# On the properties of the asymptotic incompatibility measure in
multiparameter quantum estimation

Alessandro Candeloro1,2, Matteo G.A. Paris1,2, Marco G. Genoni1,2 1 Quantum
Technology Lab, Dipartimento di Fisica Aldo Pontremoli, Università degli Studi
di Milano, I - 20133, Milano, Italy
2 Istituto Nazionale di Fisica Nucleare, Sezione di Milano, I - 20133, Milano,
Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We address the use of asymptotic incompatibility (AI) to assess the
quantumness of a multiparameter quantum statistical model. AI is a recently
introduced measure which quantifies the difference between the Holevo and the
SLD scalar bounds, and can be evaluated using only the symmetric logarithmic
derivative (SLD) operators of the model. At first, we evaluate analytically
the AI of the most general quantum statistical models involving two-level
(qubit) and single-mode Gaussian continuous-variable quantum systems, and
prove that AI is a simple monotonous function of the state purity. Then, we
numerically investigate the same problem for qudits ($d$-dimensional quantum
systems, with $2<d\leq 4$), showing that, while in general AI is not in
general a function of purity, we have enough numerical evidence to conclude
that the maximum amount of AI is attainable only for quantum statistical
models characterized by a purity larger than $\mu_{\sf min}=1/(d-1)$. In
addition, by parametrizing qudit states as thermal (Gibbs) states, numerical
results suggest that, once the spectrum of the Hamiltonian is fixed, the AI
measure is in one-to-one correspondence with the fictitious temperature
parameter $\beta$ characterizing the family of density operators. Finally, by
studying in detail the definition and properties of the AI measure we find
that: i) given a quantum statistical model, one can readily identify the
maximum number of asymptotically compatibile parameters; ii) the AI of a
quantum statistical model bounds from above the AI of any sub-model that can
be defined by fixing one or more of the original unknown parameters (or
functions thereof), leading to possibly useful bounds on the AI of models
involving noisy quantum dynamics.

††:

, ,

Keywords: quantum metrology, multiparameter quantum estimation, quantum
incompatibility.

##  1 Introduction

Quantum metrology is currenty revolutionizing the field of parameter
estimation and sensing by enhancing precision to a level that cannot be
achieved via purely classical means [1, 2, 3, 4]. Most of the paradigmatic
results in quantum metrology have been obtained for the estimation of single
parameters, typically corresponding to a phase or a frequency. In those cases,
the ultimate and in principle achievable bounds are derived in terms of the
quantum Fisher information, which is in turn obtained via the symmetric
logarithmic derivative (SLD) operator [5, 6]. The extension to the
multiparameter case is however not straightforward [7, 8, 9, 10]. In fact,
besides the expected complications due to the fact that one needs to estimate
more than one parameter, the peculiar properties of quantum mechanics make
this extension definitely non-trivial. In particular, the (potential) non-
commutativity of the optimal measurements corresponding to different
parameters poses more fundamental and interesting questions. A bound to
precision of multiparameter estimation may be obtained by generalizing the
single-parameter case, i.e. using the SLD operators corresponding to the
different parameters. On the other hand, several other bounds may be
introduced, the most relevant being the ones based on the right logarithmic
derivative operator [11], on an operator interpolating between right and
symmetric logarithmic derivative [12], and the so-called Holevo bound [13].

In general, all these bounds are not tight, i.e. it is not guaranteed that
there exists a quantum measurement that allows to saturate them. However, the
Holevo bound stands out as the most fundamental (scalar) bound for quantum
multiparameter estimation as it may be saturated by performing collective
measurements on an asymptotically large number of copies of the quantum state
encoding the set of parameters [14, 15, 16] (tighter bounds that apply in the
scenario where one allows measurements on a finite number of copies have been
derived more recently [17, 18]). Moreover, in some cases, e.g. for pure states
[19] and for the estimation of a displacement by Gaussian probes [13], the
Holevo bound can be achieved also in the standard scenario, i.e. by single-
copy measurements. These fundamental aspects have been investigated in several
multiparameter problems that may have practical applications in the quantum
regimes, such as superresolution of incoherent sources [20, 21, 22, 23],
estimation of multiple phases (or in general of unitary parameters) [24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], and estimation of phase and noise
[37, 38, 39, 32, 40, 35].

Recently, the difference between the SLD and the Holevo bound led to more
fundamental studies, focusing on the incompatibility of parameters and leading
to the definition of measures able to capture this particular aspect [41, 42].
In particular in [41], a quantity has been introduced to quantify the
quantumness of a quantum multiparameter problem. This quantity, denoted by
$\mathcal{R}_{\bm{\lambda}}$, is equal to zero if and only if the Holevo bound
coincides with the SLD based bound, and thus if the parameters to be estimated
are asymptotically compatible. Similarly, it takes its largest value
$\mathcal{R}_{\bm{\lambda}}=1$ when the difference between the Holevo and the
SLD bound is maximum, that is when the parameters are highly incompatible even
in the asymptotic regime. Given these properties, in the following we will
refer to $\mathcal{R}_{\bm{\lambda}}$ as asymptotic incompatibility (AI). The
properties of AI has been discussed in relationship with the phase diagram of
a quantum many-body system in [41], while its main properties and its
behaviour for estimation problems encoded in qubit systems have been
thoroughly investigated in [43].

In this paper, we focus on the fundamental properties of the AI measure
$\mathcal{R}_{\bm{\lambda}}$ in local quantum estimation theory. We study its
relationship with the purity of the density operators defining the quantum
statistical model, and prove that for some classes of states, the two
quantities are indeed in one-to-one correspondence. More in general, we find
evidence that the maximum amount of incompatibility may be achieved only for
purity exceeding a threshold depending on the dimension of the Hilbert space.
We conjecture that this may be a general property of quantum statistial
models. Moreover, we derive some bounds on the AI measure for quantum
statistical submodels, and provide a method to directly identify the maximum
number of compatible parameters in certain estimation problems.

The manuscript is organized as follows: in Sec. 2 we review the main aspects
of multiparameter quantum estimation and we introduce the AI measure
$\mathcal{R}_{\bm{\lambda}}$. In Sec. 3, we show our main results concerning
the relationship between the AI measure and the purity of the quantum states,
while in Sec. 4 we will derive further properties regarding the AI of
parameters of specific quantum statistical models. Sec. 5 closes the paper
with some concluding remarks.

##  2 Local quantum multiparameter estimation

In this section we briefly review the tools of local quantum multiparameter
estimation. The goal of a typical metrological problem is to estimate the
value of a set $\bm{\lambda}$ of $\mathfrak{p}$ independent parameters from a
set of $M$ outcomes $\bm{x}=\\{x_{1},x_{2},...,x_{M}\\}$ that are distributed
according to the conditional probability distribution $p(x|\bm{\lambda})$,
usualy referred to as statistical model. The estimated value of the parameters
is obtained via an estimator function $\bm{\hat{\lambda}}(\bm{x})$. If we
denote by $\mathbb{E}_{\bm{\lambda}}[\bm{\hat{\lambda}}]$ the mean value of
the function $\bm{\hat{\lambda}}$ over the probability distribution
$p(x|\bm{\lambda})$, then the precision of any unbiased estimator is
quantified by the covariance matrix
$\textup{{Cov}}_{\bm{\lambda}}[\bm{\hat{\lambda}}]_{\alpha\beta}=\mathbb{E}_{\bm{\lambda}}\left[(\bm{\hat{\lambda}}-\mathbb{E}_{\bm{\lambda}}[\bm{\hat{\lambda}}])_{\alpha}(\bm{\hat{\lambda}}-\mathbb{E}_{\bm{\lambda}}[\bm{\hat{\lambda}}])_{\beta}\right]$.
The latter satisfies the classical matrix Cramér-Rao (CCRB) bound

$\displaystyle\textup{{Cov}}_{\bm{\lambda}}[\hat{\bm{\lambda}}]\geq\frac{1}{M}\bm{\mathcal{F}}^{c}(\bm{\lambda})^{-1},$
(1)

where the classical Fisher Information matrix (CFIM) is defined via its
elements

$\displaystyle\mathcal{F}^{c}_{\alpha\beta}(\bm{\lambda})=\int_{\chi}{\rm
d}x\,p(x|\bm{\lambda})\,\left[\partial_{\alpha}\log{p(x|\bm{\lambda})}\right]\,\left[\partial_{\beta}\log{p(x|\bm{\lambda})}\right]\,.$
(2)

The inequality must be understood as the fact that
$\textup{{Cov}}_{\bm{\lambda}}[\hat{\bm{\lambda}}]-M^{-1}\bm{\mathcal{F}}^{c}(\bm{\lambda})^{-1}$
is a semi-definite positive matrix [44, 45, 46]. Classical estimation theory
studies the conditions for the attainability of the CCRB and optimal
estimators are the ones for which the inequality is saturated. In the limit of
a large number of measurements $M\to\infty$, the maximum likelihood estimator
is proven to be optimal [47].

Moving to the quantum realm, the parameters are encoded in a family of density
operators $\varrho_{\bm{\lambda}}$ referred to as a quantum statistical model.
We remark that in the following we will restrict ourselves to quantum
statistical models described by quantum states $\varrho_{\bm{\lambda}}$ whose
rank in the Hilbert space does not change by varying the parameters
$\bm{\lambda}$ in the allowed region of the parameters’ space. This will allow
us to avoid discontinuities in the behaviour of the figures of merit we will
consider [48, 49]. The conditional probabilities of observing a certain
outcome are obtained through the Born rule
$p(k|\bm{\lambda})=\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{\Pi}_{k}]$. These
depends on the measurement process involved, which is described in its
generality by a POVM, i.e by a set of operators
$\boldsymbol{\Pi}=\\{\hat{\Pi}_{k}|\hat{\Pi}_{k}\geq
0,\sum_{k}\hat{\Pi}_{k}=\hat{\mathbb{I}}\\}$. By using different approaches,
measurement-independent bounds on the matrix covariance have been introduced
[7, 9, 10]. Among them, in this paper we are going to deal first with the
bound obtained via the symmetric logarithmic derivative (SLD) operators, that
is typically referred to as SLD-bound. SLD operators $\hat{L}^{S}_{\alpha}$,
are defined implicitly as the solutions of the Lyapunov equations

$\displaystyle\partial_{\alpha}\varrho_{\bm{\lambda}}=\frac{\hat{L}^{S}_{\alpha}\varrho_{\bm{\lambda}}+\varrho_{\bm{\lambda}}\hat{L}^{S}_{\alpha}}{2}.$
(3)

The corresponding SLD quantum Fisher information matrix (QFIM) is obtained via
the formula

$\displaystyle\mathcal{Q}_{\alpha\beta}(\bm{\lambda})=\hbox{Tr}\left[\varrho_{\bm{\lambda}}\frac{\hat{L}_{\alpha}^{S}\hat{L}_{\beta}^{S}+\hat{L}_{\beta}^{S}\hat{L}_{\alpha}^{S}}{2}\right],$
(4)

and the corresponding SLD quantum Cramer-Rao (QCR) matrix bound reads [46],

$\displaystyle\textup{{Cov}}_{\bm{\lambda}}[\hat{\bm{\lambda}}]\geq\bm{\mathcal{Q}}(\bm{\lambda})^{-1}.$
(5)

It can be shown that his bound is in fact achievable for single-parameter
estimation [5, 50, 51]. This means that for a single-copy of the state
$\varrho_{\bm{\lambda}}$, a POVM whose Fisher Information is equal to the SLD-
QFI exists, and remarkably one can prove that this optimal POVM corresponds to
the projectors over the eigenstates of the SLD operator. If we now consider
the multiparameter case $\mathfrak{p}>1$, the different SLD operators may not
commute, and an optimal simultaneous estimation of all the parameters can not
be in general obtained. As a consequence, the QCR matrix bound may not always
be achievable in the single-copy scenario and more informative bounds than the
one provided by the SLD operators may exist.

