement is called
quantum discord [52, 53, 54]. They found that entanglement is not the only
quantum correlation that has no classical counterpart. Other types of quantum
correlations, such as quantum discord, may also be responsible for speeding up
some quantum algorithms while entanglement may disappear or be negligible
[55]. Since the first statements of this concept, many research efforts have
been devoted to understanding the mathematical properties and physical
meanings of discord and similar quantities. Comprehensive review of the
properties of quantum discord is available in [56] and the Refs.[57, 58]
provide recent insights into this field. Indeed, the investigation of quantum
discord presents many challenges and open questions. A major difficulty with
discord-like quantities is that they are difficult to calculate or to analyze.
Formally, quantum discord is defined as the difference between the quantum
mutual information (quantifying the total correlation) and the maximum amount
of mutual information accessible to a quantum measurement (quantifying the
classical correlation). To quantify the locally accessible (classical)
correlations, this quantification involves an optimization over all possible
local measurements. This measure is chosen to maximize the classical
correlations. Due to the complexity of the optimization process, the
computation of the quantum discord and its variants (such as the geometric
discord) is not an easy task and the analytical results are only known for
some restricted families of states. To overcome this problem, several other
versions of discord have also been proposed, including linear quantum discord
[59], local quantum uncertainty [60, 61], and local quantum Fisher information
[39]. Most of these quantum correlation quantifiers for pure quantum states
can be coincidental and can sometimes exhibit similar behavior for mixed
states.

Besides, quantum metrology is a mechanism that uses the characteristic
resources of quantum mechanics, such as entanglement and quantum discord, to
improve the parameter estimation precision via quantum measurements beyond its
classical counterpart limit. Therefore, it can be argued that quantum
resources are very important in all quantum metrology protocols [12]. In this
review, our concerns include the intrinsic connections between quantum
correlations and estimation theory and their roles in improving estimated
parameter precision. For this purpose, this review is presented with many
technical details, which could help researchers to follow and better
understand the corresponding results. Further to presenting an overview of the
main developments in estimation theory, we try to summarize and reformulate
some of the calculations scattered in a large number of works, combined of
course with our own results. To begin with, we introduce the basic concepts of
classical estimation theory before discussing the quantum advantage. We then
derive the classical Fisher information as a measure of information that can
be related to the variance of an estimated parameter. We then show how
classical Fisher information can be generalized for all possible measures to
quantum Fisher information. The motivation for studying these quantities
becomes clear when we introduce and derive the Cramér-Rao bound, which relates
the quantum and classical Fisher information to the variance. In fact, this
Cramér-Rao quantum bound is always reached in saturation in the case where
only one parameter is estimated, and in this situation, the quantum
correlations help us to improve the accuracy and efficiency of quantum
metrology protocols. Conversely, this limit is hard to saturate in the case of
multiparametric estimation due to the incompatibility between the optimal
measures of the different estimated parameters. The pertinent object in the
multiparametric estimation problem is given by the so-called quantum Fisher
information matrix. To this end, in the second part of this review, we provide
the complete techniques on the computation of this matrix, and we show that
the simultaneous multi-parameter strategy is always advantageous and can
provide better precision than the individual strategy in multiparameter
estimation procedures. Further, we intend to focus our attention on the
interaction between quantum correlations and quantum estimation theory and the
characterization of quantum correlations in terms of quantum Fisher
information will be discussed.

##  II Basic concepts of classical estimation theory

Estimation theory is an influential mathematical concept applied in various
communication and signal processing applications. This theory is useful for
estimating the desired information in received data and is therefore used in a
variety of applications ranging from radar to speech processing. In this
section, we will list the basic statistical concepts as well as the estimation
process. As we have mentioned, the main mathematical tools used by metrology
belong to statistics. In addition, we are particularly interested in
estimation theory, which shows how to correctly estimate a quantity from a
data sample [62]. The data can be of any type. For example, the data sample
can be a set of the results of random draws, or even the wavelengths of
photons coming out of a radioactive sample. Assume that a non-deterministic
process produces a set of data $x=\left\\{x_{1},x_{2},...,x_{n}\right\\}$,
which is a particular occurrence (or realization) of a set of independent and
identically distributed random variables denoted by
$X=\left\\{X_{1},X_{2},...,X_{n}\right\\}$. The probability density function
corresponding to any random variable in this set is $p\left(x;\theta\right)$,
where $x\in R$ is a possible realization of a random variable and $\theta$ is
a real number (e.g., it could be a physical parameter, such as the phase in an
optical interferometer) that is parametrized by $p\left(x;\theta\right)$.
Hence, the probability that a measurement result in our data set lies in the
interval $x_{0}\leq x\leq x_{f}$ is determined by the integral of
$p\left(x;\theta\right)$ as: $\int_{x_{0}}^{x_{f}}p\left(x;\theta\right)dx$.

Generally, the central problem in parameter estimation is the following [62,
63]: For a data set $x$ of $n$ test results, how accurately can we predict the
value of $\theta$? Indeed, in physics, the parameter $\theta$ is a physical
property of the system studied and we want to determine its value from the
experimental data $x_{i}$. For this purpose, an estimator is used to transform
the data $x$ into a prediction of $\theta$. We will denote such an estimator
by $\boldsymbol{\hat{\theta}}\left(x\right)$, which happens to be a random
variable since it depends on the values taken from the random variables in
$X$. In other words, estimating the parameter $\theta$ consists in giving an
approximate value to this parameter from a population survey. This implies
that the true value of $\theta$ is obtained if $\theta$ coincides with the
mathematical expectation of its estimator
$\boldsymbol{\hat{\theta}}\left(X\right)$, i.e.
$\theta=\textbf{E}\left[\boldsymbol{\hat{\theta}}\left(X\right)\right]$.

Figure 1: A schematic illustration of a complete classical metrological
process

We can now reformulate the estimation problem as follows; For any set of data
$x$, what is the small value of the error of our estimator? More precisely,
the minimum error is given by the variance of this estimator ${\rm
Var}_{\theta}\left(\hat{\theta}\right)$ [26]. It is defined by

$\displaystyle{\rm Var}_{\theta}\left(\hat{\theta}\right)$
$\displaystyle:=\textbf{E}\left[\left(\boldsymbol{\hat{\theta}}\left(x\right)-\theta\right)^{2}\right]$
$\displaystyle=\int
p\left(x_{1};\theta\right)p\left(x_{2};\theta\right)...p\left(x_{n};\theta\right)\left(\boldsymbol{\hat{\theta}}\left(x\right)-\theta\right)^{2}dx_{1}dx_{2}...dx_{n}$
$\displaystyle=\int
p\left(x;\theta\right)\left(\boldsymbol{\hat{\theta}}\left(x\right)-\theta\right)^{2}dx^{n},$
(1)

which is used to quantify the performance of our estimation task, with E being
a mathematical expectation. In the above equation (1), we employed the fact
that $dx^{n}=dx_{1}dx_{2}...dx_{n}$ and
$p\left(x;\theta\right)=p\left(x_{1};\theta\right)p\left(x_{2};\theta\right)...p\left(x_{n};\theta\right)$.

It is therefore clear that the law of $X$ will be completely known if the
probability distribution function $p\left(x;\theta\right)$ is known, this
necessarily implies the knowledge of the value of the unknown parameters
$\theta$. In this context, the estimation theory aims at approximating
numerically the value of the unknown parameters without knowing the
probability density and the probability law which governs our problem. In this
formalism, we make the implicit assumption that $\theta$ is knowable with
$\boldsymbol{\hat{\theta}}\left(x\right)$ and have access to a set of data
$x$.

###  II.1 Classical statistical schemes

Consider a random experiment whose results are described by a random variable
$X$, with a probability space $\left(X,x,p\left(x\right)\right)$ and a
probability density $p\left(x\right)$. The task here is to reconstruct
$p\left(x\right)$, which is called the genuine probability density, from $N$
independent sample positions or observations of $X$. There are many ways to
approach the learning problem of $p\left(x\right)$, but if the functional form
of $p\left(x\right)$ is already known, or can be assumed with reasonable
accuracy, a parametric approach is quite natural. The real probability density
$p\left(x\right)$ is assumed to belong to a parametric family of probability
densities $\left\\{p\left(x;\theta\right)\right\\}_{\theta\in\Theta}$ where
$\Theta\subset\mathbb{R}^{n}$ is the parameter space. A classical statistical
model $S_{C}$ is a family of probability densities on $X$ parametrized by $n$
real parameters $\theta\in\Theta\subset\mathbb{R}^{n}$ and noted by

$S_{C}=\left\\{\forall x\in X;p\left(x;\theta\right),\hskip
7.11317pt\left(X,x,p\left(x\right)\right)\mapsto\Theta,\hskip
7.11317pt\theta\equiv\left(\theta^{1},\theta^{2},...,\theta^{n}\right)\in\Theta\right\\},$
(2)

where the parametrization application
$\left(X,x,p\left(x\right)\right)\mapsto\Theta$ is injective and all possible
derivatives
$\partial_{\theta^{1}}...\partial_{\theta^{n}}p\left(x;\theta\right)$ exist,
i.e. we can be derivable $n^{th}$ times with respect to the estimated
parameters. It is interesting to note that the probability density
$p\left(x\right)$ is always positive and normalized. So we have two cases; If
$X$ is countable, then $p\left(x;\theta\right)$ is described in a discrete and
normalized space such that

$\sum_{x\in X}p\left(x;\theta\right)=1,\hskip
28.45274pt\forall\theta\in\Theta.$ (3)

If $X$ is uncountable, then $p\left(x;\theta\right)$ is described in a
continuous and normalized space like

$\int_{x}p\left(x;\theta\right)dx=1,\hskip 28.45274pt\forall\theta\in\Theta.$
(4)

To simplify our discussion, we employ the latter notation throughout this
review.

###  II.2 Score, Fisher Information and Cramér-Rao theorem

Measurements on physical systems produce probabilistic results, and the
estimation of physical parameters, describing a system or governing its
evolution, is a problem involving statistical inference. The source of
statistical errors may be related to imperfections in the experiment or in the
measurements, or it may be more fundamental, imposed, for example, by the
Heisenberg uncertainty relations in a quantum configuration. In the classical
case, a quantity that is key to this discussion is the Score which we denote
by ${\cal L}_{\theta}\left(x\right)$. The Score is the partial derivative of
the natural logarithm of the likelihood function, and thus indicates the
sensitivity of the system to an infinitesimal variation in the value of the
parameter $\theta$. It is defined as [64]

${\cal L}_{\theta}\left(x\right)=\frac{\partial}{\partial\theta}\log
p\left(x;\theta\right),$ (5)

which quantifies the sensitivity of the probability density function
$p\left(x;\theta\right)$ to the variations of the parameter $\theta$. This
arises from the fact that

$\displaystyle\textbf{E}\left({\cal L}_{\theta}\left(x\right)\right)$
$\displaystyle=\int p\left(x;\theta\right)\frac{\partial}{\partial\theta}\log
p\left(x;\theta\right)dx$ $\displaystyle=\int
p\left(x;\theta\right)\frac{1}{p\left(x;\theta\right)}\frac{\partial}{\partial\theta}p\left(x;\theta\right)dx$
$\displaystyle=\int\frac{\partial}{\partial\theta}p\left(x;\theta\right)dx=0.$
(6)

The classical Fisher information ${\cal F}\left(\theta\right)$ in a single
observation of a probability density function $p\left(x;\theta\right)$ is the
variance of the Score ${\cal L}_{\theta}\left(x\right)$, namely

$\displaystyle{\cal F}\left(\theta\right)$ $\displaystyle:={\cal
F}\left(p\left(x;\theta\right)\right):={\rm Var}\left({\cal
L}_{\theta}\left(x\right)\right)$
$\displaystyle=\textbf{E}\left[\left(\frac{\partial}{\partial\theta}\log
p\left(x;\theta\right)\right)^{2}\right]$ $\displaystyle=\int
p\left(x;\theta\right)\left(\frac{\partial}{\partial\theta}\log
p\left(x;\theta\right)\right)^{2}dx.$ (7)

Formally, the Fisher information (7) quantifies the information provided by a
sample on the parameter $\theta$. In addition, a Fisher information close to
zero indicates a sample with little information on the value of $\theta$. We
note here an important and useful property of classical Fisher information;
namely, non-negativity and additivity for uncorrelated processes. In other
words, if
$p\left(\boldsymbol{x};\theta\right)=p\left(x_{1};\theta\right)p\left(x_{2};\theta\right)...p\left(x_{n};\theta\right)$,
then

$\displaystyle{\cal F}\left(p\left(\boldsymbol{x};\theta\right)\right)$
$\displaystyle={\rm Var}\left[\frac{\partial}{\partial\theta}\log
p\left(x_{1};\theta\right)+...+\frac{\partial}{\partial\theta}\log
p\left(x_{n};\theta\right)\right]$ $\displaystyle=n{\rm
Var}\left[\frac{\partial}{\partial\theta}\log
p\left(x_{1};\theta\right)\right]=n{\cal F}\left(\theta\right).$ (8)

Therefore, the Fisher information in a random sample is just $n$ times the
information in a single observation. This should make intuitive sense because
it means that more observations produce more information.

The Cramér-Rao bound provides a limit on how the estimation can be performed.
It sets a fundamental limit on the variance of the estimator
$\boldsymbol{\hat{\theta}}$ regardless of how it is constructed. It is
expressed by the following formula [27, 28]

${\rm Var}\left(\boldsymbol{\hat{\theta}}\right)\geq\frac{1}{n{\cal
F}\left(\theta\right)},$ (9)

where $n$ is the trial number (the number of entries in the data set
$\boldsymbol{x}$). This bound (9) asserts that the error of any unbiased
estimator $\boldsymbol{\hat{\theta}}$ of $\theta$ is bounded by the inverse of
the Fisher information. This indicates that the variance of an estimator
$\boldsymbol{\hat{\theta}}$ is at least as high as the inverse of the Fisher
information, i.e., the greater the Fisher information, the higher the
estimation accuracy can be expected. Thus, Fisher information is the quantity
that produces an ultimate limit on the precision that can be achieved in a
parameter estimation problem. In this way, the Fisher information captures the
maximal information that can be extracted about the actual value sought for
the parameter $\theta$. Moreover, one can obviously improve the precision by
repeating the measurement (as the expression of the Cramér-Rao bound reflects
with the factor $1/n$) provided that one ensures the independence of the
repetitions. As we have seen, this bound is asymptotically attainable by
choosing the maximum estimator. For example, if the estimator is the mean
value, any Gaussian probability distribution
$p\left(\boldsymbol{X};\theta\right)$ saturates the inequality. An estimator
for which equality is satisfied is said to be efficient, but with a finite
number of measurements, there is no guarantee that such an estimator exists.

