# Imaginarity-free quantum multiparameter estimation

Jisho Miyazaki<EMAIL_ADDRESS>Shinpouin, Mikataharacho, Kita-ku,
Hamamatsu, Shizuoka, 433-8105, Japan  Keiji Matsumoto<EMAIL_ADDRESS>Quantum
Computation Group, National Institute of Informatics, 2-1-2 Hitotsubashi,
Chiyoda-ku, Tokyo 101-8430

###### Abstract

Multiparameter quantum estimation is made difficult by the following three
obstacles. First, incompatibility among different physical quantities poses a
limit on the attainable precision. Second, the ultimate precision is not
saturated until you discover the optimal measurement. Third, the optimal
measurement may generally depend on the target values of parameters, and thus
may be impossible to perform for unknown target states.

We present a method to circumvent these three obstacles. A class of quantum
statistical models, which utilizes antiunitary symmetries or, equivalently,
real density matrices, offers compatible multiparameter estimations. The
symmetries accompany the target-independent optimal measurements for pure-
state models. Based on this finding, we propose methods to implement
antiunitary symmetries for quantum metrology schemes. We further introduce a
function which measures antiunitary asymmetry of quantum statistical models as
a potential tool to characterize quantumness of phase transitions.

##  1 Introduction

Incompatibility residing in simultaneous measurements of different quantities
has been a widely acknowledged character of quantum mechanics. Besides the
philosophical implications, the measurement incompatibility imposes practical
limitations on quantum information processing.

The measurement incompatibility is particularly troubling for quantum
metrological tasks which aim at estimating parameters with ever increasing
precision [1, 2]. In its early stages, quantum metrology focused on single-
parameter estimations. Quantum correlations unlock a region of precision which
could never be reached by classical mechanics [3, 4, 5, 6, 7, 8]. The
ultimately attainable precision limits are given by quantum Cramér-Rao bounds
(QCRBs). [9, 10, 11, 12].

Quantum correlations remain advantageous for various quantum multiparameter
metrological schemes [13, 14]. Simultaneous estimations outperform
combinations of individual estimations for sensing multi-dimensional fields
[15] and imaging [16, 17]. This motivates experimental realizations of multi-
parameter metrology [18, 19, 20, 21].

Despite the rapid growth of multiparameter estimation, the status of
multiparameter QCRBs [22, 23] has always been under doubt. There are systems
which do not allow compatible estimation, rendering QCRBs unattainable [24].
Therefore, conditions to realize compatible multiparameter estimation have
been under intense scrutiny [25, 26, 18, 24, 27, 28, 15, 29, 30, 31, 32, 33].
Attainable precision limits for incompatible estimations are also avidly
studied [34, 11, 35, 36, 37, 38, 39, 27, 25, 40, 41, 42, 14, 43, 44, 45, 46].
Whether or not the QCRB is saturated, identifying attainable precision limit
is the first step of quantum multiparameter metrology.

However, even if one could identify the best precision bound, optimal
measurements to attain it can be difficult to perform, and further
investigations have been carried out in order to study the implementation of
the optimal measurements for given quantum statistical models [25, 16, 47, 29,
48, 30, 39, 49, 50]. Somewhat paradoxically, preceding methods assume
knowledge on the target point ${\bm{x}}$ to be estimated in order to identify
the optimal measurement [25, 16, 29, 30], rendering the parameter estimation
task to, strictly speaking in the case, applicable only for local parameter
estimation.

Yang _et. al._ [39] presents a method to attain the best precision limit
without any prior knowledge of the target point. The method is composed of two
steps: a rough expectation of the target-point by a tomography, and the
precise estimation with the optimal measurement depending on the rough
expectation. The measurement setting has to be adaptively updated according to
the previous measurements. Thus the measurement still depends on the target-
point.

Another drawback of proposed optimal measurements is the inclusion of
projectors onto the target state [16]. Such a projector could be highly
entangled for quantum metrology. This is why optimal measurements are
sometimes given up in multiparameter quantum metrology [15].

This article proposes the use of antiunitary symmetries to side-step the
problems of incompatibility and unfeasible measurement in multiparameter
quantum estimation. Any quantum statistical model endowed with an antiunitary
symmetry is compatible, which means that the QCRB is attainable in the
asymptotic limit. Furthermore, if the system consists only of pure states, the
antiunitary symmetry guarantees a target-independent optimal measurement. The
construction of globally optimal measurements requires a step beyond the
studies of compatibility conditions [25, 28, 15, 29, 30, 33]. Our optimal
measurement has a continuous degree of freedom, and one can choose any of them
to achieve the global optimality.

With these advantages in mind, we present methods to implement antiunitary
symmetries in multiparameter quantum metrology. The so-designed pure-state
models for quantum metrology have globally optimal measurements by
construction. Our methods do not sacrifice the attainable precisions when
implementing antiunitary symmetries.

Though unrecognized, popular models such as N00N states [51, 52, 2],
antiparallel spins [53, 54] and 3D magnetometry [15] enjoy antiunitary
symmetry. For parameter estimations of these models, we point out how to make
the full use of their antiunitary symmetry. For example, we obtain an optimal
measurement for the 3D magnetometry which is only bipartite-entangled
irrespective to the system size.

Besides practical implications, our result bridges two seemingly different
concepts, namely, the incompatibility and the imaginarity. The incompatibility
characterizes quantumness of phase transitions in quantum statistical
mechanics [55, 56, 57, 58, 59, 43]. The imaginarity is formulated as a
meaningful resource in quantum information processing [60, 61, 62]. We refine
the relation between these two concepts.

We start by reviewing theories of quantum multi-parameter estimation and
antilinear operators in Sec. 2. Section 3 presents the notion of antiunitary
symmetry of quantum statistical models, with its direct link to compatible,
optimal, and target-independent multiparameter estimations. Then we propose
methods to implement antiunitary symmetry in metrological settings in Sec. 4.
Section 5 offers discussions concerning a refinement of the link between
antiunitary symmetry and compatibility. We offer our conclusion at Sec. 6.

##  2 Preliminaries

###  2.1 Quantum multi-parameter estimation theory

Let us consider an estimation of parameters that determines density operators
of finite dimensional Hilbert space $\mathcal{H}$. The density operator
$\rho_{\bm{x}}$ is parametrized by
${\bm{x}}=(x_{1},x_{2},...,x_{n})^{\mathrm{T}}\in X\subset\Re^{n}$. The set
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is called a _quantum statistical model_ or
simply a _model_. The task is to estimate parameters ${\bm{x}}$ from results
of quantum measurements performed on multiple copies $\rho_{\bm{x}}^{\otimes
N_{Q}\times N_{C}}$. The identical measurement on $\rho_{\bm{x}}^{\otimes
N_{Q}}$ is assumed to be performed independently $N_{C}$ times. Each
measurement on $\rho_{\bm{x}}^{\otimes N_{Q}}$ can be entangled over the
$N_{Q}$ copies. An estimation process is described by the pair of a positive
operator-valued measures (POVMs)
$\Pi=\\{\Pi_{\omega}\in\mathcal{B}(\mathcal{H}^{\otimes
N_{Q}})\\}_{\omega\in\Omega}$ and an estimator
$\check{\bm{x}}:\Omega^{N_{C}}\rightarrow X$, where $\Omega$ is the set of
measurement results ($\mathcal{B}$ stands of the space of bounded linear
operators).

Precision of estimation is characterized by the following mean square error
matrix:

$\displaystyle{\bm{\Sigma}}_{\bm{x}}(\Pi,\check{\bm{x}})$
$\displaystyle:=\sum_{{\bm{\omega}}\in\Omega^{N_{C}}}p({\bm{\omega}}|{\bm{x}})\left(\check{\bm{x}}({\bm{\omega}})-{\bm{x}}\right)\left(\check{\bm{x}}({\bm{\omega}})-{\bm{x}}\right)^{\mathrm{T}},$
(1)

where
$p({\bm{\omega}}|{\bm{x}})=\prod_{k=1}^{N_{C}}\mathrm{Tr}[\Pi_{\omega_{k}}\rho_{\bm{x}}^{\otimes
N_{Q}}]$ for ${\bm{\omega}}=(\omega_{1},...,\omega_{N_{C}})^{\mathrm{T}}$. For
estimators satisfying

$\sum_{{\bm{\omega}}\in\Omega^{N_{C}}}p({\bm{\omega}}|{\bm{x}})\check{\bm{x}}({\bm{\omega}})={\bm{x}},$
(2)

the mean square error matrix is a covariance matrix around the target-point
(true value) ${\bm{x}}$. We consider locally unbiased estimators and
consistent estimators in this article. The estimator is called locally
unbiased at ${\bm{x}}$ when (2) and

$\sum_{{\bm{\omega}}\in\Omega^{N_{C}}}\check{x}_{i}({\bm{\omega}})\partial_{j}p({\bm{\omega}}|{\bm{x}})=\delta_{ij},$
(3)

holds at the point ${\bm{x}}$ ($\partial_{j}$ stands for the partial
derivative with respect to $x_{j}$). The estimator is called consistent if (2)
is satisfied at any point ${\bm{x}}$ in the asymptotic limit, namely, if

$\lim_{N_{C}\rightarrow\infty}\sum_{{\bm{\omega}}\in\Omega^{N_{C}}}p({\bm{\omega}}|{\bm{x}})\check{\bm{x}}({\bm{\omega}})={\bm{x}},\leavevmode\nobreak\
(\forall{\bm{x}}\in X)$ (4)

holds (to be more precise, consistency is defined on a sequence of estimators
$\check{\bm{x}}_{N_{C}}:\Omega^{N_{C}}\rightarrow X$ for
$N_{C}\in\mathbb{N}$). The maximum likelihood estimator

$\check{\bm{x}}=\mathrm{argmax}\left\\{\sum_{l=1}^{N_{C}}\ln
p(\omega_{l}|{\bm{x}})\middle|{\bm{x}}\in X\right\\},$ (5)

is consistent. While consistent estimators assume sufficiently many rounds of
measurements, the unbiasedness condition (4) holds globally over the parameter
space $X$.

One of the goals of quantum multiparameter estimation is to find the optimal
estimator $\check{\bm{x}}$ and measurement $\Pi$ to minimize the covariance
matrix ${\bm{\Sigma}}_{\bm{x}}(\Pi,\check{\bm{x}})$. Classical and quantum
Cramér-Rao bounds deal with the minimization over estimators and measurements,
respectively. Define the $n\times n$ classical Fisher information matrix
(CFIM) for measurement $\Pi=\\{\Pi_{\omega}\in\mathcal{B}(\mathcal{H}^{\otimes
N_{Q}})\\}_{\omega\in\Omega}$ by

$\displaystyle[\mathcal{F}_{C}(\Pi,{\bm{x}})]_{ij}$
$\displaystyle:=\sum_{\omega\in\Omega}\frac{\partial_{i}\mathrm{Tr}[\Pi_{\omega}\rho_{\bm{x}}^{\otimes
N_{Q}}]\partial_{j}\mathrm{Tr}[\Pi_{\omega}\rho_{\bm{x}}^{\otimes
N_{Q}}]}{\mathrm{Tr}[\Pi_{\omega}\rho_{\bm{x}}^{\otimes N_{Q}}]},$ (6)

where $\partial_{i}$ represents the partial derivative with respect to
$x_{i}$. CFIM provides the classical Cramér-Rao bound of the covariance matrix

$N_{C}{\bm{\Sigma}}_{\bm{x}}(\Pi,\check{\bm{x}})\geq\mathcal{F}_{C}(\Pi,{\bm{x}})^{-1}.$
(7)

At any point ${\bm{x}}\in X$, there exists an ${\bm{x}}$-dependent locally
unbiased estimator saturating the inequality for any $N_{C}$ (see e.g. Chapter
2 of [63] for an explicit construction of the estimator). Note that the
locally unbiased estimator at ${\bm{x}}$ is generally impossible to construct
if the target-point ${\bm{x}}$ is unknown. The maximum likelihood estimator
saturates the same inequality globally in the asymptotic limit
$N_{C}\rightarrow\infty$.

The remaining optimization is about the measurement. For general model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$, symmetric logarithmic derivatives (SLDs)
$L_{{\bm{x}},i}\in\mathcal{B}(\mathcal{H})$ ($i=1,...,n$) are Hermitian
operators defined indirectly by

$2\partial_{i}\rho_{\bm{x}}=L_{{\bm{x}},i}\rho_{\bm{x}}+\rho_{\bm{x}}L_{{\bm{x}},i}.$
(8)

The solution of Eq. (8) is not unique but one is given by

$L_{{\bm{x}},i}=2\int_{0}^{\infty}e^{-t\rho_{\bm{x}}}\partial_{i}\rho_{\bm{x}}e^{-t\rho_{\bm{x}}}\mathrm{d}t.$
(9)

SLDs define $n\times n$ quantum Fisher information matrix (QFIM) via [9, 10]

$[\mathcal{F}_{Q}({\bm{x}})]_{ij}:=\mathrm{Re}\left[\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]\right],$
(10)

and provides the quantum Cramér-Rao bound (QCRB) of the CFIM

$\mathcal{F}_{C}(\Pi,{\bm{x}})\leq N_{Q}\mathcal{F}_{Q}({\bm{x}}),$ (11)

which holds for any measurement $\Pi$ [22, 23]. While SLDs are not necessarily
determined uniquely by Eq. (8), the QFIM is independent of the choice of SLDs.

For multiparameter estimation $(n\geq 2)$, the QCRB is not necessarily
saturated by any measurement, unlike for single-parameter estimations. This
limitation is caused by the incompatibility between estimations of different
parameters. A necessary and sufficient condition for the saturation
$\mathcal{F}_{C}(\Pi,{\bm{x}})=N_{Q}\mathcal{F}_{Q}({\bm{x}})$ in the limit
$N_{Q}\rightarrow\infty$ is the _weak commutativity_ [25, 28]:

$[\mathcal{U}({\bm{x}})]_{ij}:=\frac{\mathrm{Im}\left[\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]\right]}{2}=0.\leavevmode\nobreak\
(\forall i,j)$ (12)

The matrix $\mathcal{U}$ is called as the mean Uhlmann curvature [55], and is
independent of the choice of SLDs. In this article, compatible models refer to
weakly commutative models.

If the saturation
$\mathcal{F}_{C}(\Pi,{\bm{x}})=N_{Q}\mathcal{F}_{Q}({\bm{x}})$ occurs for
$N_{Q}=1$, then the multiple product of the same measurement on the copied
model saturates it for any $N_{Q}$, since the CFIM is additive with respect to
the tensor product and separable measurements. For $N_{Q}=1$, a necessary
condition for the saturation
$\mathcal{F}_{C}(\Pi,{\bm{x}})=\mathcal{F}_{Q}({\bm{x}})$ is the partial
commutativity:

$\Pi_{\bm{x}}[L_{{\bm{x}},i},L_{{\bm{x}},j}]\Pi_{\bm{x}}=0,\leavevmode\nobreak\
(\forall i,j)$ (13)

where $\Pi_{\bm{x}}$ is the projector onto the support of $\rho_{\bm{x}}$
[30]. A sufficient condition for the saturation of QCRB with $N_{Q}=1$ is
quasi-classicality represented by

$[L_{{\bm{x}},i},L_{{\bm{x}},j}]=0,\leavevmode\nobreak\ (\forall i,j),$ (14)

at the point ${\bm{x}}\in X$ in question. Quasi-classical models do not demand
collective measurements for saturating the QCRBs.

All pure weakly commutative models are quasi-classical, and have measurements
to saturate QCRB (11) with $N_{Q}=1$ [25]. For a pure-state model
$\rho_{\bm{x}}=|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|$, a necessary and
sufficient condition for a POVM $\mathcal{E}$ on $\mathcal{H}$ to saturate
QCRB (11) at ${\bm{x}}\in X$ is [30],

$\displaystyle{\rm If}\leavevmode\nobreak\
\langle\psi_{\bm{x}}|E|\psi_{\bm{x}}\rangle=0\leavevmode\nobreak\ {\rm
then}\leavevmode\nobreak\ \exists\eta_{i,j}\in\Re,$
$\displaystyle\leavevmode\nobreak\
EL_{{\bm{x}},i}|\psi_{\bm{x}}\rangle=\eta_{i,j}EL_{{\bm{x}},j}|\psi_{\bm{x}}\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ (\forall i,\leavevmode\nobreak\ j),$ (15)

and

$\displaystyle{\rm If}\leavevmode\nobreak\
\langle\psi_{\bm{x}}|E|\psi_{\bm{x}}\rangle\neq 0\leavevmode\nobreak\ {\rm
then}\leavevmode\nobreak\ \exists\xi_{i}\in\Re,$ $\displaystyle
EL_{{\bm{x}},i}|\psi_{\bm{x}}\rangle=\xi_{i}E|\psi_{\bm{x}}\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ (\forall i),$ (16)

for any element $E$ of $\mathcal{E}$. Note that condition (15) should be
interpreted so that when the vector $EL_{{\bm{x}},j}|\psi_{\bm{x}}\rangle$ on
the right-hand-side of the equation is zero, condition (15) still holds with
$\eta_{i,j}$ a real infinity [64]. More precisely, condition (15) is
equivalent to

$\displaystyle{\rm If}\leavevmode\nobreak\
\langle\psi_{\bm{x}}|E|\psi_{\bm{x}}\rangle=0\leavevmode\nobreak\ {\rm
then}\leavevmode\nobreak\
\exists|\psi\rangle\in\mathcal{H},\leavevmode\nobreak\ \eta_{i}\in\Re,$
$\displaystyle\eta_{i}|\psi\rangle=EL_{{\bm{x}},i}|\psi_{\bm{x}}\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ (\forall i).$ (17)

While this interpretation of condition (15) is not obvious in the first look
at the statement of [30] Theorem 2, it can be deduced from its proof.