When dealing with matrix inequalities, an additional problem arises already in
classical estimation theory, i.e. the order of the CFIM is partial. This means
that, given two experimental strategies specified by two distinct POVM
$\mathbb{\Pi}^{(1)}$ and $\mathbb{\Pi}^{(2)}$ to which corresponds
$\bm{\mathcal{F}}^{(1)}$ and $\bm{\mathcal{F}}^{(2)}$, it might be that both
$\bm{\mathcal{F}}^{(1)}\ngeq\bm{\mathcal{F}}^{(2)}$ and
$\bm{\mathcal{F}}^{(2)}\ngeq\bm{\mathcal{F}}^{(1)}$. Thus, to understand which
strategies is better, scalar bounds have been introduced. Here, we study the
one defined in terms of a weight matrix $\bm{\mathcal{W}}$, a real and
positive definite matrix of dimension $\mathfrak{p}\times\mathfrak{p}$. The
SLD-QCR scalar bound obtainable from Equation (5) can be written as follow

$\displaystyle\hbox{Tr}\left[\bm{\mathcal{W}}\,\textup{{Cov}}_{\bm{\lambda}}[\hat{\bm{\lambda}}]\right]\geq
C^{S}(\bm{\lambda},\bm{\mathcal{W}})=\hbox{Tr}\left[\bm{\mathcal{W}}\bm{\mathcal{Q}}(\bm{\lambda})^{-1}\right].$
(6)

The role of the weight matrix $\bm{\mathcal{W}}$ is to balance the precision
of different parameters. In addition, there is a one to one correspondence
between $\bm{\mathcal{W}}$ and a change of parametrization. Given a new set of
parameter $\bm{\gamma}=\bm{\gamma}(\bm{\lambda})$ defined in terms of the
previous one, the changes in the QFI-SLD matrix is determined by the
reparametrization matrix
$\mathcal{B}_{\alpha\beta}=\partial\lambda_{\beta}/\partial\gamma_{\alpha}$.
The SLD-QFI matrix for the new parameter is [52]

$\displaystyle\bm{\mathcal{Q}}(\bm{\gamma})=\bm{\mathcal{B}}\bm{\mathcal{Q}}(\bm{\lambda})\bm{\mathcal{B}}^{T}.$
(7)

Besides, any real positive definite matrix can be decomposed as
$\bm{\mathcal{W}}=\bm{\mathcal{L}}\bm{\mathcal{L}}^{T}$, where
$\bm{\mathcal{L}}$ is unique. We see that if
$\bm{\mathcal{L}}=\bm{\mathcal{B}}^{-1}$, then a particular choice of
$\bm{\mathcal{W}}$ corresponds to a unique change of parametrization induced
by $\bm{\mathcal{B}}$.

As its matrix counterpart, the SLD-QCR scalar bound (6) is not in general
attainable due to the incompatibility of the optimal measurements
corresponding to the different parameters. The problem of finding the most
informative scalar bound was addressed by Holevo in [13]. The solution takes
the name of Holevo Cramér-Rao bound (HCRB) and it will be denoted by
$C^{H}(\bm{\lambda},\bm{\mathcal{\bm{W}}})$. It represents the most
informative bound for the asymptotic model in which a collective measurement
is performed on an asymptotically large number of copies of the state
$\varrho_{\bm{\lambda}}^{\otimes
n}=\bigotimes_{j=1}^{n}\varrho_{\bm{\lambda}}$, with $n\to\infty$ [14, 15]. In
this limit, the bound is indeed achievable and for this reason it is typically
considered as the most fundamental scalar bound for quantum multiparameter
estimation. However, the evaluation of the HCRB requires a non-trivial
minimization (see A for the formal definition of the HCRB). Nonetheless, some
results have been obtained both numerically and even analytically under some
assumptions on the quantum statistical model [38, 28, 53, 33, 34, 35, 54, 55,
56, 57]

###  2.1 Asymptotic incompatibility measure

As observed before, in a multiparameter scenario the optimal strategies for
each single parameter estimation may not be compatible and as a result the
lower bound (6) can not always be attained. A key object in this respect is
the commutator $[\hat{L}_{\alpha}^{S},\hat{L}_{\beta}^{S}]$ between the
different SLD-operators: if this commutator is equal to zero for all the
parameters in $\boldsymbol{\lambda}$, all the SLD-bounds for each single
parameter are simultaneously achievable in the single-copy scenario by
performing the same POVM, and as a result, both the matrix and scalar SLD-QCR
bound given in Eq. (5) and (6). These models are known in the litterature as
quasi-classical model [7, 55]. However, as we will describe in a moment, its
average value over the quantum statistical model
$\varrho_{\boldsymbol{\lambda}}$ plays an important role too. Typically this
quantity is introduced via the so-called Uhlmann (or Berry) curvature [58, 9],
defined via its matrix elements

$\displaystyle\mathcal{U}_{\alpha\beta}(\bm{\lambda}):=-\frac{i}{2}\hbox{Tr}\left[\varrho_{\bm{\lambda}}\left[\hat{L}^{S}_{\alpha},\hat{L}^{S}_{\beta}\right]\right].$
(8)

One proves that a necessary and sufficient condition for the attainability of
the SLD-QCR scalar bound (6) in the asymptotic limit is given by the
compatibility condition, or weak commutativity, $\bm{\mathcal{U}}=\bm{0}$
[59]. If this condition is fulfilled, that is if all the SLD operators commute
on average, then
$C^{H}(\bm{\lambda},\bm{\mathcal{W}})=C^{S}(\bm{\lambda},\bm{\mathcal{W}})$,
i.e. the SLD-QCR scalar bound can be attained in the asymptotic model
$\varrho_{\boldsymbol{\lambda}}^{\otimes n}$ with $n\to\infty$. In the
following, we will refer to models that satisfy this condition as
asymptotically classical model and parameters belonging to these models as
asymptotic compatible parameters [7, 55].

More recent results have shown how the HCRB can be bounded from above as

$\displaystyle C^{S}(\bm{\lambda},\bm{\mathcal{W}})\leq
C^{H}(\bm{\lambda},\bm{\mathcal{W}})\leq(1+\mathcal{R}_{\bm{\lambda}})C^{S}(\bm{\lambda},\bm{\mathcal{W}})\leq
2\,C^{S}(\bm{\lambda},\bm{\mathcal{W}})\,$ (9)

that is $C^{H}(\bm{\lambda},\bm{\mathcal{W}})$ cannot be larger than two times
the SLD QCR scalar bound. In the chain of inequalities above, we have
introduced the following parameter

$\displaystyle\mathcal{R}_{\bm{\lambda}}=\lvert|i\,\bm{\mathcal{Q}}(\bm{\lambda})^{-1}\bm{\mathcal{U}}(\boldsymbol{\lambda})\rvert\rvert_{\infty}\,,$
(10)

where $\lvert\lvert{\bf A}\rvert\rvert_{\infty}$ denotes the largest
eigenvalue of the matrix ${\bf A}$. In the rest of the manuscript we will
focus on this quantity, that has been introduced, as a quantumness parameter,
in [41] and studied in detail for qubit systems in [43]. As already suggested
in Eq. (9), one can prove that

$0\leq\Delta C(\bm{\lambda},\mathbf{W})\leq\mathcal{R}_{\bm{\lambda}}\leq
1\,,$

where we have defined the renormalized difference between the Holevo and SLD-
bound

$\displaystyle\Delta C(\bm{\lambda},{\bf W})=\frac{C^{\sf H}(\bm{\lambda},{\bf
W})-C^{\sf S}(\bm{\lambda},{\bf W})}{C^{\sf S}(\bm{\lambda},{\bf W})}\,.$ (11)

Moreover one also proves two relevant properties [41, 43]: (i)
$\mathcal{R}_{\bm{\lambda}}=0$ if and only if
$\bm{\mathcal{U}}(\boldsymbol{\lambda})=0$, that is when the model is
asymptotically classical; (ii) $\mathcal{R}_{\bm{\lambda}}$ is a property of
the quantum statistical model $\varrho_{\boldsymbol{\lambda}}$ only, being
also independent on possible reparametrization and, as a consequence, on the
weight matrix $\bm{\mathcal{W}}$. In [43] it was also studied its relationship
with the quantity

$\displaystyle\Delta C_{\sf max}=\max_{{\mathbf{W}}>0}\Delta
C({\bm{\lambda}},{\mathbf{W}})\,,$ (12)

for quantum statistical models encoded in qubit systems. The quantity $\Delta
C_{\sf max}$ stands out as a natural measure of asymptotic incompatibility
(AI) and it is in general upper bounded by $\mathcal{R}_{\bm{\lambda}}$. It
was shown that, while for several quantum statistical model the bound is
indeed tight, that is one finds $\Delta C_{\sf
max}=\mathcal{R}_{\bm{\lambda}}$, there exist counterexamples where $\Delta
C_{\sf max}$ is strictly smaller than $\mathcal{R}_{\bm{\lambda}}$. However,
also in these examples one observes that the two quantities have the same
general behaviour and in general that the order relations induced by them are
equivalent.
For all these reasons, and also considering the fact that the evaluation of
the quantity $\mathcal{R}_{\bm{\lambda}}$ is quite straightforward and, at
difference with $\Delta C_{\sf max}$, does not rely on the evaluation of the
Holevo bound and on its (non trivial) maximization over the weight matrices
$\mathbf{W}$, in the following we will restrict to it and we will in general
refer to $\mathcal{R}_{\bm{\lambda}}$ as a AI measure of the quantum
statistical model $\varrho_{\bm{\lambda}}$.

##  3 Asymptotic incompatibility and purity of the quantum statistical model

In this section we discuss the relationship between the AI measure
$\mathcal{R}_{\bm{\lambda}}$ and the purity of the quantum state
$\varrho_{\bm{\lambda}}$ corresponding to the quantum statistical model under
exam. We will mainly focus to scenarios where $\varrho_{\bm{\lambda}}$
describes the most general quantum state of a particular quantum system,
starting from the simplest cases of a qubit and of a single-mode Gaussian
continuous-variable quantum state, and then moving our attention to qudits,
that is to quantum states living in a $d$-dimensional Hilbert space.

