††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Optimal Local Measurements in Many-body Quantum Metrology

Jia-Xuan Liu  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Jing Yang<EMAIL_ADDRESS>Nordita, KTH Royal Institute of Technology and
Stockholm University, Hannes Alfvéns vag 12, 106 91 Stockholm, Sweden  Hai-
Long Shi  Innovation Academy for Precision Measurement Science and Technology,
Chinese Academy of Sciences, Wuhan 430071, China  INO-CNR, Largo Enrico Fermi
2, 50125 Firenze, Italy  Sixia Yu<EMAIL_ADDRESS>Hefei National Research
Center for Physical Sciences at the Microscale and School of Physical
Sciences, Department of Modern Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China

###### Abstract

Quantum measurements are key to quantum metrology. Constrained by experimental
capabilities, collective measurements on a large number of copies of
metrological probes can pose significant challenges. Therefore, the locality
in quantum measurements must be considered. In this work, we propose a method
dubbed as the “iterative matrix partition" approach to elucidate the
underlying structures of optimal local measurements, with and without
classical communications, that saturate the quantum Cramér-Rao Bound (qCRB).
Furthermore, we find that while exact saturation is possible for all two-qubit
pure states, it is generically restrictive for multi-qubit pure states.
However, we demonstrate that the qCRB can be universally saturated in an
approximate manner through adaptive coherent controls, as long as the initial
state is separable and the Hamiltonian allows for interaction. Our results
bridge the gap between theoretical proposals and experiments in many-body
metrology and can find immediate applications in noisy intermediate-scale
quantum devices.

Introduction.— Locality plays a crucial role in various branches of physics,
encompassing high energy physics [1, 2, 3], condensed matter physics [4, 5]
and quantum information theory [6, 7, 8, 9, 10]. In the context of many-body
systems, locality gives rise to the Lieb-Robinson bound [11, 12, 13], which
sets an upper limit on the spread of local operators. Despite the recent
resurgence of interest in quantum metrology using many-body Hamiltonians [14,
15, 16, 17, 18], the investigation of locality in the sensing Hamiltonian has
only been undertaken until recently [19, 20, 21, 18].

On the other hand, at the fundamental as well as the practical level, locality
in quantum measurements has been largely uncharted in many-body quantum
metrology. For example, consider a non-interacting and multiplicative sensing
Hamiltonian $H_{\lambda}=\lambda\sum_{j}h_{j}$, where $h_{j}$ is the local
Hamiltonian defined for the spin at site $j$ and $\lambda$ is the estimation
parameter. It has been show in Ref.[22] that if the initial state is prepared
in a GHZ (Greenberger–Horne–Zeilinger)-like state and the precision is
maximized among all the possible initial states and local measurements (LM)
suffice to saturate the quantum Cramér-Rao bound(qCRB). However, it is worth
to emphasize that, to our best knowledge, even for this non-interacting
Hamiltonian, little is known about whether LM can saturate the qCRB for other
initial states, not to mention that $H_{\lambda}$ in general can contain many-
body interactions and have generic parametric dependence. Additionally, for
pure states, Zhou et al [23] prove that rank$-1$ projective local measurements
with classical communications (LMCC) can be constructed to saturate the qCRB.
However, due to the classical communications between particles, the total
number of measurement basis scales exponentially with the number of particles,
which requires exponentially amount of experimental resources and thus
difficult to implement.

In contrast, the total number of basis in LM scales linearly with the number
of particles, which is feasible for experimental implementation. As such, in
this work, we present a systematic study on qCRB-saturating LM. We address the
following main questions: (i) Can LM universally saturate qCRB? (ii) If not,
in what circumstances there exists qCRB-saturating LM? (iii) If one allows
generic positive operator-valued measure (POVM) LM, the number of measurement
basis is unlimited and thus can be made as exponentially large as the LMCC.
Therefore it is natural to ask whether POVM LM can help in the saturation of
the qCRB? (iv) If exact saturation with LM is very restrictive, is it possible
to identify regimes where the approximate saturation is possible? We shall
develop a comprehensive understanding on these questions subsequently.

The Optimal Measurement Condition. — To begin with, we consider a pure quantum
state $\ket{\psi_{\lambda}}$. The quantum Fisher information (QFI) is given by
[24, 25]

$I=4\left(\braket{\partial_{\lambda}\psi_{\lambda}}{\partial_{\lambda}\psi_{\lambda}}-|\braket{\psi_{\lambda}}{\partial_{\lambda}\psi_{\lambda}}|{}^{2}\right).$
(1)

The optimal measurement condition that can saturate the qCRB is given by [23,
26, 27]

$\braket{\pi_{\omega}\big{missing}}{\mathcal{M}\big{missing}}{\pi_{\omega}}=0,$
(2)

where

$\mathcal{M}\equiv[\rho_{\lambda},\,L]=2[\rho_{\lambda},\,\partial_{\lambda}\rho_{\lambda}],$
(3)

$L$ is the symmetric logarithmic derivative defined as
$\partial_{\lambda}\rho_{\lambda}\equiv(\rho_{\lambda}L+L\rho_{\lambda})/2$
with $\rho_{\lambda}\equiv\ket{\psi_{\lambda}}\bra{\psi_{\lambda}}$ and the
POVM measurement satisfies
$\sum_{\omega}\ket{\pi_{\omega}}\bra{\pi_{\omega}}=\mathbb{I}$. Here, without
loss of generality, we only consider a set of rank$-1$ POVM operators [27]. We
would like to emphasize in Ref. [23] the optimal condition is divided into two
cases according whether
$\text{Tr}(\rho_{\lambda}\ket{\pi_{\omega}}\bra{\pi_{\omega}})$ vanishes or
not. Using the results on multi-parameter estimation [28], we argue in the
Sec. I in the Supplemental Material [27] that such a division is unnecessary
and Eq. (2) is the condition to saturate the qCRB for all types of POVM
measurements.

Figure 1: LMCC can be constructed through IMP using “block hollowization”,
where the trace of the diagonal blocks of a matrix is transformed to zero
through local unitary transformations with classical communications. The goal
is to perform a full “hollowization” procedure, where all the diagonal matrix
elements of the operator $\mathcal{M}$ are brought to zero. The IMP provides a
feasible approach, see details in the main text and the Supplemental Material
[27].