As in the case for single-parameter estimation, the optimal measurement $\Pi$
generally depend on the target point ${\bm{x}}\in X$ to be estimated. Explicit
methods to construct the optimal measurements proposed so far typically
generates parameter-dependent POVMs [16, 30]. To perform such a measurement,
one needs a prerequisite knowledge on the point ${\bm{x}}\in X$, or must start
from adaptive measurements to find ${\bm{x}}\in X$ in advance.

###  2.2 Antiunitary operators

We review basic properties of antiunitary operators on finite dimensional
Hilbert spaces. Differences between general antiunitary operators and
conjugations are relevant in this article. More advanced review on the topic
can be found in [65].

An operator $\Theta$ on $\mathcal{H}$ is said to be antilinear if

$\Theta(c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle)=c_{1}^{\ast}\Theta|\psi_{1}\rangle+c_{2}^{\ast}\Theta|\psi_{2}\rangle,$
(18)

holds for any complex coefficients $c_{1},\leavevmode\nobreak\ c_{2}$ and
vectors $|\psi_{1}\rangle,\leavevmode\nobreak\ |\psi_{2}\rangle$. The
Hermitian adjoint $\Theta^{\dagger}$ of an antilinear operator is defined
differently to linear ones by

$\langle\psi_{1}|(\Theta^{\dagger}|\psi_{2}\rangle)=\langle\psi_{2}|(\Theta|\psi_{1}\rangle),$
(19)

from which $(\Theta^{\dagger})^{\dagger}=\Theta$,
$(\Theta_{1}\Theta_{2})^{\dagger}=\Theta_{2}^{\dagger}\Theta_{1}^{\dagger}$,
$(A\Theta)^{\dagger}=\Theta^{\dagger}A^{\dagger}$ and $(\Theta
A)^{\dagger}=A^{\dagger}\Theta^{\dagger}$ follow for antilinear operators
$\Theta_{1},\leavevmode\nobreak\ \Theta_{2}$ and linear operator $A$.
Hermitian adjoints of antilinear operators are again antilinear. Antilinear
operator $\Theta$ is said to be Hermitian and unitary when
$\Theta^{\dagger}=\Theta$ and $\Theta^{\dagger}=\Theta^{-1}$ hold,
respectively. Unitary antilinear operators are called antiunitaries. Hermitian
antiunitary operators are called conjugations. Equivalently, conjugation
$\theta$ is an antiunitary satisfying $\theta^{2}=\mathbb{I}_{\mathcal{H}}$.
In this article, antiunitary operators are represented by $\Theta$, and
conjugations are by $\theta$.

A conjugation is a complex conjugation in its reference-bases. To see this,
choose a basis $\\{|\psi_{i}\rangle\\}_{i=1,...,d}$ of a $d$-dimensional
Hilbert space $\mathcal{H}$. Define a matrix representation $[\Theta]$ of
antilinear operator $\Theta$ in this basis by

$\Theta|\psi_{j}\rangle=\sum_{i=1}^{d}[\Theta]_{ij}|\psi_{i}\rangle,$ (20)

from which

$\Theta\sum_{j}c_{j}|\psi_{j}\rangle=\sum_{ij}c_{j}^{\ast}[\Theta]_{ij}|\psi_{i}\rangle,$
(21)

follows. The matrix $[\Theta]$ is unitary and symmetric when $\Theta$ is
unitary and Hermitian, respectively. A conjugation $\theta$ has a symmetric
unitary matrix representation $[\theta]$, and thus decomposed as

$[\theta]=VV^{\mathrm{T}},$ (22)

with a unitary matrix $V$. This is a direct consequence from Autonne$-$Takagi
factorization. Inserting Eq. (22) to Eq. (21), we have

$\displaystyle\theta\sum_{j}c_{j}|\psi_{j}\rangle$
$\displaystyle=\sum_{ijk}c_{j}^{\ast}V_{ik}V^{\mathrm{T}}_{kj}|\psi_{i}\rangle$
$\displaystyle=\sum_{k}$
$\displaystyle\left(\sum_{j}c_{j}V^{\dagger}_{kj}\right)^{\ast}\left(\sum_{i}V_{ik}|\psi_{i}\rangle\right).$
(23)

The action of $\theta$ is thus a complex conjugation in the $V$-rotated basis
$|\psi_{k}^{\prime}\rangle:=\sum_{i}V_{ik}|\psi_{i}\rangle$.

The reference-basis of a conjugation is not unique. If a basis
$\\{|\psi_{j}\rangle\\}_{j=1,...,\dim\mathcal{H}}$ is a reference-basis of
$\theta$, then another basis
$\\{|\psi^{\prime}_{j}\rangle\\}_{j=1,...,\dim\mathcal{H}}$ defined by

$|\psi^{\prime}_{j}\rangle:=R|\psi_{j}\rangle,\leavevmode\nobreak\
(j=1,...,\dim\mathcal{H}),$ (24)

with a real orthogonal matrix $R$ is also a reference-basis of $\theta$.

A general antiunitary operator $\Theta$ has a decomposition

$\Theta=U\theta,$ (25)

into a (linear) unitary operator $U$ and a conjugation $\theta$. $\Theta$ is a
conjugation if $U$ is symmetric with respect to the reference basis that makes
$\theta$ a complex conjugation. Otherwise $\Theta$ is an antiunitary that is
not a conjugation.

Spin-flip is an example of antiunitary operator that is not a conjugation.
Consider a 2-dimensional space $\mathcal{H}$ with a basis
$\\{|\psi^{+}\rangle,\leavevmode\nobreak\ |\psi^{-}\rangle\\}$. Spin-flip
$\Theta_{f}$ is defined by

$\Theta_{f}=\sigma_{Y}\theta,$ (26)

where $\sigma_{Y}$ is the Pauli-$Y$ operator and $\theta$ is the complex
conjugation in the same basis. Obviously $\sigma_{Y}$ is not symmetric, and
$\Theta_{f}^{2}=-\mathbb{I}_{2}\neq\mathbb{I}_{2}$.

In most part of this article antiunitary operators act in adjunction

$\Theta A\Theta^{\dagger},$ (27)

on (linear) Hermitian operators $A$ such as quantum states. The above operator
is equal to

$U\theta A\theta U^{\dagger}=UA^{\ast}U^{\dagger},$ (28)

from Eq. (25), where $A^{\ast}$ is the complex conjugation of $A$ in the
reference basis of $\theta$. The adjoint action of conjugation on a Hermitian
operator is equivalent to transposition since
$A^{\ast}=(A^{\dagger})^{\ast}=A^{\mathrm{T}}$. When $A$ is a quantum state,
namely, a density operator, the transformation $A\mapsto\Theta
A\Theta^{\dagger}$ cannot be a completely positive map.

Direct sums and tensor products of antiunitary operators are well-defined, and
both result in antiunitary operators again. The tensor product of antiunitary
operators $\Theta_{A}$ on $\mathcal{H}_{A}$ and $\Theta_{B}$ on
$\mathcal{H}_{B}$ is defined By

$\displaystyle\Theta_{A}\otimes\Theta_{B}\sum_{j}$ $\displaystyle
c_{j}|\psi^{A}_{j}\rangle\otimes|\psi^{B}_{j}\rangle$
$\displaystyle:=\sum_{j}c_{j}^{\ast}|\Theta_{A}\psi^{A}_{j}\rangle\otimes|\Theta_{B}\psi^{B}_{j}\rangle,$
(29)

for vector $\sum_{j}c_{j}|\psi^{A}_{j}\rangle\otimes|\psi^{B}_{j}\rangle$ in
the composite system $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. For arbitrary
two vectors in $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ decomposed by

$\displaystyle|\psi\rangle$
$\displaystyle=\sum_{j}c_{j}|\psi^{A}_{j}\rangle\otimes|\psi^{B}_{j}\rangle,$
(30) $\displaystyle|\xi\rangle$
$\displaystyle=\sum_{k}d_{k}|\xi^{A}_{k}\rangle\otimes|\xi^{B}_{k}\rangle,$
(31)

the inner product
$\langle\Theta_{A}\otimes\Theta_{B}\xi|\Theta_{A}\otimes\Theta_{B}\psi\rangle$
is calculated as

$\displaystyle\langle\Theta_{A}\otimes\Theta_{B}\xi|\Theta_{A}\otimes\Theta_{B}\psi\rangle$
$\displaystyle=\sum_{j,k}d_{k}c^{\ast}_{j}\langle\Theta_{A}\xi^{A}_{k}|\Theta_{A}\psi^{A}_{j}\rangle\langle\Theta_{B}\xi^{B}_{k}|\Theta_{B}\psi^{B}_{j}\rangle$
$\displaystyle=\sum_{j,k}d_{k}c^{\ast}_{j}\langle\psi^{A}_{k}|\xi^{A}_{j}\rangle\langle\psi^{B}_{k}|\xi^{B}_{j}\rangle=\langle\psi|\xi\rangle,$
(32)

by using Eq. (29). Equation
$\langle\Theta_{A}\otimes\Theta_{B}\xi|\Theta_{A}\otimes\Theta_{B}\psi\rangle=\langle\psi|\xi\rangle$
reveals that $\Theta_{A}\otimes\Theta_{B}$ is well-defined and is antiunitary.
We also have

$(U_{A}\Theta_{A})\otimes(U_{B}\Theta_{B})=(U_{A}\otimes
U_{B})(\Theta_{A}\otimes\Theta_{B}),$ (33)

where $U_{A}$ and $U_{B}$ are linear operators, and the tensor product in the
middle is of linear operators. Tensor products of pairs of an antiunitary
operator and a linear operator is not well-defined. There is no “partial
conjugation,” in contrast to the well-defined partial transposition.

##  3 Global antiunitary symmetry

This section presents a direct link between antiunitary symmetry and weak
commutativity. Theorem 2 states that quantum statistical models with “global”
antiunitary symmetries are weakly commutative at all points in the parameter
spaces. Theorem 3 ensures the target-independent optimal measurements for
pure-state models with global antiunitary symmetries. These theorems underlie
our methods to find globally optimal measurements.

###  3.1 Global antiunitary symmetry and conjugation

Let us define models with global antiunitary symmetry, and clarify the
distinction between general antiunitaries and conjugations with respect to the
definition.

###### Definition 1

A quantum statistical model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is said to
have a _global antiunitary symmetry_ (GAS) when there is an antiunitary
operator $\Theta$ such that

$\Theta\rho_{\bm{x}}\Theta^{\dagger}=\rho_{\bm{x}},$ (34)

holds for any ${\bm{x}}\in X$. Models with GASs are said to be _imaginarity-
free_.

The antiunitary operator $\Theta$ in the definition does not depend on the
parameter ${\bm{x}}$. This is why we call the antiunitary symmetry in the
sense of Def. 1 as being global.

As a trivial example, any classical model, whose states are diagonal in a
fixed basis for all ${\bm{x}}\in X$, i.e., when all the eigenvectors are
independent of ${\bm{x}}$, is imaginarity-free. The antiunitary operator is
the complex conjugation in the diagonal basis.

There also are various non-classical imaginarity-free models. This article
presents GASs of N00N states [51, 52, 2], super-dense coding [66] and 3D
magnetometry [15] in Sec. 3.4, and of antiparallel spins [53, 54], qubit
inside a disc of Bloch sphere [27] and embedding quantum simulators [67, 68,
69, 70, 71, 72] in Appx. A.

The GAS must be a conjugation for certain classes of models.

###### Lemma 1

A GAS, if exists, is given only by a conjugations if the model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ satisfies either of the following two
conditions:
(a) $\rho_{\bm{x}}$ is a pure state at all points in $X$ and the pure states
span the entire Hilbert space $\mathcal{H}$,
(b) $\rho_{\bm{x}}$ is non-degenerate at some point in $X$.

We use following two lemmas to prove lemma 1.

###### Lemma 2

For any antiunitary operator $\Theta$ and a vector $|\psi\rangle$ on a Hilbert
space $\mathcal{H}$,
$\Theta|\psi\rangle\langle\psi|\Theta^{\dagger}=|\psi\rangle\langle\psi|$
implies $\Theta^{2}|\psi\rangle=|\psi\rangle$.

_proof_)
$\Theta|\psi\rangle\langle\psi|\Theta^{\dagger}=|\psi\rangle\langle\psi|$
holds if and only if $\Theta|\psi\rangle=\epsilon|\psi\rangle$ with some
unimodular $\epsilon$. This implies
$\Theta^{2}|\psi\rangle=\epsilon^{\ast}\epsilon|\psi\rangle=|\psi\rangle$.
$\hfill\blacksquare$

###### Lemma 3

Let $A$ be an Hermitian operator on Hilbert space $\mathcal{H}$ with
eigensubspaces $\mathcal{H}_{i}$ ($\oplus_{i}\mathcal{H}_{i}=\mathcal{H}$),
and $\Theta$ be an antiunitary operator on $\mathcal{H}$ which satisfies
$\Theta A\Theta^{\dagger}=A$. Then $\Theta$ has a decomposition

$\Theta=\oplus_{i}\Theta_{i},$ (35)

where $\Theta_{i}$ is an antiunitary operator on $\mathcal{H}_{i}$ for each
$i$.

_proof_) Let us denote the eigenvector of $A$ with eigenvalue $\lambda\in\Re$
by $|\psi_{\lambda}\rangle$. Then we have

$\displaystyle\lambda\langle\psi_{\lambda}|\Theta|\psi_{\lambda^{\prime}}\rangle=\langle\psi_{\lambda}|A\Theta|\psi_{\lambda^{\prime}}\rangle$
$\displaystyle=\langle\psi_{\lambda}|\Theta A|\psi_{\lambda^{\prime}}\rangle$
$\displaystyle=$
$\displaystyle\lambda^{\prime}\langle\psi_{\lambda}|\Theta|\psi_{\lambda^{\prime}}\rangle,$
(36)

for any eigenvalues $\lambda,\leavevmode\nobreak\ \lambda^{\prime}$, where the
second equality follows from $\Theta A\Theta^{\dagger}=A$. Equation (36) is
possible only if when $\lambda\neq\lambda^{\prime}$,
$\Theta|\psi_{\lambda^{\prime}}\rangle$ and $|\psi_{\lambda}\rangle$ are
orthogonal to each other. Equivalently, $|\psi\rangle\in\mathcal{H}_{i}$
implies $\Theta|\psi\rangle\in\mathcal{H}_{i}$. Thus $\Theta$ is a sum of
antiunitaries on the eigensubspaces of $A$. $\hfill\blacksquare$
_proof of lemma 1_) (a) Let $\Theta$ be the GAS of a pure state model
$\\{|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}||{\bm{x}}\in X\\}$. We have
$\Theta^{2}|\psi_{\bm{x}}\rangle=|\psi_{\bm{x}}\rangle$ at all points from
Lem. 2. Because $\Theta^{2}$ is linear and $\mathcal{H}$ is spanned by
$\\{|\psi_{\bm{x}}\rangle|{\bm{x}}\in X\\}$, $\Theta^{2}$ must be identity.

(b) Let ${\bm{x}}_{0}\in X$ be the point where $\rho_{{\bm{x}}_{0}}$ becomes
non-degenerate. As $\rho_{{\bm{x}}_{0}}$ is Hermitian and satisfies
$\Theta\rho_{{\bm{x}}_{0}}\Theta^{\dagger}=\rho_{{\bm{x}}_{0}}$, lemma 3
reveals a decomposition of $\Theta$ given by Eq. (35). The dimension of
eigensubspaces are all $1$ because of the non-degeneracy. Then lemma 2 applies
to each $\Theta_{i}$, and we have
$\Theta_{i}^{2}=\mathbb{I}_{\mathcal{H}_{i}}$. This implies
$\Theta^{2}=\mathbb{I}_{\mathcal{H}}$, and completes the proof for the case
(b). $\hfill\blacksquare$

Even if a model has a GAS which is not a conjugation, the same model has
another symmetry provided by a conjugation:

###### Theorem 1

If a model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is imaginarity-free, there is a
conjugation which is a GAS for $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$.

This theorem remains valid even if the model is replaced by any set of
Hermitian operators, where the definition of GAS is straightforwardly
extended. In the proof presented below, $\rho_{\bm{x}}$ can be a Hermitian
operator and $X$ can be a discrete set.
_proof_) Let $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ be a set of Hermitian
operators in Hilbert space $\mathcal{H}$ with a GAS $\Theta$. We explicitly
construct a conjugation replacing $\Theta$.