###  3.1 Asymptotic incompatibility of state parameters in full tomography of
qubit systems

A generic mixed qubit state is typically written as

$\varrho_{\bm{\lambda}}=\frac{1}{2}\left(\mathbbm{1}+\sum_{j=1}^{3}\gamma_{j}\sigma_{j}\right)\,,$
(13)

where the matrices $\sigma_{j}$ denote the Pauli matrices, while
$\gamma_{1}=r\sin\theta\cos\phi$, $\gamma_{2}=r\sin\theta\sin\phi$,
$\gamma_{3}=r\cos\theta$. By considering the set of $\mathfrak{p}=3$
parameters $\bm{\lambda}=(r,\theta,\phi)$ characterizing the vector in the
Bloch sphere, one can readily derive the SLD operators by solving the
corresponding Lyapunov equations, and obtains the SLD-QFI and the Ulhman
curvature matrices

$\displaystyle\bm{\mathcal{Q}}(\bm{\lambda})$
$\displaystyle=\left(\begin{array}[]{ccc}1/(1-r^{2})&0&0\\\ 0&r^{2}&0\\\
0&0&r^{2}\sin^{2}\theta\end{array}\right)\,,$ (17)
$\displaystyle\bm{\mathcal{U}}(\bm{\lambda})$
$\displaystyle=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&r^{3}\sin\theta\\\
0&-r^{3}\sin\theta&0\end{array}\right)\,.$ (21)

The corresponding AI measure has been already derived in [43], obtaining
$\mathcal{R}_{\rm qubit}=r$. The purity of the quantum state
$\varrho_{\bm{\lambda}}$ can be easily evaluated, obtaining
$\mu=\hbox{Tr}[\varrho_{\bm{\lambda}}^{2}]=(1+r^{2})/2$, and thus leading to
the result

$\displaystyle\mathcal{R}_{\rm qubit}=\sqrt{2\mu-1}\,.$ (22)

We thus observe that the AI for the full tomography $\mathcal{R}_{\rm qubit}$
is indeed a monotonous function of the purity of the quantum state, and that
in particular it takes its limiting values $\mathcal{R}_{\rm qubit}=0$ for the
maximally mixed state, and $\mathcal{R}_{\rm qubit}=1$ for pure states only.

###  3.2 Asymptotic incompatibility for a single-mode continuous-variable
Gaussian system

As a second example we now consider a single-mode continuous-variable quantum
system, described by annihilation and creation operators satisfying the
canonical commutation relation $[\hat{a},\hat{a}^{\dagger}]=\mathbbm{1}$.
Quantum states describing such systems belongs to an infinite-dimensional
Hilbert space. A well known subclass of such states is given by Gaussian
states, typically defined as those states having a Gaussian Wigner function
[60, 61]. These states have been indeed studied in great detail, both for the
simple mathematical formulation (they can be fully described by first and
second moments of the quadrature operators
$\hat{q}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$ and
$\hat{p}=i(\hat{a}^{\dagger}-\hat{a})/\sqrt{2}$) and for practical and
fundamental reasons (they can be easily generated in the lab, and they can be
exploited for implementing quantum technology protocols as quantum
teleportation). We leave more details on Gaussian states in B, along with the
formulas needed to evaluate the SLD-QFI and the Uhlmann curvature matrices.
We here address the problem of complete estimating an arbitrary single mode
Gaussian state, which can be parametrized by $\mathfrak{p}=5$ parameters,
resulting from the application of a complex squeezing operator
$S(\xi)=\exp\left(\frac{1}{2}\xi(a^{\dagger})^{2}-\frac{1}{2}\xi^{*}a^{2}\right)$
and a complex displacement operator $D(\alpha)=\exp\left(\alpha
a^{\dagger}-\alpha^{*}a\right)$ on a thermal state
$\nu_{N}=e^{-\beta\hat{a}^{\dagger}\hat{a}}/Z$ with average photon number
$\hbox{Tr}[\nu_{N}\hat{a}^{\dagger}\hat{a}]=N$. The corresponding quantum
statistical model is thus represented by the family of density operators [61]

$\displaystyle\varrho_{\bm{\lambda}}=D(\alpha)S(\xi)\nu_{N}S(\xi)^{\dagger}D(\alpha)^{\dagger}\,,$
(23)

with a parametrization in terms of the set
$\bm{\lambda}=\\{\mathfrak{Re}{\alpha},\mathfrak{Im}{\alpha},r,\varphi,N\\}$,
and where we have written the two complex parameters as $\xi=re^{i\varphi}$
and $\alpha=\mathfrak{Re}{\alpha}+i\mathfrak{Im}{\alpha}$. The first moments
vector
${\bm{d}}=(\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{q}],\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{p}])^{T}$
has elements

$\displaystyle\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{q}]=\sqrt{2}\mathfrak{Re}{\alpha}\,,$
(24a)
$\displaystyle\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{p}]=\sqrt{2}\mathfrak{Im}{\alpha}\,,$
(24b)

while the elements of the covariance matrix (CM) (see B for details) can be
expanded in terms of symmetric $2\times 2$ matrices, i.e.
$\bm{\sigma}=s_{1}\sigma_{x}+s_{2}\bm{I}_{2}+s_{3}\sigma_{z}$ (with
$\sigma_{x}$ and $\sigma_{z}$ denoting the standard Pauli matrices), obtaining

$\displaystyle s_{1}=-(2N+1)\sinh(2r)\sin(\varphi),$ (24ya) $\displaystyle
s_{2}=(2N+1)\cosh(2r),$ (24yb) $\displaystyle
s_{3}=(2N+1)\sinh(2r)\cos(\varphi).$ (24yc)

The matrix elements for the SLD-QFI matrix and for the Uhlmann curvature
matrix can be directly evaluated via the formulas

$\displaystyle\mathcal{Q}_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\frac{\textup{Tr}[(\bm{\sigma}^{-1}\partial_{\alpha}\bm{\sigma})(\bm{\sigma}^{-1}\partial_{\beta}\bm{\sigma})]}{1+\mu^{2}}+\frac{2\partial_{\alpha}\mu\partial_{\beta}\mu}{1-\mu^{4}}$
$\displaystyle\,\,\,\,\,\,\,\,+2\left(\partial_{\alpha}\bm{d}\right)^{T}\bm{\sigma}^{-1}(\partial_{\beta}\bm{d})$
(24yz) $\displaystyle\mathcal{U}_{\alpha\beta}$
$\displaystyle=\frac{\mu^{2}}{2(\mu^{2}+1)^{2}}\textup{Tr}\left[\bm{\sigma}\Omega\left[\partial_{\alpha}\bm{\sigma}\bm{\sigma}^{-1},\partial_{\beta}\bm{\sigma}\bm{\sigma}^{-1}\right]\right]$
$\displaystyle\,\,\,\,\,\,\,\,+2\mu^{2}(\partial_{\alpha}\bm{d})\Omega(\partial_{\beta}\bm{d}),$
(24yaa)

where the purity of single-mode Gaussian states is given by $\mu=1/\sqrt{{\rm
det}\bm{\sigma}}=1/(2N+1)$. Notice that our result differ from that in [62] by
a factor $2$ in the first term of the Uhlmann matrix (for a detailed
derivation, see B).

In particular, for the quantum statistical model defined above, we obtain

$\displaystyle\mathcal{Q}_{\mathfrak{Re}{\alpha},\mathfrak{Re}{\alpha}}=4\mu\left(\cosh(2r)-\cos(\varphi)\sinh(2r)\right),$
(24yaba)
$\displaystyle\mathcal{Q}_{\mathfrak{Im}{\alpha},\mathfrak{Im}{\alpha}}=4\mu\left(\cosh(2r)+\cos(\varphi)\sinh(2r)\right),$
(24yabb)
$\displaystyle\mathcal{Q}_{\mathfrak{Re}{\alpha},\mathfrak{Im}{\alpha}}=4\mu\sinh(2r)\sin(\varphi),$
(24yabc) $\displaystyle\mathcal{Q}_{r,r}=\frac{4}{1+\mu^{2}},$ (24yabd)
$\displaystyle\mathcal{Q}_{\varphi,\varphi}=\frac{\sinh(2r)^{2}}{1+\mu^{2}},$
(24yabe) $\displaystyle\mathcal{Q}_{N,N}=\frac{4\mu^{2}}{1-\mu^{2}},$ (24yabf)

and

$\displaystyle\mathcal{U}_{\mathfrak{Re}{\alpha},\mathfrak{Im}{\alpha}}=4\mu^{2}\,,$
(24yabaca)
$\displaystyle\mathcal{U}_{\varphi,r}=\frac{4\mu\sinh(2r)}{(1+\mu^{2})^{2}},$
(24yabacb)

while the matrix elements not reported are null. We emphasize that these
values do not depend on $(\Re{\alpha},\Im{\alpha})$, and we also osberve that
the SLD-QFI matrix is block diagonal. In particular, the second block
regarding the parameters $\left(r,\varphi,N\right)$ is exactly diagonal.
Moreover, since most of the entries of the Uhlmann matrix are equal to $0$, we
conclude that most of the parameters are orthogonal globally with respect to
the SLD-QFI, with the exception of the pairs $(\Re{\alpha},\Im{\alpha})$ and
$(r,\varphi)$.

The AI measure can be readily evaluated, obtaning

$\displaystyle\mathcal{R}_{\sf GS}=\frac{2\mu}{1+\mu^{2}}\,.$ (24yabacad)

Remarkably, we see that the contribution depending on $(r,\varphi)$ cancel out
in the AI $\mathcal{R}_{\sf GS}$. Also in this case we have thus obtained that
AI is a monotonous function of purity $\mu$, and that one obtains that all the
parameters become asymptotically compatible only in the limit of a maximally
mixed state (that is $\mathcal{R}_{\sf GS}=0$ only for $\mu\rightarrow 0$),
while the maximum value of incompatibility $\mathcal{R}_{\sf GS}=1$ is
obtained if and only if $\varrho_{\bm{\lambda}}$ is a pure state.

###  3.3 Asymptotic incompatibility for of state parameters in full tomography
in qudit systems

Let us now move back to consider finite-dimensional quantum systems described
via Hilbert spaces with dimension $d$ larger than two. The most natural way to
describe the most general quantum states in this scenario is by the following
parametrization

$\varrho_{\bm{\gamma}}=\frac{1}{d}\left(\mathbbm{1}+\sum_{j=1}^{d^{2}-1}\gamma_{i}\Sigma_{i}\right)\,$
(24yabacae)

where the matrices $\Sigma_{i}$ denotes the generators of the Lie algebra
$\mathfrak{su}(d)$, and thus correspond to Pauli matrices for $d=2$ and to
Gell-Mann matrices for $d=3$. The $\mathfrak{p}=d^{2}-1$ coordinates
$\bm{\gamma}$ represent the normalized Cartesian vector in the
$d^{2}-1$-dimensional Bloch space and are also known as the components of the
$d^{2}-1$-dimensional Bloch vector or as mixture coordinates [63]. In the
following we consider the estimation of the parameters $\bm{\gamma}$, i.e. the
full tomography of the state. It is known that this model is a
$\mathcal{D}$-invariant model and hence the Holevo bound is equal to the right
logarithmic derivative scalar bound [64].
Analytical solutions for the AI measure are very hard to derive with arbitrary
dimension $d$. Hence, we address the problem numerically by randomly
generating quantum density matrices corresponding to $d=3$, $d=4$ and $d=5$.
The method we used to generate random density matrix follows two steps:

  1. 1.