We now provide a proof of the Cramér-Rao limit following an approach similar
to the Ref.[65]. In order to prove this (9), we have to make the assumption
that the estimator $\boldsymbol{\hat{\theta}}\left(x\right)$ must be unbiased.
This means that on average, the estimator $\boldsymbol{\hat{\theta}}$ is equal
to the true value of $\theta$;

$\textbf{E}\left(\hat{\theta}\left(\boldsymbol{x}\right)\right)\equiv\theta.$
(10)

By shifting $\theta$ to the left side and writing the average explicitly in
terms of the probability density function
$p\left(\boldsymbol{x};\theta\right)$. The above equation (10) becomes

$\displaystyle\int
p\left(\boldsymbol{x};\theta\right)\left(\textbf{E}\left(\hat{\theta}\left(\boldsymbol{x}\right)\right)-\theta\right)d\boldsymbol{x}^{n}=0.$
(11)

If we derive this expression with respect to $\theta$, we obtain

$\displaystyle\frac{\partial}{\partial\theta}\int
p\left(\boldsymbol{x};\theta\right)\left(\textbf{E}\left(\hat{\theta}\left(\boldsymbol{x}\right)\right)-\theta\right)d\boldsymbol{x}^{n}=\int\frac{\partial}{\partial\theta}\left(p\left(\boldsymbol{x};\theta\right)\right)\left(\textbf{E}\left(\hat{\theta}\left(\boldsymbol{x}\right)\right)-\theta\right)d\boldsymbol{x}^{n}-\int
p\left(\boldsymbol{x};\theta\right)d\boldsymbol{x}^{n}=0,$ (12)

where we used the fact that the estimator doesn’t depend explicitly on
$\theta$, hence
$\frac{\partial}{\partial\theta}\left(\hat{\theta}\left(\boldsymbol{x}\right)-\theta\right)=-1$.
Also, for any probability density function we have $\int
p\left(\boldsymbol{x};\theta\right)d\boldsymbol{x}^{n}=1$. Therefore, the
equation (12) yields

$\displaystyle\int\frac{\partial
p\left(\boldsymbol{x};\theta\right)}{\partial\theta}\left(\hat{\theta}\left(\boldsymbol{x}\right)-\theta\right)d\boldsymbol{x}^{n}=1.$
(13)

It should be noted that $\partial
p\left(\boldsymbol{x};\theta\right)/\partial\theta$ can still be written as
follows

$\displaystyle\frac{\partial}{\partial\theta}\left(\prod_{k=1}^{n}p\left(x_{k};\theta\right)\right)$
$\displaystyle=\frac{\partial
p\left(x_{1};\theta\right)}{\partial\theta}\prod_{k\neq
1}^{n}p\left(x_{k};\theta\right)+\frac{\partial
p\left(x_{2};\theta\right)}{\partial\theta}\prod_{k\neq
2}^{n}p\left(x_{k};\theta\right)+...+\frac{\partial
p\left(x_{n};\theta\right)}{\partial\theta}\prod_{k\neq
n}^{n}p\left(x_{k};\theta\right)$
$\displaystyle=\sum_{l=1}^{n}\left[\frac{\partial
p\left(x_{l};\theta\right)}{\partial\theta}\prod_{k\neq
l}^{n}p\left(x_{k};\theta\right)\right],$ (14)

where $\prod_{k\neq l}^{n}p\left(x_{k};\theta\right)$ denotes the
multiplication of all terms $p\left(x_{k};\theta\right)$ for
$k\in\left\\{1,2,...,n\right\\}$ except for the $l$-th term. When writing
$\prod_{k\neq
l}^{n}p\left(x_{k};\theta\right)=p\left(\boldsymbol{x};\theta\right)/p\left(x_{l};\theta\right)$,
we get

$\displaystyle\frac{\partial
p\left(\boldsymbol{x};\theta\right)}{\partial\theta}=\sum_{l=1}^{n}\frac{1}{p\left(x_{l};\theta\right)}\frac{\partial
p\left(x_{l};\theta\right)}{\partial\theta}p\left(\boldsymbol{x};\theta\right)=p\left(\boldsymbol{x};\theta\right)\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}.$ (15)

The previous equation could be used to re-express equation (13) as follows

$\displaystyle\int
p\left(\boldsymbol{x};\theta\right)\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\left(\hat{\theta}\left(\boldsymbol{x}\right)-\theta\right)d\boldsymbol{x}^{n}$
$\displaystyle=\int\left[\sqrt{p\left(\boldsymbol{x};\theta\right)}\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right]$
$\displaystyle\times\left[\sqrt{p\left(\boldsymbol{x};\theta\right)}\left(\hat{\theta}\left(\boldsymbol{x}\right)-\theta\right)\right]d\boldsymbol{x}^{n}=1,$
(16)

where we decided to write the probability density function
$p\left(\boldsymbol{x};\theta\right)$ as a product of its square roots. Thus
the equation (16) takes the following form

$\int
f\left(\boldsymbol{x}\right)g\left(\boldsymbol{x}\right)d\boldsymbol{x}^{n}=1,$
(17)

with

$f\left(\boldsymbol{x}\right)\equiv\sqrt{p\left(\boldsymbol{x};\theta\right)}\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta},\hskip 28.45274pt{\rm et}\hskip
28.45274ptg\left(\boldsymbol{x}\right)\equiv\sqrt{p\left(\boldsymbol{x};\theta\right)}\left(\hat{\theta}\left(\boldsymbol{x}\right)-\theta\right).$
(18)

We are going to use now the Cauchy-Schwarz inequality [66]:

$\left(\int
f\left(\boldsymbol{x}\right)g\left(\boldsymbol{x}\right)d\boldsymbol{x}^{n}\right)^{2}\leq\int
f\left(\boldsymbol{x}\right)^{2}d\boldsymbol{x}^{n}\int
g\left(\boldsymbol{x}^{\prime}\right)^{2}d{\boldsymbol{x}^{\prime}}^{n}.$ (19)

We square the two sides of the equation (16) and we apply the Cauchy-Schwarz
inequality in order to obtain

$1\leq\left[\int
p\left(\boldsymbol{x};\theta\right)\left(\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}\right]\left[\underbrace{\int
p\left(\boldsymbol{x}^{\prime};\theta\right)\left(\hat{\theta}\left(\boldsymbol{x}^{\prime}\right)-\theta\right)^{2}d{\boldsymbol{x}^{\prime}}^{n}}_{{\rm
Var}\left(\boldsymbol{\hat{\theta}}\right)}\right].$ (20)

The second term of the last equation is simply the variance of the estimator
$\boldsymbol{\hat{\theta}}$. Then

${\rm Var}\left(\boldsymbol{\hat{\theta}}\right)\geq\frac{1}{\int
p\left(\boldsymbol{x};\theta\right)\left(\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}}.$
(21)

To complete the proof, we need to show that the denominator of the equation
(21) is equal to $n{\cal F}\left(\theta\right)$. By expanding the summation
term in the denominator, we can express it as follows

$\displaystyle\int
p\left(\boldsymbol{x};\theta\right)\left(\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}$
$\displaystyle=\int
p\left(\boldsymbol{x};\theta\right)\sum_{k=1}^{n}\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}d\boldsymbol{x}^{n}$
$\displaystyle=\int
p\left(\boldsymbol{x};\theta\right)\sum_{l=1}^{n}\left(\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}$
$\displaystyle+\int p\left(\boldsymbol{x};\theta\right)\sum_{k,l=1,k\neq
l}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}d\boldsymbol{x}^{n}.$ (22)

The last term of this equation (22) can be rewritten as

$\displaystyle\int p\left(\boldsymbol{x};\theta\right)\sum_{k,l=1,k\neq
l}^{n}\frac{\partial\log p\left(x_{l};\theta\right)}{\partial\theta}$
$\displaystyle\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}d\boldsymbol{x}^{n}=\int
dx_{1}dx_{2}...dx_{n}p\left(x_{1};\theta\right)p\left(x_{2};\theta\right)...p\left(x_{n};\theta\right)$
$\displaystyle\times\sum_{k,l=1,k\neq l}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta},$ (23)

then remove the summation and integrate some of the terms as shown below

$\displaystyle\sum_{k\neq 1}\left(\int
p\left(x_{1};\theta\right)\frac{\partial\log
p\left(x_{1};\theta\right)}{\partial\theta}dx_{1}\right)\left(\int
p\left(x_{k};\theta\right)\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}dx_{k}\right)\left(\int\prod_{m\neq
1}^{n}p\left(x_{m};\theta\right)dx_{m}\right)+$ $\displaystyle\sum_{k\neq
2}\left(\int p\left(x_{2};\theta\right)\frac{\partial\log
p\left(x_{2};\theta\right)}{\partial\theta}dx_{2}\right)\left(\int
p\left(x_{k};\theta\right)\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}dx_{k}\right)\left(\int\prod_{m\neq
2}^{n}p\left(x_{m};\theta\right)dx_{m}\right)+...$ (24)

Every term above is a product of the expectation values of the Scores.
Therefore, we can rewrite this expression as

$\displaystyle\sum_{k\neq 1}\textbf{E}\left(\frac{\partial\log
p\left(x_{1};\theta\right)}{\partial\theta}\right)\textbf{E}\left(\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}\right)+\sum_{k\neq
2}\textbf{E}\left(\frac{\partial\log
p\left(x_{2};\theta\right)}{\partial\theta}\right)\textbf{E}\left(\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}\right)$ $\displaystyle\hskip
56.9055pt+...+\sum_{k\neq n}\textbf{E}\left(\frac{\partial\log
p\left(x_{n};\theta\right)}{\partial\theta}\right)\textbf{E}\left(\frac{\partial\log
p\left(x_{k};\theta\right)}{\partial\theta}\right)$ $\displaystyle=\sum_{k\neq
1}\textbf{E}\left({\cal
L}_{\theta}\left(x_{1}\right)\right)\textbf{E}\left({\cal
L}_{\theta}\left(x_{k}\right)\right)+\sum_{k\neq 2}\textbf{E}\left({\cal
L}_{\theta}\left(x_{2}\right)\right)\textbf{E}\left({\cal
L}_{\theta}\left(x_{k}\right)\right)+...+\sum_{k\neq n}\textbf{E}\left({\cal
L}_{\theta}\left(x_{n}\right)\right)\textbf{E}\left({\cal
L}_{\theta}\left(x_{k}\right)\right)$ $\displaystyle=0.$ (25)

We showed earlier that the expectation value of a score is always zero (see
equation (6)). Therefore, the whole expression is equal to zero and we can
simplify the equation (22) to read

$\displaystyle\int
p\left(\boldsymbol{x};\theta\right)\left(\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}$
$\displaystyle=\int
p\left(\boldsymbol{x};\theta\right)\sum_{l=1}^{n}\left(\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}$
$\displaystyle=\sum_{l=1}^{n}\int
dx_{1}...dx_{n}p\left(x_{1};\theta\right)...p\left(x_{n};\theta\right)\sum_{l=1}^{n}\left(\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}$
$\displaystyle=\sum_{l=1}^{n}\left[\int
p\left(x_{l};\theta\right)\left(\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}dx_{l}\right]\left[\underbrace{\int\prod_{k\neq
l}^{n}p\left(x_{k};\theta\right)dx_{k}}_{=1}\right]$
$\displaystyle=\sum_{l=1}^{n}\textbf{E}\left[\left(\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)\right].$ (26)

Consequently, each term in equation (26) is the classical Fisher information
for the random variable $X_{l}$ with probability density function
$p\left(x_{l};\theta\right)$. Since the considered random variables are
independent and identically distributed, all $p\left(x_{l};\theta\right)$ are
identical. As a result, we find

$\int
p\left(\boldsymbol{x};\theta\right)\left(\sum_{l=1}^{n}\frac{\partial\log
p\left(x_{l};\theta\right)}{\partial\theta}\right)^{2}d\boldsymbol{x}^{n}=\sum_{l=1}^{n}{\cal
F}\left(p\left(x_{l};\theta\right)\right)=n{\cal F}\left(\theta\right).$ (27)

Finally, by combining equation (27) with equation (21), we obtain the famous
Cramér-Rao bound (9). We will list here some properties of the classical
Fisher information ${\cal F}\left(\theta\right)$:

  * ($i$)

Nothing about $\theta$ can be learned when ${\cal F}\left(\theta\right)=0$,
and inversely, $\theta$ can be learned perfectly when ${\cal
F}\left(\theta\right)\longmapsto\infty$.

  * ($ii$)

Classical Fisher information is a strictly positive quantity. This can be
proved by examining a more general form of ${\cal F}\left(\theta\right)$ which
takes into account many variables $\left\\{\theta_{1},...,\theta_{d}\right\\}$
(i.e. the case of multiparameter estimation) and by examining the form of the
resulting matrices whose positive sign can be shown.

  * ($iii$)

Unlike other information measures, such as mutual information or Shannon
entropy, classical Fisher information is dimensional. We acquire units because
the probability densities $p\left(x_{l};\theta\right)$ are generally
dimensional and also because the derivative $\partial_{\theta}$ has units of
$\theta^{-1}$. This is required to relate ${\cal F}\left(\theta\right)$ to the
variance ${\rm Var}\left(\theta\right)$ through the Cramér-Rao inequality (9).

  * ($iv$)

Classical Fisher information is related to mutual information, but they are
two fundamentally dissimilar quantities. For an intuitive understanding,
${\cal F}\left(\theta\right)$ is concerned with the probability that an
estimator $\boldsymbol{\hat{\theta}}$ comes close to the ”true” value of
$\theta$ while the mutual information ${\cal I}\left(X:Y\right)$ measures the
correlation between the random variables $X$ and $Y$.

##  III Extending to classical multiparametric metrology

###  III.1 Classical Fisher Information Matrix

The main assignment here is to expand the estimation problem to the
multiparameter case by estimating a set of unknown parameters. In this case,
the various parameters we wish to estimate are constructed in the vector
defined in the parameter space
$\boldsymbol{\theta}\left(\boldsymbol{x}\right)=\left(\theta_{1}\left(\boldsymbol{x}\right),...,\theta_{d}\left(\boldsymbol{x}\right)\right)^{T}\in\mathbb{R}^{d}$.
Thereby, the realization of the estimation vector is represented by
$\boldsymbol{\hat{\boldsymbol{\theta}}}\left(\boldsymbol{x}\right)=\left(\hat{\theta}_{1}\left(\boldsymbol{x}\right),...,\hat{\theta}_{d}\left(\boldsymbol{x}\right)\right)^{T}$.
For any given estimator
$\boldsymbol{\hat{\boldsymbol{\theta}}}\left(\boldsymbol{x}\right)$ of
$\boldsymbol{\boldsymbol{\theta}}\left(\boldsymbol{x}\right)$, the
mathematical expectation is given by
$\textbf{E}\left[\left(\boldsymbol{\boldsymbol{\theta}}-\boldsymbol{\hat{\boldsymbol{\theta}}}\right)\left(\boldsymbol{\boldsymbol{\theta}}-\boldsymbol{\hat{\boldsymbol{\theta}}}\right)^{T}\right]$.
Subsequently, the mean square error is equal to the covariance matrix ${\rm
Cov}\left(\boldsymbol{\hat{\boldsymbol{\theta}}}\right)$. One of the key
results of classical probability theory, namely the Cramér-Rao inequality,
enforcing a lower bound on the covariance matrix [19]

${\rm
Cov}\left(\boldsymbol{\hat{\boldsymbol{\theta}}}\right)\geq\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)^{-1},$
(28)

where $\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)$ is the
classical Fisher information matrix having the following elements

$\displaystyle\left[\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\right]_{ij}=\int
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)\left(\frac{\partial\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\theta_{i}}\frac{\partial\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\theta_{j}}\right)dx,$
(29)

depending on the probability distribution
$p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)$ of the results
$\boldsymbol{x}$. Interestingly, the Cramér-Rao inequality only applies to
probability distributions that satisfy the following regularity condition

$\textbf{E}\left[\frac{\partial\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\theta_{i}}\right]=0,\hskip
28.45274pt\forall\theta\in\Theta.$ (30)

If such a locally unbiased estimator exists, the Cramér-Rao bound can always
be saturated. However, the identification of the saturation conditions of the
Cramér-Rao bound is a complex subject that involves a variety of technical
details. In particular, for the probability distribution satisfying equation
(30), there exists a local unbiased estimator
$\boldsymbol{\hat{\boldsymbol{\theta}}}$ saturating the Cramér-Rao bound
provided that [62]

$\frac{\partial\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\theta}=\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\left(\boldsymbol{\hat{\boldsymbol{\theta}}}-\boldsymbol{\boldsymbol{\theta}}\right).$
(31)

###  III.2 Example: Classical Estimation via Normal Distribution $X\sim
N\left(\mu,\sigma\right)$

Many practical situations can be modeled using random variables that are
governed by specific laws. It is therefore important to study probabilistic
models that can later allow us to analyze the fluctuations of certain
phenomena by evaluating, for example, the probabilities that a particular
event or outcome will be observed. In the area of statistics, normal
distributions are among the most widely used probability distributions for
modeling natural experiments involving multiple random occurrences [67, 68].
They are also called Gaussian distributions [69, 70]. More formally, a normal
distribution is an absolutely continuous probability law and depends on two
unknown parameters, namely the mathematical expectation which is a real number
denoted $\mu$, and its standard deviation or variance denoted $\sigma$. The
probability distribution function of the normal law of expectation $\mu$ and
standard deviation $\sigma$ is provided by

$\displaystyle
p\left(x;\boldsymbol{\theta}\right)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{\left(x-\mu\right)^{2}}{2\sigma^{2}}\right]\equiv\frac{1}{\sqrt{2\pi\theta_{2}}}\exp\left[-\frac{\left(x-\theta_{1}\right)^{2}}{2\theta_{2}}\right].$
(32)

The classical statistical model (2) associated with this normal distribution
is given by

$S_{N\left(\mu,\sigma\right)}=\left\\{\forall x\in
N\left(\mu,\sigma\right);p\left(x;\boldsymbol{\theta}\right)\equiv\frac{1}{\sqrt{2\pi\theta_{2}}}\exp\left[-\frac{\left(x-\theta_{1}\right)^{2}}{2\theta_{2}}\right],\hskip
7.11317pt\Theta=\left(\mu,\sigma^{2}\right)^{T},\hskip
7.11317pt\boldsymbol{\theta}=\left(\theta_{1}\equiv\mu,\theta_{2}\equiv\sigma^{2}\right)\in\Theta,\hskip
5.69046ptn=2\right\\}.$ (33)