The Iterative Matrix Partition Approach to LMCC and LM.— From now on, we shall
focus our discussion on pure states of $N$-qubit systems and search for
optimal LM and LMCC. In this case, the measurement outcome $\omega$ in Eq. (2)
becomes a string of measurement outcomes of each qubit denoted as
$\omega=(\omega_{1},\,\omega_{2},\,\cdots,\,\omega_{N})$. Zhou et al [23]
showed that the optimal projective LMCC can be constructed iteratively through

$\braket{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}{M_{\omega_{1}\cdots\omega_{j-1}}^{(j)}\big{missing}}{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}=0.$
(4)

The superscripts in basis and operators in Eq. (4) indicate the subsystems
over which they are defined and

$\displaystyle M_{\omega_{1}\cdots\omega_{j-1}}^{(j)}\equiv$
$\displaystyle\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\bra{\pi_{\omega_{j-1},\,\omega_{1}\cdots\omega_{j-2}}^{(j-1)}}\text{Tr}_{(j+1,\,\cdots
N)}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\ket{\pi_{\omega_{j-1},\,\omega_{1}\cdots\omega_{j-2}}^{(j-1)}}$
(5)

is an operator defined on the $j$-th qubit with $j\geq 2$, where the
subscripts in the “Tr” notation indicate the subsystems that are traced over.
For $j=1$, $M^{(1)}\equiv\text{Tr}_{(2\cdots N)}\mathcal{M}$ and
$\ket{\pi_{\omega_{1}}^{(1)}}$ satisfies
$\braket{\pi_{\omega_{1}}^{(1)}\big{missing}}{M^{(1)}\big{missing}}{\pi_{\omega_{1}}^{(1)}}=0$.
In Sec. II of the Supplemental Material [27], we show these properties
naturally follow from the optimal measurement condition (2) and for optimal
projective LM they reduce to

$\braket{\pi_{\omega_{j}}^{(j)}\big{missing}}{M^{(j)}\big{missing}}{\pi_{\omega_{j}}^{(j)}}=0,$
(6)

where $M^{(j)}\equiv\text{Tr}_{(1\cdots\cancel{j}\cdots N)}\mathcal{M},$ the
subscript $\cancel{j}$ indicates that the $j$-th qubit is not traced over. A
few comments in order: (i) Since $M_{\omega_{1}\cdots\omega_{j-1}}^{(j)}$ and
$M^{(j)}$ are traceless, the measurement basis in Eqs. (4, 6) can be found
through the “hollowization” process: A traceless matrix can be always brought
to a hollow matrix, i.e., a matrix with zero diagonal entries, through unitary
transformations[27, 29, 30]. (ii) While Eq. (4) is also sufficient to
guarantee the optimal measurement condition (2), this is no longer true for
Eq. (6).

To resolve this issue, we propose the “iterative matrix partition”(IMP)
approach, which not only produces the LMCC, but also illuminates the intuition
on the existence of LM. We denote the local computational basis for the $j$-th
qubit as $\ket{e_{\omega_{j}}^{(j)}}$, $\omega_{j}=1,\,2$. One can compute the
$\mathcal{M}$ operator in this basis (see a tutorial example in [27]).
Consider

$\mathcal{M}=\left[\begin{array}[]{c|l}M_{11}^{(\cancel{1})}&M_{12}^{(\cancel{1})}\\\
\hline\cr M_{21}^{(\cancel{1})}&M_{22}^{(\cancel{1})}\end{array}\right],$ (7)

where for fixed $\omega_{1}$ and $\mu_{1}$,
$M_{\omega_{1}\mu_{1}}^{(\cancel{1})}\equiv\braket{e_{\omega_{1}}^{(1)}\big{missing}}{\mathcal{M}\big{missing}}{e_{\mu_{1}}^{(1)}}$
is a $2^{N-1}\times 2^{N-1}$ matrix that acts on all the qubits except first
qubit .

Since $\mathcal{M}$ is anti-Hermitian, so is the diagonal block matrices
$M_{11}^{(\cancel{1})}$ and $M_{22}^{(\cancel{1})}$. Furthermore,
$\mathcal{M}$ is traceless, the trace of the two diagonal block matrices can
be also brought zero through a unitary transformation on the first qubit (see
Observation 3 in [27]). More precisely,

$\mathcal{M}=\sum_{\omega_{1}\mu_{1}}W_{\omega_{1}\mu_{1}}^{(\cancel{1})}\ket{\pi_{\omega_{1}}^{(1)}}\bra{\pi_{\mu_{1}}^{(1)}}$
(8)

where $|\pi_{\omega}^{(1)}\rangle\equiv U^{(1)}\ket{e_{\omega_{1}}^{(1)}}$,
$W_{\omega_{1}\mu_{1}}^{(\cancel{1})}\equiv
U^{(1)}M_{\omega_{1}\mu_{1}}^{(\cancel{1})}U^{(1)\dagger}$ and
$U^{(1)\dagger}$ is chosen such that
$\text{Tr}W_{11}^{(\cancel{1})}=\text{Tr}W_{22}^{(\cancel{1})}=0$. Note that
$W_{11}^{(\cancel{1})}$ and $W_{22}^{(\cancel{1})}$ are also anti-Hermitian
matrices.

Next, we decompose $W_{11}^{(\cancel{1})}$ and $W_{22}^{(\cancel{1})}$ in the
local computational basis of the second qubit, i.e.

$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}=\sum_{\omega_{2},\,\mu_{2}}M_{\omega_{2}\mu_{2},\,\omega_{1}}^{(\cancel{12})}\ket{e_{\omega_{2}}^{(2)}}\bra{e_{\omega_{2}}^{(2)}},$
(9)

where $M_{\omega_{2}\mu_{2},\,\omega_{1}}^{(\cancel{12})}$, analogous to
$M_{\omega_{1}\mu_{1}}^{(\cancel{1})}$, is the block matrix representation of
$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}$ in the local computational basis of
the second qubit. For fixed $\omega_{1}$, one can iterate to perform the
“block-hollowization” process for $W_{\omega_{1}\omega_{1}}^{(\cancel{1})}$,
leading to

$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}=\sum_{\omega_{2},\,\mu_{2}}W_{\omega_{2}\mu_{2},\,\omega_{1}}^{(\cancel{12})}\ket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\bra{\pi_{\mu_{2},\,\omega_{1}}^{(2)}},$
(10)

where $\ket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\equiv
U_{\omega_{1}}^{(2)}\ket{e_{\omega_{2}}^{(2)}}$ and
$W_{\omega_{2}\omega_{2},\,\omega_{1}}^{(\cancel{12})}$ is traceless and anti-
Hermitian for fixed $\omega_{1}$.