Let $\mathcal{H}_{i}$ $(i\in I)$ be the disjoint subspaces of $\mathcal{H}$
that are simultaneous eigensubspaces of all the states in
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$. The state is decomposed as

$\rho_{\bm{x}}=\Pi^{\prime}\rho_{\bm{x}}\Pi^{\prime}\oplus\bigoplus_{i\in
I}\lambda_{i}({\bm{x}})\Pi_{i},$ (37)

where $\Pi_{i}$ are projectors each onto $\mathcal{H}_{i}$, $\Pi^{\prime}$ is
a projector onto the complement subspace

$\mathcal{H}^{\prime}:=\mathcal{H}\backslash(\oplus_{i\in I}\mathcal{H}_{i}),$
(38)

and $\lambda_{i}({\bm{x}})$ are eigenvalues.

For arbitrary vector $|\psi\rangle$ orthogonal to $\mathcal{H}_{i}$, there is
a point ${\bm{x}}_{\psi}$ such that $|\psi\rangle$ is in the complement
subspace of the eigensubspace of $\rho_{{\bm{x}}_{\psi}}$ with eigenvalue
$\lambda_{i}({\bm{x}}_{\psi})$. By applying Lem. 3 to
$\rho_{{\bm{x}}_{\psi}}$, we obtain a decomposition of $\Theta$ such that
$|\psi\rangle$ belongs to the different sector from $\mathcal{H}_{i}$. As
$|\psi\rangle$ is any vector orthogonal to $\mathcal{H}_{i}$, we have a
decomposition of the antiunitary

$\Theta=\Theta_{i}^{\prime}\oplus\Theta_{i},$ (39)

where antiunitary operators $\Theta_{i}$ and $\Theta_{i}^{\prime}$ acts on
$\mathcal{H}_{i}$ and its complement, respectively. Since the decomposition
(39) exists for each $i\in I$, we have

$\Theta=\Theta^{\prime}\oplus\bigoplus_{i\in I}\Theta_{i},$ (40)

where $\Theta^{\prime}$ is an antiunitary operator on $\mathcal{H}^{\prime}$.

The defining equation $\Theta\rho_{\bm{x}}\Theta^{\dagger}=\rho_{\bm{x}}$ of
the GAS is rewritten by the decompositions of the state (37) and the
antiunitary (40) to

$\Theta^{\prime}\rho_{\bm{x}}^{\prime}\Theta^{\prime\dagger}=\rho_{\bm{x}}^{\prime},$
(41)

for the truncated (unnormalized) state
$\rho_{\bm{x}}^{\prime}:=\Pi^{\prime}\rho_{\bm{x}}\Pi^{\prime}$, and

$\Theta_{i}\Pi_{i}\Theta_{i}^{\dagger}=\Pi_{i}.\leavevmode\nobreak\ (\forall
i\in I)$ (42)

Any antiunitary operator on $\mathcal{H}_{i}$ satisfies Eq. (42) instead of
$\Theta_{i}$. Thus all $\Theta_{i}$ in $\Theta$ can be replaced by
conjugations.

We now show that $\Theta^{\prime}$ is a conjugation. To do this, we consider
the following step-by-step procedure to decompose $\mathcal{H}^{\prime}$ and
$\Theta^{\prime}$. The procedure starts from $N=1$ with the trivial
decomposition $\mathcal{H}^{(1)}_{1}=\mathcal{H}^{\prime}$ and
$\Theta^{(1)}_{1}=\Theta^{\prime}$, and ends at $N=\dim\mathcal{H}^{\prime}$
with a decomposition of $\mathcal{H}^{\prime}$ into $1$-dimensional subspaces.

At the step $N$, a disjoint subspace decomposition $\mathcal{H}^{(N)}_{j}$
($j=1,...,N$) of $\mathcal{H}^{\prime}$ and corresponding decomposition
$\Theta^{\prime}=\oplus_{j=1}^{N}\Theta^{(N)}_{j}$ of antiunitary is given.
Choose a subspace with dimension more than $2$, say $\mathcal{H}^{(N)}_{k}$,
and a unit vector $|\psi\rangle\in\mathcal{H}^{(N)}_{k}$ therein. By
definition of the truncated state $\rho_{\bm{x}}^{\prime}$, there is a point
${\bm{x}}$ such that $|\psi\rangle$ is not an eigenvector of
$\rho_{{\bm{x}}}$. From Lem. 3, $\Theta^{\prime}$ is decomposed as

$\Theta^{\prime}=\Theta_{\psi}^{a}\oplus\Theta_{\psi}^{b},$ (43)

where $\Theta_{\psi}^{a}$ and $\Theta_{\psi}^{b}$ are antiunitary operator on
disjoint subspaces $\mathcal{H}_{\psi}^{a}$, $\mathcal{H}_{\psi}^{b}$
($\mathcal{H}^{\prime}=\mathcal{H}_{\psi}^{a}\oplus\mathcal{H}_{\psi}^{b}$),
respectively, such that $|\psi\rangle$ is not orthogonal to either
$\mathcal{H}_{\psi}^{a}$ or $\mathcal{H}_{\psi}^{b}$. For the two
decompositions (43) and $\Theta^{\prime}=\oplus_{j=1}^{N}\Theta^{(N)}_{j}$ to
be consistent, we necessarily have the subspace decomposition
$\mathcal{H}^{(N+1)}_{j}$ ($j=1,...,N+1$) of $\mathcal{H}^{\prime}$ defined by

$\displaystyle\mathcal{H}^{(N+1)}_{j}$
$\displaystyle=\mathcal{H}^{(N)}_{j},\leavevmode\nobreak\ (j\neq
k,\leavevmode\nobreak\ N+1)$ (44) $\displaystyle\mathcal{H}^{(N+1)}_{k}$
$\displaystyle=\mathcal{H}^{(N)}_{k}\wedge\mathcal{H}_{\psi}^{a},$ (45)
$\displaystyle\mathcal{H}^{(N+1)}_{N+1}$
$\displaystyle=\mathcal{H}^{(N)}_{k}\wedge\mathcal{H}_{\psi}^{b},$ (46)

and the corresponding decomposition
$\Theta^{\prime}=\oplus_{j=1}^{N+1}\Theta^{(N+1)}_{j}$ of antiunitary.

At the end of the procedure, we obtain the decomposition of antiunitary

$\Theta^{\prime}=\oplus_{j=1}^{\dim\mathcal{H}^{\prime}}\Theta^{\dim\mathcal{H}^{\prime}}_{j},$
(47)

into $1$-dimensional antiunitaries. Since any $1$-dimensional antiunitary must
be a conjugation, $\Theta^{\dim\mathcal{H}^{\prime}}_{j}$ is a conjugation for
all $j=1,...,N$, and hence $\Theta^{\prime}$ is, too.

Now that $\Theta^{\prime}$ is a conjugation and $\Theta_{i}$ can be replaced
by a conjugation $\theta_{i}$ on $\mathcal{H}_{i}$ for all $i\in I$,

$\theta:=\Theta^{\prime}\oplus\bigoplus_{i\in I}\theta_{i},$ (48)

is a conjugation on $\mathcal{H}$ which satisfies Eqs. (41) and (42). Thus
$\theta$ is a conjugation that gives the GAS for $\\{\rho_{\bm{x}}|{\bm{x}}\in
X\\}$. $\hfill\blacksquare$

When a conjugation $\theta$ provides a GAS for model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$, there is a basis that represents
$\rho_{\bm{x}}$ as a real matrix for all ${\bm{x}}\in X$. The model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is “free from imaginarity” in this sense.
Conversely, any model consisting only of real density matrices have a
conjugation as a GAS. Theorem 1, by identifying antiunitary-symmetric models
with conjugation-symmetric models, justifies the term “imaginarity-free” to be
used on models with GASs.

####  3.1.1 Antiunitary symmetry defined on horizontal lift

A lift of a quantum statistical model is its orbit in the associated fiber
bundle. For pure-state models, the defined GAS is equivalent to the
antiunitary symmetry on the so-called horizontal lift of the model. We briefly
explain this equivalence without any prerequisite knowledge on geometry here.
The content of this section is independent on other parts of this article. See
[73, 74] for complete expositions.

For a pure-state model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$, a lift is
identified with a parameterized unit vector $|\psi_{\bm{x}}\rangle$ in
$\mathcal{H}$ such that

$|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|=\rho_{\bm{x}},$ (49)

for any ${\bm{x}}\in X$. The lift is said to be horizontal, if it satisfies

$\mathrm{Im}\left[\langle\psi_{\bm{x}}|\partial_{i}\psi_{\bm{x}}\rangle\right]=0,$
(50)

for any $i$ and ${\bm{x}}\in X$. The horizontal lift exists if and only if the
model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is quasi-classical.

The global antiunitary symmetry (GAS) of a horizontal lift is defined as the
antiunitary operator $\Theta$ such that

$\Theta|\psi_{\bm{x}}\rangle=|\psi_{\bm{x}}\rangle,$ (51)

holds for any ${\bm{x}}\in X$. Now it is readily seen that $\Theta$ becomes a
GAS of the model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$. Conversely, if a model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ has a GAS $\Theta$, then there is a lift
satisfying (51). This lift is horizontal since $|\psi_{\bm{x}}\rangle$ and
$|\partial_{i}\psi_{\bm{x}}\rangle$ are in the same real subspace of
$\mathcal{H}$ invariant by the antiunitary $\Theta$.

The model has a GAS if its horizontal lift does, and they share the same
antiunitary operator.

###  3.2 Global antiunitary symmetry and weak commutativity

The weak commutativity is ensured by a GAS.

###### Lemma 4

If model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ has a GAS $\Theta$, there is an
SLD $L_{{\bm{x}},j}$ satisfying $\Theta
L_{{\bm{x}},j}\Theta^{\dagger}=L_{{\bm{x}},j}$ for any point ${\bm{x}}\in X$
and for any parameter $x_{j}$.

_proof_) Define the SLD by Eq. (9). Then we have

$\displaystyle\Theta L_{{\bm{x}},j}\Theta^{\dagger}$
$\displaystyle=2\int_{0}^{\infty}\Theta
e^{-t\rho_{\bm{x}}}\Theta^{\dagger}\Theta\partial_{j}\rho_{\bm{x}}\Theta^{\dagger}\Theta
e^{-t\rho_{\bm{x}}}\Theta^{\dagger}\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}e^{-t\Theta\rho_{\bm{x}}\Theta^{\dagger}}\partial_{j}\rho_{\bm{x}}e^{-t\Theta\rho_{\bm{x}}\Theta^{\dagger}}\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}e^{-t\rho_{\bm{x}}}\partial_{j}\rho_{\bm{x}}e^{-t\rho_{\bm{x}}}\mathrm{d}t=L_{{\bm{x}},j},$
(52)

where the second equality follows from
$\Theta\partial_{j}\rho_{\bm{x}}\Theta^{\dagger}=\partial_{j}\rho_{\bm{x}}$.
$\hfill\blacksquare$

###### Theorem 2

Any imaginarity-free model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is weakly
commutative at any point in $X$.

_proof_) Let $\Theta$ be the GAS of model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$.
From Lem. 4, there are SLDs satisfying $\Theta
L_{{\bm{x}},j}\Theta^{\dagger}=L_{{\bm{x}},j}$ for all $j$. For these SLDs we
have

$\displaystyle\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]^{\ast}$
$\displaystyle=\mathrm{Tr}[\Theta\rho_{\bm{x}}\Theta^{\dagger}\Theta
L_{{\bm{x}},i}\Theta^{\dagger}\Theta L_{{\bm{x}},j}\Theta^{\dagger}]$
$\displaystyle=\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}].$ (53)

Therefore the Uhlmann curvature (12) is zero for any pairs of $(i,j)$.
$\hfill\blacksquare$

According to Thm. 2, parameter encoding schemes are not crucial for compatible
multiparameter estimations. There are physical systems which provide only
weakly commutative models, irrespective to the parameter encoding schemes. For
example, Majorana fermions have antiunitary particle-hole symmetry. Thus any
models constituting of Majorana fermions are weakly commutative.

Quantum mechanics in real Hilbert spaces lacks an uncertainty principle
between observables of a specific form [75]. If $A$ and $B$ are observables
such that $A^{\ast}=A$ and $B^{\ast}=B$, the expectation value
$\mathrm{Tr}[\rho[A,B]]$ is zero for any real density operator
$\rho=\rho^{\ast}$. The Heisenberg uncertainty principle $\langle\Delta
A\rangle\langle\Delta B\rangle\geq|\langle[A,B]\rangle|/2$ is trivialized.
Nevertheless there is an uncertainty principles of a different form [75, 76].

We are not sure how Thm. 2 relates to the absence of Heisenberg uncertainty
principle. The link between uncertainty principles and incompatibility of
parameters should be clarified in more detail. Such an attempt has started
only recently [77].

Theorem 2 directly relates geometric phases and the phases of off-diagonal
elements in density operators. The mean Uhlmann curvature is the Uhlmann
geometric phase (per unit area) accumulated by infinitesimal loops around the
point [55]. The Uhlmann geometric phase coincides to Berry phase for pure
states [78]. If the phases of off-diagonal elements of the density operator
are all zero at the neighborhood of ${\bm{x}}\in X$, the geometric phases
accumulated by infinitesimal loops around the point is also zero.

Antiunitary symmetries are sometimes easier to check than the zero of mean
Uhlmann curvature. Consider thermal state of a parametrized Hamiltonian
$H_{\bm{x}}$ presented by a canonical ensemble $e^{-H_{\bm{x}}}/Z$. Here we
assume that the temperature is contained in the Hamiltonian as a parameter.
Such a model is rather studied in quantum statistical mechanics, than in
quantum metrology. Non-zero mean Uhlmann curvature characterizes quantum phase
transitions of the thermal state [55, 56, 57, 58, 59, 43]. Lemma 3 indicates
that $e^{-H_{\bm{x}}}/Z$ is imaginarity-free if and only if $H_{\bm{x}}$ does.
That is, if the Hamiltonian endows with an antiunitary symmetry such as time-
reverse, the thermal state has zero mean Uhlmann curvature.

One might wonder if QFIMs of imaginarity-free models have a particular form.
As we will see in Sec. 4.2, it is possible to construct an imaginarity-free
model with QFIM $\mathcal{F}_{Q}({\bm{x}})$ from arbitrary model with QFIM
$\mathcal{F}_{Q}({\bm{x}})/2$. GASs do not pose any restriction on the form of
QFIMs. As a consequence, imaginarity-freeness does not imply the stricter
notion of “compatibility” referring to the saturation of QCRBs with diagonal
QFIMs [28].

As a final remark, there are weakly commutative models without any GAS. This
implies that the converse of Thm. 2 is not true in general: weak commutativity
does not necessarily imply imaginarity-freeness. Even some single-parameter
models, such as off-equator phase estimation presented in Appx. A.1.2, do not
have any GAS.

###  3.3 Optimal target-independent measurements

Any imaginarity-free pure-state model is accompanied by a set of globally
optimal POVMs. This means that the QCRB is saturated by the corresponding
measurement at any target-point.

In this section, we set the number $N_{Q}$ of copies in joint measurements to
$1$, because we focus mainly on pure imaginarity-free models which are
automatically quasi-classical [25].

A set of optimal POVMs is known for antiparallel spins [54] (see Appx. A.1.3
for details). These POVM operators are first proposed in [53], and proven to
maximize a fidelity measure [79]. The POVMs must be independent from the
target point ${\bm{x}}$ in the parameter space, in order to be optimal for the
fidelity measure. Later the same set of operators is shown to saturate QCRB
[54], namely, it is optimal in terms of a precision bound as well.

The target-independent POVMs are fascinating because they are optimal all over
the parameter space. In general, around different regions in the parameter
space, we require different POVMs to achieve the best precision limit in the
region. This is the case even for single-parameter estimations. If the optimal
POVMs are independent of the target point, the precise measurement can be
carried out without any prerequisite knowledge on the source.

The following theorem presents target-independent optimal POVMs.

###### Theorem 3

Let $\Theta$ be an antiunitary on $\mathcal{H}$. Any POVM
$\mathcal{E}_{\Theta}$ satisfying conditions

$\displaystyle{\rm rank}E=1\leavevmode\nobreak\ $ $\displaystyle(\forall
E\in\mathcal{E}_{\Theta}),$ (54) $\displaystyle\Theta
E\Theta^{\dagger}=E\leavevmode\nobreak\ $ $\displaystyle(\forall
E\in\mathcal{E}_{\Theta}),$ (55)

saturates the QCRB (11) of a pure-state model $\\{\rho_{\bm{x}}|{\bm{x}}\in
X\\}$ with $N_{Q}=1$ everywhere in $X$ if $\Theta$ is a GAS of
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$.

Since CFIM is additive with respect to tensor products and product
measurements, the $N_{Q}$ product of antiunitary-invariant rank-1 measurements
saturates the QCRB (11) for any $N_{Q}$.

For proving Thm. 3, we use the following lemma to restrict GASs to
conjugations. Recall that any imaginarity-free model has a conjugation as a
GAS (Thm. 1).