First, we generate the eigenvalues of the density matrix
$\varrho_{\bm{\gamma}}$, which belongs to the probability simplex
$\bm{X}_{d}=\\{\bm{x}=(x_{1},...,x_{d})|\sum_{i=1}^{d}x_{i}=1\\}$.

  2. 2.

Second, we randomly generate unitary matrix $U$. We remind that different $U$
corresponds to different eigenvectors of $\varrho_{\bm{\gamma}}$ and thus to
different quantum states and more importantly to different values of the
parameters $\bm{\gamma}$.

For each random state we evaluate the AI measure from the definition (10) by
numerically solving the Lyapunov equation for the SLD operators and then by
evaluating the SLD-QFI and Uhlmann curvature matrices. The results are
reported in Fig. 1 as function of the purity of the state. We clearly see that
there is no one-to-one correspondence between the value of $\mu$ and the value
of $\mathcal{R}^{(d)}$. However, we see that the AI is strictly less than one
whenever the purity of the state is lower than $\mu=1/(d-1)$. Thus, we can
state the following conjecture:

Conjecture 1. Given a quantum statistical model described by a family quantum
states $\varrho_{\bm{\lambda}}$ living in a $d$-dimensional Hilbert space, the
maximum amount of the AI measure $\mathcal{R}_{\bm{\lambda}}=1$ may be
observed only if the purity of the quantum state is larger than a minimum
threshold $\mu_{\sf min}=1/(d-1)$.

This conjecture is clearly consistent for the two cases discussed above, i.e.
qubit and single-mode Gaussian states, whose AI was exactly derived, and it is
surrogated by the numerical evidence in Fig. 1.

We now introduce a second way to parametrize a $d$-dimensional quantum system
that will be particularly useful fo our purposes, i.e.

$\varrho_{\bm{\lambda}}=\frac{e^{-\beta H}}{\hbox{Tr}\left[e^{-\beta
H}\right]},\quad\quad H=\sum_{j=1}^{d^{2}-1}\lambda_{j}\Sigma_{j}$ (24yabacaf)

The coordinates $\bm{\lambda}$ are known as exponential coordinates [63] while
the parameter $\beta$ play the role of the inverse temperature of the state
with respect to the Hamiltonian operator $H$, clearly recalling the Gibbs
state form used in statistical mechanics. However, this is not a _true_
coordinate since it can be included in the definition of the $\bm{\lambda}$,
hence must be considered more as a rescaling parameter which allow us to
constrain the exponential coordinates to be in a finite interval
($|\lambda_{i}|\leq 1$ in our simulations).

Figure 1: Left panel: scatter plot of the AI measure (y-axis) vs purity
(x-axis) for qutrit systems. The black lines are the AI for fixed spectrum of
$H$, see main text for details. The most left line refers to a choice of $H$
with $2$ degenerate eigenvalues.
Right panel: scatter plot of the AI measure (y-axis) vs purity (x-axis) for
$4$-dimensional quantum systems. Again, the black lines are the AI for a fixed
spectrum of $H$. The red-dashed lines refers to choice of $H$ with at least
$2$ degenerate eigenvalues.

In general, a closed formula for the AI is hard to derive also with this
parameterization. Hence, we investigate the problem by considering a
simplified case, i.e. a diagonal
$H=\textup{diag}\left\\{\Delta_{0},\Delta_{1},...,\Delta_{d-1}\right\\}$, with
the constraint $|\Delta_{i}|\leq 1$. In other words, we set $\lambda_{j}=0$
for the indexes $j$ corresponding to non-diagonal operators $\Sigma_{j}$, and
we address the estimation for all the parameters corresponding to the
exponential coordinates $\bm{\lambda}$. The reason why we study this class of
states is twofold: the first is that the calculations are greatly simplified,
the second is that the results we find seem to be valid more generally, as we
will explain later in more detail.
We have evaluated analytically the AI measure corresponding to the estimation
of all the $d^{2}-1$ parameters encoded in this particular quantum statistical
model, finding

$\mathcal{R}_{\sf qudit}=\tanh\left(\frac{\beta\Delta_{M}}{2}\right)$
(24yabacag)

where $\Delta_{M}=\max_{i}{\Delta_{i}}-\min_{j}{\Delta_{j}}$, i.e. the maximum
difference between the eigenvalues of $H$ (we remark that the calculations are
much simpler by evaluating the matrices for the parameters $\bm{\gamma}$, and
exploiting the fact that $\mathcal{R}_{\bm{\lambda}}$ is invariant under
reparametrization). The result in (24yabacag) is consistent with the AI for a
qubit states, as for $d=2$ the purity $\mu$ and the argument of the $\tanh$
are in one-to-one correspondence. Indeed, the purity can be written in terms
of $\bm{\lambda}$ as
$\mu=(1+\tanh(\beta|\bm{\lambda}\cdot\bm{\lambda}|)^{2})/2$. In the case of a
diagonal $H$, we have $\lambda_{1}=\lambda_{2}=0$ and
$\Delta_{M}=2|\lambda_{3}|$, and thus

$\mu=\frac{1}{2}(1+\tanh(\beta|\lambda_{3}|)^{2}=\frac{1}{2}\left(1+\tanh\left(\frac{\beta\Delta_{M}}{2}\right)^{2}\right)\,.$
(24yabacah)

By simply using (24yabacag) we indeed obtain the result in Eq. (22).
Remarkably, also the Gaussian case might be reduced to this expression if we
consider that the purity in terms of the average number of excitation $N$ is
$\mu=(2N+1)^{-1}$. By replacing the value of $N$ by the formula for the Bose-
Einstein statistics $N=1/(\exp{\beta\omega}-1)$, we obtain that the purity can
be written as $\mu=\tanh(\beta\omega/2)$. Then, the AI (24yabacad) becomes
exactly equal to (24yabacag) by fixing $\Delta_{M}=2\omega$.

This line of reasoning of course does not prove that the AI is equal to
(24yabacag) for arbitrary $d$-dimensional density matrix
$\varrho_{\bm{\lambda}}$. To try to answer this question, we addressed the
problem by studying these quantities for the numerically generated random
quantum states of dimension $d=3,4,5$. In more detaiil

  1. 1.

We evaluate the quantity $\beta\Delta_{M}$ for each random states
$\varrho_{\bm{\lambda}}$ we generate (see previous paragraph for the details
of the random generation of $\varrho_{\bm{\lambda}}$) via the formulas
$\beta^{(rand)}\Delta^{(rand)}_{i}=-\log(x_{i})$ , and
$\beta^{(rand)}\Delta_{M}^{(rand)}=\max_{i}\beta^{(rand)}\Delta^{(rand)}_{i}-\min_{j}\beta^{(rand)}\Delta^{(rand)}_{j}$
(we remind that the quantities $\\{x_{i}\\}$ correspond to the eigenvalues of
$\varrho_{\bm{\lambda}}$).

  2. 2.

For each random state we evaluate the AI measure corresponding to its full
estimation via Eq. (10), and we compare it with the formula (24yabacag)
evaluated via the parameter $\beta^{(rand)}\Delta_{M}^{(rand)}$.

  3. 3.

We find that the two quantities match for all the quantum states that we have
numerical generated.

We conclude that for $d=3,4,5$ the AI is fully determined only by
$\beta\Delta_{M}$, i.e. it corresponds to a property of the spectrum of $H$
only (to be more precise it depends only on the maximum and minimum
eigenvalues of $H$). For these reasons, we state a second conjecture here

Conjecture 2. Given a quantum statistical model described by a family quantum
states $\varrho_{\bm{\lambda}}$ living in a $d$-dimensional Hilbert space, the
AI corresponding to the full estimation of the state is given by (24yabacag),
where $\beta$ is the fictious temperature of the state with respect to the
Hamiltonian operator $H$ in the exponential coordinates (24yabacaf).

A further element in favour of this conjecture comes from Fig. 1, in which we
clearly see that the AI for a fixed spectrum (black lines) span the whole area
covered by the AI measure values corresponding to the random states generated.
In addition, we see that the AI for a $4$-dimensional system with $H$ having
two eigenvalues with degeneracy $2$ reaches the maximum value of
$\mathcal{R}=1$ when the purity is $\mu=1/2$, since in the case in which
$\Delta_{0}=\Delta_{1}$ and $\Delta_{2}=\Delta_{3}$, we have that $\mu\to 1/2$
for $\beta\to\infty$.
We finally remark again that, as we described above, the formula in Eq.
(24yabacag) can be readily evaluated, after diagonalizing the quantum state
describing the quantum statistical model
$\varrho_{\bm{\lambda}}=\sum_{i}x_{i}|\psi_{i}\rangle\langle\psi_{i}|$, and by
observing that $\beta\Delta_{i}=-\log(x_{i})$.