Iterating this process to the $N$-th qubit, we arrive at

$W_{\omega_{N-1}\omega_{N-1},\,\omega_{1}\cdots\omega_{N-2}}^{(\cancel{1\cdots
N-1})}=\sum_{\omega_{N},\,\mu_{N}}M_{\omega_{N}\mu_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots
N})}\ket{e_{\omega_{N}}^{(N)}}\bra{e_{\mu_{N}}^{(N)}},$ (11)

where for fixed $\omega_{1},\,\cdots,\,\omega_{N-1}$,
$M_{\omega_{N}\mu_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots N})}$
is a $2\times 2$ anti-Hermitian traceless matrix. Finally, we perform the
“hollowization” and obtain

$\displaystyle
W_{\omega_{N-1}\omega_{N-1},\,\omega_{1}\cdots\omega_{N-2}}^{(\cancel{1\cdots
N-1})}$ $\displaystyle=$
$\displaystyle\sum_{\omega_{N},\,\mu_{N}}W_{\omega_{N}\mu_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots
N})}\ket{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}\bra{\pi_{\mu_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}},$
(12)

where $\ket{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}\equiv
U_{\omega_{1}\cdots\omega_{N-1}}^{(N)}\ket{e_{\omega_{N}}^{(N)}}$ and
$W_{\omega_{N}\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots
N})}=0$ for $\omega_{N}=1,\,2$. The procedure is pictorially depicted in Fig.
1.

If $U_{\omega_{1}\cdots\omega_{j-1}}^{(j)}$ is independent of
$\omega_{1}\cdots\omega_{j-1}$ for all $j\in[2,N]$ (the case of $j=1$ is
obvious), then we call the IMP is degenerate. We prove in Sec. IV in [27] the
following two observations: (i) a generic IMP gives rise to an LMCC
measurement, as shown in Fig. 1. (ii) a degenerate IMP is equivalent to the
existence of optimal LM.

An immediate application of the IMP approach is that it illuminates on the
“self-similar” structure of the GHZ states, which guarantees the existence of
local optimal measurements. Such a structure, to our best knowledge, has not
been appreciated previously in the literature. To elaborate, consider the
sensing Hamiltonian

$H=\lambda S_{z},$ (13)

and an initial GHZ state $\ket{\psi_{0}}=(|0\rangle^{\otimes
N}+|1\rangle^{\otimes N})/\sqrt{2}$, where $S_{z}=\sum_{j}\sigma_{z}^{(j)}/2$
and $\ket{0},\ket{1}$ are the excited and ground states of $\sigma_{z}$,
respectively. As time evolves, the state remains at a GHZ state, but with
parameter-dependent relative phase, i.e.,
$|\psi_{\lambda}\rangle=(|0\rangle^{\otimes N}+e^{i\lambda
Nt}|1\rangle^{\otimes N})/\sqrt{2}$ The $\mathcal{M}$ operator is given by

$\mathcal{M}=\text{i}Nt(|0^{\otimes N}\rangle\langle 0^{\otimes
N}|-|1^{\otimes N}\rangle\langle 1^{\otimes N}|).$ (14)

In the first iteration, we observe that
$M_{12}^{(\cancel{1})}=M_{21}^{(\cancel{1})}=0$ in the computational basis
$\ket{0^{(1)}}$ and $\ket{1^{(1)}}$ vanishes, which simplifies the iteration
dramatically and leads to [27]

$U^{(1)}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&1\\\
e^{i\phi^{(1)}}&-e^{i\phi^{(1)}}\end{array}\right].$ (15)

Consequently, we immediately obtain

$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}=\frac{1}{2}(M_{11}^{(\cancel{1})}+M_{22}^{(\cancel{1})})=\frac{\text{i}Nt}{2}(|0^{\otimes
N-1}\rangle\langle 0^{\otimes N-1}|-|1^{\otimes N-1}\rangle\langle 1^{\otimes
N-1}|),$ (16)

where $\omega_{1}=1,\,2$. A few observations can be drawn: (i) The first
iteration of the IMP process leads to exact the same diagonal blocks
$W_{11}^{(1)}=W_{22}^{(1)}$. (i) The matrix structure of
$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}$ , apart from the dimensionality and
an irrelevant prefactor, is identical to Eq. (14) . The consequence of such a
self-similar structure is that $U_{\omega_{1}}^{(2)}$, more generally
$U_{\omega_{1}\cdots\omega_{k-1}}^{(k)}$, does not depend on the previous
measurement outcomes. We note that $U^{(k)}$ is the the same as $U^{(1)}$
apart from the phase factor $e^{\text{i}\phi^{(k)}}.$ At the $k$-th iteration,
we arrive at $W_{\omega_{k}\omega_{k}}^{(\cancel{1\cdots
k})}=\text{i}Nt/2^{k}(|0^{\otimes N-k}\rangle\langle 0^{\otimes
N-k}|-|1^{\otimes N-k}\rangle\langle 1^{\otimes N-k}|).$ The IMP for the GHZ
state is degenerate and LMCC reduces to LM.

Fundamental Theorems on Optimal LM.— Now we present several theorems on
optimal LM. Our first theorem is the following:

######  Theorem 1.

The qCRB of a 2-qubit pure state is universally saturated by projective LM.

###### Proof.

We perform the IMP procedure. After the first round, we obtain two anti-
Hermitian matrix $W_{\omega_{1}\omega_{1}}^{(\cancel{1})}$ of the dimension
$2\times 2$. According to Observation 2 in the Supplementary Material [27],
$W_{11}^{(\cancel{1})}$ and $W_{22}^{(\cancel{1})}$ are simultaneously
hollowizable. ∎

In [27], we provide a tutorial explanation demonstrating the application of
the IMP approach using a two-qubit example and construct the corresponding
optimal LM.