It is interesting to note that

$\log
p\left(x;\boldsymbol{\theta}\right)=-\frac{1}{2}\log\left(2\pi\sigma^{2}\right)-\frac{\left(x-\mu\right)^{2}}{2\sigma^{2}},$
(34)

we then obtain

$\displaystyle\frac{\partial^{2}\log
p\left(x;\boldsymbol{\theta}\right)}{\partial^{2}\sigma^{2}}=\frac{1}{2\sigma^{4}}-\frac{\left(x-\mu\right)^{2}}{\sigma^{6}},\hskip
56.9055pt\frac{\partial^{2}\log
p\left(x;\boldsymbol{\theta}\right)}{\partial^{2}\mu}=-\frac{1}{\sigma^{2}}.$
(35)

If we address the scenario in which $\mu$ and $\sigma^{2}$ are unknown
parameters, then the classical multiparameter estimation strategy is required.
Using equation (29), it is easy to verify that

$\displaystyle\left[\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\right]_{\mu\mu}$
$\displaystyle=\int
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)\left(\frac{\partial\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\mu}\right)^{2}dx$
$\displaystyle=\int\frac{1}{p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}\left(\frac{\partial
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\mu}\right)^{2}dx$
$\displaystyle=-\int
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)\frac{\partial^{2}\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial^{2}\mu}dx$
$\displaystyle=-\textbf{E}\left[\frac{\partial^{2}\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial^{2}\mu}\right].$
(36)

Thus, we calculate all the components of the classical Fisher information
matrix in the same way, and eventually we find

$\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)_{N\left(\mu,\sigma\right)}=\begin{pmatrix}\left[\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\right]_{\mu\mu}&\left[\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\right]_{\mu\sigma^{2}}\\\
\left[\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\right]_{\sigma^{2}\mu}&\left[\mathbb{F}\left(\boldsymbol{\boldsymbol{\theta}}\right)\right]_{\sigma^{2}\sigma^{2}}\end{pmatrix}=\begin{pmatrix}-\textbf{E}\left[\frac{\partial^{2}\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial^{2}\mu}\right]&-\textbf{E}\left[\frac{\partial^{2}\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\mu\partial\sigma^{2}}\right]\\\
-\textbf{E}\left[\frac{\partial^{2}\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial\sigma^{2}\partial\mu}\right]&-\textbf{E}\left[\frac{\partial^{2}\log
p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)}{\partial^{2}\sigma^{2}}\right]\end{pmatrix}=\begin{pmatrix}\frac{1}{\sigma^{2}}&0\\\
0&\frac{1}{2\sigma^{4}}\end{pmatrix}.$

At this step, we have learned that the Fisher information depends not only on
the properties of the system but also on the process that causes its evolution
and on the chosen measurement due to the probability law that gives us the
expression of $p\left(\boldsymbol{x};\boldsymbol{\boldsymbol{\theta}}\right)$
(see Eq.7). Then, one of the goals of metrology being to maximize the Fisher
information, it is therefore necessary to determine which is the best protocol
for measuring the system. For this reason, it would be good to know what is
the best precision achievable if we use the optimal measurement protocol, even
if we do not know what it consists of. We will therefore extend the classical
discussion we have had so far to quantum metrology by introducing the notion
of quantum Fisher information and quantum estimation theory.

##  IV Quantum estimation theory

The universal probabilistic nature of quantum mechanics leads us to work
regularly with probabilities. Moreover, if we investigate areas related to
experiments or physical realizations, this probabilistic nature of quantum
mechanics becomes even more visible. Furthermore, some exotic features such as
entanglement arise from quantum mechanics, which are directly related to the
probabilistic properties of the quantum system. In the quantum estimation
framework, the background of the parameter estimation problem remains
virtually unchanged from the classical framework. Here, an unknown parameter
$\theta$ encoded in a quantum state $\rho_{\theta}$ builds a quantum
statistical model, then individual qubits can be measured using POVM
measurements and these results $\left\\{\theta_{1},...,\theta_{n}\right\\}$
are used for the construction of an estimate $\boldsymbol{\theta}$. The major
distinction from the classical case is that the probability density function
$p\left(x;\boldsymbol{\theta}\right)$ is not arbitrary, but is a function that
depends on both the quantum state $\rho$ and the applied measurement.
Moreover, the results of the POVM measurements
$\left\\{\theta_{1},...,\theta_{n}\right\\}$, analogous to
$\left\\{x_{1},...,x_{n}\right\\}$ in the classical framework, are not
necessarily independent of each other due to the quantum correlations existing
in the probe quantum state. This allows estimates to be made with a super-
classical precision known as the Heisenberg limit [44, 19].

The purpose of this section is to describe quantum systems from the point of
view of metrology. We present a complete review of the principal theoretical
tools needed to address the parameters estimation problem via quantum measures
and quantum states, starting from the original wording developed by Holevo
[71, 72, 73] and Helstrom [74, 66], up to the most recent analytical results
concerning the two estimation strategies.

###  IV.1 Quantum statistical schemes

By analogy with the classical case, a quantum statistical model $S_{Q}$ is
defined by a family of density operators in a Hilbert space $\cal{H}$ and is
parameterized by $n$ real parameters $\theta\in\Theta\subset\mathbb{R}^{n}$,
where $\Theta$ is an estimated parameter space. This means that

$S_{Q}=\left\\{\rho_{\theta}=\Lambda_{\theta}\left(\rho\right):\hskip
14.22636pt\theta=\left(\theta^{1},\theta^{2},...,\theta^{n}\right)\in\Theta\right\\},$
(37)

where the application $\Lambda_{\theta}$ is injective (i.e., there is a
reverse of $\Lambda_{\theta}$). In fact, a quantum statistical model typically
looks like this: The considered system $S$ is prepared at time $t=0$ in an
initial state $\rho\equiv\rho\left(t=0\right)$, and then passes through a
quantum channel $\Lambda_{\theta}$ which depends on the true value of one or
more parameters $\theta$. So the associated model is defined as
$\rho_{\theta}:=\Lambda_{\theta}\left(\rho\right)$. A typical example of
application $\Lambda_{\theta}$ is the unit channel generated by the
Hamiltonian of the system $S$, namely $\rho_{\theta}=U_{t}\rho
U_{t}^{\dagger}$, and $U_{t}=\exp\left(-itH_{\theta}\right)$ where the
parameter $\theta$ is called the Hamiltonian parameter.

As an illustration, we consider here the thermal model which describes the
equilibrium state of a quantum system interacting with a thermal reservoir,
with

$\rho_{\beta}=\frac{\exp\left(-\beta H\right)}{Z},\hskip 28.45274pt{\rm
and}\hskip 28.45274ptZ={\rm Tr}\left(\exp\left(-\beta H\right)\right),$ (38)

where the interest parameter, conventionally referred to as $\beta$, is the
inverse of the reservoir temperature and $H$ is the Hamiltonian of the system.

Usually, any quantum statistical model
$S=\left\\{\rho_{\boldsymbol{\theta}}\right\\}_{\boldsymbol{\theta}\in\Theta}$,
performing a measurement with the operators $\left\\{\Pi_{x}\right\\}_{x\in
X}$, leads to a classical statistical model via the formula
$p_{\theta}\left(x\right)=p\left(x;\theta\right)={\rm
Tr}\left(\rho_{\theta}\Pi_{x}\right)$, where the sample space $X$ is expected
to be countable. To illustrate, let us take a state of a two-level quantum
system that we can decompose on the eigenstate basis
$\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$ in terms of two
angles $\left\\{\theta^{1}\equiv\theta,\theta^{2}\equiv\varphi\right\\}$ as
follows

$\left|\psi\right\rangle=\cos\left(\frac{\theta}{2}\right)\left|0\right\rangle+e^{i\varphi}\sin\left(\frac{\theta}{2}\right)\left|1\right\rangle,$
(39)

with $0\leq\theta\leq\pi$, $0\leq\varphi\leq 2\pi$ and evolving in a two
dimensional Hilbert space. The state of the qubit is described by the Bloch
vector $\vec{r}$ as shown below:

$\hat{\rho}=\frac{1}{2}\sum_{i=0}^{3}r_{i}\hat{\sigma_{i}},\hskip
28.45274pt\hat{\sigma_{0}}=\hat{\openone}_{2\times 2},\hskip
28.45274ptr_{i}={\rm Tr}\left(\hat{\sigma_{i}}\hat{\rho}\right),$ (40)

where $\sigma_{i}$ is the Pauli operator in the direction $i$ ($i=x,y,z$).
Then, the state of a two-level system is visualized on the Bloch sphere by the
Bloch vector

$\vec{r}=\left({\begin{array}[]{*{20}{c}}\left\langle{\hat{\sigma_{x}}}\right\rangle\\\
\left\langle{\hat{\sigma_{y}}}\right\rangle\\\
\left\langle{\hat{\sigma_{z}}}\right\rangle\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}\cos\theta\sin\varphi\\\
\sin\theta\sin\varphi\\\ \cos\varphi\end{array}}\right).$ (41)

If the state $\hat{\rho}$ is a pure state, the density matrix (40) reduces to

$\hat{\rho}=\frac{1}{2}\left({\begin{array}[]{*{20}{c}}1+\cos\frac{\theta}{2}&1-i\cos\frac{\theta}{2}\sin\frac{\theta}{2}\\\
1+i\cos\frac{\theta}{2}\sin\frac{\theta}{2}&1-\cos\frac{\theta}{2}\end{array}}\right).$
(42)

Assuming that we perform the projection value measurement (PVM) of the
operator $\sigma_{z}$ on the state (42), which is described by the set of
projection operators

$\hat{\Pi}_{+}\equiv\frac{1}{2}\left(\hat{\openone}+\hat{\sigma_{z}}\right),\hskip
28.45274pt{\rm and}\hskip
28.45274pt\hat{\Pi}_{-}\equiv\frac{1}{2}\left(\hat{\openone}-\hat{\sigma_{z}}\right),$
(43)

then the probability distribution of the result is obtained by the following
calculation

$p\left(+,\theta\right)\equiv{\rm
Tr}\left(\hat{\Pi}_{+}\hat{\rho}\right)=\frac{1}{2}\left(1+\cos\frac{\theta}{2}\right),\hskip
28.45274ptp\left(-,\theta\right)\equiv{\rm
Tr}\left(\hat{\Pi}_{-}\hat{\rho}\right)=\frac{1}{2}\left(1-\cos\frac{\theta}{2}\right).$
(44)

Consequently, this statistical model is parameterized by

$x=\left\\{+,-\right\\},\hskip
28.45274ptp\left(\pm,\theta\right)=\frac{1}{2}\left(1\pm\cos\frac{\theta}{2}\right),$
(45)

and the parameter space $\Theta$ becomes

$\Theta=\left\\{\boldsymbol{\theta}=\left(\theta,\varphi\right),\hskip
14.22636pt0\leq\theta\leq\pi,\hskip 14.22636pt0\leq\varphi\leq 2\pi,\hskip
14.22636ptn=2\right\\}.$ (46)

###  IV.2 Quantum Fisher Information

The Fisher information defines a limit on the precision of unknown parameters
related to a specific measurement strategy. It is obvious that different
strategies extract different amounts of information. Typically, by employing a
particular measurement scheme that extracts information about a given
characteristic, we exclude the possibility of measuring complementary
observables in the same experiment. For a quantum system, the variation of the
unknown parameter $\theta$ in the system state $\rho_{\theta}$ provides an
upper quantum limit on the classical Fisher information, ${\cal
F}\left(p\left(x;\theta\right)\right)\leq{\cal
F}_{Q}\left(\rho_{\theta}\right)$, with the quantum Fisher information ${\cal
F}_{Q}\left(\rho_{\theta}\right)$ is given by

${\cal F}_{Q}\left(\rho_{\theta}\right)={\rm
Tr}\left(\hat{L}_{\theta}^{2}\rho_{\theta}\right),$ (47)

where the symmetric logarithmic derivative $\hat{L}_{\theta}$ is a self-
adjoint operator that relates to the variation of the state $\rho_{\theta}$
with infinitesimal changes in the value of the unknown parameter $\theta$.
This is determined by solving the following Lyapunov equation

$\frac{\partial\rho_{\theta}}{\partial\theta}=\frac{1}{2}\left(\hat{L}_{\theta}\rho_{\theta}+\rho_{\theta}\hat{L}_{\theta}\right).$
(48)

In other words, the quantum Fisher information (47) can be formally derived by
maximizing the classical Fisher information (7) over all possible measurements
that can be made on the state $\rho$, and qualitatively, is a measure of the
amount of information that a state contains about the parameter $\theta$. A
general solution for the symmetric logarithmic derivative $L_{\theta}$ can be
found as

$\hat{L}_{\theta}=2\int_{0}^{\infty}\exp\left[-\rho_{\theta}t\right]\frac{\partial\rho_{\theta}}{\partial\theta}\exp\left[-\rho_{\theta}t\right]dt,$
(49)

which is not an easy equation to handle. We will derive here an alternative
form explicitly in terms of eigenvalues $\lambda_{i}$ and eigenvectors
$\left|\vartheta_{i}\right\rangle$ of the density matrix $\rho_{\theta}$. By
writing $\rho_{\theta}$ in its diagonal basis;
$\rho_{\theta}\left|\vartheta_{i}\right\rangle=\lambda_{i}\left|\vartheta_{i}\right\rangle$.
Then we can write

$\displaystyle\left(\frac{\partial\rho_{\theta}}{\partial\theta}\right)_{ij}$
$\displaystyle=\left\langle\vartheta_{i}\right|\frac{\partial\rho_{\theta}}{\partial\theta}\left|\vartheta_{j}\right\rangle$
$\displaystyle=\frac{1}{2}\left[\left\langle\vartheta_{i}\right|\hat{L}_{\theta}\rho_{\theta}\left|\vartheta_{j}\right\rangle+\left\langle\vartheta_{i}\right|\rho_{\theta}\hat{L}_{\theta}\left|\vartheta_{j}\right\rangle\right]$
$\displaystyle=\frac{1}{2}\left[\lambda_{j}\left(\hat{L}_{\theta}\right)_{ij}+\lambda_{i}\left(\hat{L}_{\theta}\right)_{ij}\right].$
(50)

Next, we solve to find $L_{\theta}$ as

$\left(\hat{L}_{\theta}\right)_{ij}=\frac{2}{\lambda_{i}+\lambda_{j}}\left\langle\vartheta_{i}\right|\frac{\partial\rho_{\theta}}{\partial\theta}\left|\vartheta_{j}\right\rangle,$
(51)

where the denominator includes only those terms that satisfy
$\lambda_{i}+\lambda_{j}\neq 0$.

####  IV.2.1 Analytical expression of QFI for pure states

Here, we shall focus on the much simpler scenario in which our quantum state
remains a pure state, such that
$\rho_{\theta}^{2}=\rho_{\theta}=\left|\psi_{\theta}\right\rangle\left\langle\psi_{\theta}\right|$.
Therefore, we have

$\frac{\partial\rho_{\theta}}{\partial\theta}=\frac{\partial\rho_{\theta}^{2}}{\partial\theta}=\rho_{\theta}\frac{\partial\rho_{\theta}}{\partial\theta}+\frac{\partial\rho_{\theta}}{\partial\theta}\rho_{\theta}.$
(52)

When we compare this with the equation (48), we get
$\hat{L}_{\theta}=2\frac{\partial\rho_{\theta}}{\partial\theta}$. We therefore
find

$\displaystyle{\cal F}_{Q}\left(\rho_{\theta}\right)=4{\rm
Tr}\left[\rho_{\theta}\left(\frac{\partial\rho_{\theta}}{\partial\theta}\right)^{2}\right]=4\left[\left\langle\hat{\psi_{\theta}}|\hat{\psi_{\theta}}\right\rangle-|\left\langle\hat{\psi_{\theta}}|\psi_{\theta}\right\rangle|^{2}\right],$
(53)

where
$\left|\hat{\psi_{\theta}}\right\rangle=\partial_{\theta}\left|\psi_{\theta}\right\rangle$.
The last expression, combined with the assumption of unitary transformations
${\cal
U}_{\theta}\left|\psi_{0}\right\rangle\equiv\left|\psi_{\theta}\right\rangle$
where ${\cal U}_{\theta}=\exp\left[-i\theta\hat{H}_{\theta}\right]$ and with
$\hat{H}_{\theta}$ is a hermitian operator, brings us to the last simplified
form of quantum Fisher information. For these states, the analytical
expression of the quantum Fisher information is expressed as

${\cal
F}_{Q}\left(\left|\psi_{\theta}\right\rangle\left\langle\psi_{\theta}\right|\right)\equiv{\cal
F}_{Q}\left(\left|\psi_{0}\right\rangle,\hat{H}_{\theta}\right)=4{\rm
Var}\left(\hat{H}_{\theta}\right),$ (54)

which is proportional to the variance of the Hermitian operator
$\hat{H}_{\theta}$. Moreover, an important property of the quantum Fisher
information is its convexity. This property implies that ${\cal
F}_{Q}\left(\rho\right)$ achieves its maximum value on pure states and hence,
the mixed quantum states cannot increase the achievable estimation
sensitivity. We can therefore generalize the last formula as

${\cal F}_{Q}\left(\rho\right)\leq 4{\rm Var}\left(\hat{H}_{\theta}\right),$
(55)

with equality applied to the pure states.