###### Lemma 5

There exists a POVM $\mathcal{E}_{\Theta}$ on $\mathcal{H}$ satisfying Eqs.
(54) and (55) if and only if $\Theta$ is a conjugation.

_proof_) If $\Theta$ is a conjugation, projectors onto vectors of $\Theta$’s
reference-basis form a POVM with required properties.

Conversely, let $\mathcal{E}_{\Theta}=\\{q_{\omega}|E_{\omega}\rangle\langle
E_{\omega}|\\}_{\omega\in\Omega}$ ($q_{\omega}\in(0,1]$) be a POVM satisfying
conditions (54) and (55). From Lem. 2, we have
$\Theta^{2}|E_{\omega}\rangle=|E_{\omega}\rangle$ for all $\omega$. Because
$\\{|E_{\omega}\rangle\\}_{\omega\in\Omega}$ spans the entire Hilbert space
$\mathcal{H}$, we have $\Theta^{2}=\mathbb{I}_{\mathcal{H}}$. In other words,
$\Theta$ is a conjugation. $\hfill\blacksquare$
_proof of Thm. 3_) From Lem. 5, we can assume that the antiunitary operator is
a conjugation $\theta$.

Let $\mathcal{E}_{\theta}$ be a POVM satisfying conditions (54) and (55). From
Lem. 4, there are SLDs obeying the antiunitary symmetry $\theta
L_{{\bm{x}},j}\theta=L_{{\bm{x}},j}$. It suffices to show that conditions (16)
and (17) (Theorems 1 and 2 of [30]) hold for these antiunitary-invariant SLDs
and for any element in $\mathcal{E}$.

For this purpose, we choose a particular pure-state vector for
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$. Let $|\psi_{\bm{x}}\rangle$ be the
continuously parametrized unit vector such that
$|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|=\rho_{\bm{x}}$ and
$\theta|\psi_{\bm{x}}\rangle=|\psi_{\bm{x}}\rangle$ hold everywhere.
Equivalently, $\\{|\psi_{\bm{x}}\rangle\leavevmode\nobreak\
|\leavevmode\nobreak\ {\bm{x}}\in X\\}$ is defined to be the horizontal lift
of $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$.

Let $E$ be an element of POVM $\mathcal{E}$. Take a reference-basis
$\\{|\Pi^{E}_{k}\rangle\\}_{k=1,...,\dim\mathcal{H}}$ of $\theta$ so that

$|\Pi^{E}_{1}\rangle\langle\Pi^{E}_{1}|=q_{E}E,\leavevmode\nobreak\
\theta|\Pi^{E}_{k}\rangle=|\Pi^{E}_{k}\rangle,$ (56)

for $k=1,...,\dim\mathcal{H}$, where $q_{E}\in(0,1]$ is a positive
coefficient. SLDs are represented by real matrices

$L_{{\bm{x}},j}=\sum_{k,l=1}^{\dim\mathcal{H}}\ell^{j}_{kl}(E,{\bm{x}})|\Pi^{E}_{k}\rangle\langle\Pi^{E}_{l}|,$
(57)

in the basis $\\{|\Pi^{E}_{k}\rangle\\}_{k=1,...,\dim\mathcal{H}}$, where
$\ell^{j}_{kl}(E,{\bm{x}})$ are real coefficients. Since
$\\{|\Pi^{E}_{k}\rangle\\}_{k=1,...,\dim\mathcal{H}}$ spans the real Hilbert
subspace of $\mathcal{H}$ invariant under $\theta$, the vector
$|\psi_{\bm{x}}\rangle$ of the horizontal lift have the decomposition

$|\psi_{\bm{x}}\rangle=\sum_{k=1}^{\dim\mathcal{H}}r_{k}(E,{\bm{x}})|\Pi^{E}_{k}\rangle,$
(58)

with real coefficients $r_{k}(E,{\bm{x}})$.

Now we check the conditions (16) and (17) on $E$. Equation (58) alone implies

$E|\psi_{\bm{x}}\rangle=q_{E}r_{1}(E,{\bm{x}})|\Pi^{E}_{1}\rangle,$ (59)

and Eqs. (57) and (58) together imply

$EL_{{\bm{x}},j}|\psi_{\bm{x}}\rangle=q_{E}\left(\sum_{k=1}^{\dim\mathcal{H}}\ell^{j}_{1k}(E,{\bm{x}})r_{k}(E,{\bm{x}})\right)|\Pi^{E}_{1}\rangle.$
(60)

We find that $E|\psi_{\bm{x}}\rangle$ and
$EL_{{\bm{x}},j}|\psi_{\bm{x}}\rangle$ are all proportional to
$|\Pi^{E}_{k}\rangle$ by real coefficients. This implies that $E$ satisfies
conditions (16) and (17) at any point ${\bm{x}}\in X$.

Because $E$ is an arbitrary element in $\mathcal{E}$, conditions (16) and (17)
are satisfied by all elements in $\mathcal{E}$ everywhere.
$\hfill\blacksquare$

The optimality of antiunitary-invariant POVM is independent not only on the
target point, but also on how the parameters are encoded in the state. The
optimal POVMs depend only on the GAS, which in turn depends only on the states
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ as a set. Thus re-parametrization change
neither the GAS nor the optimal POVMs. One does not have to use different
POVMs according to different parametrization.

The condition on POVMs indicated by Thm. 3 is so loose that continuously many
such POVMs exist. To see this, let us consider a pure-state model whose GAS is
a conjugation $\theta$. The conjugation defines a
$\dim\mathcal{H}$-dimensional real Hilbert subspace of $\mathcal{H}$ spanned
by vectors satisfying $\theta|\psi\rangle=|\psi\rangle$. You can take any
vectors from this $\theta$-invariant real subspace to construct the optimal
POVM by $E=|\psi\rangle\langle\psi|$. For example, the reference basis for
representing a conjugation $\theta$ as a complex conjugation is not unique. In
Appx. A.1.3, we find two different sets of optimal POVMs for antiparallel
spins from its GAS. One of them is given in [53] and the other one is new. As
another example, rank-1 projectors distributed randomly over the real subspace
also saturates the QCRB [74].

While any antiunitary-invariant measurements work equally well, there should
be experimentally preferable ones, such as those requiring less entanglement.
Consider, for example, an estimation of parameters ${\bm{x}}$ encoded as a
unitary evolution $e^{-iH_{\bm{x}}}$. A typical method of quantum metrology
utilizes multiple copies of the same system driven by
$e^{-i\sum_{n=1}^{N}H^{(n)}_{\bm{x}}}=\otimes_{n=1}^{N}e^{-iH^{(n)}_{\bm{x}}}$.
If the $1$-particle Hamiltonian $H_{\bm{x}}$ has an antiunitary skew-symmetry
$\Theta H_{\bm{x}}\Theta^{\dagger}=-H_{\bm{x}}$, the overall Hamiltonian has
an antiunitary skew-symmetry under $\Theta^{\otimes N}$. We show in Sec. 4.1
how to construct imaginarity-free models based on Hamiltonians’ antiunitary
skew-symmetries. If you construct an imaginarity-free model based on the GAS
$\Theta^{\otimes N}$, the POVM must be invariant under $\Theta^{\otimes N}$.
You would prefer the measurement in the product POVM, each being invariant
under $\Theta$.

We apply the above reasoning to find the optimal measurement basis for 3D
magnetometry [15] in Sec. 3.4.3. There the GAS provides a parameter-
independent optimal measurement basis. The entanglement required for the
measurement is reduced from $N$-partite to $2$-partite. The search of
reasonable measurements is possible thanks to the freedom of the optimal
measurement for imaginarity-free models.

We close this section by extending the globally optimal POVMs to globally
optimal estimation strategies. Even if a target-independent measurement
saturates the QCRB everywhere, the optimal locally unbiased estimator could
still depend on the target-point.

Maximum likelihood estimators are independent on the target-point to be
estimated. The only condition for the maximum likelihood estimator to work
globally is the identifiability of the parameter. More precisely, the
probability distribution $p(\Omega|{\bm{x}})$ generated by the measurement
must be in one-to-one correspondence with the point ${\bm{x}}$. Projectors
onto a single reference-basis for $\theta$ will not produce sufficiently
informatic probability distributions to identify the parameters in general.

Fortunately, the condition on the optimal POVMs described in Thm. 3 is loose
enough to admit the POVMs to identify the parameters. Under the restrictions
(54) and (55), it is possible to construct a POVM
$\\{E_{\omega}\\}_{\omega\in\Omega}$ such that the map
$\rho\mapsto(\mathrm{Tr}[E_{\omega}\rho])_{\omega\in\Omega}$ from
$\theta$-invariant density operators
$\\{\rho\in\mathcal{B}(\mathcal{H})\leavevmode\nobreak\ |\leavevmode\nobreak\
\rho\geq 0,\leavevmode\nobreak\ \mathrm{Tr}[\rho]=1,\leavevmode\nobreak\
\theta\rho\theta=\rho\\}$ to $\Re^{\Omega}$ is one-to-one. An example of such
_real_ -informationally-complete POVMs associated to $\theta$ is the rank-1
projectors distributed randomly over the real subspace defined by $\theta$
[74]. Because the space of real density operators associated to $\theta$
contains $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ if $\theta$ is its GAS, the
probability distribution generated by $\\{E_{\omega}\\}_{\omega\in\Omega}$
identifies the parameter. In this way we obtain the following corollary:

###### Corollary 1

Let $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ be an imaginarity-free pure-state
model. Suppose that the model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is
identifiable in that the map ${\bm{x}}\mapsto\rho_{\bm{x}}$ is invertible.
Then there is a POVM $\Pi=\\{\Pi_{k}\\}_{k\in\Omega}$ on $\mathcal{H}$ with
which the maximum likelihood estimator
$\check{\bm{x}}:\Omega^{N_{C}}\rightarrow X$ satisfies

$\lim_{N_{C}\rightarrow\infty}N_{C}{\bm{\Sigma}}_{\bm{x}}(\Pi,\check{\bm{x}})=\mathcal{F}_{Q}({\bm{x}})^{-1},$
(61)

at any point ${\bm{x}}\in X$.

We can globally saturate QCRB by a fixed measurement and a maximum likelihood
estimator without any prior knowledge on the target-point, if the pure-state
model has a GAS. In practice, there is no need to stick to the real-
informationally-complete POVMs. One can employ POVMs that identify the state
from the model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ only.

###  3.4 Examples of imaginarity-free models

In this section, we introduce three examples of imaginarity-free models. These
are N00N states (Sec. 3.4.1), super-dense coding (Sec. 3.4.2), 3D magnetometry
(Sec. 3.4.3). The Venn diagram in FIG. 1 summarizes known classifications of
weakly commutative models together with imaginarity-free models.

Figure 1: Classifications of weakly commutative models. A model is said to be
classical when its states are simultaneously diagonalizable. Example models
provided in this article are classified to either of the marked regions. The
triangle contains off-equator phase estimation of a spin (Appx. A.1.2). The
square contains N00N states (Sec. 3.4.1), super-dense coding (Sec. 3.4.2), 3D
magnetometry (Sec. 3.4.3), and antiparallel spins (Appx. A.1.3). The circle
contains depolarized antiparallel spins (Appx. A.1.3) and qubit inside a disc
of Bloch sphere (Appx. A.2).

The examples presented in this section all correspond to the square symbol in
FIG. 1. To ensure the existences of models at each region of the diagram, we
present some other examples in Appendix A.

In this section again, we set the number $N_{Q}$ of copies in joint
measurements to $1$.

####  3.4.1 N00N states

The first example is a single-parameter model. The N00N state of optical
interferometry is a milestone of quantum metrology and is still under
theoretical and experimental studies. See, e.g. [51, 52, 2] for theoretical
reviews and [80] for a review including experiments.

The pure-state model with a phase parameter $\phi$ is given by

$|\psi_{\phi}\rangle:=\frac{|N,0\rangle+e^{-iN\phi}|0,N\rangle}{\sqrt{2}},$
(62)

where $|N_{1},N_{2}\rangle$ represents a two-mode state having $N_{1}$ and
$N_{2}$ photons in mode 1 and mode 2, respectively. We can readily see that
the N00N state distributes on the equator of Bloch sphere with poles
$|N,0\rangle$ and $|0,N\rangle$, and thus can be represented by real density
matrices in a suitable basis. Any antiunitary operator $\Theta$ satisfying

$\displaystyle\Theta c|0,N\rangle=c^{\ast}|N,0\rangle,$
$\displaystyle\leavevmode\nobreak\ \Theta c|N,0\rangle=c^{\ast}|0,N\rangle,$
(63) $\displaystyle\hskip 56.9055pt(\forall c\in\mathbb{C})$

is a GAS for N00N state model (62). From the GAS (63) we recover a target-
independent optimal measurement basis including vectors

$\frac{|N,0\rangle\pm|0,N\rangle}{\sqrt{2}}.$ (64)

The measurement on this basis is carried out by a combination of a beam-
splitter and a photon-number-counting measurement.

Note that not all the pure-state single-parameter models are imaginarity-free
in general. A counterexample is provided in Appx. A.1.2 by an off-equator
phase estimation of a spin. Even for single-parameter models, the globally
optimal measurement is not guaranteed if the model does not have a GAS.

####  3.4.2 Super-dense coding

Fujiwara [66] considered an estimation of SU$(2)$ unitary $U$ represented on
$2$-dimensional space. The model is given by

$\left\\{\rho_{U}=(U\otimes\mathbb{I})\rho(U\otimes\mathbb{I})^{\dagger}\middle|U\in\mathrm{SU}(2)\right\\},$
(65)

where $\rho$ is a $2$-qubit state. The unitary is represented by
$3$-parameters. We consider pure input state
$\rho=|\psi_{r}\rangle\langle\psi_{r}|$ defined by
$|\psi_{r}\rangle=\sqrt{r}|0,0\rangle+\sqrt{1-r}|1,1\rangle$ with a constant
$r\in[0,1]$. This is a parameter estimation version of super-dense coding
[81]. The model is weakly commutative if and only if $r=1/2$, namely, when
$\rho$ is maximally entangled [66].

Model $\\{\rho_{U}|U\in\mathrm{SU}(2)\\}$ is indeed imaginarity-free for
maximally entangled $\rho$. For conjugation $\theta$ in the reference basis
$\\{|0\rangle,\leavevmode\nobreak\ |1\rangle\\}$ we have

$\displaystyle(\theta\otimes\theta)(U\otimes\mathbb{I})$
$\displaystyle|\psi_{1/2}\rangle=(\theta
U\theta\otimes\mathbb{I})|\psi_{1/2}\rangle$
$\displaystyle=-(\sigma_{Y}U\sigma_{Y}\otimes\mathbb{I})|\psi_{1/2}\rangle$
(66)
$\displaystyle=(\sigma_{Y}\otimes\sigma_{Y})(U\otimes\mathbb{I})|\psi_{1/2}\rangle$
(67)

where we have used identities
$(\sigma_{Y}\otimes\mathbb{I})|\psi_{1/2}\rangle=(\mathbb{I}\otimes\sigma_{Y}^{\mathrm{T}})|\psi_{1/2}\rangle$
and $\theta U\theta=-\sigma_{Y}U\sigma_{Y}$ holding for any SU$(2)$ unitary
$U$. Thus the model is symmetric under the product conjugation
$\theta\otimes\theta$ followed by $\sigma_{Y}\otimes\sigma_{Y}$.

With the aid of GAS we can construct the target-independent optimal
measurement with $N_{Q}=1$ which is not addressed in [66]. Since Eq. (67)
holds for $U=i\sigma_{k}$ ($k=X,Y,Z$, $\sigma_{k}$ represents the Pauli-$k$
operator), projectors onto the Bell measurement basis
$\\{|\psi_{1/2}\rangle,\leavevmode\nobreak\
(i\sigma_{k}\otimes\mathbb{I})|\psi_{1/2}\rangle\leavevmode\nobreak\
|\leavevmode\nobreak\ k=X,Y,Z\\}$ forms a rank-1 antiunitary-invariant optimal
POVM. The measurement used by the receiver of original super-dense coding
scheme [81] is an optimal measurement for the parameter-estimation version.

If the input state is not maximally entangled, namely, if $r\neq 1/2$, there
is no unitary $V$ on $\mathbb{C}^{2}$ such that
$(\sigma_{Y}\otimes\mathbb{I})|\psi_{r}\rangle=(\mathbb{I}\otimes
V)|\psi_{r}\rangle$ holds. Therefore it is impossible to transform Eq. (66) to
Eq. (67) by replacing $\sigma_{Y}\otimes\sigma_{Y}$ to $\sigma_{Y}\otimes V$.
Indeed, there is no GAS when $r\neq 1/2$ because the model (65) is not weakly
commutative for these $r$ [66].

Note that we did not require the explicit expression of parametrized unitaries
to find a GAS. Theorem 2 enables the shortcut in deducing weak commutativity.