##  4 Further properties of asymptotic incompatibility measure

In this section we analyze in more detail some further properties of the AI
measure $\mathcal{R}_{\bm{\lambda}}$, sheding more light on the relationship
between a given quantum statistical model $\varrho_{\bm{\lambda}}$ and
possible sub-models that arise when one or more parameters of the set
$\bm{\lambda}$ are considered known and thus not to be estimated.
In the definition of $\mathcal{R}_{\bm{\lambda}}$ in Eq. (10), only the the
largest eigenvalue of the matrix
$\bm{\mathcal{I}}=i\,\bm{\mathcal{Q}}(\bm{\lambda})^{-1}\bm{\mathcal{U}}(\boldsymbol{\lambda})$
is considered. In the following we will study more in detail the role of the
spectrum of $\bm{\mathcal{I}}$ in the characterization of the quantum
statistical model. To do so, we first introduce the Cauchy interlace theorem
[65]:

######  Theorem 4.1 (Cauchy interlace theorem).

Consider a $N\times N$ hermitian matrix $\bm{\mathcal{A}}$ and any
$(N-1)\times(N-1)$ principal sub-matrix $\bm{\mathcal{B}}$ of
$\bm{\mathcal{A}}$. Consider the corresponding eigenvalues in decreasing order
$\bm{a}=\\{a_{1},\ldots,a_{N}\\}$ and $\bm{b}=\\{b_{1},\ldots,b_{N-1}\\}$.
Then, $\bm{a}$ interlace $\bm{b}$, that is

$\displaystyle a_{N}\leq b_{N-1}\leq a_{N-1}\leq\ldots\leq b_{2}\leq a_{2}\leq
b_{1}\leq a_{1}.$ (24yabacai)

This result can now be applied for our purposes to the matrix
$\bm{\mathcal{I}}$, leading to the following theorem:

######  Theorem 4.2 (Bound on AI measure for quantum statistical sub-models).

Given a quantum statistical model defined by a set of
$\mathfrak{p}-$parameters $\bm{\lambda}$, and any other possible sub-model
defined by a set of (possibly reparametrized) $(\mathfrak{p}-1)$ parameters
$\tilde{\bm{\lambda}}$, then the two corresponding AI measures
$\mathcal{R}^{(\mathfrak{p})}_{\bm{\lambda}}$ and
$\mathcal{R}^{(\mathfrak{p-1})}_{\tilde{\bm{\lambda}}}$ satisfy the inequality

$\displaystyle
r_{2}\leq\mathcal{R}^{(\mathfrak{p}-1)}_{\tilde{\bm{\lambda}}}\leq\mathcal{R}^{(\mathfrak{p})}_{\bm{\lambda}}\,,$
(24yabacaj)

where $r_{2}$ denotes the second largest eigenvalue of the matrix
$\bm{\mathcal{I}}$ corresponding to the quantum statistical model
$\varrho_{\bm{\lambda}}$.
Proof. As observed in [41], the eigenvalues of $\bm{\mathcal{I}}$ are either
$0$ or given in pairs $h_{i}=\pm r_{i}$, with $0<r_{i}\leq
1,i=1,\ldots,\delta$ and
$\delta\leq\left\lfloor{(\mathfrak{p}+1)/2}\right\rfloor$. The thesis of the
theorem is thus a simple corollary of the Cauchy interlace theorem 4.1 stated
above.

One can further show that, if we consider smaller sub-matrices of
$\bm{\mathcal{I}}$, we can recursively apply this argument to smaller
statistical sub-model $\bar{\bm{\lambda}}$ with $(\mathfrak{p}-j)$ parameters.
Eventually one obtains that the AI measure for the corresponding statistical
sub-model is bounded as

$\displaystyle
r_{j+1}\leq\mathcal{R}^{(\mathfrak{p}-j)}_{\bar{\bm{\lambda}}}\leq\mathcal{R}^{(\mathfrak{p})}_{\bm{\lambda}}.$
(24yabacak)

In particular we see that any statistical sub-model with $(\mathfrak{p}-j)$
parameters is incompatible if we restrict ourselves to $j\leq\delta-1$, since
for those $j$ we have that $r_{i}>0$. This observation leads to the following
corollary:

Corollary (Upper bound on the number of compatible parameters of a quantum
statistical model). By denoting with $\delta$ the number of strictly positive
eigenvalues of $\bm{\mathcal{I}}$, then the quantum statistical model
$\varrho_{\bm{\lambda}}$ contains at number of compatible parameters that is
upper bounded as

$\displaystyle\mathfrak{p}_{\sf comp}\leq\mathfrak{p}^{(bound)}_{\sf
comp}=\mathfrak{p}-\delta\,.$ (24yabacal)

This result can be easily applied to the full estimation problem we studied in
the previous section. Indeed, we can use the same evidence we have for the
Conjecture 2 to conjecture that the eigenvalues of the matrix
$\bm{\mathcal{I}}$ are

$\text{Eig}(\bm{\mathcal{I}})=\left\\{0,...,0,\pm r_{(1,0)},\pm r_{(2,0)},\pm
r_{(2,1)},...,\pm r_{(d-1,d-3)},\pm r_{(d-1,d-2)}\right\\}$ (24yabacam)

where

$r_{i,j}=\tanh\left(\frac{\beta(\Delta_{i}-\Delta_{j})}{2}\right)$ (24yabacan)

and $(\Delta_{0},...,\Delta_{d-1})$ are the $\beta$-normalized eigenvalues in
ascending order of $H$, defined in Eq. (24yabacaf). If the eigenvalues of $H$
are non-degenerate, the number of strictly positive eigenvalues of
$\bm{\mathcal{I}}$ is simply

$\delta_{\sf ndg}=\begin{pmatrix}d\\\ 2\end{pmatrix}=\frac{d(d-1)}{2}.$
(24yabacao)

Instead, for $H$ with degenerate spectrum, the result slightly change. Indeed,
if we denote with $\kappa$ the number of distinct degenerate eigenvalues and
with $\eta_{i}$ the corresponding degeneracies (with $i=1,...,\kappa$), then
the number of strictly positive eigenvalues is

$\delta_{\sf dg}=\begin{pmatrix}d\\\
2\end{pmatrix}-\sum_{i=1}^{\kappa}\begin{pmatrix}\eta_{i}\\\
2\end{pmatrix}=\frac{d(d-1)}{2}-\sum_{i=1}^{\kappa}\frac{\eta_{i}(\eta_{i}-1)}{2}.$
(24yabacap)

Thus, given the numerical evidence we obtained from our simulations, we
conjecture the following

Conjecture 3 (Upper bound on the number of compatible parameters for the full
estimation of a $d$-dimensional quantum system). By denoting with $d$ the
dimension of the quantum system $\varrho_{\bm{\lambda}}$, the number of
compatible parameters $\mathfrak{p}_{\sf comp}$ in the full estimation of the
$d^{2}-1$-parameters describing $\varrho_{\bm{\lambda}}$ is upper bounded by
the quantity

$\displaystyle\mathfrak{p}^{(bound)}_{\sf comp}$
$\displaystyle=d^{2}-1-\delta_{\sf dg}$
$\displaystyle=\frac{(d+2)(d-1)}{2}+\sum_{i=1}^{\kappa}\frac{\eta_{i}(\eta_{i}-1)}{2},$
(24yabacaq)

where $\kappa$ is the number of distinct degenerate eigenvalues and $\eta_{i}$
the degeneracy degree, with $i=1,...,\kappa$.

This conjecture shows that the value of $\mathfrak{p}^{(bound)}_{\sf comp}$
depends on the values of the parameters, and in particular on the values that
makes the corresponding density matrix degenerate. Indeed, the larger is the
number of degenerate eigenvalues, the larger is the maximum number of possible
compatible parameters. In the limit of full degeneracy, i.e. $\kappa=1$ and
$\eta_{i}=d$, we see that $\mathfrak{p}^{(bound)}_{\sf comp}\to d^{2}-1$, i.e
equals the number the number of parameters of the model. Instead, in the case
with no degeneracy in the spectrum of $H$, we have the minimum value, i.e.
$\mathfrak{p}^{(bound)}_{\sf comp}=(d+2)(d-1)/2$. We would also like to stress
that the model is always full-rank as far as the value of $\beta$ is finite,
independently on the number of degenerate eigenvalues of $H$.

An upper bound on the number of compatible parameters has been also derived
recently in [66] by following a different approach and exploiting the
algebraic structure of the quantum statistical model. We have compared this
bound with our bound for the full estimation of a qubit, that is by
considering the estimation of the Bloch sphere parameters
$\bm{\lambda}=(r,\theta,\phi)$. We found that the matrix ${\bm{\mathcal{I}}}$
has eigenvalues ${\rm Eig}(\bm{\mathcal{I}})=\\{r,-r,0\\}$, leading to the
upper bound $\mathfrak{p}^{(bound)}_{\sf comp}=2$. This results does indeed
coincide with the one obtained in [66].

We also remark that, while the bound in [66] was derived for finite
dimensional Hilbert space, in our case we made no assumption on the dimension
of the Hilbert space. Hence we can apply our result also for infinite
dimensional system. For instance, let us consider the paradigmatic case of the
estimation of the parameters characterizing a single-mode Gaussian state
defined as in Eq. (23) and already treated in Sec. 3. The spectrum of the
corresponding matrix $\bm{\mathcal{I}}$ is given by

$\displaystyle\textup{Eig}(\bm{\mathcal{I}})=\left\\{\frac{2\mu}{1+\mu^{2}},\mu,0,-\mu,-\frac{2\mu}{1+\mu^{2}}\right\\}.$
(24yabacar)

This leads to the AI measure for the complete statistical model
$\mathcal{R}^{(5)}_{\bm{\lambda}}=2\mu/(1+\mu^{2})$ we discussed above. By
further inspecting the set of eigenvalues above we can however also conclude
that for any subset of $\mathfrak{p}=4$ parameters $\tilde{\bm{\lambda}}$, the
corresponding AI parameter is bounded as

$\displaystyle\mu\leq\mathcal{R}^{(4)}_{\tilde{\bm{\lambda}}}\leq\frac{2\mu}{1+\mu^{2}},$
(24yabacas)

and that there is a maximum of $\mathfrak{p}_{\sf comp}^{(bound)}=3$
compatible parameters. In general, by observing the form of the Uhlmann
curvature matrix we also see that the only incompatible models are the one
which deals with the simultaneous estimation of $\\{r,\varphi\\}$ or
$\\{\mathfrak{Re}{\alpha},\mathfrak{Im}{\alpha}\\}$.
In the examples above, if we restrict to subsets of the original parameters
$\bm{\lambda}=\\{\mathfrak{Re}{\alpha},\mathfrak{Im}{\alpha},r,\varphi,N\\}$,
all the results are directly availabe from the explicit solution of the SLD-
QFIM (24yaba) – (24yabf) and the Uhlmann matrix (24yabaca) – (24yabacb). Here
below we rather present an example where the results cannot be obtained
straightforwardly in analytical form, while the properties discussed above can
still provide a first insight without needing to solve exactly the estimation
problem.
We address the estimation of two dynamical parameters, the frequency and the
loss rate $\ddot{\bm{\lambda}}=\\{\omega,\gamma\\}$ that characterize the
evolution due to the Markovian master equation

$\displaystyle\dot{\varrho}=-i\frac{\omega}{2}[\hat{q}^{2}+\hat{p}^{2},\varrho]+\gamma\mathcal{D}[\hat{a}]\varrho,$
(24yabacat)

with
$\mathcal{D}[\hat{a}]\varrho=\hat{a}\varrho\hat{a}^{\dagger}-\frac{1}{2}\left(\hat{a}^{\dagger}\hat{a}\varrho+\varrho\hat{a}^{\dagger}\hat{a}\right)$.
By consider an initial Gaussian state $\varrho(0)$ as the one in Eq. (23), the
dynamics remains Gaussian and thus can be fully described by means of the
first and second moments of its quadrature operators. In particular one can
readily evaluate analytically the purity of the quantum state, obtaining

$\displaystyle\mu(t)=\sqrt{(1-e^{-\gamma t})(1-e^{-\gamma
t}(1-2s_{2}(0)))+e^{-2\gamma t}\mu(0)^{2}},$ (24yabacau)

¿From this result and via Eq. (24yabacad), we can directly evaluate
$\mathcal{R}^{(5)}_{\bm{\lambda}}$, whereas the AI measure
$\mathcal{R}^{(2)}_{\ddot{\bm{\lambda}}}$ for the quantum statistical model
defined above is evaluated numerically by exploiting the techniques reported
in B. To present the results, we choose the initial state to be a pure state
$N=0$ and we parametrize it in terms of its average excitations number
$\langle\hat{a}^{\dagger}\hat{a}\rangle$ and its fraction of squeezing $\eta$
defined as

$\displaystyle\langle\hat{a}^{\dagger}\hat{a}\rangle=|\alpha|^{2}+\sinh(r)^{2},$
(24yabacav)
$\displaystyle\eta=\frac{\sinh(r)^{2}}{\langle\hat{a}^{\dagger}\hat{a}\rangle}\,.$
(24yabacaw)

We plot the results in Figure 2 and we see that
$\mathcal{R}^{(5)}_{\bm{\lambda}}$ is, as expected, always larger than
$\mathcal{R}^{(2)}_{\ddot{\bm{\lambda}}}$, confirming that
$\mathcal{R}^{(5)}_{\bm{\lambda}}$ can be used as an upper bound for the
quantumness for any other sub-statistical model. Besides, as we see from the
left panel in Figure 2, the upper bound may not be much informative, since the
evolution of $\mathcal{R}^{(5)}_{\bm{\lambda}}$ and
$\mathcal{R}^{(2)}_{\ddot{\bm{\lambda}}}$ in time are opposite. In addition,
this example shows that the condition on the number of eigenvalue is not
sufficient to have a compatible model.