Beyond two qubits, as we will show later by a counterexample, universal
saturation of the qCRB is not possible. Nevertheless, as an alternative method
to the IMP approach, one can use the following theorem to determine explicitly
whether an $N$-qubit pure state can saturate the qCRB.

######  Theorem 2.

For $N$-qubit system labeled by $\mathscr{X}_{N}=\\{1,\,2,\,\cdots,N\\}$, the
qCRB of a pure state can be saturated by LM if and only if for each non-empty
subset $\alpha\subseteq\mathscr{X}_{N}$ there exists Bloch vectors
$\\{\bm{n}^{(j)}\\}_{j\in\alpha}$ such that

$\text{Tr}\mathcal{M}\mathcal{N}_{\alpha}=0,\quad\mathcal{N}_{\alpha}\equiv\otimes_{n\in\alpha}\bm{n}^{(j)}\cdot\bm{\sigma}^{(j)}.$
(17)

where the projectors of the projective LM is given by

$\Pi_{\omega_{j}}^{(j)}=\frac{\mathbb{I}^{(j)}+(-1)^{\omega_{j}}\bm{n}^{(j)}\cdot\bm{\sigma}^{(j)}}{2},\,\omega_{j}=\pm
1.$ (18)

Having discussed the projective LM, let us now come to generic non-projective
POVM LM. For such measurements, unlike projective LM, the number of
measurement basis is not necessarily bounded by two for each local spin and
can become as many as the projective LMCC. Then a natural question arises: Are
POVM LM more powerful than projective LM? The answer is no, according to the
following theorem.

######  Theorem 3.

For $N$-qubit pure-state, if there is a generic qCRB-saturating POVM LM, then
there must exists a qCRB-saturating projective LM.

By the virtue of Theorem 3, it suffices to focus on projective LM. If optimal
projective LM cannot be found, then it is impossible to reach the qCRB by
using POVM LM with a large number of measurement basis. In this sense, generic
POVM LM does not help in reaching the qCRB. However, this does not exclude
their other possible utilities. As we have shown before, in the projective LM
basis, applying IMP to the GHZ state leads to the property of self-similarity.
It is an interesting open question to search for states that display self-
similarity in generic POVM LM basis, which could lead to non-GHZ-like many-
body states that saturate the qCRB.

We consider a pure state
$\ket{\psi_{\lambda}(t)}=U_{\lambda}(t)\ket{\psi_{0}}$ that is generated from
a unitary parameter-dependent quantum channel $U_{\lambda}(t)$ and an initial
pure state $\ket{\psi_{0}}$, where $U_{\lambda}(t)$ satisfied the Schrödinger
equation $\text{i}\dot{U}_{\lambda}(t)=H_{\lambda}(t)U_{\lambda}(t)$. In this
case, the quantum Fisher information is given by

$I_{\lambda}=4\text{Var}\left(G_{\lambda}(t)\right){}_{\ket{\psi_{0}}},$ (19)

and $\mathcal{M}$ can be rewritten as

$\mathcal{M}=-2\text{i}U_{\lambda}(t)[\rho_{0},\,[G_{\lambda}(t),\,\rho_{0}]]U_{\lambda}^{\dagger}(t),$
(20)

where the metrological generator is defined as [14, 31]

$G_{\lambda}(t)\equiv\text{i}U_{\lambda}^{\dagger}(t)\partial_{\lambda}U_{\lambda}(t)=\int_{0}^{t}U_{\lambda}^{\dagger}(s)\partial_{\lambda}H_{\lambda}(s)U_{\lambda}(s)ds.$
(21)

So we have the following theorem [27]:

######  Theorem 4.

Given a pair of initial state $\ket{\psi_{0}}$ and a unitary channel
$U_{\lambda}(t)$, the qCRB of $\ket{\psi_{\lambda}(t)}$ can be saturated at
the instantaneous time $t$ by LM if and only

$\text{Cov}\left(\mathcal{N}_{\alpha}^{(\text{H})}(t)G_{\lambda}(t)\right)_{\ket{\psi_{0}}}=0,\,\forall\alpha\subseteq\mathscr{X}_{N},$
(22)

where the set $\mathscr{X}_{N}$ is same as in Theorem 2 and
$\mathcal{N}_{\alpha}^{(\text{H})}(t)\equiv
U_{\lambda}^{\dagger}(t)\mathcal{N}_{\alpha}U_{\lambda}(t)$ is the Heisenberg
evolution of $\mathcal{N}_{\alpha}$ and
$\text{Cov}(AB)_{\ket{\psi_{0}}}\equiv\frac{1}{2}\langle\\{A,\,B\\}\rangle_{\ket{\psi_{0}}}-\langle
A\rangle_{\ket{\psi_{0}}}\langle B\rangle_{\ket{\psi_{0}}}$.

One can check immediately that the GHZ state with $\sigma_{x}$-LM satisfies
Theorem 4. Now we are in a position to give a minimum 3-qubit counter-example
that fails to saturating the qCRB under LM. Consider $H_{\lambda}=\lambda
H_{0},$where
$H_{0}\equiv\sum_{a=x,\,y}(\sigma_{a}^{(1)}\sigma_{a}^{(2)}+\sigma_{a}^{(2)}\sigma_{a}^{(3)})$,
the initial state is the W state, i.e.,
$|\psi_{0}\rangle=(|100\rangle+|010\rangle+|001\rangle)/\sqrt{3}$. We assume
the true value of $\lambda$ is zero so that
$|\psi_{\lambda}(t)\rangle=|\psi_{0}\rangle$. It should be clarified that in
this case despite the state does not change over time, it does not mean the
parameter cannot be estimated accurately. In fact, it is straightforward to
see QFI is $4t^{2}\text{Var}[H_{0}]_{\ket{\psi_{0}}}=32t^{2}/9,$ independent
of the value of $\lambda$. In [27], using symmetry arguments, we show that the
set of equations determined by Eq. (22) can not be consistent with each other.
Therefore, neither projective LM nor generic POVM LM exists according to
Theorem 3.