####  IV.2.2 Analytical expression of QFI for mixed states

In this part, we provide a derivation of the quantum Fisher information for
mixed quantum states. For mixed states, we can rewrite the state
$\rho_{\theta}\equiv\rho$ as a decomposition
$\rho=\sum_{i}p_{i}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|$
where $p_{i}$ and $\left|\psi_{i}\right\rangle$ are respectively the
eigenstates and the eigenvectors of the density matrix $\rho$. The quantum
Fisher information in this case can generally be expressed as follows

$\displaystyle{\cal
F}_{Q}\left(\rho\right)=4\sum_{i=0}^{s}\left(\left(\partial_{\theta}\sqrt{p_{i}}\right)^{2}+p_{i}\left\langle\partial_{\theta}\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle-
p_{i}|\left\langle\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle|^{2}\right)-8\sum_{i\neq
j}\frac{p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2},$
(56)

where the Hermitian operator $\hat{H}_{\theta}$ is defined by
$\hat{H}_{\theta}=-i{\cal U}_{\theta}^{\dagger}\partial_{\theta}{\cal
U}_{\theta}$. The symmetric logarithmic derivative (51) then becomes

$\hat{L}_{\theta}=2\sum_{ij}\frac{\left\langle\psi_{i}\right|\partial_{\theta}\rho\left|\psi_{j}\right\rangle}{p_{i}+p_{j}}\left|\psi_{i}\right\rangle\left\langle\psi_{j}\right|.$
(57)

Subsequently, by superimposing the definition of the symmetric logarithmic
derivative (48) with two states of the eigenbasis, one gets

$\displaystyle\left\langle\psi_{i}\right|\partial_{\theta}\rho\left|\psi_{j}\right\rangle$
$\displaystyle=\frac{1}{2}\left(\left\langle\psi_{i}\right|\hat{L}_{\theta}\rho\left|\psi_{j}\right\rangle+\left\langle\psi_{i}\right|\rho\hat{L}_{\theta}\left|\psi_{j}\right\rangle\right)$
$\displaystyle=\frac{1}{2}\left(\left\langle\psi_{i}\right|\hat{L}_{\theta}\sum_{k}p_{k}\left|\psi_{k}\right\rangle\left\langle\psi_{k}\right|\left|\psi_{j}\right\rangle+\left\langle\psi_{i}\right|\sum_{k}p_{k}\left|\psi_{k}\right\rangle\left\langle\psi_{k}\right|\hat{L}_{\theta}\left|\psi_{j}\right\rangle\right)$
$\displaystyle=\frac{1}{2}\left(\left\langle\psi_{i}\right|\hat{L}_{\theta}p_{j}\left|\psi_{j}\right\rangle+\left\langle\psi_{i}\right|p_{i}\hat{L}_{\theta}\left|\psi_{j}\right\rangle\right)$
$\displaystyle=\frac{1}{2}\left(p_{i}+p_{j}\right)\left(\hat{L}_{\theta}\right)_{ij},$
(58)

wherein we have determined
$\left(\hat{L}_{\theta}\right)_{ij}=\left\langle\psi_{i}\right|\hat{L}_{\theta}\left|\psi_{j}\right\rangle$.
Therefore, we can write the quantum Fisher information as

$\displaystyle{\cal F}_{Q}\left(\rho\right)$
$\displaystyle=\sum_{i}\left\langle\psi_{i}\right|\hat{L}_{\theta}^{2}\rho\left|\psi_{i}\right\rangle$
$\displaystyle=\sum_{ij}\left\langle\psi_{i}\right|\hat{L}_{\theta}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\hat{L}_{\theta}\rho\left|\psi_{i}\right\rangle$
$\displaystyle=\sum_{ij}\left\langle\psi_{i}\right|\hat{L}_{\theta}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\hat{L}_{\theta}\left|\psi_{i}\right\rangle
p_{i}$
$\displaystyle=\sum_{ij}\left(\hat{L}_{\theta}\right)_{ij}\left(\hat{L}_{\theta}\right)_{ji}p_{i}.$
(59)

Now, the symmetric logarithmic derivative $\left(\hat{L}_{\theta}\right)_{ij}$
can be rewritten as follows

$\left(\hat{L}_{\theta}\right)_{ij}=\frac{2\left\langle\psi_{i}\right|\partial_{\theta}\rho\left|\psi_{j}\right\rangle}{p_{i}+p_{j}}.$
(60)

Using this expression, the quantum Fisher information is expressed by the
following formula

${\cal
F}_{Q}\left(\rho\right)=\sum_{i=0}^{s}\sum_{j=0}^{t}\frac{4p_{i}}{\left(p_{i}+p_{j}\right)^{2}}|\left\langle\psi_{i}\right|\partial_{\theta}\rho\left|\psi_{j}\right\rangle|^{2}.$
(61)

Here we have entered the two indices $s$ and $t$. The state supports only $s$,
so all indices $p_{t}=0$ for $t\geq s$, because the operator
$\partial_{\theta}\rho$ is Hermitian. Afterward, we can show from the spectral
decomposition of $\rho$ that

$\displaystyle\left\langle\psi_{i}\right|\partial_{\theta}\rho\left|\psi_{j}\right\rangle$
$\displaystyle=\sum_{k}\left(\left\langle\psi_{i}\right|\partial_{\theta}p_{k}\left|\psi_{k}\right\rangle\left\langle\psi_{k}|\psi_{j}\right\rangle+p_{k}\left\langle\psi_{i}|\partial_{\theta}\psi_{k}\right\rangle\left\langle\psi_{k}|\psi_{j}\right\rangle+p_{i}\left\langle\psi_{i}|\psi_{k}\right\rangle\left\langle\partial_{\theta}\psi_{k}|\psi_{j}\right\rangle\right)$
$\displaystyle=\partial_{\theta}p_{i}\delta_{ij}+p_{j}\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle+p_{i}\left\langle\partial_{\theta}\psi_{i}|\psi_{j}\right\rangle.$
(62)

We use the fact that the resolution of the identity becomes zero when it is
differentiated, i.e.,

$\partial_{\theta}\openone\equiv\sum_{i}\left|\partial_{\theta}\psi_{i}\right\rangle\left\langle\psi_{i}\right|+\left|\psi_{i}\right\rangle\left\langle\partial_{\theta}\psi_{i}\right|=0,$
(63)

this implies
$\left\langle\partial_{\theta}\psi_{i}|\psi_{j}\right\rangle=-\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle$.
So we find that

$\left\langle\psi_{i}\right|\partial_{\theta}\rho\left|\psi_{j}\right\rangle=\partial_{\theta}p_{i}\delta_{ij}+\left(p_{j}-p_{i}\right)\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle.$
(64)

We now insert this into the above equation to find

$\displaystyle{\cal F}_{Q}\left(\rho\right)$
$\displaystyle=\sum_{i=0}^{s}\sum_{j=0}^{t}\frac{4p_{i}}{\left(p_{i}+p_{j}\right)^{2}}|\partial_{\theta}p_{i}\delta_{ij}+\left(p_{j}-p_{i}\right)\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}$
$\displaystyle=\sum_{i=0}^{s}\frac{1}{p_{i}}\left(\partial_{\theta}p_{i}\right)^{2}+\sum_{i=0}^{s}\sum_{j=0}^{t}\frac{4p_{i}\left(p_{i}-p_{j}\right)^{2}}{\left(p_{i}+p_{j}\right)^{2}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}.$
(65)

We notice that the second sum can be divided into two sums; one that runs on
the support until $s$ and the other $j$ the support from $s+1$ to $t$. Hence

$\displaystyle\sum_{i=0}^{s}\sum_{j=0}^{t}\frac{4p_{i}\left(p_{i}-p_{j}\right)^{2}}{\left(p_{i}+p_{j}\right)^{2}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}=\sum_{i,j=0}^{s}\frac{4p_{i}\left(p_{i}-p_{j}\right)^{2}}{\left(p_{i}+p_{j}\right)^{2}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}+\sum_{i=0}^{s}\sum_{j=s+1}^{t}4p_{i}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}.$
(66)

It should be noted that all $p_{s+1}=0$, thus the second sum can be written
like this

$\sum_{j=s+1}^{t}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|=\openone-\sum_{j=0}^{s}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|.$
(67)

Putting this expression into the above expression gives the following result

$\displaystyle\sum_{i=0}^{s}\sum_{j=s+1}^{t}4p_{i}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}$
$\displaystyle=\sum_{i=0}^{s}\sum_{j=s+1}^{t}4p_{i}\left\langle\partial_{\theta}\psi_{i}|\psi_{j}\right\rangle\left\langle\psi_{j}|\partial_{\theta}\psi_{i}\right\rangle$
$\displaystyle=\sum_{i=0}^{s}4p_{i}\left\langle\partial_{\theta}\psi_{i}\right|\left[\openone-\sum_{j=0}^{s}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\right]\left|\partial_{\theta}\psi_{i}\right\rangle$
$\displaystyle=\sum_{i=0}^{s}4p_{i}\left\langle\partial_{\theta}\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle-\sum_{i,j=0}^{s}4p_{i}|\left\langle\psi_{j}|\partial_{\theta}\psi_{i}\right\rangle|^{2}.$
(68)

If we put all these elements together, we come up with

$\displaystyle{\cal F}_{Q}\left(\rho\right)$
$\displaystyle=\sum_{i=0}^{s}\frac{1}{p_{i}}\left(\partial_{\theta}p_{i}\right)^{2}+\sum_{i,j=0}^{s}\frac{4p_{i}\left(p_{i}-p_{j}\right)^{2}}{\left(p_{i}+p_{j}\right)^{2}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}+\sum_{i=0}^{s}4p_{i}\left\langle\partial_{\theta}\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle-\sum_{i,j=0}^{s}4p_{i}|\left\langle\psi_{j}|\partial_{\theta}\psi_{i}\right\rangle|^{2}$
$\displaystyle=\sum_{i=0}^{s}\frac{1}{p_{i}}\left(\partial_{\theta}p_{i}\right)^{2}+\sum_{i=0}^{s}4p_{i}\left\langle\partial_{\theta}\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle-8\sum_{i,j=0}^{s}\frac{p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}.$
(69)

Furthermore, we can also write the first part of this expression in the form

$\sum_{i=0}^{s}\frac{1}{p_{i}}\left(\partial_{\theta}p_{i}\right)^{2}=4\sum_{i=0}^{s}\left(\partial_{\theta}\sqrt{p_{i}}\right)^{2}.$
(70)

We now remark that the third sum of the equation (69) holds both diagonal and
non-diagonal values. As a result, this expression can be further simplified
and the quantum Fisher information is finally expressed as:

$\displaystyle{\cal F}_{Q}\left(\rho\right)$
$\displaystyle=4\sum_{i=0}^{s}\left(\partial_{\theta}\sqrt{p_{i}}\right)^{2}+\sum_{i=0}^{s}4p_{i}\left\langle\partial_{\theta}\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle-8\left(\sum_{i}\delta_{ij}+\sum_{i\neq
j}\right)\frac{p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}$
$\displaystyle=4\sum_{i=0}^{s}\left(\left(\partial_{\theta}\sqrt{p_{i}}\right)^{2}+p_{i}\left\langle\partial_{\theta}\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle-
p_{i}|\left\langle\psi_{i}|\partial_{\theta}\psi_{i}\right\rangle|^{2}\right)-8\sum_{i\neq
j}\frac{p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}|\partial_{\theta}\psi_{j}\right\rangle|^{2}.$
(71)

When the states $\left|\psi_{i}\right\rangle$ are evolving over time, we
obtain $\left|\psi_{i}\left(t\right)\right\rangle={\cal
U}_{\theta}\left(t\right)\left|\psi_{i}\left(0\right)\right\rangle$, which
involves
$\partial_{\theta}\left|\psi_{i}\left(t\right)\right\rangle=\partial_{\theta}{\cal
U}_{\theta}\left(t\right)\left|\psi_{i}\left(0\right)\right\rangle$. In order
to simplify the handling of this expression, we define the generator
$\hat{H}_{\theta}=-i{\cal U}_{\theta}^{\dagger}\partial_{\theta}{\cal
U}_{\theta}$. By applying the identity property ${\cal
U}_{\theta}^{\dagger}{\cal U}_{\theta}=\openone$, we observe that

${\cal U}_{\theta}^{\dagger}\partial_{\theta}{\cal
U}_{\theta}=-\left(\partial_{\theta}{\cal U}_{\theta}^{\dagger}\right){\cal
U}_{\theta}.$ (72)

In addition, we have

$\hat{H}_{\theta}^{2}=-{\cal U}_{\theta}^{\dagger}\left(\partial_{\theta}{\cal
U}_{\theta}\right){\cal U}_{\theta}^{\dagger}\partial_{\theta}{\cal
U}_{\theta}=\left(\partial_{\theta}{\cal
U}_{\theta}\right)^{\dagger}\partial_{\theta}{\cal U}_{\theta}.$ (73)

In the case of a unitary evolution operator ${\cal U}_{\theta}$, the classical
contribution vanishes ($\partial_{\theta}\sqrt{p_{i}}=0$). Thereby we proceed
to write

$\displaystyle{\cal F}_{Q}\left(\rho\right)=4\sum_{i=0}^{s}p_{i}$
$\displaystyle\left(\left\langle\psi_{i}\left(0\right)\right|\hat{H}_{\theta}^{2}\left|\psi_{i}\left(0\right)\right\rangle-\left\langle\psi_{i}\left(0\right)\right|\hat{H}_{\theta}\left|\psi_{i}\left(0\right)\right\rangle\right)$
$\displaystyle-8\sum_{i\neq
j}\frac{p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}\left(0\right)\right|\hat{H}_{\theta}\left|\psi_{i}\left(0\right)\right\rangle|^{2}.$
(74)

The following table summarizes the properties of classical and quantum
estimation and the relationship between them:

Estimation Theory
---
Classical Strategy | Quantum Strategy
Probability density $p\left(x;\theta\right)$ | Density matrix $\rho_{\theta}=\Lambda_{\theta}\left(\rho_{0}\right)$
Score ${\cal L}_{\theta}\left(x\right)=\frac{\partial}{\partial\theta}\log p\left(x;\theta\right)$ | Symmetric logarithmic derivative $\hat{L}_{\theta}$
$\frac{\partial p\left(x;\theta\right)}{\partial\theta}=\frac{1}{2}\left({\cal L}_{\theta}p\left(x;\theta\right)+p\left(x;\theta\right){\cal L}_{\theta}\right)$ | $\frac{\partial\rho_{\theta}}{\partial\theta}=\frac{1}{2}\left(\hat{L}_{\theta}\rho_{\theta}+\rho_{\theta}\hat{L}_{\theta}\right)$
${\rm Var}\left({\cal L}_{\theta}\right)=\sum_{x}p\left(x;\theta\right){\cal L}_{\theta}^{2}={\cal F}\left(\theta\right)$ | ${\rm Var}\left(\hat{L}_{\theta}\right)={\rm Tr}\left(\rho\hat{L}_{\theta}^{2}\right)={\cal F}_{Q}\left(\rho\right)$
$\bar{{\cal L}_{\theta}}=\sum_{x}p\left(x;\theta\right){\cal L}_{\theta}=0$ | $\left\langle\hat{L}_{\theta}\right\rangle={\rm Tr}\left(\rho\hat{L}_{\theta}\right)=\frac{\partial}{\partial\theta}{\rm Tr}\left(\rho\right)=0$
Table 1: Basic properties of classical and quantum estimation theory

Interestingly, there’s always an optimal measure for any $\rho$ such that
${\cal F}\left(\theta\right)={\cal F}_{Q}\left(\rho_{\theta}\right)$. This
optimal measure is given by the projection on the eigenstates
$\left|l\right\rangle$ of $\hat{L}_{\theta}$. In this case, we have
$p\left(l;\theta\right)=\left\langle
l\right|\rho_{\theta}\left|l\right\rangle$. Then

$\frac{\partial p\left(l;\theta\right)}{\partial\theta}=\left\langle
l\right|\frac{\partial\rho_{\theta}}{\partial\theta}\left|l\right\rangle=lp\left(l;\theta\right).$
(75)

Therefore, it is easy to verify that

${\cal
F}\left(\theta\right)=\sum_{l}\frac{1}{p\left(l;\theta\right)}\left(\frac{\partial
p\left(l;\theta\right)}{\partial\theta}\right)^{2}=\sum_{l}l^{2}p\left(l;\theta\right)={\rm
Tr}\left(\rho_{\theta}\hat{L}_{\theta}^{2}\right)={\cal
F}_{Q}\left(\rho_{\theta}\right).$ (76)

##  V Multiparametric quantum metrology

So far, we have considered only one parameter to be estimated. However, there
are tasks for which it is important to estimate several parameters. These
scenarios include, for example, the simultaneous estimation of several phases
and the estimation of several spins pointing in different directions or, in
general, of parameters corresponding to non-commutable unitary generators. The
theory described in the previous sections can be naturally generalized to the
multiparameter estimation strategy. To determine the accuracy in this case, we
need to compute the Fisher quantum information matrix, denoted
$\boldsymbol{{\cal F}}_{Q}\left(\boldsymbol{\theta}\right)$, which describes
the limits of distinguishing infinitely close quantum states
$\rho_{\boldsymbol{\theta}}$ and
$\rho_{\boldsymbol{\theta}+d\boldsymbol{\theta}}$ where
$\boldsymbol{\theta}=\left\\{\theta_{1},..\theta_{\mu},\theta_{\nu},..\theta_{n}\right\\}$.
In general, larger elements of this matrix predict better distinguishability,
which leads to better accuracy in estimating the parameter vector
$\boldsymbol{\theta}$.