Figure 2: Plot of $\mathcal{R}^{(5)}_{\bm{\lambda}}$ (red line) and
$\mathcal{R}^{(2)}_{\ddot{\bm{\lambda}}}$ (blue line). Left panel:
$\langle\hat{a}^{\dagger}\hat{a}\rangle=4$; thick line: $\eta=1$; dashed line:
$\eta=0.5$. Right panel: $\eta=0.1$; thick line
$\langle\hat{a}^{\dagger}\hat{a}\rangle=4$; dashed line:
$\langle\hat{a}^{\dagger}\hat{a}\rangle=20$. Both panel: $\gamma=\omega=1$.

##  5 Conclusions

In this paper, we have studied in details the properties of the AI measure
$\mathcal{R}_{\bm{\lambda}}$ of incompatibility and discussed its use in
assessing the quantumness of multiparameter estimation problems. At first, we
have focused on the estimation of the full set of parameters characterizing a
given quantum system, showing that for qubits and single-mode Gaussian systems
$\mathcal{R}_{\bm{\lambda}}$ is a simple monotonous function of the purity
$\mu$ of the state. We have then considered a generic $d$-dimensional quantum
systems and, using analytical and numerical tools, we found that in general
the one-to-one correspondence between $\mathcal{R}_{\bm{\lambda}}$ and $\mu$
does not hold anymore. However, numerical results suggest that the maximum
number of AI is attainable only for quantum statistical models having a purity
$\mu\geq 1/(d-1)$. We conjecture that this may be a general property of
$d$-dimensional quantum systems. Upon considering quantum statistical models
described as Gibbs state of a given Hamiltonian, we have also shown that the
AI measure can be written as a simple function of the fictitious temperature
parameter and the spectrum of the Hamiltonian.

We have then studied in detail the role of the spectrum of the matrix
$\bm{\mathcal{I}}=i\bm{\mathcal{Q}}(\bm{\lambda})^{-1}\bm{\mathcal{U}}(\bm{\lambda})$
and have determined bounds on $\mathcal{R}_{\bm{\lambda}}$ for quantum
statistical submodels. In particular, we have shown that the number of
strictly positive eigenvalues of $\bm{\mathcal{I}}$ determines the maximum
number of compatible parameters in a given statistical model.

Very recently there has been much interest in deriving bounds that apply when
one allows measurements on a finite number of copies [17, 18]; we thus expect
that our approach can be extended from the asymptotic model, identifying and
studying a hierarchy of incompatibility measures in this finite copies
scenario. More in general our results pave the way to a deeper understanding
of the fundamental properties of multiparameter quantum estimation, and
provide potentially useful tools to approach multiparameter problems that
cannot be addressed analytically.

## Acknowledgements

The authors acknowledge several useful discussions with Francesco Albarelli.
MGAP is member of INdAM-GNFM.

##  Appendix A The Holevo Cramér-Rao bound

The Holevo Bound $C^{H}(\bm{\lambda},\bm{\mathcal{W}})$ can be evluated via
the following minimization

$\displaystyle
C^{H}(\bm{\lambda},\bm{\mathcal{W}})=\min_{\hat{\bm{\mathcal{X}}}\in\mathbb{X}_{\bm{\lambda}}}\left[\hbox{Tr}\left[\bm{\mathcal{W}}\mathfrak{Re}{\bm{\mathcal{Z}}[\hat{\bm{\mathcal{X}}}]}\right]+\|\sqrt{\bm{\mathcal{W}}}\mathfrak{Im}{\bm{\mathcal{Z}}[\hat{\bm{\mathcal{X}}}]}\sqrt{\bm{\mathcal{W}}}\|_{1}\right],$
(24yabacax)

where the elements of the hermitian $d\times d$ matrix $\bm{\mathcal{Z}}$ are

$\displaystyle\mathcal{Z}[\hat{\bm{\mathcal{X}}}]_{\mu\nu}=\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{\mathcal{X}}_{\mu}\hat{\mathcal{X}}_{\nu}],$
(24yabacay)

while $\hat{\bm{\mathcal{X}}}$ is an array of $d$ Hermitian operators such
that

$\displaystyle\hbox{Tr}[\varrho_{\bm{\lambda}}\hat{\mathcal{X}}_{\mu}]=0$
(24yabacaz)
$\displaystyle\hbox{Tr}[\hat{\mathcal{X}}_{\mu}\partial_{\nu}\varrho_{\bm{\lambda}}]=\frac{1}{2}\hbox{Tr}[\varrho_{\bm{\lambda}}\\{\hat{\mathcal{X}}_{\mu},\hat{L}^{S}_{\nu}\\}]=\delta_{\mu\nu}$
(24yabacba)

##  Appendix B Multiparameter estimation for continuous-variable Gaussian
states

In this section we focus on estimation problems in bosonic continuous variable
systems. In particular, we study $n$ modes Gaussian state [60, 61], which are
properly described in terms of quadrature operators
$\hat{\bm{r}}=\\{\hat{q}_{1},\hat{p}_{1},\ldots,\hat{q}_{n},\hat{p}_{n}\\}$.
They satisfy the canonical commutation relation

$\displaystyle\left[\hat{r}_{j},\hat{r}_{k}\right]=i\Omega_{jk},$ (24yabacbb)

where $\Omega=i\sigma_{y}^{\oplus n}$. A Gaussian state $\varrho$ is
completely determined by its vector of first moments
$\bm{d}=\hbox{Tr}[\varrho\bm{\hat{r}}]$ and its covariance matrix (CM)
$\bm{\sigma}=\hbox{Tr}[\varrho\\{(\bm{\hat{r}}-\bm{d}),(\bm{\hat{r}}-\bm{d})^{T}\\}]$,
with
$\\{\hat{\bm{l}},\hat{\bm{l}}^{T}\\}_{jk}=\hat{l}_{j}\hat{l}_{k}+\hat{l}_{k}\hat{l}_{j}$
[60, 61].

In order to derive the SLD-QFI matrix elements, several approaches have been
proposed in the literature [67, 68, 69, 61, 62, 70] . As we are interested
also in the Uhlmann curvature matrix, in the following we will focus on the
derivations pursued in [61, 62] that are indeed based on the writing the SLD
operators in terms of the moments of the Guassian states defining the quantum
statistical model. Let us then assume that a set of parameters $\bm{\lambda}$
defines a quantum statistical models in a family of Gaussian quantum states
$\varrho_{\bm{\lambda}}$. One shows that the SLD for a parameter
$\alpha\in\bm{\lambda}$ is at most quadratic in the moments [61] and can be
written as

$\displaystyle\hat{L}^{S}_{\alpha}=L^{(0)}_{\alpha}\mathbb{I}+\bm{L}_{\alpha}^{(1)}\cdot\hat{\bm{r}}+\hat{\bm{r}}^{T}\cdot\bm{L}^{(2)}_{\alpha}\cdot\bm{\hat{r}},$
(24yabacbc)

where $L^{(0)}_{\alpha}$ is a real number, $\bm{L}^{(1)}_{\alpha}$ is $2n$
real vector and $\bm{L}^{(2)}_{\alpha}$ is a $2n\times 2n$ real symmetric
matrix. After some algebra, one can find that the SLD-QFI matrix elements are
given by [61, 62]

$\displaystyle\mathcal{Q}_{\alpha\beta}=\frac{1}{2}\textup{Tr}[L^{(2)}_{\alpha}(\partial_{\beta}\bm{\sigma})]+2(\partial_{\alpha}\bm{d}^{T})\bm{\sigma}^{-1}(\partial_{\beta}\bm{d})$
(24yabacbd)

Here below we derive the analogous result for the Uhlmann curvature matrix.
First, we evaluate the commutator of $\hat{L}^{S}_{\alpha}$ and
$\hat{L}^{S}_{\beta}$, that is

$\displaystyle\left[\hat{L}^{S}_{\alpha},\hat{L}^{S}_{\beta}\right]=\left[\bm{L}^{(1)^{T}}_{\alpha}\hat{\bm{r}},\bm{L}^{(1)^{T}}_{\beta}\hat{\bm{r}}\right]+\left[\bm{L}^{(1)^{T}}_{\alpha}\hat{\bm{r}},\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\beta}\hat{\bm{r}}\right]+\left[\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\alpha}\hat{\bm{r}},\bm{L}^{(1)^{T}}_{\beta}\hat{\bm{r}}\right]+$
$\displaystyle+\left[\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\alpha}\hat{\bm{r}},\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\beta}\hat{\bm{r}}\right].$
(24yabacbe)

By recalling that $\left[\hat{r}_{j},\hat{r}_{k}\right]=i\Omega_{jk}$, the
first term of the latter is (we use Einstein convention on indexes summations)

$\displaystyle\left[\bm{L}^{(1)^{T}}_{\alpha}\hat{\bm{r}},\bm{L}^{(1)^{T}}_{\beta}\hat{\bm{r}}\right]=L^{(1)}_{\alpha,i}L^{(1)}_{\beta,j}\left[\hat{r}_{i},\hat{r}_{j}\right]=i\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}\mathbb{I};$
(24yabacbf)

the second term is ( we recall that $L^{(2)}_{\alpha,jk}=L^{(2)}_{\alpha,kj}$)