Universal Approximate Saturation with Adaptive Control. — As one can see from
Theorem 4, the saturation of the qCRB with LM can be very restrictive.
Nevertheless, we observe that if

$\mathcal{N}_{\alpha}^{(\text{H})}(t)\ket{\psi_{0}}\propto\ket{\psi_{0}},\,\,\forall\alpha\subseteq\mathscr{X}_{N}$
(23)

is satisfied at time $t$, then Eq. (22) holds. Note that the case where
$\ket{\psi_{0}}$ is an eigenstate of $G_{\lambda}(t)$ is trivial as it leads
to a vanishing QFI.

To this end, when the initial state is a product of pure states, one can first
choose $\mathcal{N}_{\alpha}(0)$ such that Eq. (23) hold at $t=0$. As time
evolves, $\mathcal{N}_{\alpha}(t)$ will the spread and Eq. (23) will no longer
hold. However, one can take advantage of our prior knowledge and apply a
proper control Hamiltonian such that dynamics is frozen or at least very slow.
That is,

$\delta H(t)=H_{\lambda}(t)+H_{\text{1}}(t),$ (24)

where the control Hamiltonian $H_{\text{1}}(t)=-H_{\lambda_{*}}(t)$ and
$\lambda_{*}$ is our priori knowledge on the estimation parameter. Then
$\mathcal{N}_{\alpha}^{(\text{H})}(t)$ remains close to
$\mathcal{N}_{\alpha}(0)$ for quite long time as long as $\lambda_{*}$ is
close to $\lambda$. It is worth to note that in local estimation theory,
adaptive estimation is usually exploited where some refined knowledge of the
estimation parameter is known a priori [32, 33, 34]. Quantum control was
explored in quantum metrology before, but aiming to boosting the QFI [35, 36,
18, 31, 37] and overcome the measurement noise. [38, 39]. It is remarkable
that quantum controls here, which facilities LM to saturate the qCRB is fully
consistent with the QFI-boosting controls in Ref. [35, 31, 36]. Finally, we
note that as long as $\lambda_{*}$ close to $\lambda$, the metrological
generator associated with the dynamics generated by Eq. (24) becomes
$G_{\lambda}(t)=\int_{0}^{t}\partial_{s}H_{\lambda}(s)ds$ and QFI is still
given by Eq. (19). Let us consider the following an example, where

$H_{\lambda}=\lambda S_{z}^{2},$ (25)

and the initial state is a spin coherent state [40] parameterized by

$|\psi_{0}\rangle=\bigotimes_{k=1}^{N}\left[\cos\frac{\theta}{2}|0\rangle^{(k)}+e^{i\phi}\sin\frac{\theta}{2}|1\rangle^{(k)}\right].$
(26)

Equation (25) is nonlinear and non-local. It has been shown previously that
precision beyond the shot-noise scaling in classical sensing [41, 14, 19] can
be achieved. However, the optimal LM that reaches such a non-classical
precision is still missing in the literature. To this end, we apply coherent
control $H_{\text{1}}=-\lambda_{*}S_{z}^{2}$ so that $\delta H=\delta\lambda
S_{z}^{2}$ where $\delta\lambda\equiv\lambda-\lambda_{*}$ state. The QFI
corresponding to the initial state Eq. (26) is [27]

$I=4t^{2}\text{Var}[S_{z}^{2}]_{\ket{\psi_{0}}}=4t^{2}\sum_{k=1}^{3}f_{k}(\cos\theta)N^{k},$
(27)

and scales cubically in $N$, surpassing the Heisenberg limit. In Fig. 2, the
comparison between the QFI and classical Fisher information (CFI) associated
with the LM (18), where
$\bm{n}^{(j)}=(\sin\theta\cos\phi,\,\sin\theta\sin\phi,\,\cos\theta)$ are
plotted. One can readily see that qCRB is asymptotically saturated as
$\lambda_{*}$ approaches $\lambda$.

Figure 2: QFI and CFI associated with local measurements with the Hamiltonian
(25) and initial state (26). The value of parameters are:
$\theta=\frac{\pi}{3}$, $\phi=\frac{\pi}{2}$ and $\lambda=2$.(a) The variation
of the saturation with the number of qubits $N$ for a fixed evolution time
$t=1/0.01\lambda=50$. The red solid dots is the numerical calculations for the
QFI while the dashed line is plotted according to Eq. (27). The blue
triangles, yellow stars, and brown diamonds represent the numerically obtained
CFI $/N^{2}$ under the application of different control strategies,
respectively. (b) The saturation behavior of the qCRB under time-dependent
evolution for $N=8$. The red solid line is plotted according to Eq. (27),
while the yellow dash-dotted line, blue dashed line, and brown solid line
respectively depict the CFI under different control strategies, respectively.

Conclusion and outlook. — We systematically study optimal LMCC and LM that can
saturate the qCRB in many-body sensing. We propose an IMP approach that
illuminates the structure of the optimal LMCC and LM and provide several
fundamental theorems on the qCRB-saturating optimal LM. We show that under LM,
the qCRB can be universally saturated in an approximate way with adaptive
control, regardless of the form of the sensing Hamiltonian.

Currently, in the protocols of many-body sensing [42, 43, 17, 18, 16, 14, 15],
there is not yet a systematic construction of the optimal LM. Our results fill
the gap between theoretical proposal of many-body sensing and its experimental
realization. We expect to see their near-term implementation in noisy
intermediate scale quantum devices [44, 45, 46]. Future works include
generalization to qudits, continuous variable systems, and qubit-cavity
systems, application to entanglement detection [47, 48, 49] and spin-squeezing
[40, 50, 51], investigation of the effect of decoherence, etc.

Acknowledgement. —We thank Sisi Zhou for useful communications. JY was funded
by the Wallenberg Initiative on Networks and Quantum Information (WINQ). HLS
was supported by the NSFC key grants No. 12134015 and No. 92365202. SY was
supported by Key-Area Research and Development Program of Guangdong Province
Grant No. 2020B0303010001.