On the other hand, how to increase the quantum Fisher information matrix is a
challenging topic in quantum metrology. In fact, the estimation of a single
parameter plays an important role in many ways, due to the existence of an
optimal final state containing a maximum amount of the quantum Fisher
information. Realistic problems can usually involve multiple parameters
because there is no optimal state in which the quantum Fisher information
matrix is larger than the other states. This is related to the fact that the
optimal measures for the estimation of different parameters are not
necessarily commutable. In addition, the Cramér-Rao inequality is not always
saturable because the measurements of different parameters may be
incompatible. All these restrictions make the simultaneous estimation of
several parameters an important task in quantum metrology. In the same
context, techniques for computing the quantum Fisher information matrix have
seen rapid development in various scenarios and models. However, there is no
literature that summarizes these techniques in a structured way. Therefore,
this section provides extensive techniques for calculating the matrix
$\boldsymbol{{\cal F}}_{Q}\left(\boldsymbol{\theta}\right)$ across a variety
of scenarios.

###  V.1 Matrix representation of the quantum Fisher information

Let us start by considering the general case, i.e. a family of quantum states
$\rho_{\boldsymbol{\theta}}$ which depend on a set of $n$ different parameters
$\boldsymbol{\theta}=\left\\{\theta_{\mu}\right\\}$, $\mu=1,..,n$. For each
parameter involved, we can derive the operators of the symmetric logarithmic
derivative as follows

$\frac{\partial\rho_{\boldsymbol{\theta}}}{\partial\theta_{\mu}}=\frac{1}{2}\left(\hat{L}_{\theta_{\mu}}\rho_{\boldsymbol{\theta}}+\rho_{\boldsymbol{\theta}}\hat{L}_{\theta_{\mu}}\right).$
(77)

The components of the quantum Fisher information matrix are defined as [19]

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}\left(\rho_{\boldsymbol{\theta}}\right):=\frac{1}{2}{\rm
Tr}\left(\rho_{\boldsymbol{\theta}}\left\\{\hat{L}_{\theta_{\mu}},\hat{L}_{\theta_{\nu}}\right\\}\right),$
(78)

where $\left\\{,\right\\}$ stands for anti-commutation and
$\hat{L}_{\theta_{\mu}}\left(\hat{L}_{\theta_{\nu}}\right)$ is the symmetric
logarithmic derivative for the parameter
$\theta_{\mu}\left(\theta_{\nu}\right)$, which is derived by equation (77). It
is generally recognized that ${\rm
Tr}\left(\rho_{\boldsymbol{\theta}}\hat{L}_{\theta_{\mu}}\right)$, therefore
using the above equation (78), $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}\left(\rho_{\boldsymbol{\theta}}\right)$ can
also be expressed as below

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}\left(\rho_{\boldsymbol{\theta}}\right)={\rm
Tr}\left(\hat{L}_{\theta_{\nu}}\partial_{\theta_{\mu}}\rho_{\boldsymbol{\theta}}\right)=-{\rm
Tr}\left(\rho_{\boldsymbol{\theta}}\partial_{\theta_{\mu}}\hat{L}_{\theta_{\nu}}\right).$
(79)

According to equation (78), the diagonal entry of the matrix
$\boldsymbol{{\cal F}}\left(\rho_{\boldsymbol{\theta}}\right)$ reads

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\mu}}\left(\rho_{\boldsymbol{\theta}}\right)={\rm
Tr}\left(\rho_{\boldsymbol{\theta}}\hat{L}_{\theta_{\mu}}^{2}\right),$ (80)

which is exactly the quantum Fisher information for the parameter
$\theta_{\mu}$. Besides, some properties of the quantum Fisher information
have been well presented above. Similarly, the quantum Fisher information
matrix also has some powerful properties that have been widely applied in
practice. We present these properties here as follows

*

$\boldsymbol{{\cal F}}\left(\rho_{\boldsymbol{\theta}}\right)$ is real
symmetric, namely $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\boldsymbol{{\cal
F}}_{\theta_{\nu}\theta_{\mu}}\in\mathbb{R}^{2}$.

*

$\boldsymbol{{\cal F}}\left(\rho_{\boldsymbol{\theta}}\right)$ is positive
semi-definite, i.e. $\boldsymbol{{\cal
F}}\left(\rho_{\boldsymbol{\theta}}\right)\geq 0$. If $\boldsymbol{{\cal
F}}>0$, then $\left[\boldsymbol{{\cal
F}}^{-1}\right]_{\theta_{\mu}\theta_{\mu}}=1/\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\mu}}$ for any $\theta_{\mu}$.

*

$\boldsymbol{{\cal
F}}\left(\rho_{\boldsymbol{\theta}}\right)=\boldsymbol{{\cal
F}}\left(U\rho_{\boldsymbol{\theta}}U^{\dagger}\right)$ for a unitary
operation $U$.

*

$\boldsymbol{{\cal F}}$ is a monotonic function under a completely positive
and trace preserving map $\Lambda$, this means that $\boldsymbol{{\cal
F}}\left(\Lambda\left(\rho\right)\right)\leq\boldsymbol{{\cal
F}}\left(\rho\right)$.

*

Convexity property: $\boldsymbol{{\cal
F}}\left(p\rho_{1}+\left(1-p\right)\rho_{2}\right)\leq p\boldsymbol{{\cal
F}}\left(\rho_{1}\right)+\left(1-p\right)\boldsymbol{{\cal
F}}\left(\rho_{2}\right)$ for $p\in\left[0,1\right]$.

###  V.2 Analytical expression of the quantum Fisher information matrix

Here we review the derivation techniques of the quantum Fisher information
matrix and some analytical results for specific cases. The standard derivation
of this matrix generally assumes that the rank of the density matrix is
complete, meaning that all eigenvalues of the $\rho$ density matrix are
positive. More precisely, when we write
$\rho=\sum_{i}\lambda_{i}\left|\vartheta_{i}\right\rangle\left\langle\vartheta_{i}\right|$,
where $\lambda_{i}$ and $\left|\vartheta_{i}\right\rangle$ are the
corresponding eigenvalue and eigenstate, we generally assume that
$\lambda_{i}>0$, for any $0\leq i\leq{\rm dim}\left(\rho\right)-1$. With this
assumption, the quantum Fisher information matrix elements can be written as
follows

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\sum_{i,j=0}^{d-1}\frac{2\textsf{Re}\left(\left\langle\vartheta_{i}\right|\partial_{\theta_{\mu}}\rho\left|\vartheta_{j}\right\rangle\left\langle\vartheta_{j}\right|\partial_{\theta_{\nu}}\rho\left|\vartheta_{i}\right\rangle\right)}{\lambda_{i}+\lambda_{j}},$
(81)

with Re denotes the real part and $d$ is the dimension of the density matrix.
One can easily see that if the density matrix is not of full rank, there may
be divergent terms in the above equation. To extend it to general density
matrices, we can remove the divergent terms as follows

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\sum_{i,j=0,\lambda_{i}+\lambda_{j}\neq
0}^{d-1}\frac{2\textsf{Re}\left(\left\langle\vartheta_{i}\right|\partial_{\theta_{\mu}}\rho\left|\vartheta_{j}\right\rangle\left\langle\vartheta_{j}\right|\partial_{\theta_{\nu}}\rho\left|\vartheta_{i}\right\rangle\right)}{\lambda_{i}+\lambda_{j}}.$
(82)

By substituting the spectral decomposition of $\rho$ in the above equation, we
can rewrite it as

$\displaystyle\boldsymbol{{\cal F}}_{\theta_{\mu}\theta_{\nu}}$
$\displaystyle=\sum_{i=0}^{d-1}\frac{\left(\partial_{\theta_{\mu}}\lambda_{i}\right)\left(\partial_{\theta_{\nu}}\lambda_{i}\right)}{\lambda_{i}}+\sum_{i\neq
j,\lambda_{i}+\lambda_{j}\neq
j}^{d-1}\frac{2\left(\lambda_{i}-\lambda_{j}\right)^{2}}{\left(\lambda_{i}+\lambda_{j}\right)}\textsf{Re}\left(\left\langle\vartheta_{i}|\partial_{\theta_{\mu}}\vartheta_{j}\right\rangle\left\langle\partial_{\theta_{\nu}}\vartheta_{j}|\vartheta_{i}\right\rangle\right)$
$\displaystyle=\sum_{i}\frac{\left(\partial_{\theta_{\mu}}\lambda_{i}\right)\left(\partial_{\theta_{\nu}}\lambda_{i}\right)}{\lambda_{i}}+\sum_{i}4\lambda_{i}\textsf{Re}\left(\left\langle\partial_{\theta_{\mu}}\vartheta_{i}|\partial_{\theta_{\nu}}\vartheta_{i}\right\rangle\right)+\sum_{i\neq
j}\frac{8\lambda_{i}\lambda_{j}}{\lambda_{i}+\lambda_{j}}\left(\left\langle\partial_{\theta_{\mu}}\vartheta_{i}|\vartheta_{j}\right\rangle\left\langle\vartheta_{j}|\partial_{\theta_{\nu}}\vartheta_{i}\right\rangle\right).$
(83)

Due to the relationship between the quantum Fisher information matrix and the
quantum Fisher information, one can easily obtain

$\displaystyle\boldsymbol{{\cal F}}_{\theta_{\mu}\theta_{\mu}}=$
$\displaystyle{\cal
F}_{Q}\left(\rho_{\theta_{\mu}}\right)=\sum_{i}\frac{\left(\partial_{\theta_{\mu}}\lambda_{i}\right)^{2}}{\lambda_{i}}+\sum_{i}4\lambda_{i}\left(\left\langle\partial_{\theta_{\mu}}\vartheta_{i}|\partial_{\theta_{\mu}}\vartheta_{i}\right\rangle\right)$
$\displaystyle-\sum_{i\neq
j}\frac{8\lambda_{i}\lambda_{j}}{\lambda_{i}+\lambda_{j}}|\left\langle\partial_{\theta_{\mu}}\vartheta_{i}|\vartheta_{j}\right\rangle|^{2}.$
(84)

The Bloch representation is another tool also well used in quantum estimation
theory. For a $d$-dimensional density matrix, it can be expressed by

$\rho=\frac{1}{d}\left(\openone+\sqrt{\frac{d\left(d-1\right)}{2}}\vec{r}.\vec{K}\right),$
(85)

where $\vec{r}=\left(r_{1},r_{2},..,r_{m},..\right)^{T}$ is the Bloch vector
and $\vec{K}$ is a ($d^{2}-1$)-dimensional vector of the generator of
$SU\left(d\right)$ which satisfies ${\rm Tr}\left(K_{i}\right)=0$. The anti-
commutation relation for them is
$\left\\{K_{i},K_{j}\right\\}=\frac{4}{d}\delta_{ij}\openone+\sum_{m=1}^{d^{2}-1}v_{ijm}K_{m}$,
and the commutation relation being
$\left[K_{i},K_{j}\right]=i\sum_{m=1}^{d^{2}-1}\epsilon_{ijm}K_{m}$, where
$v_{ijm}$ and $\epsilon_{ijm}$ are the symmetric and antisymmetric structure
constants.

Very recently, Watanabe and his co-workers provided a simple analytical
formula of the quantum Fisher information matrix for a general Bloch vector by
considering the Bloch vector itself as parameters to be estimated. Using the
Bloch representation of a $d$-dimensional density matrix, the quantum Fisher
information matrix can be expressed as follows [26]

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\left(\partial_{\theta_{\nu}}\vec{r}\right)^{T}\left(\frac{d}{2\left(d-1\right)}G-\vec{r}\vec{r}^{T}\right)\partial_{\theta_{\mu}}\vec{r},$
(86)

with $G$ is a real symmetric matrix whose components are

$G_{ij}=\frac{1}{2}{\rm
Tr}\left(\rho\left\\{K_{i},K_{j}\right\\}\right)=\frac{2}{d}\delta_{ij}+\sqrt{\frac{d-1}{2d}}\sum_{m}v_{ijm}r_{m}.$
(87)

In an analytical point of view, the most widely used scenario of this theorem
concerns one-qubit systems, where
$\rho=\left(\openone+\vec{r}.\vec{\sigma}\right)$ with the vector of Pauli
matrices $\vec{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$. In such
circumstances, the quantum Fisher information matrix reduces to

$\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\left(\partial_{\theta_{\mu}}\vec{r}\right)\left(\partial_{\theta_{\nu}}\vec{r}\right)+\frac{\left(\vec{r}.\partial_{\theta_{\mu}}\vec{r}\right)\left(\vec{r}.\partial_{\theta_{\nu}}\vec{r}\right)}{1-|\vec{r}|^{2}},$
(88)

where $|\vec{r}|$ is the norm of $\vec{r}$. For a single qubit pure state, we
find $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\left(\partial_{\theta_{\mu}}\vec{r}\right)\left(\partial_{\theta_{\nu}}\vec{r}\right)$.

For a general two-qubit state, the derivation of the quantum Fisher
information matrix requires the diagonalization of a $4\times 4$ density
matrix, which is difficult to solve analytically. Nevertheless, some special
two-qubit states, such as the $X$-state, can be analytically diagonalized.
Indeed, the two-qubit state $\rho_{X}$ can be rewritten as a block diagonal in
the form $\rho_{X}=\rho^{(0)}\oplus\rho^{(1)}$, where $\oplus$ represents the
direct sum and

$\rho^{(0)}=\left({\begin{array}[]{*{20}{c}}\rho_{11}&\rho_{14}\\\
\rho_{41}&\rho_{44}\end{array}}\right),\hskip 28.45274pt{\rm and}\hskip
28.45274pt\rho^{(1)}=\left({\begin{array}[]{*{20}{c}}\rho_{22}&\rho_{23}\\\
\rho_{32}&\rho_{33}\end{array}}\right).$ (89)

It is worth noting that $\rho^{(0)}$ and $\rho^{(1)}$ are not density matrices
because their trace are not normalized. The quantum Fisher information matrix
for this state can be written as $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}=\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}^{(0)}+\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}^{(1)}$, where $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}^{(0)}(\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}^{(1)})$ is the quantum Fisher information
matrix for $\rho^{(0)}(\rho^{(1)})$. The eigenvalues of $\rho^{(i)}$ are

$\lambda_{\pm}^{(i)}=\frac{1}{2}\left({\rm Tr}\rho^{(i)}\pm\sqrt{\left({\rm
Tr}\rho^{(i)}\right)^{2}-4{\rm det}\rho^{(i)}}\right),$ (90)

and the corresponding eigenstates are

$\left|\vartheta_{\pm}^{(i)}\right\rangle={\cal
N}_{\pm}^{(i)}\left(\frac{1}{2{\rm
Tr}\left(\rho^{(i)}\sigma_{+}\right)}\left[{\rm
Tr}\left(\rho^{(i)}\sigma_{+}\right)\pm\sqrt{\left({\rm
Tr}\rho^{(i)}\right)^{2}-4{\rm det}\rho^{i}}\right],1\right)^{T},$ (91)

with ${\cal N}_{\pm}^{(i)}$ ($i=0,1$) is the normalization coefficient. In
this case, the specific form of $\sigma_{z}$ et $\sigma_{+}$ are

$\sigma_{z}=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&-1\end{array}}\right),\hskip
56.9055pt\sigma_{+}=\left({\begin{array}[]{*{20}{c}}0&1\\\
0&0\end{array}}\right).$ (92)

Based on the above mentioned information, $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}^{(i)}$ can be written specifically as follows

$\displaystyle\boldsymbol{{\cal F}}_{\theta_{\mu}\theta_{\nu}}^{(i)}=$
$\displaystyle\sum_{k=\pm}\frac{\left(\partial_{\theta_{\mu}}\lambda_{k}^{(i)}\right)\left(\partial_{\theta_{\nu}}\lambda_{k}^{(i)}\right)}{\lambda_{k}^{(i)}}+\lambda_{k}^{(i)}\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}\left(\left|\vartheta_{k}^{(i)}\right\rangle\right)$
$\displaystyle-\frac{16{\det\rho^{(i)}}}{{\rm
Tr}\rho^{(i)}}\textsf{Re}\left(\left\langle\partial_{\theta_{\mu}}\vartheta_{+}^{(i)}|\vartheta_{-}^{(i)}\right\rangle\left\langle\vartheta_{-}^{(i)}|\partial_{\theta_{\nu}}\vartheta_{+}^{(i)}\right\rangle\right),$
(93)

where $\boldsymbol{{\cal
F}}_{\theta_{\mu}\theta_{\nu}}\left(\left|\vartheta_{k}^{(i)}\right\rangle\right)$
is the quantum Fisher information matrix for the state
$\left|\vartheta_{k}^{(i)}\right\rangle$.