####  3.4.3 3D magnetometry

3D magnetometry has an $N$ spin system interacting with the external magnetic
field [15]. The Hamiltonian is

$H_{\bm{\varphi}}:=\sum_{i=1}^{N}{\bm{\varphi}}\cdot{\bm{\sigma}}_{i},$ (68)

where ${\bm{\varphi}}=(\varphi_{X},\varphi_{Y},\varphi_{Z})\in\Re^{3}$ is
proportional to the magnetic field, and ${\bm{\sigma}}_{i}$ is the vector of
Pauli matrices for $i$th spin. The parameter to be estimated is
${\bm{\varphi}}$. The $N$-spin system undergoes unitary transformation
$e^{-iH_{\bm{\varphi}}}$. The model of 3D magnetometry is

$\left\\{e^{-iH_{\bm{\varphi}}}|\Psi\rangle\langle\Psi|e^{iH_{\bm{\varphi}}}\middle|{\bm{\varphi}}\in\Re^{3}\right\\},$
(69)

where $|\Psi\rangle$ is the initial state to be optimized.

Inspired by the single-parameter phase estimation, in [15] they considered

$|\Psi\rangle:=\frac{|\Psi_{X}^{GHZ}\rangle+e^{i\delta_{Y}}|\Psi_{Y}^{GHZ}\rangle+e^{i\delta_{Z}}|\Psi_{Z}^{GHZ}\rangle}{\mathcal{N}},$
(70)

as the initial state. Here $\mathcal{N}$ is the normalization factor,
$\delta_{Y},\delta_{Z}\in[0,2\pi]$ are constants to be chosen for making the
model weakly commutative. State
$|\Psi_{k}^{GHZ}\rangle=(|\psi_{k}^{+}\rangle^{\otimes
N}+|\psi_{k}^{-}\rangle^{\otimes N})/\sqrt{2}$ is a $N$-partite GHZ state with
eigenstates of Pauli operators
$\sigma_{k}|\psi_{k}^{\pm}\rangle=\pm|\psi_{k}^{\pm}\rangle$. For later
convenience, we choose the phases of $|\psi_{k}^{\pm}\rangle$ to have

$\displaystyle|\psi_{X}^{+}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
1\end{array}\right),\leavevmode\nobreak\ $
$\displaystyle|\psi_{X}^{-}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-1\end{array}\right),$ (75)
$\displaystyle|\psi_{Y}^{+}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
i\end{array}\right),\leavevmode\nobreak\ $
$\displaystyle|\psi_{Y}^{-}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}i\\\
1\end{array}\right),$ (80)
$\displaystyle|\psi_{Z}^{+}\rangle=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\leavevmode\nobreak\ $
$\displaystyle|\psi_{Z}^{-}\rangle=\left(\begin{array}[]{c}0\\\
1\end{array}\right).$ (85)

We consider the symmetry under spin-flip on $N$ systems,

$\Theta_{f}^{\otimes N},$ (86)

where $\Theta_{f}=\sigma_{Y}\theta$ and $\theta$’s reference basis is
$\\{|\psi^{+}_{Z}\rangle,\leavevmode\nobreak\ |\psi^{-}_{Z}\rangle\\}$. Since
the spin-flip changes the sign of Pauli operators
${\bm{\sigma}}\mapsto-{\bm{\sigma}}$, the unitary evolution
$e^{-iH_{\bm{\varphi}}}$ is invariant under spin-flip. Thus the spin-flip is a
GAS of the model if the initial state $|\Psi\rangle$ is also invariant under
the spin-flip.

Now we derive the values of $\delta_{Y}$ and $\delta_{Z}$ to make the model
imaginarity-free. From our choice of $|\psi_{k}^{\pm}\rangle$ follow

$\displaystyle\Theta_{f}|\psi_{k}^{+}\rangle=i|\psi_{k}^{-}\rangle,\leavevmode\nobreak\
$ $\displaystyle\Theta_{f}|\psi_{k}^{-}\rangle=-i|\psi_{k}^{+}\rangle,$ (87)
$\displaystyle(k=X,Y,Z)$

and hence

$\displaystyle\Theta_{f}^{\otimes
N}|\Psi_{k}^{GHZ}\rangle=\left\\{\begin{array}[]{lr}\frac{|\psi_{k}^{+}\rangle^{\otimes
N}-|\psi_{k}^{-}\rangle^{\otimes N}}{(-i)^{N}\sqrt{2}}&(N:\mathrm{odd}),\\\
(-1)^{N/2}|\Psi_{k}^{GHZ}\rangle&(N:\mathrm{even}),\end{array}\right.$ (90)

for any integer $N$. Therefore $\Theta_{f}^{\otimes
N}|\Psi\rangle\propto|\Psi\rangle$ is the case when
$\delta_{Y}=0,\leavevmode\nobreak\ \pi$ and $\delta_{Z}=0,\leavevmode\nobreak\
\pi$ for $N=4m$, and $\delta_{Y}=\pi/2,\leavevmode\nobreak\ 3\pi/2$ and
$\delta_{Z}=\pi/2,\leavevmode\nobreak\ 3\pi/2$ for $N=4m+2$ with some integer
$m$. The model $e^{-iH_{\bm{\varphi}}}|\Psi\rangle$ has a GAS and thus is
weakly commutative for these $\delta_{Y}$ and $\delta_{Z}$. We are not sure if
the difference between our solutions on $(\delta_{Y},\delta_{Z})$ with those
found in [15] is entirely due to the choice of phases since vectors
$|\psi_{k}^{\pm}\rangle$ are not explicitly indicated in [15].

Baumgratz and Datta [15] found an optimal measurement basis including the
target state
$e^{-iH_{\bm{\varphi}}}|\Psi\rangle\langle\Psi|e^{iH_{\bm{\varphi}}}$, which
is $N$-partite entangled. They also considered projectors

$\Pi_{k,\pm}=\frac{\mathbb{I}\pm\sigma_{k}^{\otimes
N}}{6},\leavevmode\nobreak\ (k=1,2,3)$ (91)

and numerically exhibited that these POVMs are close to the optimal. The
projector is invariant under the GAS (86) since

$\Theta_{f}^{\otimes N}\Pi_{k,\pm}\left(\Theta_{f}^{\otimes
N}\right)^{\dagger}=\frac{\mathbb{I}\pm(-1)^{N}\sigma_{k}^{\otimes
N}}{6}=\Pi_{k,\pm},$ (92)

holds for all $k=1,2,3$ and even $N$. That the high-rank projectors
$\Pi_{k,\pm}$ almost achieves the ultimate precision limit is still non-
trivial.

We derive different measurement bases which are guaranteed to be optimal from
the GAS. The GAS $\Theta_{f}^{\otimes N}$ is a tensor product of $N/2$
conjugations $\Theta_{f}^{\otimes 2}$. Equation (87) implies that for
$k=X,Y,Z$, the basis

$\left\\{\frac{|\psi_{k}^{-},\psi_{k}^{-}\rangle\pm|\psi_{k}^{+},\psi_{k}^{+}\rangle}{-\sqrt{2}i},\leavevmode\nobreak\
\frac{|\psi_{k}^{-},\psi_{k}^{+}\rangle\pm|\psi_{k}^{+},\psi_{k}^{-}\rangle}{\sqrt{2}}\right\\},$
(93)

is a reference basis for the bipartite conjugation $\Theta_{f}^{\otimes 2}$,
and thus $\Theta_{f}^{\otimes N}$ has the $N/2$-product of (93) as a reference
basis. This implies that the bipartite measurements in the basis (93),
performed independently on $N/2$ copies, is optimal.

##  4 Methods to implement global antiunitary symmetries

In this section we consider methods to construct imaginarity-free models. The
benefit of implementing the GASs is twofold. First, the weak commutativity of
the model is ensured at any point (Thm. 2). Second, the models have parameter-
independent optimal POVMs (Thm. 3 and Cor. 1), when the models are pure.

We focus on quantum metrology, and thus assume some ability to modify the
physical systems to adjust the models. If the model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ cannot be adjusted, the attainable
precision does not improve even if we implement antiunitary symmetry on
$\rho_{\bm{x}}$ by pre-measurement operations. The assumed ability is to
replace the initial states before interactions.

Even if you can adjust the models, you may still wonder if the restriction to
GASs is worth taking. GASs inevitably restrict available models. There may be
a model without GAS but still offers better precisions for all the parameters.
In short, you may wonder if compatible models offer maximum precisions.
Unfortunately, previous methods to design compatible models for quantum
metrology did not consider optimizations on the attainable precision [25, 28,
33].

This question has almost been ignored since it was difficult to calculate the
optimal precision bounds for incompatible models. To propose a metrology
scheme, you have to evaluate the precisions attained by the scheme. If the
model is compatible, it suffices to calculate the QCRB. Otherwise you have to
calculate the so-called Holevo Cramér-Rao bound [34, 11, 35, 36, 37, 38, 39],
whose closed-form expression is missing. This has been one of the reasons for
researchers to seek for compatible models. However, methods to calculate the
Holevo Cramér-Rao bound has been developed [27, 25, 40, 41, 42, 44], and a
semi-definite programming is available now [14]. The complexity is further
reduced when you require only a lower bound of the Holevo Cramér-Rao bound
[45]. The final blow on the QCRB is its modest advantage: the Holevo Cramér-
Rao bound secures at least half of the QCRB even in the worst case [43]. This
encourages the pursuit for better precisions in the realm of incompatible
models.

Our answer to the question on the simultaneous realizability of best
precisions and GASs is twofold.

In Sec. 4.1, we consider a particular generator of unitary transformation. The
optimal initial state for the phase estimation inevitably leads to
imaginarity-free models. The GAS coexists with the optimal initial state, and
provides optimal global measurements at the same time.

In Sec. 4.2, we propose a different method to implement antiunitary symmetry
by generalizing antiparallel spins. The method implements GASs without
deteriorating the precision limit of a given model, under some symmetry
conditions.

###  4.1 Antiunitary skew-symmetric Hamiltonians

The first method is inspired by 3D magnetometry. Consider a model whose state
given by

$e^{-iH_{\bm{x}}}\rho_{\rm ini}e^{iH_{\bm{x}}},$ (94)

where $H_{\bm{x}}$ is a parametrized Hamiltonian. The Hamiltonian is assumed
to have a skew-symmetry

$\Theta H_{\bm{x}}\Theta^{\dagger}=-H_{\bm{x}},$ (95)

under an antiunitary $\Theta$. If there is a state $|\psi_{\rm ini}\rangle$
satisfying

$\Theta|\psi_{\rm ini}\rangle=\epsilon|\psi_{\rm ini}\rangle,$ (96)

with some unimodular $\epsilon$, $\Theta$ becomes a GAS of the model (94) by
setting the initial state to $\rho_{\rm ini}=|\psi_{\rm
ini}\rangle\langle\psi_{\rm ini}|$. The existence of antiunitary skew-symmetry
(95) is the key for implementing the GAS.

Now we question the optimality of the antiunitary-invariant initial state (96)
for estimating ${\bm{x}}$. The N00N state reviewed in Sec. 3.4.1 is
antiunitary-invariant and, at the same time, the best initial state for the
phase estimation on interferometry. We show in the following that the
simultaneous realization of the optimal initial state and the GAS is not a
mere coincidence.

Optimal initial states are known for particular single-parameter estimations.
Consider a quantum system driven by a traceless generator $H$. Now we are
given a task to estimate the single parameter $t$ from the state

$e^{-iHt}\rho_{\rm ini}e^{iHt},$ (97)

where the initial state $\rho_{ini}$ is of our choice. The best initial state
that maximizes the quantum Fisher information for $t$ of the state (97) is
given by [82]

$\rho_{\rm ini}=|\psi_{\rm ini}\rangle\langle\psi_{\rm
ini}|,\leavevmode\nobreak\ \leavevmode\nobreak\ |\psi_{\rm
ini}\rangle=\frac{|\lambda_{+}\rangle+e^{i\phi}|\lambda_{-}\rangle}{\sqrt{2}},$
(98)

where $\phi\in[0,2\pi]$ is arbitrary, and $|\lambda_{+}\rangle$ and
$|\lambda_{-}\rangle$ are eigenstates of $H$ with maximum and minimum
eigenvalues $\lambda_{+}$ and $\lambda_{-}$, respectively. The optimal state
maximizes the quantum Fisher information by maximizing the uncertainty in
expecting $H$ [83, 84].

We can readily see that the resulting model

$e^{-iHt}|\psi_{\rm
ini}\rangle=\frac{e^{-i\lambda_{+}t}|\lambda_{+}\rangle+e^{i(\phi-\lambda_{-})t}|\lambda_{-}\rangle}{\sqrt{2}},$
(99)

is imaginarity-free. The state lies on a great circle of the Bloch sphere of
the qubit system spanned by $|\lambda_{+}\rangle$ and $|\lambda_{-}\rangle$.
Therefore the state is represented with real coefficients in a suitable basis.

When the Hamiltonian has a specific form, a GAS of the model (99) is derived
from the antiunitary skew-symmetry of the Hamiltonian. Suppose that the
spectrum of the generator $H$ distributes symmetrically about the origin. We
further assume that the eigensubspaces $\mathcal{H}_{\lambda}$ and
$\mathcal{H}_{-\lambda}$ belonging to eigenvalues $\lambda$ and $-\lambda$,
respectively, have the same dimension. Then an operator $\Theta$ on
$\mathcal{H}$ defined by

$\displaystyle\Theta\sum_{\lambda}\sum_{d=1}^{\dim\mathcal{H}_{\lambda}}c_{\lambda,d}|\lambda,d\rangle=\sum_{\lambda}\sum_{d=1}^{\dim\mathcal{H}_{\lambda}}c_{\lambda,d}^{\ast}|-\lambda,d\rangle,$
(100) $\displaystyle\hskip
28.45274pt\left(|\lambda_{+},1\rangle:=|\lambda_{+}\rangle,\leavevmode\nobreak\
|\lambda_{-},1\rangle:=|\lambda_{-}\rangle\right)$

is an antiunitary. This antiunitary is a symmetry of the initial state. We
have $\Theta|\psi_{\rm ini}\rangle=e^{-i\phi}|\psi_{\rm ini}\rangle$ and hence

$\Theta\rho_{\rm ini}\Theta^{\dagger}=\rho_{\rm ini},$ (101)

by definition. Moreover, the skew-symmetry

$\Theta H\Theta^{\dagger}=-H,$ (102)

of the generator under the action of $\Theta$ follows from the definition.

Combining Eqs. (101) and (102), we arrive at the equality

$\Theta e^{-iHt}\rho_{\rm ini}e^{iHt}\Theta^{\dagger}=e^{-iHt}\rho_{\rm
ini}e^{iHt},$ (103)

meaning that $\Theta$ is a GAS of the model (97) for any optimal initial
state. Its optimal global measurements are lead by Thm. 3 and Cor. 1.

The optimization of initial state presented above underlies single-parameter
metrology schemes including the N00N state reviewed in Sec. 3.4.1.

In summary, if the spectrum and the eigensubspaces of generator $H$ distribute
symmetrically about the origin, the model (97) with any optimal initial state
(98) is imaginarity-free. The resulting model with optimal initial state
provides the target-independent optimal measurement to attain the best QCRB.
For a special generator of this kind, the optimization of initial state
naturally leads to imaginarity-free models.

This observation dispels a doubt that GASs spoil advantages of quantum
statistical models. Imaginarity-free models look close to classical ones in
that the freedom of imaginary numbers disappears, that all the parameters are
compatible, and that global optimal measurements exist. The classical behavior
can, however, coexist with the most effective metrological schemes offered by
quantum models. In this sense, the imaginarity-free estimations benefits from
both of classical and quantum estimations.

###  4.2 Antiparallel model

Our second method requires the state $\rho_{\bm{x}}$ of the original model and
its conjugation $\rho_{\bm{x}}^{\ast}$. This is a non-trivial requirement on
the physical system since there is no completely-positive map to transform
given state $\rho_{\bm{x}}$ to its conjugation $\rho_{\bm{x}}^{\ast}$. We
consider how to produce mutually conjugate state pairs later. See FIG. 2 for a
schematic representation of our second proposal.

Figure 2: Methods and advantages to implement GASs by antiparallel models. The
task is to estimate the values of parameters encoded in a Hamiltonian
$H_{\bm{x}}$ driving the system. The initial states you prepare go through the
parameter-dependent unitary evolution $e^{-iH_{\bm{x}}}$. They are output as
the parametrized states $\rho_{\bm{x}}$, on which you perform measurements to
estimate the parameters. (a) If the parametrized states do not form a weakly
commutative model, QCRB cannot be saturated. Even if the model is weakly
commutative, optimal measurements may depend on the output state, and require
adaptive measurements $\Pi_{1},\Pi_{2},...$ in general. (b) If the model is
imaginarity-free, the weak commutativity is guaranteed. If the outputs are
pure states, there is a target-independent optimal measurement $\Pi$. Sometime
it is possible to produce an antiparallel model, with implemented GAS. This is
the case at least when the Hamiltonian has an antiunitary skew-symmetry
$\Theta H_{\bm{x}}\Theta=-H_{\bm{x}}$ (Sec. 4.2.1), and when the inverse state
evolution $e^{iH_{\bm{x}}}$ is available (Sec. 4.2.2).

We propose to use, if possible, half-conjugated copies
$\rho_{\bm{x}}\otimes\rho_{\bm{x}}^{\ast}$ instead of the normal copies
$\rho_{\bm{x}}\otimes\rho_{\bm{x}}$.