## References

  * Peskin and Schroeder [2015] M. E. Peskin and D. V. Schroeder,  _An Introduction To Quantum Field Theory, Student Economy Edition_ , 1st ed. (Westview Press, 2015).
  * Huang [2010] K. Huang,  _Quantum Field Theory: From Operators to Path Integrals, 2nd Edition_ , 2nd ed. (Wiley-VCH, Weinheim, 2010).
  * Coleman [2011] S. Coleman, arXiv:1110.5013 [hep-ph, physics:hep-th, physics:physics] (2011), arxiv:1110.5013 [hep-ph, physics:hep-th, physics:physics] .
  * Hastings and Wen [2005] M. B. Hastings and X.-G. Wen, Physical Review B 72, 045141 (2005).
  * Chen  _et al._ [2010] X. Chen, Z.-C. Gu, and X.-G. Wen, Physical Review B 82, 155138 (2010).
  * Nielsen  _et al._ [2006] M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Science 311, 1133 (2006).
  * Sels and Polkovnikov [2017] D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences 114, E3909 (2017).
  * Carlini  _et al._ [2006] A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Physical Review Letters 96, 060503 (2006).
  * Yang and del Campo [2022] J. Yang and A. del Campo, Minimum-Time Quantum Control and the Quantum Brachistochrone Equation (2022), arxiv:2204.12792 [quant-ph] .
  * Chen  _et al._ [2023] C.-F. Chen, A. Lucas, and C. Yin, Speed limits and locality in many-body quantum dynamics (2023), arxiv:2303.07386 [cond-mat, physics:math-ph, physics:quant-ph] .
  * Lieb and Robinson [1972] E. H. Lieb and D. W. Robinson, Communications in Mathematical Physics 28, 251 (1972).
  * Nachtergaele  _et al._ [2009] B. Nachtergaele, H. Raz, B. Schlein, and R. Sims, Communications in Mathematical Physics 286, 1073 (2009).
  * Bravyi  _et al._ [2006] S. Bravyi, M. B. Hastings, and F. Verstraete, Physical Review Letters 97, 050401 (2006).
  * Boixo  _et al._ [2007] S. Boixo, S. T. Flammia, C. M. Caves, and JM. Geremia, Physical Review Letters 98, 090401 (2007).
  * Roy and Braunstein [2008] S. M. Roy and S. L. Braunstein, Physical Review Letters 100, 220501 (2008).
  * Beau and del Campo [2017] M. Beau and A. del Campo, Physical Review Letters 119, 010403 (2017).
  * Yang  _et al._ [2022a] J. Yang, S. Pang, A. del Campo, and A. N. Jordan, Physical Review Research 4, 013133 (2022a).
  * Yang  _et al._ [2022b] J. Yang, S. Pang, Z. Chen, A. N. Jordan, and A. del Campo, Phys. Rev. Lett. 128, 160505 (2022b).
  * Shi  _et al._ [2023] H.-L. Shi, X.-W. Guan, and J. Yang, Universal shot-noise limit for quantum metrology with local Hamiltonians (2023), arxiv:2308.03696 [cond-mat, physics:math-ph, physics:nlin, physics:quant-ph] .
  * Yin and Lucas [2023] C. Yin and A. Lucas, Heisenberg-limited metrology with perturbing interactions (2023), arxiv:2308.10929 [math-ph, physics:quant-ph] .
  * Chu  _et al._ [2023] Y. Chu, X. Li, and J. Cai, Physical Review Letters 130, 170801 (2023).
  * Giovannetti  _et al._ [2006] V. Giovannetti, S. Lloyd, and L. Maccone, Physical Review Letters 96, 010401 (2006).
  * Zhou  _et al._ [2020] S. Zhou, C.-L. Zou, and L. Jiang, Quantum Science and Technology 5, 025005 (2020).
  * Helstrom [1976] C. W. Helstrom,  _Quantum Detection and Estimation Theory_ (Academic Press, 1976).
  * Holevo [2011] A. S. Holevo,  _Probabilistic and Statistical Aspects of Quantum Theory_ (Springer Science & Business Media, 2011).
  * Braunstein and Caves [1994] S. L. Braunstein and C. M. Caves, Physical Review Letters 72, 3439 (1994).
  * [27] See Supplemental Material.
  * Yang  _et al._ [2019] J. Yang, S. Pang, Y. Zhou, and A. N. Jordan, Physical Review A 100, 032104 (2019).
  * Horn and Johnson [2012] R. A. Horn and C. R. Johnson,  _Matrix Analysis_ , 2nd ed. (Cambridge University Press, New York, NY, 2012).
  * Fillmore [1969] P. A. Fillmore, The American Mathematical Monthly 76, 167 (1969), 2317264 .
  * Pang and Jordan [2017] S. Pang and A. N. Jordan, Nature Communications 8, 14695 (2017).
  * Fujiwara [2006] A. Fujiwara, Journal of Physics A: Mathematical and General 39, 12489 (2006).
  * Hayashi [2005] M. Hayashi, ed.,  _Asymptotic Theory Of Quantum Statistical Inference: Selected Papers_ (World Scientific Pub Co Inc, Singpore, 2005).
  * Paris [2009] M. G. A. Paris, International Journal of Quantum Information 07, 125 (2009).
  * Yuan and Fung [2015] H. Yuan and C.-H. F. Fung, Physical Review Letters 115, 110401 (2015).
  * Yang  _et al._ [2017] J. Yang, S. Pang, and A. N. Jordan, Physical Review A 96, 020301 (2017).
  * Liu and Yuan [2017] J. Liu and H. Yuan, Physical Review A 96, 012117 (2017).
  * Len  _et al._ [2022] Y. L. Len, T. Gefen, A. Retzker, and J. Kołodyński, Nature Communications 13, 6971 (2022).
  * Zhou  _et al._ [2023] S. Zhou, S. Michalakis, and T. Gefen, Optimal protocols for quantum metrology with noisy measurements (2023), arxiv:2210.11393 [quant-ph] .
  * Ma  _et al._ [2011] J. Ma, X. Wang, C. Sun, and F. Nori, Physics Reports 509, 89 (2011).
  * Boixo  _et al._ [2008] S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan, and C. M. Caves, Physical Review A 77, 012317 (2008).
  * Niezgoda and Chwedeńczuk [2021] A. Niezgoda and J. Chwedeńczuk, Physical Review Letters 126, 210506 (2021).
  * Czajkowski  _et al._ [2019] J. Czajkowski, K. Paw\lowski, and R. Demkowicz-Dobrzański, New Journal of Physics 21, 053031 (2019).
  * Hou  _et al._ [2019] Z. Hou, R.-J. Wang, J.-F. Tang, H. Yuan, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Physical Review Letters 123, 040501 (2019).
  * Liu  _et al._ [2021] R. Liu, Y. Chen, M. Jiang, X. Yang, Z. Wu, Y. Li, H. Yuan, X. Peng, and J. Du, npj Quantum Information 7, 1 (2021).
  * Riedel  _et al._ [2010] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, Nature 464, 1170 (2010).
  * Hyllus  _et al._ [2012] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Physical Review A 85, 022321 (2012).
  * Tóth [2012] G. Tóth, Physical Review A 85, 022322 (2012).
  * Li and Luo [2013] N. Li and S. Luo, Physical Review A 88, 014301 (2013).
  * Tóth and Apellaniz [2014] G. Tóth and I. Apellaniz, Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
  * Kitagawa and Ueda [1993] M. Kitagawa and M. Ueda, Physical Review A 47, 5138 (1993).