###  V.3 Quantum Fisher Information Matrix via Vectorization Method

Unlike the previous analytical expression of the quantum Fisher information
matrix which required the diagonalization of the density matrix, Šafránek [75]
has recently provided another analytical expression using the density matrix
in the Liouville space formalism. Technically, by employing the vectorization
of the density matrix $\rho$, the Lyapunov equations (77) are converted to a
linear system that can be solved without diagonalizing the matrix $\rho$. This
approach has proven to be successful and can be used for any finite
dimensional density matrix. This is based on using the vec-operator which
allows us to transform a density matrix into a column vector. Indeed, the
operator ${\rm vec}\left[A\right]$ for any matrix $A\in\emph{M}^{n\times n}$
in the Liouville space, where $\emph{M}^{n\times n}$ refers to the space of
$n\times n$ martrices, is given by

${\rm
vec}\left[A\right]=\left(a_{11},...,a_{n1},a_{12},...,a_{n2},a_{1n},...,a_{nn}\right)^{T}.$
(94)

Moreover, through the expression
$A=\sum_{k,l=1}^{n}a_{kl}\left|k\right\rangle\left\langle l\right|$, the ${\rm
vec}$-operator becomes

${\rm vec}\left[A\right]=\left(\openone_{n\times n}\otimes
A\right)\sum_{i=1}^{n}e_{i}\otimes e_{i}.$ (95)

where $e_{i}$ stands for the computational components of $\emph{M}^{n\times
n}$. Taking advantage of the Kronecker product properties [76], we can find
the results

$\displaystyle{\rm vec}\left[AB\right]$ $\displaystyle=\left(\openone_{n\times
n}\otimes A\right){\rm vec}\left[B\right]=\left(B^{T}\otimes\openone_{n\times
n}\right){\rm vec}\left[A\right],$ $\displaystyle{\rm
Tr}\left(A^{T}B\right)={\rm vec}\left[A\right]^{\dagger}{\rm
vec}\left[B\right],$ $\displaystyle{\rm
vec}\left[AXB\right]=\left(B^{\dagger}\otimes A\right){\rm
vec}\left[X\right],$ (96)

that apply to any matrix $A$, $B$ and $X$. With the above properties (96), it
is straight forward to verify that the quantum Fisher information matrix can
be expressed as follows

$\boldsymbol{{\cal F}}_{\theta_{\mu}\theta_{\nu}}=2{\rm
vec}\left[\partial_{\theta_{\mu}}\rho\right]^{\dagger}\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}{\rm
vec}\left[\partial_{\theta_{\nu}}\rho\right],$ (97)

with $\rho^{*}$ being the complex conjugate of $\rho$. The symmetric
logarithmic derivative operators in Liouville space, denoted by ${\rm
vec}\left[\hat{L}_{\theta_{\mu}}\right]$, are reduced to

${\rm
vec}\left[\hat{L}_{\theta_{\mu}}\right]=2\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}{\rm
vec}\left[\partial_{\theta_{\mu}}\rho\right].$ (98)

The advantage of this method would be that it can be computed analytically for
an arbitrary system and is only based on the computation of the inverse of the
matrix $\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)$. For high-
dimensional systems, efficient methods such as Cholesky decomposition may be
required [77]. It is worth noting that the above formula is only applicable if
the density matrix $\rho$ has full rank and thus is invertible. If $\rho$ is
not invertible, Šafránek gives another formal solution of quantum Fisher
information matrix by using the invertible matrix
$\rho_{s}\equiv\left(1-s\right)\rho+\frac{s}{d}\openone$ corresponding to
$\rho$ such that the equation (78) becomes

$\boldsymbol{{\cal F}}_{\theta_{\mu}\theta_{\nu}}=2\lim_{s\rightarrow 0}{\rm
vec}\left[\partial_{\theta_{\mu}}\rho_{s}\right]^{\dagger}\left(\rho_{s}\otimes\openone+\openone\otimes\rho_{s}^{*}\right)^{-1}{\rm
vec}\left[\partial_{\theta_{\nu}}\rho_{s}\right].$ (99)

To illustrate how the derived expressions can be used, we take the example of
an isotropic Heisenberg $XY$ model submitted to an external magnetic field.
Here, we estimated the temperature and the magnetic field of this quantum
Heisenberg model in which the Hamiltonian describing the system can be written
as [78, 79, 80, 81]

$H=\sum_{n=1}^{2}J\left(S_{n}^{x}S_{n+1}^{x}+S_{n}^{y}S_{n+1}^{y}\right)+B\sum_{n=1}^{2}S_{n}^{z},$
(100)

where $S_{n}^{i}=\frac{1}{2}\sigma_{n}^{i}$ (with $i=x,y,z$) and
$\sigma_{n}^{i}$ are the spin-$\frac{1}{2}$ operators and the familiar Pauli
matrices at site $n$ respectively, $B$ is the constant external magnetic field
acting on the spin-$\frac{1}{2}$ particle oriented in the $z$-direction, and
the parameter $J$ denotes the linear strength of the Heisenberg coupling
coefficient for the spin interaction. Notice that the Heisenberg spin chain is
antiferromagnetic for $J$ positive, and becomes ferromagnetic for $J$
negative. In terms of the raising and lowering operator
$\sigma_{n}^{\pm}=\sigma_{n}^{x}\pm\sigma_{n}^{y}$, the Hamiltonian (100)
becomes

$H=J\left(\sigma_{1}^{+}\sigma_{2}^{-}+\sigma_{2}^{+}\sigma_{1}^{-}\right)+\frac{B}{2}\left(\sigma_{1}^{z}+\sigma_{2}^{z}\right),$
(101)

and the eigenvalues and eigenvectors associated to the Hamiltonian are easily
obtained as

$H\left|00\right\rangle=B\left|00\right\rangle,\hskip
28.45274ptH\left|11\right\rangle=-B\left|11\right\rangle,\hskip 28.45274pt{\rm
and}\hskip
28.45274ptH\left|\psi^{\pm}\right\rangle=B\left|\psi^{\pm}\right\rangle,$
(102)

where
$\left|\psi^{\pm}\right\rangle=\left(\left|00\right\rangle\pm\left|11\right\rangle\right)/\sqrt{2}$
are the Bell maximally entangled states. For a system in equilibrium at
temperature $T$, the state is described by density operator
$\rho\left(T\right)=\exp\left(-\beta H\right)/Z$, where $Z={\rm
Tr}\left[\exp\left(-\beta H\right)\right]$ with $\beta=1/k_{B}t$ and $k_{B}$
is the Boltzmann’s constant. In the standard basis
$\left\\{\left|00\right\rangle,\left|01\right\rangle,\left|10\right\rangle,\left|11\right\rangle\right\\}$,
the resulting density matrix of the system can be written as

$\rho\left(T\right)=\frac{1}{2\left(\cosh\left(\beta B\right)+\cosh\left(\beta
J\right)\right)}\left({\begin{array}[]{*{20}{c}}\exp\left(-\beta
B\right)&0&0&0\\\ 0&\cosh\left(\beta J\right)&-\sinh\left(\beta J\right)&0\\\
0&-\sinh\left(\beta J\right)&\cosh\left(\beta J\right)&0\\\
0&0&0&\exp\left(\beta B\right)\end{array}}\right).$ (103)

To determine the bounds on simultaneous estimation of parameters $B$ and $T$,
we need to derive the formula

$\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}=\left({\begin{array}[]{*{20}{c}}\varGamma_{11}&0_{4\times
4}&0_{4\times 4}&0_{4\times 4}\\\ 0_{4\times
4}&\varGamma_{22}&\varGamma_{23}&0_{4\times 4}\\\ 0_{4\times
4}&\varGamma_{32}&\varGamma_{33}&0_{4\times 4}\\\ 0_{4\times 4}&0_{4\times
4}&0_{4\times 4}&\varGamma_{44}\end{array}}\right),$ (104)

with the quantities $\varGamma_{ij}$ ($i,j=1,2,3,4$) are

$\varGamma_{11}=\left({\begin{array}[]{*{20}{c}}\alpha&0&0&0\\\
0&\xi&\delta&0\\\ 0&\delta&\xi&0\\\ 0&0&0&\lambda\end{array}}\right),\hskip
56.9055pt\varGamma_{22}=\varGamma_{33}=\left({\begin{array}[]{*{20}{c}}\xi&0&0&0\\\
0&\kappa&\eta&0\\\ 0&\eta&\kappa&0\\\ 0&0&0&\vartheta\end{array}}\right),$
(105)

and

$\varGamma_{44}=\left({\begin{array}[]{*{20}{c}}\lambda&0&0&0\\\
0&\vartheta&\tau&0\\\ 0&\tau&\vartheta&0\\\
0&0&0&\alpha\end{array}}\right),\hskip
56.9055pt\varGamma_{23}=\varGamma_{32}=\left({\begin{array}[]{*{20}{c}}\mu&0&0&0\\\
0&\eta&\varpi&0\\\ 0&\varpi&\eta&0\\\ 0&0&0&\tau\end{array}}\right),$ (106)

with the entries given by

$\alpha={e^{\beta B}}\left({\cosh\left(\beta B\right)+\cosh\left(\beta
J\right)}\right),\hskip 42.67912pt\xi=1+{e^{\beta B}}\cosh\left(\beta
J\right),$ (107) $\delta={e^{\beta B}}\sinh\left(\beta J\right),\hskip
28.45274pt\lambda=1+\frac{{\cosh\left(\beta J\right)}}{{\cosh\left(\beta
B\right)}},\hskip 28.45274pt\tau=\left({\cosh\left(\beta
B\right)-\sinh\left(\beta B\right)}\right)\sinh\left(\beta J\right),$ (108)
$\kappa=\frac{1}{4}\left({\cosh\left(\beta B\right)+\cosh\left(\beta
J\right)}\right)\left({3+\cosh\left(2\beta J\right)}\right){\mathop{\rm
sech}\nolimits}\left(\beta J\right),$ (109)
$\eta=\frac{1}{2}\left({\cosh\left(\beta B\right)+\cosh\left(\beta
J\right)}\right){\mathop{\rm\sinh}\nolimits}\left(\beta J\right),\hskip
42.67912pt\vartheta=1+{e^{-\beta B}}\cosh\left(\beta J\right),$ (110)
$\mu={e^{\beta B}}\sinh\left(\beta J\right),\hskip
42.67912pt\varpi=\frac{1}{2}\left({\cosh\left(\beta B\right)+\cosh\left(\beta
J\right)}\right)\tanh\left(\beta J\right)\sinh\left(\beta J\right).$ (111)

In the estimated parameter space basis
$\left\\{\left|B\right\rangle,\left|T\right\rangle\right\\}$, the quantum
Fisher information matrix via the vectorization method is established as
follows

$\displaystyle\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}=\left[{\begin{array}[]{*{20}{c}}{2{\rm
vec}{{\left[{{\partial_{B}}\rho}\right]}^{T}}{\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}}{\rm
vec}\left[{{\partial_{B}}\rho}\right]}&{2{\rm
vec}{{\left[{{\partial_{B}}\rho}\right]}^{T}}{\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}}{\rm
vec}\left[{{\partial_{T}}\rho}\right]}\\\ {2{\rm
vec}{{\left[{{\partial_{T}}\rho}\right]}^{T}}{\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}}{\rm
vec}\left[{{\partial_{B}}\rho}\right]}&{2{\rm
vec}{{\left[{{\partial_{T}}\rho}\right]}^{T}}{\left(\rho\otimes\openone+\openone\otimes\rho^{*}\right)^{-1}}{\rm
vec}\left[{{\partial_{T}}\rho}\right]}\end{array}}\right].$ (114)

By a straightforward calculation, one can check that

$\displaystyle{\rm
vec}\left[{{\partial_{T}}\rho}\right]={\left({{\partial_{T}}\chi,0,0,0,0,{\partial_{T}}\gamma,{\partial_{T}}\varsigma,0,0,{\partial_{T}}\varsigma,{\partial_{T}}\gamma,0,0,0,0,{\partial_{T}}\Pi}\right)^{T}},$
(115)

and

$\displaystyle{\rm
vec}\left[{{\partial_{B}}\rho}\right]={\left({{\partial_{B}}\chi,0,0,0,0,{\partial_{B}}\gamma,{\partial_{B}}\varsigma,0,0,{\partial_{B}}\varsigma,{\partial_{B}}\gamma,0,0,0,0,{\partial_{B}}\Pi}\right)^{T}},$
(116)

where the quantities $\chi$, $\Pi$, $\zeta$ and $\gamma$ in the above
equations are given by

$\chi=\frac{{{e^{-\beta B}}}}{{2\left({\cosh\left(\beta
B\right)+\cosh\left(\beta J\right)}\right)}},\hskip
42.67912pt\Pi=\frac{{{e^{\beta B}}}}{{2\left({\cosh\left(\beta
B\right)+\cosh\left(\beta J\right)}\right)}},$ (117)
$\zeta=\frac{{-\sinh\left(\beta J\right)}}{{2\left({\cosh\left(\beta
B\right)+\cosh\left(\beta J\right)}\right)}},\hskip
42.67912pt\gamma=\frac{{\cosh\left(\beta J\right)}}{{2\left({\cosh\left(\beta
B\right)+\cosh\left(\beta J\right)}\right)}}.$ (118)

Insertion of the expressions given in Eq.(115) and Eq.(116) into Eq.(9) yields
the inverse of the quantum Fisher information matrix

$\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}^{-1}=\frac{1}{{\det\left(\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}\right)}}\left[{\begin{array}[]{*{20}{c}}\boldsymbol{{\cal
F}}_{TT}&{-{\boldsymbol{{\cal F}}_{BT}}}\\\ {-{\boldsymbol{{\cal
F}}_{BT}}}&{{\boldsymbol{{\cal F}}_{BB}}}\end{array}}\right],$ (119)

where their entries coefficients are given by

$\boldsymbol{{\cal F}}_{TT}=\frac{{{{\rm{e}}^{-2\beta
B}}\left({1+{{\rm{e}}^{2\beta B}}+2{{\rm{e}}^{\beta B}}\cosh\left(\beta
J\right)}\right)}}{{4{T^{4}}{{\left({\cosh\left(\beta
B\right)+\cosh\left(\beta
J\right)}\right)}^{3}}}}\left(\begin{array}[]{l}\left({1+{{\rm{e}}^{2\beta
B}}}\right)\left({{B^{2}}+{J^{2}}}\right)\cosh\left(\beta J\right)+\\\
2\left({{{\rm{e}}^{\beta
B}}\left({{B^{2}}+{J^{2}}}\right)-B\left({-1+{{\rm{e}}^{2\beta
B}}}\right)J\sinh\left(\beta J\right)}\right)\end{array}\right),$ (120)
$\boldsymbol{{\cal F}}_{BB}=\frac{{2{{\rm{e}}^{\beta
B}}\left({2{{\rm{e}}^{\beta B}}+\left({1+{{\rm{e}}^{2\beta
B}}}\right)\cosh\left(\beta
J\right)}\right)}}{{{T^{2}}{{\left({1+{{\rm{e}}^{2\beta B}}+2{{\rm{e}}^{\beta
B}}\cosh\left(\beta J\right)}\right)}^{2}}}},$ (121)

and

$\boldsymbol{{\cal F}}_{BT}={F_{TB}}=\frac{{2{{\rm{e}}^{\beta
B}}\left({2B{{\rm{e}}^{\beta B}}+B\left({1+{{\rm{e}}^{2\beta
B}}}\right)\cosh\left(\beta J\right)-\left({-1+{{\rm{e}}^{2\beta
B}}}\right)J\sinh\left(\beta
J\right)}\right)}}{{{T^{3}}{{\left({1+{{\rm{e}}^{2\beta B}}+2{{\rm{e}}^{\beta
B}}\cosh\left(\beta J\right)}\right)}^{2}}}}.$ (122)