$\displaystyle\left[\bm{L}^{(1)^{T}}_{\alpha}\hat{\bm{r}},\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\beta}\hat{\bm{r}}\right]$
$\displaystyle=L^{(1)}_{\alpha,i}L^{(2)}_{\beta,jk}\left[\hat{r}_{i},\hat{r}_{j}\hat{r}_{k}\right]=$
(24yabacbg)
$\displaystyle=L^{(1)}_{\alpha,i}L^{(2)}_{\beta,jk}\left(\left[\hat{r}_{i},\hat{r}_{j}\right]\hat{r}_{k}+\hat{r}_{j}\left[\hat{r}_{i},\hat{r}_{k}\right]\right)=$
$\displaystyle=L^{(1)}_{\alpha,i}L^{(2)}_{\beta,jk}\left(i\Omega_{ij}\hat{r}_{k}+\hat{r}_{j}i\Omega_{ik}\right)=$
$\displaystyle=2i\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\hat{\bm{r}};$

while the third term is

$\displaystyle\left[\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\alpha}\hat{\bm{r}},\bm{L}^{(1)^{T}}_{\beta}\hat{\bm{r}}\right]=2i\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(1)}_{\beta};$
(24yabacbh)

and eventually the fourth term is (we used the symmetry of $L^{(2)}_{\alpha}$
and the skew-symmetry of $\Omega$)

$\displaystyle\left[\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\alpha}\hat{\bm{r}},\hat{\bm{r}}^{T}\bm{L}^{(2)}_{\beta}\hat{\bm{r}}\right]$
$\displaystyle=L^{(2)}_{\alpha,ij}L^{(2)}_{\beta,kl}\left[\hat{r}_{i}\hat{r}_{j},\hat{r}_{k}\hat{r}_{l}\right]=$
(24yabacbi)
$\displaystyle=L^{(2)}_{\alpha,ij}L^{(2)}_{\beta,kl}\left(i\Omega_{jk}\hat{r}_{i}\hat{r}_{l}+i\Omega_{ik}\hat{r}_{j}\hat{r}_{l}+i\Omega_{jl}\hat{r}_{k}\hat{r}_{i}+i\Omega_{il}\hat{r}_{k}\hat{r}_{j}\right)=$
$\displaystyle=2i\bm{\hat{r}}^{T}\left(\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}-\bm{L}^{(2)}_{\beta}\Omega\bm{L}^{(2)}_{\alpha}\right)\bm{\hat{r}}.$

So eventually we obtain

$\displaystyle\left[\hat{L}^{S}_{\alpha},\hat{L}^{S}_{\beta}\right]=i\left(\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}\mathbb{I}+2\bm{B}^{T}\hat{\bm{r}}+2\bm{\hat{r}}^{T}\bm{A}\bm{\hat{r}}\right)$
(24yabacbj)

where $\bm{A}$ and $\bm{B}$ are respectively a $d\times d$ symmetric matrix
and a $d$ vector

$\displaystyle\bm{A}=\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}-\bm{L}^{(2)}_{\beta}\Omega\bm{L}^{(2)}_{\alpha}=2\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta},$
(24yabacbk)
$\displaystyle\bm{B}=\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}-\bm{L}^{(2)}_{\beta}\Omega\bm{L}^{(1)}_{\alpha}$
(24yabacbl)

Now we can evaluate

$\displaystyle\mathcal{U}_{\alpha\beta}=-\frac{i}{2}\hbox{Tr}\left[\varrho_{\bm{\lambda}}\left[\hat{L}^{S}_{\alpha},\hat{L}^{S}_{\beta}\right]\right]=$
(24yabacbm)
$\displaystyle=\frac{1}{2}\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}+\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{d}+\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}+\hbox{Tr}\left[\varrho_{\bm{\lambda}}\bm{\hat{r}}^{T}\bm{A}\bm{\hat{r}}\right]$
(24yabacbn)

Since $A_{ij}\Omega_{ij}=0$, we obtain that

$\displaystyle\hbox{Tr}\left[\varrho_{\bm{\lambda}}\bm{\hat{r}}^{T}\bm{A}\bm{\hat{r}}\right]$
$\displaystyle=A_{ij}\hbox{Tr}\left[\varrho_{\bm{\lambda}}\hat{r}_{i}\hat{r}_{j}\right]=$
(24yabacbo)
$\displaystyle=A_{ij}\left(\frac{\sigma_{ij}}{2}+\frac{i\Omega_{ij}}{2}+d_{i}d_{j}\right)=$
$\displaystyle=\frac{1}{2}\hbox{Tr}\left[\bm{A}\bm{\sigma}\right]+\bm{d}^{T}\bm{A}\bm{d}=$
$\displaystyle=\hbox{Tr}\left[\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{\sigma}\right]+2\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{d}$

¿From [61] we recall that
$\bm{L}^{(1)}_{\alpha}=2\bm{\sigma}^{-1}(\partial_{\alpha}\bm{d})-2\bm{L}^{(2)}_{\alpha}\bm{d}$
and so we derive the following

$\displaystyle\frac{1}{2}\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}$
$\displaystyle=2\bigg{(}(\partial_{\alpha}\bm{d})^{T}\bm{\sigma}^{-1}\Omega\bm{\sigma}^{-1}(\partial_{\beta}\bm{d})-\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{\sigma}^{-1}(\partial_{\beta}\bm{d})+$
(24yabacbp)
$\displaystyle\quad-(\partial_{\alpha}\bm{d})^{T}\bm{\sigma}^{-1}\Omega
L^{(2)}_{\beta}\bm{d}+\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{d}\bigg{)}$
(24yabacbq)
$\displaystyle\bm{L}^{(1)^{T}}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{d}$
$\displaystyle=2\left((\partial_{\alpha}\bm{d})^{T}\bm{\sigma}^{-1}\Omega\bm{L}^{(2)}_{\beta}\bm{d}-\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{d}\right)$
(24yabacbr)
$\displaystyle\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(1)}_{\beta}$
$\displaystyle=2\left(\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{\sigma}^{-1}(\partial_{\beta}\bm{d})-\bm{d}^{T}\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{d}\right)$
(24yabacbs)

By inserting these last calculations, we end up with the following Uhlmann
matrix

$\displaystyle\mathcal{U}_{\alpha\beta}=\hbox{Tr}\left[\bm{L}^{(2)}_{\alpha}\Omega\bm{L}^{(2)}_{\beta}\bm{\sigma}\right]+2(\partial_{\alpha}\bm{d})\bm{\sigma}^{-1}\Omega\bm{\sigma}^{-1}(\partial_{\beta}\bm{d}).$
(24yabacbt)

Please notice that our result differ from that in [62] by a factor $2$ in the
first term of the Uhlmann matrix.

If we now focus on single-mode Gaussian state, we can write (24yabacbd) and
(24yabacbt) only in terms of the covariance matrix $\bm{\sigma}$, the vector
$\bm{d}$ and the purity of the state $\mu=1/\sqrt{\det{\bm{\sigma}}}$ as

$\displaystyle\mathcal{Q}_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\frac{\textup{Tr}[(\bm{\sigma}^{-1}\partial_{\alpha}\bm{\sigma})(\bm{\sigma}^{-1}\partial_{\beta}\bm{\sigma})]}{1+\mu^{2}}+\frac{2\partial_{\alpha}\mu\partial_{\beta}\mu}{1-\mu^{4}}$
$\displaystyle\,\,\,\,\,\,\,\,+2\left(\partial_{\alpha}\bm{d}\right)^{T}\bm{\sigma}^{-1}(\partial_{\beta}\bm{d})$
(24yabacbu) $\displaystyle\mathcal{U}_{\alpha\beta}$
$\displaystyle=\frac{\mu^{2}}{2(\mu^{2}+1)^{2}}\textup{Tr}\left\\{\bm{\sigma}\Omega\left[\partial_{\alpha}\bm{\sigma}\bm{\sigma}^{-1},\partial_{\beta}\bm{\sigma}\bm{\sigma}^{-1}\right]\right\\}$
$\displaystyle\,\,\,\,\,\,\,\,+2\mu^{2}(\partial_{\alpha}\bm{d})\Omega(\partial_{\beta}\bm{d}).$
(24yabacbv)