###### Definition 2

Let $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ be a quantum statistical model.
_Antiparallel model_ for $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is defined by

$\left\\{\rho_{\bm{x}}\otimes\rho_{\bm{x}}^{\ast}\middle|{\bm{x}}\in
X\right\\},$ (104)

where $\cdot^{\ast}$ represents a complex conjugation in a fixed basis.

The antiparallel model has a GAS

$S\theta^{\otimes 2},$ (105)

_independently_ to the original model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$,
where $S$ represents the swap operator
$S|\psi,\psi^{\prime}\rangle=|\psi^{\prime},\psi\rangle$. Thus the
antiparallel model is weakly commutative at any point in the parameter space
(Thm. 2) _independently_ to the original model $\\{\rho_{\bm{x}}|{\bm{x}}\in
X\\}$.

The antiparallel model does not deteriorate the attainable precision. QFIMs of
the original model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$
($\mathcal{F}_{Q}({\bm{x}})$), of the antiparallel model (104)
($\mathcal{F}_{Q}^{\mathrm{anti}}({\bm{x}})$), and of “parallel model”
$\\{\rho_{\bm{x}}\otimes\rho_{\bm{x}}|{\bm{x}}\in X\\}$
($\mathcal{F}_{Q}^{\mathrm{para}}({\bm{x}})$) are related by

$\mathcal{F}_{Q}^{\mathrm{para}}({\bm{x}})=2\mathcal{F}_{Q}({\bm{x}})=\mathcal{F}_{Q}^{\mathrm{anti}}({\bm{x}})$
(106)

at any point ${\bm{x}}\in X$. The second equality is obtained directly from
SLDs for the antiparallel model

$L_{{\bm{x}},i}^{\rm anti}:=L_{{\bm{x}},i}\otimes\mathbb{I}+\mathbb{I}\otimes
L_{{\bm{x}},i}^{\ast},\leavevmode\nobreak\ (i=1,...,n),$ (107)

where $L_{{\bm{x}},i}$ are SLDs of the original model. Since the antiparallel
model is weakly commutative, the QCRB (106) is asymptotically
($N_{Q}\rightarrow\infty$) attainable on the antiparallel model. The same
bound is not attainable on the original incompatible model and the parallel
model.

A reference basis to represent the GAS (105) as a complex conjugation is

$\left\\{|i,i\rangle,\frac{|i,j\rangle+|j,i\rangle}{\sqrt{2}},\frac{|i,j\rangle-|j,i\rangle}{\sqrt{2}i}\middle|i\neq
j\right\\},$ (108)

where $|i,j\rangle$ are the product basis for the complex conjugation. When
the model is pure, the corresponding projectors form an optimal set of POVMs
to achieve QCRB (106) with $N_{Q}=1$ _independently_ to the original model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ (Thm. 3).

A notable theoretical aspect of “antiparallelization” is its model-
independence. According to the precision bound (106), antiparallel state pair
$\rho_{\bm{x}}\otimes\rho_{\bm{x}}^{\ast}$ always surpasses the parallel one
$\rho_{\bm{x}}\otimes\rho_{\bm{x}}$ with respect to the attainable precision
limit. Mutually conjugated state pairs have revealed their intrinsic utility
both theoretically [53, 79, 85, 86, 87, 88, 54] and experimentally [89]. Since
these works all focus on restricted models such as spins, the extendible
advantages of mutually conjugated state pairs has been kept unknown. The
reduction of incompatibility manifests a model-independent benefit of
conjugate state pairs.

A potential intuitive reason why the conjugated state pair offers better mean
square errors than the mere pair is that the former distributes over larger
operator-space compared to the latter. Volumes of spin ensembles characterized
by entropy or code dimension are considered to be positively correlated to the
fidelity in estimating the direction of spins from the sources of ensembles
[53, 90, 91, 88]. We can define a state ensemble from a quantum statistical
model, by, for example, assuming uniform distribution over the parameter
space. Note, however, that mean square errors are defined locally at each
point unlike the ensemble volumes and the fidelity. Therefore mean square
error should be compared to certain local quantity such as rate of volume
growth around each point, rather than to the ensemble volume itself. We leave
the investigation on the ensemble volume of general antiparallel models as a
future work.

Note that the antiparallel model is not the only method to implement a GAS
independently of the original model. Appendix A.3 introduces a different kind
of compatible models which are inspired by embedding quantum simulators [67,
68, 69, 70, 71, 72].

There are instances where the mutually conjugate states $\rho_{\bm{x}}$ and
$\rho_{\bm{x}}^{\ast}$ are available at the same time. Here we consider
quantum metrological schemes to estimate parameters encoded in a Hamiltonian
$H_{\bm{x}}$. Tasks of quantum metrology abstract to estimations of parameters
from states $e^{-iH_{\bm{x}}}|\psi\rangle$ with the initial state
$|\psi\rangle$ of your choice. The antiparallel models additionally require
the conjugated states $\theta e^{-iH_{\bm{x}}}|\psi\rangle=(\theta
e^{-iH_{\bm{x}}}\theta)|\theta\psi\rangle$. If it is possible to produce
arbitrary initial states, the remaining task is to realize conjugate state
evolution $\theta e^{-iH_{\bm{x}}}\theta=e^{i\theta H_{\bm{x}}\theta}$.

In the following subsections we introduce two major instances where the
conjugate state evolution is available.

####  4.2.1 Hamiltonians’ skew-symmetry

Let us assume that a Hamiltonian $H_{\bm{x}}$ flips its sign

$\Theta H_{\bm{x}}\Theta^{\dagger}=-H_{\bm{x}},$ (109)

under an antiunitary $\Theta=U\theta$. In this case, the conjugated state
evolution $e^{i\theta H_{\bm{x}}\theta}$ can be decomposed to

$e^{i\theta H_{\bm{x}}\theta}=U^{\dagger}e^{i\Theta
H_{\bm{x}}\Theta^{\dagger}}U=U^{\dagger}e^{-iH_{\bm{x}}}U.$ (110)

Namely, the conjugate state evolution is unitarily equivalent to the original
evolution $e^{-iH_{\bm{x}}}$. Since the original evolution is available, the
conjugated state

$U^{\dagger}e^{-iH_{\bm{x}}}U|\theta\psi\rangle,$ (111)

can be produced by implementing unitary transformations $U$ and $U^{\dagger}$
before and after the evolution.

Note that if there is a state $|\psi\rangle$ such that
$\Theta|\psi\rangle=\epsilon|\psi\rangle$ with some unimodular $\epsilon$, the
model $e^{-iH_{\bm{x}}}|\psi\rangle$ itself obtains the GAS $\Theta$ because
$\Theta e^{-iH_{\bm{x}}}|\psi\rangle=\epsilon e^{-iH_{\bm{x}}}|\psi\rangle$
holds (see Sec. 4.1 for details). This is how 3D magnetometry and models in
Sec. 4.1 acquire their GASs.

####  4.2.2 Inverse transformation

Suppose that the inverse transformation $e^{iH_{\bm{x}}}$ is available in
addition to $e^{-iH_{\bm{x}}}$. Let $\theta$ be any conjugation that leaves
$H_{\bm{x}}$ invariant. For example, one may take the eigenbasis of
$H_{\bm{x}}$ as the reference basis for conjugation. Then for any initial
state $|\psi\rangle$,

$e^{iH_{\bm{x}}}|\theta\psi\rangle,$ (112)

is the conjugate state of $e^{-iH_{\bm{x}}}|\psi\rangle$.

##  5 Discussions

###  5.1 Antiunitary invariant POVMs on mixed states

The antiunitary-invariant POVMs do not necessarily saturate QCRBs for mixed
models. This is partly because multipartite measurements ($N_{Q}\geq 2$) are
necessary for saturating QCRBs in general. Any single-copy measurement cannot
saturate the QCRB for such a model. Theorem 3 does not generalizes to models
with mixed states.

Now you may wonder what if the imaginarity-free mixed-state model is quasi-
classical. Any quasi-classical model is accompanied by a (potentially target-
dependent) single-copy measurement to saturate the QCRB. Does our antiunitary-
invariant POVM saturate QCRBs for quasi-classical mixed models?

The answer is “no.” Even if a quasi-classical mixed state model has a GAS, the
antiunitary-invariant POVM does not necessarily form the optimal POVM. A
counterexample is given in Appx. A.1.4 by the off-equator estimation on
depolarized antiparallel spins. Our antiunitary-invariant POVM is not optimal
to this quasi-classical mixed model.

The antiunitary-invariant POVMs do not seem to have any special property in
the mixed state regime. In Appx. A.1.4, we demonstrate that different
antiunitary-invariant POVMs give different CFIMs for the off-equator phase
estimation on depolarized antiparallel spins. This implies that the
antiunitary-invariant POVMs do not share anything in terms of the measurement
precisions at the mixed state regime. Their coincidence is only at the pure
state limit.

###  5.2 Local antiunitary symmetry

We have seen that the converse of Thm. 2 is not true. There are weakly
commutative models without any GAS. To refine Thm. 2, we compare weak
commutativity and antiunitary symmetry at each point in the parameter spaces.

We are motivated from the following geometric perspective. Consider
application of Thm. 2 to subspaces of the whole parameter space. If $X_{1}$ is
a subspace of the parameter space $X$ and if there is a GAS
$\Theta_{1}\rho_{\bm{x}}\Theta_{1}^{\dagger}=\rho_{\bm{x}}$ inside $X_{1}$,
then the model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is weakly commutative at
all points in $X_{1}$. If there are such “small” GASs
$\Theta_{i}\rho_{\bm{x}}\Theta_{i}^{\dagger}=\rho_{\bm{x}}$ for covering
subspaces $X_{i}$ $(i\in I,\leavevmode\nobreak\ X=\cup_{i\in I}X_{i})$, then
the model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is weakly commutative at all
points in $X$. Note that $\Theta_{i}$s do not necessarily coincide.

Now a question is, whether any weakly commutative models have such covering
subspaces endowed with antiunitary symmetries. While global weak commutativity
does not imply GAS, it may imply patchworks of the small antiunitary
symmetries. If so, weak commutativity and the existence of antiunitary
symmetries should be equivalent at sufficiently small regions. In the light of
Thm. 3, such a “local” antiunitary symmetry might be useful for constructing
target-_dependent_ optimal POVMs.

The answer to the question is positive for pure-state models and negative in
general. We show that all weakly commutative pure-state models have local
antiunitary symmetries. However, this result does not extend to mixed-state
models. The weakly commutativity does not imply local antiunitary symmetries.

We define the local antiunitary symmetry (LAS) as a straightforward limitation
of the GAS to neighborhoods of a point.

###### Definition 3

A model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is said to have a LAS at
${\bm{x}}\in X$, if there is an antiunitary operator $\Theta$ such that
$\Theta\rho_{\bm{x}}\Theta^{\dagger}=\rho_{\bm{x}}$ and
$\Theta\partial_{i}\rho_{\bm{x}}\Theta^{\dagger}=\partial_{i}\rho_{\bm{x}}$
hold for all $i$.

To see how LAS relates to the GAS, consider the first-order approximation of
state $\rho_{{\bm{x}}+\Delta{\bm{x}}}$ around the point ${\bm{x}}\in X$:

$\rho_{{\bm{x}}+\Delta{\bm{x}}}\approx\rho_{\bm{x}}+\sum_{i=1}^{n}\Delta
x_{i}\partial_{i}\rho_{\bm{x}},$ (113)

where $\Delta x_{i}$ are small real numbers. Then the LAS $\Theta$ satisfies

$\Theta\rho_{{\bm{x}}+\Delta{\bm{x}}}\Theta^{\dagger}\approx\rho_{{\bm{x}}+\Delta{\bm{x}}},$
(114)

in the first-order approximation. Thus if there is a LAS at ${\bm{x}}\in X$,
the states in a small neighborhood around ${\bm{x}}$ share an approximately
good GAS.

LAS does imply weak commutativity at each point. Furthermore, these two are
equivalent for pure-state models.

###### Theorem 4

If a model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ has a LAS at ${\bm{x}}\in X$,
it is weakly commutative at ${\bm{x}}\in X$. Pure models have LASs at points
where the models are weakly commutative.

The proof of Thm. 4 is delegated to Appx. B.1. The off-equator phase
estimation on spin, for example, have LASs at all points, despite having no
GAS. This implies that the piecewise antiunitary symmetries do not necessarily
joint together to form a single antiunitary symmetry.

The LAS accompanies the optimal POVMs to saturate QCRB for pure-state models
(see Appx. B.2 for details). Similarly to the case of GAS (Thm. 3), they are
the rank-$1$ POVM invariant under the LAS. Since the LAS may depend on the
target point, the optimal POVMs are also target-dependent in general.

Regarding mixed states, there are weakly commutative models without any LAS.
We can further say that there are quasi-classical models without any LAS when
$\dim\mathcal{H}\geq 3$, but not when $\dim\mathcal{H}=2$. Examples are
presented in Appx. B.3. The structure of compatibility is more involved than
to be explained by antiunitary symmetries.

###  5.3 Quantifying antiunitary asymmetry

In this section, we try quantifying antiunitary asymmetry of quantum
statistical models. Theorems 2 and 4 both states that incompatibility, a
unique property of quantum mechanics, is absent for models with antiunitary
symmetry. Conversely, a model behaves quantum mechanically when it does not
have antiunitary symmetry.

There will be many functions measuring antiunitary asymmetry of a quantum
statistical model. For a model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ at a point
${\bm{x}}\in X$ define

$\displaystyle M(\rho_{\bm{x}})$
$\displaystyle:=\min_{\begin{subarray}{c}\Theta\in\mathcal{U}^{\ast}(\mathcal{H})\\\
\Theta\rho_{\bm{x}}\Theta^{\dagger}=\rho_{\bm{x}}\end{subarray}}\frac{1}{n}\mathrm{Tr}\left[\sum_{i=1}^{n}(\partial_{i}\rho_{\bm{x}}-\Theta\partial_{i}\rho_{\bm{x}}\Theta^{\dagger})^{2}\right],$
(115)

where $\mathcal{U}^{\ast}(\mathcal{H})$ denotes the set of antiunitary
operators on $\mathcal{H}$. This measure is faithful in that it takes zero if
and only if there is a _local_ antiunitary symmetry at ${\bm{x}}\in X$. It is
also invariant under isometric coordinate transformations ${\bm{x}}\mapsto
R{\bm{x}}$ ($R$: isometric matrix, see Appx. 5.3 for details).

We are inspired by imaginarity measures formulated in the resource theory to
define the asymmetry measure (115). Originally, the imaginarity of a single
quantum state is measured with respect to a fixed basis
$\\{|k\rangle\\}_{k=1,...,d}$ [60, 61, 62], or equivalently to a fixed
conjugation $\theta$. The formalisms resemble resource theories of coherence
[92, 93, 94], in that they both require a fixed reference basis.

In contrast, the weak commutativity is implied by any conjugation. Such a
basis-independent definition of imaginarity has not been studied yet. However,
there is a basis-independent definition of coherence termed “set coherence”
[95]. In Appx. C we define “set imaginarity” by following the procedure for
set coherence, and extend it to asymmetry measures for quantum statistical
models.

The faithful measure of antiunitary asymmetry characterizes quantumness of the
model, similarly to an incompatibility measure. Carollo _et. al._ [43]
introduced a faithful measure of incompatibility. As being faithful, it takes
zero only if the mean Uhlmann curvature is zero. The incompatibility measure
is used to detect quantumness of phase transitions through the incompatibility
of parametrized thermal states [43]. Belliardo and Giovannetti [33] introduced
another faithful measure of incompatibility as a ratio between attainable
precision and the precision based on SLD QFIMs.

Faithful asymmetry measures could be more sensitive to systems’ quantumness
than the incompatibility is. As is presented in Appx. B.3, there are models
with zero incompatibility but with positive antiunitary asymmetry.
Furthermore, imaginarity can be defined on single-parameter models, while the
incompatibility requires more than $2$ parameters. Thus measures of
antiunitary asymmetry could be more useful than the incompatibility measure
for characterizing quantum phase transitions. We hope to study this in more
detail on a different occasion.

##  6 Conclusion

We introduced a class of quantum statistical models endowed with symmetries
under antiunitary operators. A model of this class consists only of real
density matrices. The null phase of density matrices implies the null mean
Uhlmann curvature, enabling the compatible multiparameter estimation. A
variety of existing models have antiunitary symmetries.

If a pure-state model has an antiunitary symmetry, any rank-$1$ antiunitary-
invariant POVM saturates the QCRB. Once you find the antiunitary symmetry, you
discover an optimal measurement at the same time. The antiunitary-invariant
measurement is optimal independently of the target point to be estimated. To
perform such a measurement, you require neither prerequisite knowledge on the
target, nor the adaptive measurement to guess the target. By utilizing the
freedom of the optimal measurements, we derived an antiunitary-invariant
measurement for 3D magnetometry, which requires only bipartite entanglement
irrespective to the number of spins. The antiunitary-invariant measurement is,
however, not necessarily optimal for mixed-state models.