Supplemental Material

In this Supplemental Material, we present detailed discussions on (1) the
optimal measurement condition, (2) properties of LMCC and LM (3) hollowization
and block hollowization, (4) iterative matrix partition, (5) proofs of
Theorems 2-4, (6) the three-qubit counter-example that cannot saturate the
qCRB, and (7) the analytic calculations of the QFI in Eq. (27).

###### Contents

  1. I Revisiting the QCRB-saturating optimal measurements
    1. I.1 Optimal measurement condition for POVM operators
    2. I.2 Rank$-1$ POVM measurements are sufficient
  2. II  Properties of optimal LMCC and LM
    1. II.1 Properties of the optimal LMCC
    2. II.2 Properties of LM
  3. III The procedure of hollowization and block hollowization
  4. IV Proofs and examples related to the “iterative matrix partition” (IMP) approach
    1. IV.1 Generic IMP leads to an optimal LMCC
    2. IV.2 A degenerate IMP leads to an optimal LM
    3. IV.3 An optimal LM allows a degenerate IMP
    4. IV.4 A two-qubit tutorial example for the “iterative matrix partition” approach
  5. V Proof of Theorem 2
  6. VI Proof of Theorem 3
  7. VII Proof of Theorem 4
  8. VIII Details on the three-qubit counter-example
  9. IX Analytical calculations of Eq. (27)

##  I Revisiting the QCRB-saturating optimal measurements

###  I.1 Optimal measurement condition for POVM operators

In this section, we revisit the optimal measurement condition for saturating
the QCRB and find a simplified yet still necessary and sufficient condition,
compared to Ref. [23]. To begin with, let us note the classical Fisher
information associated with a POVM measurement is given by

$F_{\omega}=\frac{(\text{Tr}[\partial_{\lambda}\rho_{\lambda}\Pi_{\omega}])^{2}}{\text{Tr}[\rho_{\lambda}\Pi_{\omega}]}.$
(S1)

Following the procedure in Ref. [26], one can find that

$F_{\omega}\leq I_{\omega},$ (S2)

where

$I_{\omega}=\text{Tr}[\rho_{\lambda}L\Pi_{\omega}L].$ (S3)

The inequality is saturated provided [28]

$\Pi_{\omega}\ket{\psi_{n\lambda}}=\xi_{\omega}\Pi_{\omega}L\ket{\psi_{n\lambda}},\,\xi_{\omega}\in\mathbb{R},\,\forall\omega,$
(S4)

where $\ket{\psi_{n\lambda}}$ is the eigenvector of $\rho_{\lambda}$
corresponding to strictly positive eigenvalues. Summing over all the
measurement outcome $\omega$ on both sides of Eq. (S2), we obtain

$F\leq I,$ (S5)

where

$I\equiv\sum_{\omega}I_{\omega}\equiv\text{Tr}[\rho_{\lambda}L^{2}].$ (S6)

Ref. [23] further recast Eq. (S4) into the following condition

$\sqrt{\Pi_{\omega}}\mathcal{M}\sqrt{\Pi_{\omega}}=0,\,\forall\omega,$ (S7)

if $\text{Tr}[\rho_{\lambda}\Pi_{\omega}]\neq 0$ and

$\sqrt{\Pi_{\omega}}L\ket{\psi_{n\lambda}}=0,$ (S8)

if $\text{Tr}[\rho_{\lambda}\Pi_{\omega}]=0$, where the POVM measurement
satisfies $\sum_{\omega}\Pi_{\omega}=\mathbb{I}$.

However, we would like to point out the condition Eq. (S8) is redundant. This
can be seen as follows: As shown in Ref. [28], when
$\text{Tr}[\rho_{\lambda}\Pi_{\omega}]=0$ is satisfied,
$\ket{\psi_{n\lambda}}$ must lies in the kernel of $\Pi_{\omega}$. As a
consequence,

$\text{Tr}[\partial_{\lambda}\rho_{\lambda}\Pi_{\omega}]=0,\qquad\text{Tr}[\Pi_{\omega}L\rho_{\lambda}]=0,\>\text{if}\>\text{Tr}[\rho_{\lambda}\Pi_{\omega}]=0.$
(S9)

Furthermore, Eq. (S8) will also be satisfied automatically, if one substitute
the expression of the SLD operator. In fact, $F_{\omega}$ should be calculated
via the L’hospital rule, which turns out for that Eq. (S2) is always saturated
for single-parameter estimation (see Theorem 2 in Ref. [28]). Therefore the
constraint (S8) is unnecessary.

Finally, Eq. (S7) implies that

$\Pi_{\omega}\mathcal{M}\Pi_{\omega}=0,\,\forall\omega.$ (S10)

To prove the converse direction, one just needs to note that $\Pi_{\omega}$
and $\sqrt{\Pi}_{\omega}$ has the same support.