## References

## References

  * [1] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone.  Advances in quantum metrology.  Nat. Photonics, 5(4):222–229, apr 2011.
  * [2] Rafał Demkowicz-Dobrzański, Marcin Jarzyna, and Jan Kołodyński.  Quantum Limits in Optical Interferometry.  In Emil Wolf, editor, Prog. Opt. Vol. 60, chapter 4, pages 345–435. Elsevier, Amsterdam, 2015.
  * [3] C. L. Degen, F Reinhard, and Paola Cappellaro.  Quantum sensing.  Rev. Mod. Phys., 89(3):035002, jul 2017.
  * [4] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd.  Advances in photonic quantum sensing.  Nature Photonics, 12(12):724–733, 2018.
  * [5] Carl W Helstrom and Carl W Helstrom.  Quantum detection and estimation theory, volume 84.  Academic press New York, 1976.
  * [6] Matteo G. A. Paris.  Quantum estimation for quantum technology.  Int. J. Quantum Inf., 07(supp01):125–137, jan 2009.
  * [7] Francesco Albarelli, Marco Barbieri, Marco G Genoni, and Ilaria Gianani.  A perspective on multiparameter quantum metrology: from theoretical tools to applications in quantum imaging.  Physics Letters A, 384(12):126311, 2020.
  * [8] Magdalena Szczykulska, Tillmann Baumgratz, and Animesh Datta.  Multi-parameter quantum metrology.  Adv. Phys. X, 1(4):621–639, jul 2016.
  * [9] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang.  Quantum fisher information matrix and multiparameter estimation.  Journal of Physics A: Mathematical and Theoretical, 53(2):023001, 2019.
  * [10] Jasminder S Sidhu and Pieter Kok.  Geometric perspective on quantum parameter estimation.  AVS Quantum Science, 2(1):014701, 2020.
  * [11] Horace P. Yuen and Melvin Lax.  Multiple-parameter quantum estimation and measurement of nonselfadjoint observables.  IEEE Trans. Inf. Theory, 19(6):740–750, nov 1973.
  * [12] K. Yamagata.  Maximum logarithmic derivative bound on quantum state estimation as a dual of the holevo bound.  Journal of Mathematical Physics, 62(6):062203, 2021.
  * [13] Alexander S Holevo.  Probabilistic and statistical aspects of quantum theory, volume 1.  Springer Science & Business Media, 2011.
  * [14] Masahito Hayashi and Keiji Matsumoto.  Asymptotic performance of optimal state estimation in qubit system.  J. Math. Phys., 49(10):102101, oct 2008.
  * [15] Mădălin Guţă and Jonas Kahn.  Local asymptotic normality for qubit states.  Phys. Rev. A, 73(5):052108, may 2006.
  * [16] Yuxiang Yang, Giulio Chiribella, and Masahito Hayashi.  Attaining the ultimate precision limit in quantum state estimation.  Communications in Mathematical Physics, 368(1):223–293, 2019.
  * [17] Lorcán O. Conlon, Jun Suzuki, Ping Koy Lam, and Syed M. Assad.  Efficient computation of the nagaoka–hayashi bound for multiparameter estimation with separable measurements.  npj Quantum Information, 7(1):110, 2021.
  * [18] Hongzhen Chen, Yu Chen, and Haidong Yuan.  Hierarchical incompatibility measures in multi-parameter quantum estimation.  arXiv:2109.05807 [quant-ph], September 2021.  arXiv: 2109.05807.
  * [19] Keiji Matsumoto.  A new approach to the Cramér-Rao-type bound of the pure-state model.  J. Phys. A, 35(13):3111–3123, apr 2002.
  * [20] Andrzej Chrostowski, Rafał Demkowicz-Dobrzański, Marcin Jarzyna, and Konrad Banaszek.  On super-resolution imaging as a multiparameter estimation problem.  Int. J. Quantum Inf., 15(08):1740005, dec 2017.
  * [21] Zhixian Yu and Sudhakar Prasad.  Quantum Limited Superresolution of an Incoherent Source Pair in Three Dimensions.  Phys. Rev. Lett., 121(18):180504, oct 2018.
  * [22] Carmine Napoli, Samanta Piano, Richard Leach, Gerardo Adesso, and Tommaso Tufarelli.  Towards Superresolution Surface Metrology: Quantum Estimation of Angular and Axial Separations.  Phys. Rev. Lett., 122(14):140505, apr 2019.
  * [23] Lukas J. Fiderer, Tommaso Tufarelli, Samanta Piano, and Gerardo Adesso.  General expressions for the quantum fisher information matrix with applications to discrete quantum imaging.  PRX Quantum, 2:020308, Apr 2021.
  * [24] Manuel A. Ballester.  Estimation of unitary quantum operations.  Phys. Rev. A, 69(2):022303, 2004.
  * [25] Cyril Vaneph, Tommaso Tufarelli, and Marco G. Genoni.  Quantum estimation of a two-phase spin rotation.  Quantum Meas. Quantum Metrol., 1(3):12–20, jan 2013.
  * [26] Marco G. Genoni, Matteo G. A. Paris, Gerardo Adesso, Hyunchul Nha, Peter L. Knight, and M. S. Kim.  Optimal estimation of joint parameters in phase space.  Phys. Rev. A, 87(1):012107, jan 2013.
  * [27] Peter C. Humphreys, Marco Barbieri, Animesh Datta, and Ian A. Walmsley.  Quantum Enhanced Multiple Phase Estimation.  Phys. Rev. Lett., 111(7):070403, 2013.
  * [28] Tillmann Baumgratz and Animesh Datta.  Quantum Enhanced Estimation of a Multidimensional Field.  Phys. Rev. Lett., 116(3):030801, jan 2016.
  * [29] Christos N. Gagatsos, Dominic Branford, and Animesh Datta.  Gaussian systems for quantum-enhanced multiple phase estimation.  Phys. Rev. A, 94(4):042342, oct 2016.
  * [30] Paul A. Knott, Timothy J. Proctor, Anthony J. Hayes, Jason F. Ralph, Pieter Kok, and Jacob A. Dunningham.  Local versus global strategies in multiparameter estimation.  Phys. Rev. A, 94(6):062312, dec 2016.
  * [31] Luca Pezzè, Mario A. Ciampini, Nicolò Spagnolo, Peter C. Humphreys, Animesh Datta, Ian A. Walmsley, Marco Barbieri, Fabio Sciarrino, and Augusto Smerzi.  Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases.  Phys. Rev. Lett., 119(13):130504, sep 2017.
  * [32] Emanuele Roccia, Ilaria Gianani, Luca Mancino, Marco Sbroscia, Fabrizia Somma, Marco G. Genoni, and Marco Barbieri.  Entangling measurements for multiparameter estimation with two qubits.  Quantum Sci. Technol., 3(1):01LT01, jan 2018.
  * [33] Mark Bradshaw, Syed M Assad, and Ping Koy Lam.  A tight cramér–rao bound for joint parameter estimation with a pure two-mode squeezed probe.  Physics Letters A, 381(32):2598–2607, 2017.
  * [34] Mark Bradshaw, Ping Koy Lam, and Syed M Assad.  Ultimate precision of joint quadrature parameter estimation with a gaussian probe.  Physical Review A, 97(1):012106, 2018.
  * [35] Francesco Albarelli, Jamie F Friel, and Animesh Datta.  Evaluating the holevo cramér-rao bound for multiparameter quantum metrology.  Physical review letters, 123(20):200503, 2019.
  * [36] Jamie Friel, Pantita Palittapongarnpim, Francesco Albarelli, and Animesh Datta.  Attainability of the holevo-cramér-rao bound for two-qubit 3d magnetometry, 2020.
  * [37] Mihai D. Vidrighin, Gaia Donati, Marco G. Genoni, Xian-Min Jin, W. Steven Kolthammer, M. S. Kim, Animesh Datta, Marco Barbieri, and Ian A. Walmsley.  Joint estimation of phase and phase diffusion for quantum metrology.  Nat. Commun., 5(1):3532, dec 2014.
  * [38] Philip J. D. Crowley, Animesh Datta, Marco Barbieri, and Ian A. Walmsley.  Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry.  Phys. Rev. A, 89(2):023845, feb 2014.
  * [39] Matteo Altorio, Marco G. Genoni, Mihai D. Vidrighin, Fabrizia Somma, and Marco Barbieri.  Weak measurements and the joint estimation of phase and phase diffusion.  Phys. Rev. A, 92(3):032114, 2015.
  * [40] Emanuele Roccia, Valeria Cimini, Marco Sbroscia, Ilaria Gianani, Ludovica Ruggiero, Luca Mancino, Marco G. Genoni, Maria Antonietta Ricci, and Marco Barbieri.  Multiparameter approach to quantum phase estimation with limited visibility.  Optica, 5(10):1171, oct 2018.
  * [41] Angelo Carollo, Bernardo Spagnolo, Alexander A Dubkov, and Davide Valenti.  On quantumness in multi-parameter quantum estimation.  Journal of Statistical Mechanics: Theory and Experiment, 2019(9):094010, 2019.
  * [42] Federico Belliardo and Vittorio Giovannetti.  Incompatibility in quantum parameter estimation.  New Journal of Physics, 23(6):063055, jun 2021.
  * [43] Sholeh Razavian, Matteo G. A. Paris, and Marco G Genoni.  On the quantumness of multiparameter estimation problems for qubit systems.  Entropy, 22(11):1197, 2020.
  * [44] Shun-ichi Amari and Hiroshi Nagaoka.  Methods of information geometry, volume 191.  American Mathematical Soc., 2007.
  * [45] Harald Cramer.  Mathematical methods of statistics.  Princeton U. Press, Princeton, 500, 1946.
  * [46] Masahito Hayashi.  Quantum Information Theory: Mathematical Foundation.  Springer, 2016.
  * [47] Erich L Lehmann and George Casella.  Theory of point estimation.  Springer Science & Business Media, 2006.
  * [48] Dominik Šafránek.  Discontinuities of the quantum fisher information and the bures metric.  Phys. Rev. A, 95:052320, May 2017.
  * [49] Luigi Seveso, Francesco Albarelli, Marco G Genoni, and Matteo G A Paris.  On the discontinuity of the quantum fisher information for quantum statistical models with parameter dependent rank.  Journal of Physics A: Mathematical and Theoretical, 53(2):02LT01, dec 2019.
  * [50] Samuel L Braunstein and Carlton M Caves.  Statistical distance and the geometry of quantum states.  Physical Review Letters, 72(22):3439, 1994.
  * [51] Matteo G. A. Paris.  Quantum estimation for quantum technology.  International Journal of Quantum Information, 7(supp01):125–137, 2009.
  * [52] Marco G. Genoni, Paolo Giorda, and Matteo G. A. Paris.  Optimal estimation of entanglement.  Phys. Rev. A, 78:032303, Sep 2008.
  * [53] Jun Suzuki.  Explicit formula for the holevo bound for two-parameter qubit-state estimation problem.  Journal of Mathematical Physics, 57(4):042201, 2016.
  * [54] Jasminder S Sidhu, Yingkai Ouyang, Earl T Campbell, and Pieter Kok.  Tight bounds on the simultaneous estimation of incompatible parameters.  Physical Review X, 11(1):011028, 2021.
  * [55] Jun Suzuki.  Information geometrical characterization of quantum statistical models in quantum estimation theory.  Entropy, 21(7):703, 2019.
  * [56] Wojciech Gorecki, Sisi Zhou, Liang Jiang, and Rafał Demkowicz-Dobrzański.  Quantum error correction in multi-parameter quantum metrology.  arXiv:1901.00896, 2019.
  * [57] Syed M. Assad, Jiamin Li, Yuhong Liu, Ningbo Zhao, Wen Zhao, Ping Koy Lam, Z. Y. Ou, and Xiaoying Li.  Accessible precisions for estimating two conjugate parameters using gaussian probes.  Phys. Rev. Research, 2:023182, May 2020.
  * [58] Angelo Carollo, Bernardo Spagnolo, and Davide Valenti.  Uhlmann curvature in dissipative phase transitions.  Scientific reports, 8(1):1–16, 2018.
  * [59] Sammy Ragy, Marcin Jarzyna, and Rafał Demkowicz-Dobrzański.  Compatibility in multiparameter quantum metrology.  Physical Review A, 94(5):052108, 2016.
  * [60] Alessandro Ferraro, Stefano Olivares, and Matteo G. A. Paris.  Gaussian states in continuous variable quantum information.  arXiv preprint quant-ph/0503237, 2005.
  * [61] Alessio Serafini.  Quantum continuous variables: a primer of theoretical methods.  CRC press, 2017.
  * [62] Rosanna Nichols, Pietro Liuzzo-Scorpo, Paul A Knott, and Gerardo Adesso.  Multiparameter gaussian quantum metrology.  Physical Review A, 98(1):012114, 2018.
  * [63] Ingemar Bengtsson and Karol Życzkowski.  Geometry of quantum states: an introduction to quantum entanglement.  Cambridge university press, 2017.
  * [64] Zhu Huangjun.  Quantum state estimation and symmetric informationally complete POMs.  PhD thesis, PhD thesis, National University of Singapore, 2012. 1, 5, 2012\.
  * [65] Suk-Geun Hwang.  Cauchy’s interlace theorem for eigenvalues of hermitian matrices.  The American Mathematical Monthly, 111(2):157–159, 2004.
  * [66] Shingo Kukita.  An upper bound on the number of compatible parameters in simultaneous quantum estimation.  Journal of Physics A: Mathematical and Theoretical, 53(9):095303, February 2020.  Publisher: IOP Publishing.
  * [67] O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun.  Quantum parameter estimation using general single-mode gaussian states.  Phys. Rev. A, 88:040102, Oct 2013.
  * [68] Zhang Jiang.  Quantum fisher information for states in exponential form.  Phys. Rev. A, 89:032128, Mar 2014.
  * [69] Leonardo Banchi, Samuel L. Braunstein, and Stefano Pirandola.  Quantum fidelity for arbitrary gaussian states.  Phys. Rev. Lett., 115:260501, Dec 2015.
  * [70] Dominik Šafránek.  Estimation of gaussian quantum states.  Journal of Physics A: Mathematical and Theoretical, 52(3):035304, dec 2018.