We proposed methods to implement antiunitary symmetries in several settings of
quantum metrology.

For phase estimations of particular unitary evolutions, all the optimal
initial states lead to imaginarity-free models. Antiunitary symmetries can
coexist with optimal metrological schemes in this setting.

Mutually conjugate state pair $\rho_{\bm{x}}\otimes\rho_{\bm{x}}^{\ast}$ has
better precision bounds than $\rho_{\bm{x}}\otimes\rho_{\bm{x}}$ does,
independently of the model $\rho_{\bm{x}}$. The conjugate pair has an
antiunitary symmetry independent of the original model $\rho_{\bm{x}}$. We
consider how to produce the conjugated state $\rho_{\bm{x}}^{\ast}$ when a
parametrized unitary evolution generates $\rho_{\bm{x}}$. This is possible
especially if the Hamiltonian has an antiunitary skew-symmetry, or if the
inverse state evolution is also available.

Finally, we studied antiunitary symmetries varying along the target points of
a model. A pure-state model is compatible if and only if it has the target-
dependent antiunitary symmetry. Based on the study, we introduced a function
to measure antiunitary asymmetry of quantum statistical models at given target
points. The measure is faithful and invariant under isometric coordinate
transformations on the parameter space.

Our results have room for extension. Despite no mention here, infinite
dimensional Hilbert spaces have unique uncertainty relations. Target-
independent optimal measurements are, including their existence, still unknown
for mixed state models. Extending our results in these untouched regions will
help develop various metrology schemes such as superresolution [47, 96, 97,
48, 98, 31].

We are currently relying on physicists’ intuition to discover the antiunitary
symmetry of quantum statistical models. While intuition sometimes works
perfectly on physics-born models, a general algorithm to find the antiunitary
symmetry (if it exists) would be much more effective. Such an algorithm will
be useful not only for quantum metrology, but also for understanding quantum
phase transitions via asymmetry measures.

## Acknowledgement

JM would like to thank M. T. Quintino, M. Araújo and J. Sidhu for valuable
discussions.

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##  Appendix A Additional examples of imaginarity-free models

###  A.1 Antiparallel spins

Some pure quasi-classical models cease to be quasi-classical when they pass
depolarizing channels. We demonstrate this phenomena with antiparallel spins
in A.1.3. In contrast, depolarizing channels preserve the existence and non-
existence of GAS. For $\delta\in(0,1)$, the depolarized model

$\left\\{(1-\delta)\rho_{\bm{x}}+\delta\frac{\mathbb{I}}{\dim\mathcal{H}}\middle|{\bm{x}}\in
X\right\\},$ (116)

is imaginarity-free if and only if the original model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ is. In fact,
$\Theta\rho_{\bm{x}}\Theta^{\dagger}=\rho_{\bm{x}}$ and
$\Theta\mathbb{I}\Theta^{\dagger}=\mathbb{I}$ implies
$\Theta(p\rho_{\bm{x}}+q\mathbb{I})\Theta^{\dagger}=p\rho_{\bm{x}}+q\mathbb{I}$
for any real numbers $p$ and $q$. Depolarizing channels break quasi-
classicality while preserving GASs.

####  A.1.1 Spin

Before proceeding to the weakly commutative models, we review a model that is
itself not weakly commutative. For spherical coordinates $\eta\in[0,\pi]$ and
$\phi\in[0,2\pi]$ define a qubit state on $2$-dimensional space
$\mathcal{H}_{2}$

$\displaystyle\rho_{(\eta,\phi)}=\frac{\mathbb{I}_{2}+{\mathbf{n}}_{(\eta,\phi)}\cdot{\bm{\sigma}}}{2},$
(117)
$\displaystyle\left({\mathbf{n}}_{(\eta,\phi)}=(\sin\eta\cos\phi,\sin\eta\sin\phi,\cos\eta)\right)$

where ${\bm{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of
unitary Pauli matrices. The Bloch sphere representation of a spin is given in
FIG. 3 (a). The mean Uhlmann curvature of model

$\left\\{\rho_{(\eta,\phi)}\middle|\eta\in[0,\pi],\leavevmode\nobreak\
\phi\in[0,2\pi]\right\\},$ (118)

is given by

$\displaystyle\mathcal{U}=\left[\begin{array}[]{cc}\mathcal{U}_{\eta\eta}&\mathcal{U}_{\eta\phi}\\\
\mathcal{U}_{\phi\eta}&\mathcal{U}_{\phi\phi}\\\
\end{array}\right]=\frac{1}{2}\left[\begin{array}[]{cc}0&\sin\eta\\\
-\sin\eta&0\\\ \end{array}\right].$ (123)

Thus (118) is weakly commutative only at the north and south poles
$\eta=0,\pi$. We call the state $\rho_{(\eta,\phi)}$ and the model (118) as a
spin and a spin model, respectively. The spin model turns weakly commutative
with some modifications presented in FIG. 3 (b) and (c).

Figure 3: Bloch sphere representations of the spin model and its variants. (a)
A (single) spin model. The state is represented by the Bloch vector, which is
parametrized by the spherical coordinates (with unit radius). (b) The off-
equator phase estimation on spin model. The latitude $\eta$ is regarded as a
constant. (c) The antiparallel spins. The Bloch vectors are directing opposite
to each other.

####  A.1.2 Off-equator phase estimation

Fixing the latitude $\eta$ of the spin to a constant $c$ results in a single-
parameter model of phase $\phi\in[0,2\pi]$. States of the single-parameter
model

$\left\\{\rho_{(c,\phi)}\middle|\phi\in[0,2\pi]\right\\},$ (124)

form a circle in the Bloch sphere with a constant latitude $c\in(0,\pi)$ (a
schematic representation is in FIG. 3 (b)).

The model does not have any GAS except at the equator $c=\pi/2$. At the
equator, the spin-flip followed by $\pi$-rotation along the $z$-axis becomes a
GAS. Outside the equator, the circular orbits are passed to the other
hemisphere by the spin-flip. There is no unitary to return the whole circles
to the original hemispheres [86].

####  A.1.3 Antiparallel spins

Antiparallel spins are two spins directing the opposite to each other. A
schematic representation is given in FIG. 3 (c). The antiparallel spin model
is represented by

$\left\\{\rho_{(\eta,\phi)}\otimes\Theta_{f}\rho_{(\eta,\phi)}\Theta_{f}^{\dagger}\middle|\eta\in[0,\pi],\leavevmode\nobreak\
\phi\in[0,2\pi]\right\\},$ (125)

since the spin-flip operator $\Theta_{f}$ flips a spin to its opposite
direction.

Gisin and Popescu [53] showed that the spin direction
${\mathbf{n}}_{(\eta,\phi)}$ can be better guessed from the antiparallel spins
than from parallel spins. Later Chang _et. al._ [54] demonstrated the
advantage of antiparallel spins for the estimation of spherical coordinates
$(\eta,\phi)$. The weak commutativity of antiparallel spins is shown there.
Since the model consists of pure states, it must be quasi-classical [25].

The antiparallel spins has a GAS

$S\Theta_{f}^{\otimes 2},$ (126)

where $\Theta_{f}^{\otimes 2}$ is the tensor product of spin-flips, and $S$ is
the operator which swaps the sides of bipartite system
$\mathcal{H}_{2}\otimes\mathcal{H}_{2}$. The antiunitary operator
$\Theta_{f}^{\otimes 2}$ flips the directions of spins on both sides, and then
unitary $S$ swaps the spins to recover the original state. The GAS (126) is a
conjugation despite $\Theta_{f}$ is not.

Depolarized antiparallel spins

$(1-\delta)\rho_{(\eta,\phi)}\otimes\Theta_{f}\rho_{(\eta,\phi)}\Theta_{f}^{\dagger}+\delta\frac{\mathbb{I}_{4}}{4}.$
(127)

are no longer quasi-classical, nor partially commutative (see Appx. A.1.4 for
details). Nevertheless, the GAS (126) is preserved by the depolarized channel.

Let us consider the optimal POVMs for antiparallel spins. Vectors

$|\Pi_{k}\rangle=\frac{\sqrt{3}+1}{2\sqrt{2}}|{\mathbf{n}}_{k},-{\mathbf{n}}_{k}\rangle+\frac{\sqrt{3}-1}{2\sqrt{2}}|-{\mathbf{n}}_{k},{\mathbf{n}}_{k}\rangle,$
(128)

with properly chosen four Bloch vectors ${\mathbf{n}}_{k}$ ($k=1,...,4$) are
known to form an optimal measurement basis for antiparallel spins [54]. Here,
$|{\mathbf{n}}\rangle$ represents a state vector with Bloch vector
${\mathbf{n}}$ (see [53, 54] for details on the phase factors). This set of
vectors indeed satisfy

$S\Theta_{f}^{\otimes 2}|\Pi_{k}\rangle=-|\Pi_{k}\rangle,$ (129)

and thus $\\{i|\Pi_{k}\rangle\\}_{k=1,...,4}$ is a reference basis that
represents $S\Theta_{f}^{\otimes 2}$ as a mere complex conjugation.

There are different reference bases for the same conjugation
$S\Theta_{f}^{\otimes 2}$. An example is

$\left\\{i|0,1\rangle,\leavevmode\nobreak\ i|1,0\rangle,\leavevmode\nobreak\
\frac{|0,0\rangle+|1,1\rangle}{\sqrt{2}},\leavevmode\nobreak\
\frac{|0,0\rangle-|1,1\rangle}{\sqrt{2}i}\right\\},$ (130)

where $\\{|0,0\rangle,\leavevmode\nobreak\ |0,1\rangle,\leavevmode\nobreak\
|1,0\rangle,\leavevmode\nobreak\ |1,1\rangle\\}$ now represents the
computational basis. Later we see that our basis (130) sometimes outperforms
(128) in terms of the precision for depolarized antiparallel spins (Sec. 5.1).
This is unexpected because the basis (128) provided by Gisin and Popescu [53]
looks much specialized to the estimation of ${\mathbf{n}}_{(\eta,\phi)}$. In
fact, each component of (128) is the optimal covariant measurement vector to
maximize a fidelity measure for this specific problem [79].

####  A.1.4 Classical and quantum Fisher information matrices for antiparallel
spins

We first show that the depolarized antiparallel spins is not quasi-classical.
Then we present the deviation $\Delta\phi^{2}$ of single-parameter $\phi$
stemming from POVM measurements on depolarized antiparallel spins. The latter
corresponds to the precision of the off-equator phase estimation on the
depolarized antiparallel spins.

The state of depolarized antiparallel spins is defined by Eq. (127). A pair of
SLDs for depolarized antiparallel spins is given by

$L^{\rm
anti}_{x}=(1-\delta)\left(L_{x}\otimes\mathbb{I}_{2}+\mathbb{I}_{2}\otimes\Theta_{f}L_{x}\Theta_{f}^{\dagger}\right),$
(131)

for $x=\eta,\phi$, where $L_{x}$ is the SLD for single spin, given by

$\displaystyle L_{\eta}$
$\displaystyle=\left[\begin{array}[]{cc}-\sin\eta&e^{-i\phi}\cos\eta\\\
e^{i\phi}\cos\eta&\sin\eta\\\ \end{array}\right],$ (134) $\displaystyle
L_{\phi}$ $\displaystyle=\left[\begin{array}[]{cc}0&-ie^{-i\phi}\sin\eta\\\
ie^{i\phi}\sin\eta&0\\\ \end{array}\right].$ (137)

The SLD operators for the depolarized antiparallel spins is unique since the
state is of full-rank. The commutator $[L^{\rm anti}_{\eta},L^{\rm
anti}_{\phi}]$ of SLD operators reduces to

$(1-\delta)^{2}\left([L_{\eta},L_{\phi}]\otimes\mathbb{I}_{2}+\mathbb{I}_{2}\otimes\Theta_{f}[L_{\eta},L_{\phi}]\Theta_{f}\right).$
(138)

This never becomes zero since $[L_{\eta},L_{\phi}]$ is not proportional to
$\mathbb{I}_{2}$. Thus the depolarized antiparallel spins is not quasi-
classical. It is not partially commutative as being of full-rank.

The deviation $\Delta\phi^{2}$ can be calculated as the inverse of the
classical Fisher information about $\phi$. We here provide a full analysis on
the two CFIMs for reference-basis measurements (128) and (130). Note that for
general depolarized state
$(1-\delta)\rho_{\bm{x}}+\delta\mathbb{I}/\dim\mathcal{H}$ and a projector
$\Pi_{k}$, probability to have the corresponding result $k$ is

$\displaystyle\mathrm{Tr}\left[\left((1-\delta)\rho_{\bm{x}}+\frac{\delta\mathbb{I}}{\dim\mathcal{H}}\right)\Pi_{k}\right]$
(139)
$\displaystyle=(1-\delta)\mathrm{Tr}[\rho_{\bm{x}}\Pi_{k}]+\frac{\delta}{\dim\mathcal{H}}.$
(140)

The $i,j$-component of the CFIM is

$[\mathcal{F}_{C}({\bm{x}})]_{ij}=(1-\delta)\sum_{k}\frac{\partial_{i}\mathrm{Tr}[\rho_{\bm{x}}\Pi_{k}]\partial_{j}\mathrm{Tr}[\rho_{\bm{x}}\Pi_{k}]}{\mathrm{Tr}[\rho_{\bm{x}}\Pi_{k}]+\frac{\delta}{4(1-\delta)}}.$
(141)

The CFIM is calculated from the probabilities
$\mathrm{Tr}[\rho_{\bm{x}}\Pi_{k}]$ to obtain the result $k$ by the
measurement on the original state.

Let us first consider the basis (128) given by Gisin and Popescu [53]. The
probabilities to obtain result $k$ ($k=1,...,4$) can be found, for example, in
the appendix of [54]:

$\displaystyle\langle\Pi_{k}|\rho_{(\eta,\phi)}\otimes\Theta_{f}\rho_{(\eta,\phi)}\Theta_{f}^{\dagger}|\Pi_{k}\rangle=\frac{1}{3}A_{k}(\eta,\phi)^{2},$
(142) $\displaystyle
A_{k}(\eta,\phi):=\sin\eta\cos\left(\phi-\frac{(k-1)2\pi}{3}\right)$
$\displaystyle\hskip
85.35826pt-\frac{\sqrt{2}}{4}\cos\eta+\frac{\sqrt{6}}{4}.$ (143)

The corresponding elements of CFIM for the depolarized antiparallel spins are
given by

$\displaystyle[\mathcal{F}_{C}]_{ij}=$ $\displaystyle\frac{4(1-\delta)}{3}$
$\displaystyle\times$
$\displaystyle\sum_{k=1}^{4}\frac{A_{k}(\eta,\phi)^{2}B_{k}^{i}(\eta,\phi)B_{k}^{j}(\eta,\phi)}{A_{k}(\eta,\phi)^{2}+\frac{3\delta}{4(1-\delta)}},$
(144) $\displaystyle B_{k}^{\eta}(\eta,\phi):=$
$\displaystyle\cos\eta\cos\left(\phi-\frac{(k-1)2\pi}{3}\right)$
$\displaystyle\hskip 71.13188pt-\frac{\sqrt{2}}{4}\sin\eta,$ (145)
$\displaystyle B_{k}^{\phi}(\eta,\phi):=$
$\displaystyle-\sin\eta\sin\left(\phi-\frac{(k-1)2\pi}{3}\right).$ (146)

The deviation $\Delta\phi^{2}$ for this measurement is given by
$[\mathcal{F}_{C}({\bm{x}})]_{\phi\phi}^{-1}$.

If our measurement (130) is applied on the pure antiparallel spins, the
probabilities are

$\displaystyle\langle
0,1|\rho_{(\eta,\phi)}\otimes\Theta_{f}\rho_{(\eta,\phi)}\Theta_{f}^{\dagger}|0,1\rangle=\frac{(1+\cos\eta)^{2}}{4},$
(147) $\displaystyle\langle
1,0|\rho_{(\eta,\phi)}\otimes\Theta_{f}\rho_{(\eta,\phi)}\Theta_{f}^{\dagger}|1,0\rangle=\frac{(1-\cos\eta)^{2}}{4},$
(148) $\displaystyle\frac{(\langle 0,0|\pm\langle
1,1|)\rho_{(\eta,\phi)}\otimes\Theta_{f}\rho_{(\eta,\phi)}\Theta_{f}^{\dagger}(|0,0\rangle\pm|1,1\rangle)}{2}$
$\displaystyle=\frac{\sin^{2}\eta(1\mp\cos 2\phi)}{4}.$ (149)

The corresponding elements of CFIM are given by

$\displaystyle\left[\mathcal{F}_{C}\right]_{\eta\eta}=$
$\displaystyle(1-\delta)\left\\{\frac{\sin^{2}\eta(1+\cos\eta)^{2}}{(1+\cos\eta)^{2}+\frac{\delta}{1-\delta}}+\frac{\sin^{2}\eta(1-\cos\eta)^{2}}{(1-\cos\eta)^{2}+\frac{\delta}{1-\delta}}\right\\}$
$\displaystyle\hskip
28.45274pt+(1-\delta)\left\\{\frac{\sin^{2}\eta\cos^{2}\eta(1+\cos
2\phi)^{2}}{\sin^{2}\eta(1+\cos
2\phi)+\frac{\delta}{1-\delta}}+\frac{\sin^{2}\eta\cos^{2}\eta(1-\cos
2\phi)^{2}}{\sin^{2}\eta(1-\cos 2\phi)+\frac{\delta}{1-\delta}}\right\\},$
(150) $\displaystyle\left[\mathcal{F}_{C}\right]_{\eta\phi}=$
$\displaystyle(1-\delta)\left\\{\frac{\sin^{3}\eta\cos\eta\sin 2\phi(1-\cos
2\phi)}{\sin^{2}\eta(1-\cos
2\phi)+\frac{\delta}{1-\delta}}-\frac{\sin^{3}\eta\cos\eta\sin 2\phi(1+\cos
2\phi)}{\sin^{2}\eta(1+\cos 2\phi)+\frac{\delta}{1-\delta}}\right\\},$ (151)
$\displaystyle\left[\mathcal{F}_{C}\right]_{\phi\phi}=$
$\displaystyle(1-\delta)\left\\{\frac{\sin^{4}\eta\cos^{2}2\phi}{\sin^{2}\eta(1+\cos
2\phi)+\frac{\delta}{1-\delta}}+\frac{\sin^{4}\eta\cos^{2}2\phi}{\sin^{2}\eta(1-\cos
2\phi)+\frac{\delta}{1-\delta}}\right\\}.$ (152)

The deviation $\Delta\phi^{2}$ is again given by the inverse of
$[\mathcal{F}_{C}({\bm{x}})]_{\phi\phi}$.