In any given quantum multiparameter metrology strategy, the quantum Cramer-Rao
bound provided us with the lower limit of the covariance matrix of estimators
$\hat{\boldsymbol{\theta}}=\left(B,T\right)$. This limit reads as follows

${\rm
Cov}\left(\hat{\boldsymbol{\theta}}\right)\equiv\left[{\begin{array}[]{*{20}{c}}{\rm
Var}\left(B\right)&{\rm Cov}\left(B,T\right)\\\ {\rm Cov}\left(T,B\right)&{\rm
Var}\left(T\right)\end{array}}\right]\geq\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}^{-1}.$
(123)

By comparing the coefficients on both sides of the quantum Cramer-Rao
inequality, we then get

$\displaystyle{\rm Var}\left(B\right)\geq\frac{{{\boldsymbol{{\cal
F}}_{TT}}}}{{\det\left(\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}\right)}},\hskip
56.9055pt{\rm and}\hskip 56.9055pt{\rm
Var}\left(T\right)\geq\frac{{{\boldsymbol{{\cal
F}}_{BB}}}}{{\det\left(\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}\right)}}.$
(124)

As previously shown, the saturation of these above inequalities provides us
the highest accuracy on the estimation of the parameters $B$ and $T$.
Therefore, the optimal analytical expressions of the precision are coincident
with the minimum variances of the estimated parameters. Under these
conditions, ${\mathop{\rm Var}}\left(B\right)_{\min}$ for the external
magnetic field parameter and ${\mathop{\rm Var}}\left(T\right)_{\min}$ for the
temperature parameter yield the following results

${\mathop{\rm Var}}\left(T\right)_{\min}=\frac{{{{\rm{e}}^{-\beta
B}}{T^{4}}\left({2{{\rm{e}}^{\beta B}}+\left({1+{{\rm{e}}^{2\beta
B}}}\right)\cosh\left(\beta J\right)}\right)}}{{2{J^{2}}}}.$ (125)

and

${\rm{Var}}{\left(B\right)_{\min}}=\frac{{{{\rm{e}}^{-4\beta
B}}{T^{2}}{{\left(1+{{\rm{e}}^{2\beta B}}+2{{\rm{e}}^{\beta
B}}\cosh\left(\beta
J\right)\right)}^{3}}}}{{16{J^{2}}{{\left({\cosh\left(\beta
B\right)+\cosh\left(\beta
J\right)}\right)}^{3}}}}\left(\begin{array}[]{l}\left({1+{{\rm{e}}^{2\beta
B}}}\right)\left({{B^{2}}+{J^{2}}}\right)\cosh\left(\beta J\right)+\\\
2\left({{{\rm{e}}^{\beta
B}}\left({{B^{2}}+{J^{2}}}\right)-B\left({-1+{{\rm{e}}^{2\beta
B}}}\right)J\sinh\left(\beta J\right)}\right)\end{array}\right).$ (126)

Straightforward calculations leads to find also the analytical expressions of
the symmetric logarithmic derivatives corresponding respectively to the
parameters $T$ and $B$ as

${L_{T}}=2\left[{\begin{array}[]{*{20}{c}}{\alpha\,{\partial_{T}}\,\chi}&0&0&0\\\
0&{(\kappa+\varpi){\mkern 1.0mu}{\partial_{T}}\,\gamma+2\eta{\mkern
1.0mu}{\partial_{T}}\zeta}&{(\kappa+\varpi){\mkern
1.0mu}{\partial_{T}}\varsigma+2\eta{\mkern 1.0mu}{\partial_{T}}\gamma}&0\\\
0&{(\kappa+\varpi){\mkern 1.0mu}{\partial_{T}}\varsigma+2\eta{\mkern
1.0mu}{\partial_{T}}\gamma}&{(\kappa+\varpi){\mkern
1.0mu}{\partial_{T}}\gamma+2\eta{\mkern 1.0mu}{\partial_{T}}\varsigma}&0\\\
0&0&0&{\alpha{\mkern 1.0mu}{\partial_{T}}\Pi}\end{array}}\right],$ (127)

and

${L_{B}}=2\left[{\begin{array}[]{*{20}{c}}{\alpha\,{\partial_{B}}\,\chi}&0&0&0\\\
0&{(\kappa+\varpi){\mkern 1.0mu}{\partial_{B}}\,\gamma+2\eta{\mkern
1.0mu}{\partial_{B}}\varsigma}&{(\kappa+\varpi){\mkern
1.0mu}{\partial_{B}}\varsigma+2\eta{\mkern 1.0mu}{\partial_{B}}\gamma}&0\\\
0&{(\kappa+\varpi){\mkern 1.0mu}{\partial_{B}}\varsigma+2\eta{\mkern
1.0mu}{\partial_{B}}\gamma}&{(\kappa+\varpi){\mkern
1.0mu}{\partial_{B}}\gamma+2\eta{\mkern 1.0mu}{\partial_{B}}\varsigma}&0\\\
0&0&0&{\alpha{\mkern 1.0mu}{\partial_{B}}\Pi}\end{array}}\right].$ (128)

Generally, the multi-parameter estimation protocol is harder than the single-
parameter estimation protocol since we cannot always achieve the optimal
estimation of multiple parameters simultaneously encrypted in a given quantum
system. This is mainly due to the incompatibility of quantum measurements
[82]. Consequently, the quantum limits of accuracy cannot usually be achieved.
To saturate the quantum Cramér-Rao bound in multiparameter estimation
problems, we need to know the influence of this incompatibility on our
estimation problem. This is efficiently done by using a recently introduced
quantity named quantumness [83, 84]. It is a scalar quantity that evaluates
the asymptotic incompatibility of a simultaneous estimate of several
parameters and is expressed as follows

${\cal R}_{\boldsymbol{\theta}}:=\parallel
2i\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}^{-1}U_{\boldsymbol{\theta}}\parallel_{\infty},$
(129)

where $\parallel Z\parallel_{\infty}$ stands for the largest eigenvalue of a
matrix $Z$ and $U_{\boldsymbol{\theta}}$ is the mean Uhlmann curvature matrix
defined by

$U_{\theta_{\mu}\theta_{\nu}}=-\frac{i}{4}{\rm
Tr}\left(\rho_{\theta}\left[\hat{L}_{\theta_{\mu}},\hat{L}_{\theta_{\nu}}\right]\right).$
(130)

The values of the quantumness ${\cal R}_{\boldsymbol{\theta}}$ are included in
the range $0\leq{\cal R}_{\boldsymbol{\theta}}\leq 1$. The Cramer-Rao
inequality (9) is not saturated when the incompatibility between the
measurements of the simultaneously estimated parameters is maximal, which
corresponds to the quantumness maximal with ${\cal
R}_{\boldsymbol{\theta}}=1$. Contrary, the condition ${\cal
R}_{\boldsymbol{\theta}}=0$ ensures the saturation of Cramér-Rao bound and the
parameterization model is asymptotically compatible. Furthermore, we get

$U_{\theta_{\mu}\theta_{\nu}}=0\hskip 14.22636pt\Leftrightarrow\hskip
14.22636pt{\cal R}_{\boldsymbol{\theta}}=0,$ (131)

and in the special case of a two parameter simultaneous estimation, equation
(1129) becomes

${\cal R}_{\boldsymbol{\theta}}=\sqrt{\frac{{\rm
Det\left(2U_{\theta_{\mu}\theta_{\nu}}\right)}}{{\rm
Det\left(\mathbb{F}_{\boldsymbol{\boldsymbol{\theta}}}\left(\theta_{\mu},\theta_{\nu}\right)\right)}}}.$
(132)

##  VI Interferometric phase estimation and non-classical correlations

Very recently, a major effort has been undertaken to evaluate the dynamics of
quantum Fisher information to establish the validity of quantum entanglement
in various quantum metrology scenarios and models. In unitary processes, it
has been shown that entanglement leads to a significant improvement in the
accuracy of parameter estimation. In fact, the quantum entanglement can be
exploited as a quantum resource to exceed the standard quantum limit and
achieve the Heisenberg limit [16]. Furthermore, the gain of information that
becomes noise-resistant is essential to promising entanglement in quantum
estimation theory [85, 86]. This fact raises an important question; Can
quantum correlations beyond entanglement be related to the estimation
precision in quantum metrology protocols? Is it possible to quantify the
quantum correlations in terms of quantum Fisher information? Several studies
have recently been conducted in this direction. Indeed, a quantum correlation
quantifier called the local quantum Fisher information has been recently
introduced to understand the role of quantum correlation beyond entanglement
in a black-box quantum metrology task [39]. This quantifier is related
directly to the quantum Fisher information and also provides the accuracy of
an interferometric phase estimate. Local quantum uncertainty [60], as another
quantum correlation measure, can also be related to the precision in phase
estimation protocols. This supports the idea that quantum Fisher information
is adding a new tool to the existing list of quantum correlation quantifiers
in bipartite quantum systems.

Figure 2: Conceptual scheme of a complete quantum phase estimation
metrological process. The various parameters are defined in the main text.

A complete quantum metrology process generally includes four steps (Fig.2);
First, the preparation of the probe state. Second, the estimated parameter
parameterization via unitary evolution. Third, a measurement on the final
state via POVM measurements and fourth step consists of data analysis via
classical estimation. This last step has been well studied in classical
statistics, therefore, the main concern of quantum estimation is the first
three steps. Here, we focus on the first step in which we examine how non-
classical correlations in the input state improve the accuracy of the
estimated parameters in the interferometric phase estimation protocol.

###  VI.1 Local quantum Fisher information as a measure of non-classical
correlations

A major challenge for information theorists is to find a new measure of
quantum correlations via quantum Fisher information. Actually, local quantum
Fisher information has been introduced to treat discord-type quantum
correlations in interferometric phase estimation [39]. This quantifier allows
us to better understand how non-classical correlations contribute to
establishing accuracy in quantum estimation theory. It meets the criteria
identifying discord type quantifiers that any good quantum correlation
quantifier should satisfy. Indeed, it is non-negative and vanishes only for
the classically correlated states. Also, it is invariant under any local
unitary operation and coincides with the geometric quantum discord for any
pure quantum states.

Consider first a bipartite quantum state $\rho^{AB}$ of dimension $d_{A}\times
d_{B}$ acting on a Hilbert space $H_{A}\otimes H_{B}$, and let
$K_{A}=K_{A}\otimes\openone_{B}$ be a local observable on a single part $A$.
The dynamics of the density matrix $\rho^{AB}$ is governed by the local phase
shift transformation $e^{-i\theta K_{A}}$, i.e.,
$\rho_{\theta}^{AB}=e^{-i\theta K_{A}}\rho^{AB}e^{i\theta K_{A}}$. Quantum
Fisher information via local measurements and via local observables, ${\cal
F}\left(\rho^{AB},K_{A}\right)$, is called local quantum Fisher information on
the sub-system $A$. When ${\cal F}\left(\rho^{AB},K_{A}\right)=0$, then no
information can be obtained by measurement because no change in the evolution
will be observed. Building on this, the local quantum Fisher information
captures the amount of quantum correlations that exist in a given quantum
state and allows to quantify the non-classical correlations in terms of
quantum Fisher information [39]. It is defined as the minimum quantum Fisher
information over all local observables with respect to the subsystem $A$ and
given by

${\cal Q}_{F}\left(\rho^{AB}\right)=\min_{K_{A}}{\cal
F}\left(\rho^{AB},K_{A}\right).$ (133)

The advantage of using this measure compared to other quantum discord
quantifiers relies on the possibility to obtain the closed analytical form.
For $2\times d$ quantum systems with ${\rm dim}\left({\cal H}_{A}\right)=2$
and ${\rm dim}\left({\cal H}_{B}\right)=d$, it is simple to show that the
local quantum Fisher information (133) reduces to

${\cal Q}_{F}\left(\rho^{AB}\right)=\min_{K_{A}}\left[{\rm Tr}\left(\rho
K_{A}^{2}\right)-\sum_{i\neq
j}\frac{2p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}\right|K_{A}\left|\psi_{j}\right\rangle|^{2}\right],$
(134)

where we used the spectral decomposition of the density matrix $\rho^{AB}$,
i.e.,
$\rho^{AB}=\sum_{i}p_{i}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|$
with $p_{i}\geq 0$ and $\sum_{i}p_{i}=1$.

Basically, any qubit can be represented geometrically in the Bloch sphere and
the general form of a local Hamiltonian can be described as
$K_{A}=\vec{\sigma}\vec{r}$, with $|\vec{r}|=1$ and
$\vec{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ are the usual
Pauli matrices (the generators of the $su(2)$ algebra). Therefore, we can show
that ${\rm Tr}\left(\rho K_{A}^{2}\right)=1$ and the second part of the
equation (134) is expressed as below

$\displaystyle\sum_{i\neq
j}\frac{2p_{i}p_{j}}{p_{i}+p_{j}}|\left\langle\psi_{i}\right|K_{A}\left|\psi_{j}\right\rangle|^{2}$
$\displaystyle=\sum_{i\neq
j}\sum_{l,k=1}^{3}\frac{2p_{i}p_{j}}{p_{i}+p_{j}}\left\langle\psi_{i}\right|\sigma_{l}\otimes\openone_{B}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\sigma_{k}\otimes\openone_{B}\left|\psi_{i}\right\rangle$
$\displaystyle=\vec{r}^{\dagger}.M.\vec{r},$ (135)

where the components of the $3\times 3$ symmetric matrix $M$ are determined by

$M_{lk}=\sum_{i\neq
j}\frac{2p_{i}p_{j}}{p_{i}+p_{j}}\left\langle\psi_{i}\right|\sigma_{l}\otimes\openone_{B}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\sigma_{k}\otimes\openone_{B}\left|\psi_{i}\right\rangle.$
(136)

The minimization of ${\cal F}\left(\rho^{AB},K_{A}\right)$ requires maximizing
the quantity $\vec{r}^{\dagger}.M.\vec{r}$ on all unit vectors $\vec{r}$.
Here, its maximum value corresponds to the maximum eigenvalue of $M$.
Therefore, the analytical expression of the local quantum Fisher information
${\cal Q}_{F}\left(\rho^{AB}\right)$, which quantifying the pairwise quantum
correlation in the given quantum bipartite state, yields as

${\cal Q}_{F}\left(\rho^{AB}\right)=1-\lambda_{\rm max}\left(M\right),$ (137)

with $\lambda_{\rm max}$ denotes the maximal eigenvalue of the symmetric
matrix $M$ whose entries are given in equation (136)

###  VI.2 Local quantum uncertainty as a discord-like quantity

The physicists study nature by making measurements and predicting their
results. In a classical world, the error bands are exclusively due to
technological limitations, so it is possible to measure any two observables
with arbitrary precision. However, such a type of measurement is not always
possible in quantum systems since quantum mechanics stipulates that two non-
commutable observables cannot be measured jointly with arbitrary precision
even if one could access a flawless measuring device. Moreover, the
measurement of a quantum system is completely different from that of a
classical system, because not only the result obtained is probabilistic, but
also the state of the quantum system is modified during this process, and the
objective is to determine, in a more precise way, the result of this
measurement. The idea of accuracy is directly related to the uncertainty
associated with the result of this measurement and the uncertainty relation
gives the statistical nature of the errors in this kind of measurement.