The QFIM of pure antiparallel model is

$\mathcal{F}_{Q}^{\rm anti}=2\left[\begin{array}[]{cc}1&0\\\
0&\sin^{2}\eta\end{array}\right]=2\mathcal{F}_{Q}^{\rm spin},$ (153)

where $\mathcal{F}_{Q}^{\rm spin}$ is the QFIM of single spin. While this
relation can be derived by a direct calculation, it is expected by Eq. (106).
The depolarization deteriorates QFIM to

$\mathcal{F}_{Q}^{\rm anti}=2(1-\delta)^{2}\left[\begin{array}[]{cc}1&0\\\
0&\sin^{2}\eta\end{array}\right].$ (154)

Again the minimum of deviation $\Delta\phi^{2}$ is the inverse
$[\mathcal{F}_{Q}^{\rm anti}]_{\phi\phi}=1/2(1-\delta)^{2}\sin^{2}\eta$. This
bound is the best attained by the (potentially target-dependent) optimal
measurements each on a single copy of the depolarized antiparallel spin pairs.

We take the latitude $\eta$ as a constant and consider the single-parameter
estimation of phase $\phi$, which is trivially quasi-classical. This is the
antiparallel version of off-equator phase estimation. The deviation
$\Delta\phi^{2}$ based on the measurements in reference bases (128) and (130)
is compared to the optimal deviation in FIG. 4.

Figure 4: Performances of the reference-basis measurements for the off-equator
phase estimation on depolarized antiparallel spins. The deviation
$\Delta\phi^{2}=[\mathcal{F}_{C}]_{\phi\phi}^{-1}$ stemming from measurements
in the bases (128) (dotted solid lines) and (130) (dashed lines) are
represented as functions of $\delta$ and compared to the optimal bound
$[\mathcal{F}_{Q}]_{\phi\phi}^{-1}$ (black solid line). The target points are
$(\eta,\phi)=(3\pi/4,\pi/8)$ (red lines) and $(3\pi/4,\pi/4)$ (blue lines),
respectively. These measurements are optimal only at the pure limit
$\delta=0$. See Appendix A.1.4 for analytic expressions of $\Delta\phi^{2}$.

The deviations are equal to the optimal value only at the pure limit
($\delta=0$), and otherwise lager. As the coloured lines in FIG. 4 show, there
is a separation of CFIMs calculated from two measurement bases (128) and
(130).

###  A.2 Qubit inside a disc

Suzuki [27] gave an weakly commutative model of a qubit

$\displaystyle\left\\{\rho_{\bm{x}}=\frac{\mathbb{I}_{2}+{\mathbf{n}}_{\bm{x}}\cdot{\bm{\sigma}}}{2}\middle|\begin{array}[]{l}{\mathbf{n}}_{\bm{x}}=f_{1}({\bm{x}}){\mathbf{n}}_{1}+f_{2}({\bm{x}}){\mathbf{n}}_{2},\\\
{\bm{x}}\in X\end{array}\right\\},$ (157)

where $f_{1}$ and $f_{2}$ are arbitrary real functions of the parameter
${\bm{x}}\in X$ such that $|{\mathbf{n}}_{\bm{x}}|\leq 1$, and
${\mathbf{n}}_{1}$ and ${\mathbf{n}}_{2}$ are arbitrary (not necessarily
orthogonal) unit vectors on $3$-dimensional real space. In the Bloch sphere
representation, the qubit is in a disc spanned by two vectors
${\mathbf{n}}_{1}$ and ${\mathbf{n}}_{2}$. The disc cross the origin $(0,0,0)$
and cuts the sphere into halves. This model is not quasi-classical [27].

The qubit inside a disc (157) is imaginarity-free. The state transforms under
the spin-flip $\Theta_{f}$ as

$\frac{\mathbb{I}_{2}+{\mathbf{n}}_{\bm{x}}\cdot{\bm{\sigma}}}{2}\mapsto\frac{\mathbb{I}_{2}-{\mathbf{n}}_{\bm{x}}\cdot{\bm{\sigma}}}{2}.$
(158)

Mutually opposite vectors ${\mathbf{n}}_{\bm{x}}$ and $-{\mathbf{n}}_{\bm{x}}$
are always in the same plane which includes the origin $(0,0,0)$. Thus the
spin-flip followed by the unitary $\pi$-rotation along this plane takes any
state $\rho_{\bm{x}}$ to itself.

It is crucial for the disc to cross the origin. Otherwise the model does not
have a GAS.

###  A.3 Embedding quantum simulators

Here we introduce a class of model inspired by embedding quantum simulators
[67, 68, 69, 70, 71, 72]. _EQS model_ for pure state model
$\\{|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}||{\bm{x}}\in X\\}$ is defined by

$\left\\{\rho_{\bm{x}}^{\mathrm{EQS}}:=|\Psi_{\bm{x}}^{\mathrm{EQS}}\rangle\langle\Psi_{\bm{x}}^{\mathrm{EQS}}|\in\mathcal{B}(\mathcal{H}\otimes\mathcal{H}_{2})\middle|{\bm{x}}\in
X\right\\},$ (159)

where

$|\Psi_{\bm{x}}^{\mathrm{EQS}}\rangle:=\frac{|\psi_{\bm{x}}\rangle\otimes|0\rangle+(\theta|\psi_{\bm{x}}\rangle)\otimes|1\rangle}{\sqrt{2}},$
(160)

with some conjugation $\theta$ on $\mathcal{H}$, where $\mathcal{H}_{2}$ is a
two dimensional ancillary system with an orthonormal basis
$\\{|0\rangle,|1\rangle\\}$. The transformation
$|\psi_{\bm{x}}\rangle\mapsto|\Psi_{\bm{x}}^{\mathrm{EQS}}\rangle$ is not
realized as a completely positive map. Extra resources are required to produce
the state $|\Psi_{\bm{x}}^{\mathrm{EQS}}\rangle$.

The global phase subtlety matters when modifying to EQS models. The
modifications to EQS models result in different models from
$|\psi_{\bm{x}}\rangle$ and from $|e^{i\phi({\bm{x}})}\psi_{\bm{x}}\rangle$
where $\phi({\bm{x}})$ is a parameter-dependent global phase. While these two
original models are equivalent, their EQS models demonstrates different
precision bounds.

Embedding quantum simulators typically utilize the encoding of state
$|\psi_{\bm{x}}\rangle$ in the form

$\frac{|\mathrm{Re}\psi_{\bm{x}}\rangle\otimes|0\rangle+|\mathrm{Im}\psi_{\bm{x}}\rangle\otimes|1\rangle}{\sqrt{2}}$
(161)

instead of $|\Psi_{\bm{x}}^{\mathrm{EQS}}\rangle$ [67]. With (161) we arrive
at the same precision bound since it is unitarily equivalent to
$|\Psi_{\bm{x}}^{\mathrm{EQS}}\rangle$.

A GAS of EQS model (159) is

$(\mathbb{I}_{\mathcal{H}}\otimes\sigma_{X})(\theta\otimes\theta_{2}),$ (162)

where $\theta\otimes\theta_{2}$ is the conjugation in the product basis
$|i\rangle\otimes|j\rangle$ such that $\theta|i\rangle=|i\rangle$
($i=1,...,\dim\mathcal{H},\leavevmode\nobreak\ j=0,1$). A reference basis to
represent (162) as a complex conjugation is

$\left\\{|i\rangle\otimes\frac{|0\rangle+|1\rangle}{\sqrt{2}},\leavevmode\nobreak\
|i\rangle\otimes\frac{|0\rangle-|1\rangle}{\sqrt{2}i}\middle|\leavevmode\nobreak\
i=1,...,\dim\mathcal{H}\right\\}.$ (163)

Thus a product measurement saturates QCRB.

QFIMs of EQS models are sometimes improved from the original models. QFIM of
an EQS model relates to that of the original model by

$\mathcal{F}_{Q}^{\mathrm{EQS}}({\bm{x}})=\mathcal{F}_{Q}({\bm{x}})+{\bm{X}}{\bm{X}}^{\mathrm{T}},$
(164)

where

${\bm{X}}^{\mathrm{T}}=2i\left(\langle\partial_{1}\psi_{\bm{x}}|\psi_{\bm{x}}\rangle\leavevmode\nobreak\
...\leavevmode\nobreak\
\langle\partial_{n}\psi_{\bm{x}}|\psi_{\bm{x}}\rangle\right),$ (165)

is a real vector. It is futile to modify EQS models again to their EQS models
to obtain even better precision bound. The second modifications are equivalent
to mere unitary transformations.

##  Appendix B Local antiunitary symmetry

This section technically supports the discussion of Sec. 5.2 on LAS by
providing proofs. We prove Thm. 4 in Sec. B.1. Section B.2 derives optimal
local measurements from LAS. Section B.3 gives examples of quasi-classical
models without any antiunitary symmetry.

Following lemma is frequently employed throughout this section.

###### Lemma 6

A model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ has a LAS $\Theta$ at ${\bm{x}}\in
X$ if and only if (1) $\Theta\rho\Theta^{\dagger}=\rho_{\bm{x}}$ holds and (2)
there are set of SLD operators satisfying

$\Theta L_{{\bm{x}},j}\Theta^{\dagger}=L_{{\bm{x}},j},\leavevmode\nobreak\
(j=1,...,m)$ (166)

at ${\bm{x}}$.

_proof_) Suppose $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ has a LAS $\Theta$ at
${\bm{x}}\in X$. A set of SLD operators is defined by Eq. (9). These SLD
operators satisfy Eq. (166) because
$\Theta\rho_{\bm{x}}\Theta^{\dagger}=\rho_{\bm{x}}$ and
$\Theta\partial\rho_{\bm{x}}\Theta^{\dagger}=\partial\rho_{\bm{x}}$ holds by
the assumption.

Conversely, if $\rho_{\bm{x}}$ and $L_{{\bm{x}},j}$ are $\Theta$-invariant,
the defining equation (8) implies that $\Theta$ is a LAS of
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ at ${\bm{x}}\in X$. $\hfill\blacksquare$
Note that if the density operator is of full-rank, then the SLDs are in one-
to-one correspondence with the derivatives $\partial_{j}\rho_{\bm{x}}$.

###  B.1 Proof of Theorem 4

We start by proving that any model $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ with a
LAS $\Theta$ at ${\bm{x}}\in X$ is weakly commutative at ${\bm{x}}$. From Lem.
6, there are SLD operators $L_{{\bm{x}},j}$ satisfying Eq. (166). Then we have

$\displaystyle
2\mathrm{Im}\left[\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]\right]$
$\displaystyle=\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]-\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]^{\ast}$
$\displaystyle=\mathrm{Tr}[\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}]-\mathrm{Tr}[\Theta\rho_{\bm{x}}L_{{\bm{x}},i}L_{{\bm{x}},j}\Theta^{\dagger}]$
$\displaystyle=0,$ (167)

because $L_{{\bm{x}},i}$, $L_{{\bm{x}},j}$ and $\rho_{\bm{x}}$ are all
invariant under $\Theta$. This reveals the weak commutativity of
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ at ${\bm{x}}$.

Conversely, assume that a pure-state model
$\\{|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}||{\bm{x}}\in X\\}$ is weakly
commutative at point ${\bm{x}}\in X$. Since all pure weakly commutative models
are quasi-classical [25], there is a set of SLDs satisfying

$[L_{{\bm{x}},i},L_{{\bm{x}},j}]=0,$ (168)

for any pairs of $i,j$. Let $\\{|\Pi_{k}\rangle\\}_{k=1,...,d}$ be a basis of
$\mathcal{H}$ in which the SLDs are diagonalized

$L_{{\bm{x}},i}=\sum_{k=1}^{d}a^{(i)}_{k}|\Pi_{k}\rangle\langle\Pi_{k}|,$
(169)

where $a^{(i)}_{k}$ are the real eigenvalues. We represent the state vector

$|\psi_{\bm{x}}\rangle=\sum_{k=1}^{d}b_{k}e^{i\phi_{k}}|\Pi_{k}\rangle,$ (170)

with the real coefficients $b_{k}$ and phase $\phi_{k}$ ($k=1,...,d$).

Define a new basis vectors $|\Pi_{k}^{\prime}\rangle$ by

$|\Pi_{k}^{\prime}\rangle:=e^{i\phi_{k}}|\Pi_{k}\rangle,$ (171)

for $k=1,...,d$. The state $|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|$ is
represented by a real matrix in this basis since
$|\psi_{\bm{x}}\rangle=\sum_{k=1}^{d}b_{k}|\Pi_{k}^{\prime}\rangle$. The SLDs
are still diagonalized by the new basis. The derivative
$\partial_{i}(|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|)$ is also real in
this basis since

$\displaystyle 2\partial_{i}(|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|)$
$\displaystyle=L_{{\bm{x}},i}|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|+|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}|L_{{\bm{x}},i}$
$\displaystyle=\sum_{k,l=1}^{d}(a^{(i)}_{k}+a^{(i)}_{l})b_{k}b_{l}|\Pi_{k}^{\prime}\rangle\langle\Pi_{l}^{\prime}|,$
(172)

holds for all $i$. The complex conjugation in the basis
$\\{|\Pi_{k}^{\prime}\rangle\\}_{k=1,...,d}$ is a LAS of
$\\{|\psi_{\bm{x}}\rangle\langle\psi_{\bm{x}}||{\bm{x}}\in X\\}$ at
${\bm{x}}\in X$.

###  B.2 Optimal measurements from local antiunitary symmetry

Here we prove that rank-$1$ POVM that is invariant under LASs forms optimal
measurement for local multiparameter estimation.

###### Theorem 5

If a POVM $\mathcal{E}_{\Theta}$ satisfies conditions (54) and (55), and if
$\Theta$ is a LAS of $\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ at ${\bm{x}}\in X$,
$\mathcal{E}_{\Theta}$ saturates the QCRB (11) of model
$\\{\rho_{\bm{x}}|{\bm{x}}\in X\\}$ at ${\bm{x}}\in X$.

_proof_) From Thm. 3, the POVM $\mathcal{E}_{\Theta}$ satisfying conditions
(54) and (55) exists only if $\Theta$ is a conjugation. Therefore we assume
that the antiunitary $\Theta$ is a conjugation $\theta$.

From Lem. 6 there is a set of SLD operators $\\{L_{{\bm{x}},j}\\}_{j=1,...,n}$
satisfying $\theta L_{{\bm{x}},j}\theta=L_{{\bm{x}},j}$. The remaining
procedure of proof is the same to the proof of Thm. 3, except that all the
statements are about the specific point ${\bm{x}}$. $\hfill\blacksquare$

###  B.3 Quasi-classical models without local antiunitary symmetries

We construct quasi-classical models without any LASs. According to Thm. 4,
these examples are inevitably mixed-state models. We give two kinds of
examples in Sec. B.3.1 and Sec. B.3.2. We also show that any quasi-classical
qubit model has a LAS in Sec. B.3.3.

The following lemma is used in exhibiting the non-existence of antiunitary
symmetries for both of the examples.

###### Lemma 7

Let $D$ be a real non-degenerate $d\times d$ diagonal matrix. If
$U^{\dagger}DU$ remains real for a unitary $U$, then $U$ is decomposed to

$U=\Phi R,$ (173)

where $R$ is an orthogonal matrix and

$\Phi={\rm diag}(e^{i\phi_{1}},...,e^{i\phi_{d}}),$ (174)

with some set of phases $\phi_{j}\in[0,2\pi]$.

_proof_) Let us denote the real matrix $U^{\dagger}DU$ by $Q$. Since $Q$ is