In standard practice and in the framework of the quantum mechanical formalism,
the uncertainty of an observable $K$ in a quantum state $\rho$ is generally
quantified by the variance which is given by the following relation

${\rm Var}\left(\rho,K\right):={\rm Tr}\left(\rho K_{0}^{2}\right)={\rm
Tr}\left(\rho K^{2}\right)-{\rm Tr}\left(\rho K\right)^{2},$ (138)

where ${\rm Tr}$ denotes the trace and $K_{0}=K-{\rm Tr}\rho K$. For the pure
states $\rho=\left|\psi\right\rangle\left\langle\psi\right|$, the variance
reduces to

${\rm
Var}\left(\left|\psi\right\rangle\left\langle\psi\right|,K\right)=\left\langle\psi\right|K^{2}\left|\psi\right\rangle-\left\langle\psi\right|K\left|\psi\right\rangle^{2}.$
(139)

Indeed, the quantification of uncertainty in terms of variance is well adapted
when the states are pure. However, this uncertainty includes classical
ignorance and a part of quantum nature for mixed states. The mixture of states
is responsible for the classical part and the quantum part resulting from the
non-commutativity between the studied state and the measured observable. Thus,
the variance can have contributions from both quantum and classical sources,
i.e., ${\rm Var}\left(\rho\right)\equiv{\rm Var}_{q}\left(\rho\right)+{\rm
Var}_{c}\left(\rho\right)$. There are still several methods to quantify the
uncertainty, such as those based on entropy measurements which are widely used
as indicators of uncertainty in quantum information theory. This type of
measurement is still insufficient and limited because these quantifiers are
always influenced by the mixed state of the studied state. In order to
determine the quantum part of the variance (138), Wigner and Yanase introduced
the notion of skew information as [87, 88]

${\cal I}\left({\rho,K}\right):={{\cal
I}_{\rho}}\left(K\right):=\frac{1}{2}{\rm
Tr}\left\\{{{{\left({i\left[{\sqrt{\rho},K}\right]}\right)}^{2}}}\right\\},$
(140)

where $\left[.,.\right]$ stands for the commutator. Importantly, unlike the
variance, the Wigner-Yanase skew information is not affected by classical
mixing. The latter quantifies the degree of non-commutativity between a
quantum state $\rho$ and an observable $K$ that can be considered as a
Hamiltonian or any other conserved quantity. Furthermore, the skew information
provides an alternative measure of the information content of the studied
state with respect to non-commutative observable with the conserved quantity
$K$. In other words, it represents the amount of information contained in the
quantum state $\rho$ that is not accessible by the measurement of observable
$K$. The equation (140) can be simplified as

$\displaystyle{\cal I}\left({\rho,K}\right)$ $\displaystyle=-\frac{1}{2}{\rm
Tr}\\{\left(\sqrt{\rho}K-K\sqrt{\rho}\right)^{2}\\}$
$\displaystyle=-\frac{1}{2}\left\\{{{\rm
Tr}\left({\sqrt{\rho}K\sqrt{\rho}K}\right)-{\rm Tr}\left({K\rho K}\right)-{\rm
Tr}\left({\sqrt{\rho}{K^{2}}\sqrt{\rho}}\right)+{\rm
Tr}\left({K\sqrt{\rho}K\sqrt{\rho}}\right)}\right\\}$ $\displaystyle={\rm
Tr}\left({\rho{K^{2}}}\right)-{\rm
Tr}\left({\sqrt{\rho}K\sqrt{\rho}K}\right).$ (141)

Very recently, Girolami and co-workers defined the local quantum uncertainty
as the minimum skew information that can be obtained by a single local
measurement [60]. More precisely, for a bipartite quantum state $\rho_{AB}$,
we consider the local observable
$K^{\Lambda}=K_{A}^{\Lambda}\otimes\openone_{B}$ such that $K_{A}^{\Lambda}$
is a Hermitian operator acting on states of the $A$ subsystem and having a
non-degenerate spectrum $\Lambda$. The local quantum uncertainty with respect
to the subsystem $A$ is given by

${\cal U}_{A}^{\Lambda}\left(\rho_{AB}\right):=\min_{K_{A}^{\Lambda}}{\cal
I}\left(\rho_{AB},K_{A}^{\Lambda}\otimes\openone_{B}\right),$ (142)

where the minimization involves all the local observables that act on the
subsystem $A$. Indeed, $\Lambda$ here is non-degenerate because it corresponds
to the maximal informational observables of subsystem $A$. This quantifier has
strong reasons to be considered as a faithful measure of the quantum
correlations of bipartite quantum states. Indeed, the local quantum
uncertainty is invariant under local unitary transformations, is not
increasing under local operations on the subsystem $B$, vanish if and only if
the quantum state is classically correlated [60]. For pure states, the local
quantum uncertainty is a monotonic entanglement. As a result, the local
quantum uncertainty satisfies all the physical requirements for a quantum
correlation measure.

To evaluate the minimum in equation (142), there is an important requirement
for the optimization observables that minimize the skew information, namely
that they must have a fixed non-degenerate spectrum. In practice, the
observables with this spectrum acting on the subsystem $A$ can be
parameterized by

$K_{A}^{\Lambda}=V_{A}\Lambda V_{A}^{\dagger},$ (143)

where $V_{A}$ defines the measurement basis that can be arbitrarily modified
on the special unit group of the subsystem $A$. If we restrict our discussion
to the case of bipartite quantum systems of qubit-qudit type where the
subsystem $A$ has dimension $2$ and the subsystem $B$ has dimension $d$, the
non-degenerate fixed-spectrum qubit observable $\sigma_{3}$ can be
parametrized by $K_{A}=V_{A}\sigma_{3}V_{A}^{\dagger}=\vec{s}.\vec{\sigma}$,
where $\Arrowvert\vec{s}\Arrowvert=1$ and $\sigma_{i}$ are the Pauli matrices,
also representing the generators of the $su\left(2\right)$ algebra. By
selecting this spectrum, the local quantum uncertainty is normalized to unity
for maximally entangled pure states. Then, the $\Lambda$ spectrum can be
removed in the equation (142) and the local quantum uncertainty is equivalent
to

$\displaystyle{\cal U}_{A}\left(\rho_{AB}\right)$
$\displaystyle=\min_{\vec{s}}{\cal
I}\left(\rho_{AB},\vec{s}.\vec{\sigma}\otimes\openone_{d}\right)$
$\displaystyle=\min_{\vec{s}}\sum_{ij}s_{i}s_{j}\left[{\rm
Tr}\\{\rho_{AB}\sigma_{i}\sigma_{j}-\sqrt{\rho_{AB}}\left(\sigma_{i}\otimes\openone_{d}\right)\sqrt{\rho_{AB}}\left(\sigma_{j}\otimes\openone_{d}\right)\\}\right]$
$\displaystyle=1-\max\sum_{ij}s_{i}s_{j}\left[{\rm
Tr}\\{\sqrt{\rho_{AB}}\left(\sigma_{i}\otimes\openone_{d}\right)\sqrt{\rho_{AB}}\left(\sigma_{j}\otimes\openone_{d}\right)\\}\right]$
$\displaystyle=1-\max\left[\vec{s}^{\dagger}{\cal W}\vec{s}\right],$ (144)

where ${\cal W}$ is a $3\times 3$ symmetric matrix whose elements are

$w_{ij}={\rm
Tr}\\{\sqrt{\rho_{AB}}\left(\sigma_{i}\otimes\openone_{B}\right)\sqrt{\rho_{AB}}\left(\sigma_{j}\otimes\openone_{B}\right)\\},$
(145)

with $i,j=1,2,3$. As we have already seen in the minimization procedure of the
local quantum Fisher information, it is necessary to maximize the quantity
${\cal W}$ on all unit vectors $\vec{s}$ to minimize the skew information. Its
value coincides with the maximal eigenvalue of the matrix ${\cal W}$.
Consequently, a compact formula for the local quantum uncertainty is given by

${\cal U}_{A}\left(\rho_{AB}\right)=1-\max\\{\xi_{1},\xi_{2},\xi_{3}\\},$
(146)

where $\xi_{i}$ represent the eigenvalues of the matrix ${\cal
W}=\left(w_{ij}\right)_{3\times 3}$. Hence, it is clear that the calculation
of the local quantum uncertainty requires the calculation of the elements
$w_{ij}$ of the matrix ${\cal W}$. Thus, this measure has the advantage of
being simple when it comes to calculating the maximum on the parameters
related to the measurements as it must be done to calculate the quantum
discord. Moreover, the local quantum uncertainty is reduced to the monotonic
entanglement for pure states. In fact, it reduces to the linear entropy of the
measured reduced subsystem (i.e., concurrence). This interesting coincidence
occurs only for quantum systems of dimension $2\otimes d$. However, for
multipartite quantum embedded systems (for a large Hilbert space dimension),
it is difficult to obtain a compact form of the local quantum uncertainty
without complex optimization. To clarify this issue, we consider a pure state
$\left|\psi\right\rangle=\sum_{mn}C_{mn}\left|mn\right\rangle$, the elements
of the matrix $\cal W$ become

$w_{ij}=\left\langle\psi\right|\sigma_{i}\otimes\openone_{d}\left|\psi\right\rangle\left\langle\psi\right|\sigma_{j}\otimes\openone_{d}\left|\psi\right\rangle.$
(147)

The first term of the above equation becomes as follows

$\displaystyle\left\langle\psi\right|\sigma_{i}\otimes\openone_{d}\left|\psi\right\rangle=\sum_{mk}\left(C{C}^{\dagger}\right)_{km}\left\langle
m\right|\sigma_{i}\left|k\right\rangle.$ (148)

The reduced density matrix of subsystem $A$ can be rewritten in terms of the
coefficients $C_{mn}$ as

$\displaystyle\rho_{A}=\sum_{mkl}C_{ml}C_{lk}^{\dagger}\left|m\right\rangle\left\langle
k\right|=\sum_{mkl}\left(CC^{\dagger}\right)_{mk}\left|m\right\rangle\left\langle
k\right|.$ (149)

We then obtain

$\left\langle
m\right|\rho_{A}\left|k\right\rangle=\left(CC^{\dagger}\right)_{mk}.$ (150)

We return to the expression (148) and using the equation (150), we can easily
see that it reduces to the average value of $\sigma_{i}$ with respect to the
base of the subsystem $A$. So we have

$\displaystyle\left\langle\psi\right|\sigma_{i}\otimes\openone_{d}\left|\psi\right\rangle=\sum_{k}\left\langle
k\right|\rho_{A}\sigma_{i}\left|k\right\rangle={\rm
Tr}_{A}\left(\rho_{A}\sigma_{i}\right)=\left\langle\sigma_{i}\right\rangle_{A}.$
(151)

This leads to
$w_{ij}=\left\langle\sigma_{i}\right\rangle_{A}\left\langle\sigma_{j}\right\rangle_{A}$,
and the matrix ${\cal W}$ can be written as

${\cal
W}=\left({\begin{array}[]{*{20}{c}}{\left\langle\sigma_{1}\right\rangle_{A}^{2}}&{\left\langle\sigma_{1}\right\rangle_{A}\left\langle\sigma_{2}\right\rangle_{A}}&{\left\langle\sigma_{1}\right\rangle_{A}\left\langle\sigma_{3}\right\rangle_{A}}\\\
{\left\langle\sigma_{2}\right\rangle_{A}\left\langle\sigma_{1}\right\rangle_{A}}&{\left\langle\sigma_{2}\right\rangle_{A}^{2}}&{\left\langle\sigma_{2}\right\rangle_{A}\left\langle\sigma_{3}\right\rangle_{A}}\\\
{\left\langle\sigma_{3}\right\rangle_{A}\left\langle\sigma_{1}\right\rangle_{A}}&{\left\langle\sigma_{2}\right\rangle_{A}\left\langle\sigma_{3}\right\rangle_{A}}&{\left\langle\sigma_{3}\right\rangle_{A}^{2}}\end{array}}\right).$
(152)

By diagonalizing the matrix ${\cal W}$, we find a single non-zero eigenvalue
which is written

$\xi_{\max}=\left\langle\sigma_{1}\right\rangle_{A}^{2}+\left\langle\sigma_{2}\right\rangle_{A}^{2}+\left\langle\sigma_{3}\right\rangle_{A}^{2},$
(153)

and the local quantum uncertainty for pure states becomes

${\cal
U}_{A}\left(\left|\psi\right\rangle\left\langle\psi\right|\right)=1-\left[\left\langle\sigma_{1}\right\rangle_{A}^{2}+\left\langle\sigma_{2}\right\rangle_{A}^{2}+\left\langle\sigma_{3}\right\rangle_{A}^{2}\right].$
(154)

Besides, the Fano-Bloch representation of the matrix (149) is written
$\rho_{A}=\frac{1}{2}\left(1+\vec{r}\vec{\sigma_{i}}\right)$, where $\vec{r}$
denotes the Bloch vector with

$\left\langle\sigma_{i}\right\rangle_{A}={\rm
Tr}\left(\rho_{A}\sigma_{i}\right)=\frac{1}{2}\left({\rm
Tr}\sigma_{i}+r_{j}{\rm Tr}\left(\sigma_{i}\sigma_{j}\right)\right)=r_{i}.$
(155)

Similarly, we also have

${\rm
Tr}\left(\rho_{A}^{2}\right)=\frac{1}{2}\left(1+r_{i}^{2}\right)=\frac{1}{2}\left(1+\xi_{\max}\right).$
(156)

Therefore, for pure states
$\rho_{AB}=\left|\psi\right\rangle\left\langle\psi\right|$, the local quantum
uncertainty coincides with the linear entropy of entanglement. We have then

${\cal
U}_{A}\left(\left|\psi\right\rangle\left\langle\psi\right|\right)=2\left[1-{\rm
Tr}\left(\rho_{A}^{2}\right)\right]=S_{2}\left(\rho_{A}\right).$ (157)

Interestingly, the analytical expression of the local quantum uncertainty
(144) can be applied to multi-qubit systems without any major difficulty.
Indeed, we can always consider the multi-qubit system (with $N$ qubits) as a
system of dimension $2\otimes d$ where $d=2\otimes...\otimes 2$ can represent
the remaining $N-1$ qubits as a quantum system with a Hilbert space of
dimension $d$. Since there are several bipartitions for a multi-qubit system,
it is possible that some bipartitions are quantum correlated, some classically
correlated and some may be completely uncorrelated with other qubits.
Therefore, we need to define the local quantum uncertainty for each
bipartition. For each case, we obtain a symmetric matrix ${\cal W}_{k}$, so in
total we obtain $N$ symmetric matrices. After calculating all these local
quantum uncertainties, we can calculate the average local quantum uncertainty
for a given multi-qubit state. For this purpose, we consider an arbitrary
density matrix $\rho_{N}$ associated to a system describing $N$ qubits and we
apply local measurements on each qubit. Then, the matrix elements of each
bipartition $\rho_{k}$ (with $k=1,2,...,N$) are given by the following
expressions

$\displaystyle{\hat{w}}_{ij}^{1}={\rm
Tr}\\{\sqrt{\rho_{N}}\left(\sigma_{i}\otimes\openone_{2}...\otimes\openone_{2}\right)\sqrt{\rho_{N}}\left(\sigma_{j}\otimes\openone_{2}...\otimes\openone_{2}\right)\\},$
$\displaystyle{\hat{w}}_{ij}^{2}={\rm
Tr}\\{\sqrt{\rho_{N}}\left(\openone_{2}\otimes\sigma_{i}...\otimes\openone_{2}\right)\sqrt{\rho_{N}}\left(\openone_{2}\otimes\sigma_{j}...\otimes\openone_{2}\right)\\},$
$\displaystyle\hskip 5.69046pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip
28.45274pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip
28.45274pt:$ $\displaystyle\hskip 5.69046pt:\hskip 28.45274pt:\hskip
28.45274pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip
28.45274pt:\hskip 28.45274pt:$ $\displaystyle{\hat{w}}_{ij}^{N}={\rm
Tr}\\{\sqrt{\rho_{N}}\left(\openone_{2}\otimes\openone_{2}...\otimes\sigma_{i}\right)\sqrt{\rho_{N}}\left(\openone_{2}\otimes\openone_{2}...\otimes\sigma_{j}\right)\\}.$
(158)

Then, the local quantum uncertainties associated with each bipartition are
determined as follows

$\displaystyle{\cal
U}_{1/23...N}\left(\rho_{N}\right)=1-\max\\{\xi_{1}^{1},\xi_{2}^{1},\xi_{3}^{1}\\},$
$\displaystyle{\cal
U}_{2/13...N}\left(\rho_{N}\right)=1-\max\\{\xi_{1}^{2},\xi_{2}^{2},\xi_{3}^{2}\\},$
$\displaystyle\hskip 5.69046pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip
28.45274pt:\hskip 28.45274pt:\hskip 28.45274pt:$ $\displaystyle\hskip
5.69046pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip 28.45274pt:\hskip
28.45274pt:\hskip 28.45274pt:$ $\displaystyle{\cal
U}_{N/12...N-1}\left(\rho_{N}\right)=1-\max\\{\xi_{1}^{N},\xi_{2}^{N},\xi_{3}^{N}\\},$
(159)

where $\xi_{i}^{k}$ (with $i=1,2,3$) are the eigenvalues of the symmetric
matrices $\left({\cal W}_{k}\right)_{3\times 3}$ which can be determined
easily. The local quantum uncertainty, which provides us the quantity of the
global quantum correlations in the global state $\rho_{N}$, can be viewed as
the average value of the local quantum uncertainties of each bipartition,
i.e.,

${\cal U}\left(\rho_{N}\right)=\frac{1}{N}\left(\sum_{k=A}^{N}{\cal
U}_{k/N_{k}}\right)\equiv\frac{1}{N}\left(\sum_{k=A}^{N}{\cal
U}_{k/12...\bar{k}...N}\right),$ (160)

with $N_{k}=12...\bar{k}...N$ are the remaining ($N-1$)-qubits excluding the
$k$-qubit.

Admittedly, the quantum correlation quantification in qudit-qudit systems
($d_{1}\otimes d_{2}$) remains an open problem. In this sense, a glimmer of
hope comes from the Ref.[89], in which the authors emphasize that the closed
form of the local quantum uncertainty can be obtained for quantum states of