###  I.2 Rank$-1$ POVM measurements are sufficient

Now we shall prove if there exists optimal POVM measurements
$\\{\Pi_{\omega}\\}$ that are not of rank-$1$, i.e.,

$\Pi_{\omega}=\sum_{s}\alpha_{\omega s}\ket{\pi_{\omega s}}\bra{\pi_{\omega
s}},$ (S11)

then one can break them into rank-$1$ measurement operators $\\{\Pi_{\omega
s}\\}$ with

$\Pi_{\omega s}\equiv\alpha_{\omega s}\ket{\pi_{\omega s}}\bra{\pi_{\omega
s}}.$ (S12)

Since $\Pi_{\omega}$ satisfies Eq. (S4), we find

$\sum_{s}\alpha_{\omega s}\ket{\pi_{\omega s}}\braket{\pi_{\omega
s}\big{missing}}{\psi_{n\lambda}}=\xi_{\omega}\ket{\pi_{\omega
s}}\braket{\pi_{\omega s}\big{missing}}{L\big{missing}}{\psi_{n\lambda}}.$
(S13)

For fixed $\omega$, $\\{\ket{\pi_{\omega s}}\\}$ are orthonormal are linearly
independent, we must have

$\alpha_{\omega s}\braket{\pi_{\omega
s}\big{missing}}{\psi_{n\lambda}}=\xi_{\omega}\braket{\pi_{\omega
s}\big{missing}}{L\big{missing}}{\psi_{n\lambda}},$ (S14)

which apparently leads to

$\Pi_{\omega s}\ket{\psi_{n\lambda}}=\xi_{\omega}\Pi_{\omega
s}L\ket{\psi_{n\lambda}}.$ (S15)

This implies that $\\{\Pi_{\omega s}\\}$ is also an also optimal POVM
measurement with rank-$1$. Note that

$\Pi_{\omega s}\Pi_{\omega r}=\delta_{sr},$ (S16)

but

$\Pi_{\omega s}\Pi_{\mu r}\neq 0,\,\text{for}\,\omega\neq\mu.$ (S17)

In summary, without loss of generality, one just need to consider rank$-1$
optimal measurements such that Eq. (2) is satisfied.

##  II  Properties of optimal LMCC and LM

In this section, we observe useful properties of the optimal LMCC from optimal
measurement condition. For LMCC, these properties in turn guarantee the
optimal measurement conditions. We point out that this is the intuition and
motivation that underlies the construction recipe by Zhou et al [23].

When it comes to LM, one can observe similar properties for the optimal LM
from the optimal measurement condition. However, these properties alone cannot
guarantee the optimal measurement condition. Thus, one needs to resort to the
“bi-partition” intuition in the main text.

###  II.1 Properties of the optimal LMCC

We would like to find LMCC such that

$\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{1},\,\cdots\omega_{N}}^{(N)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\ket{\pi_{\omega_{1},\,\cdots\omega_{N}}^{(N)}}=0,$
(S18)

where

$\sum_{\omega_{j}}\ket{\pi_{\omega_{1}\cdots\omega_{j}}^{(j)}}\bra{\pi_{\omega_{1}\cdots\omega_{j}}^{(j)}}=\mathbb{I}^{(j)},\,\forall
j\in N.$ (S19)

Interesting properties of the LMCC can be observed from the optimal
measurement condition (S18).

We emphasize that the LMCC, if they exist, they must satisfy the following
properties. First, we take the summation over
$\omega_{2},$$\omega_{3},\cdots\omega_{N}$ in Eq. (S18) and find

$\braket{\pi_{\omega_{1}}^{(1)}\big{missing}}{M^{(1)}\big{missing}}{\pi_{\omega_{1}}^{(1)}}=0,$
(S20)

where

$\displaystyle M^{(1)}$
$\displaystyle\equiv\sum_{\omega_{2}}\cdots\sum_{\omega_{N}}\bra{\pi_{\omega_{2},\omega_{1}}^{(2)}}\otimes\cdots\otimes\bra{\pi_{\omega_{N},\omega_{1}\,\cdots\omega_{N-1}}^{(N)}}\mathcal{M}\ket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\otimes\cdots\otimes\ket{\pi_{\omega_{N},\omega_{1}\,\cdots\omega_{N-1}}^{(N)}}$
(S21) $\displaystyle=\text{Tr}_{(2\cdots N)}\mathcal{M}.$ (S22)

Note that we have used Eq. (S19) and the sum must be performing in the order
from $\omega_{N}$ up to $\omega_{1}$ in Eq. (S21). Given
$\text{Tr}_{1}M^{(1)}=\text{Tr}\mathcal{M}=0,$it is possible to find
$\ket{\pi_{\omega_{1}}^{(1)}}$ by hollowizing $M^{(1)}$ according to Sec. III.

Once $\ket{\pi_{\omega_{1}}^{(1)}}$ is found, one can use it as an input and
take the summation over $\omega_{3},\cdots\omega_{N}$ in Eq. (S18) leads

$\braket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}\big{missing}}{M_{\omega_{1}}^{(2)}\big{missing}}{\pi_{\omega_{2},\omega_{1}}^{(2)}}=0,$
(S23)

where

$\displaystyle M_{\omega_{1}}^{(2)}$
$\displaystyle\equiv\bra{\pi_{\omega_{1}}^{(1)}}\otimes\sum_{\omega_{3}}\cdots\sum_{\omega_{N}}\left(\bra{\pi_{\omega_{3},\,\omega_{1}\omega_{2}}^{(3)}}\otimes\cdots\otimes\bra{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}\mathcal{M}\ket{\pi_{\omega_{3},\,\omega_{1}\omega_{2}}^{(3)}}\otimes\cdots\otimes\ket{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}\right)\otimes\ket{\pi_{\omega_{1}}^{(1)}}$
$\displaystyle=\bra{\pi_{\omega_{1}}^{(1)}}\text{Tr}_{(3\dots
N)}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}.$ (S24)

From last step, we know

$\text{Tr}_{2}M_{\omega_{1}}^{(2)}=\braket{\pi_{\omega_{1}}^{(1)}\big{missing}}{M^{(1)}\big{missing}}{\pi_{\omega_{1}}^{(1)}}=0.$
(S25)

So hollowizing $M_{\omega_{1}}^{(2)}$ is also possible. In this procedure, we
have the recursive relation

$M_{\omega_{1}\cdots\omega_{j-1}}^{(j)}=\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{j-1},\,\omega_{1}\cdots\omega_{j-2}}^{(j-1)}}\text{Tr}_{j+1\cdots
N}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\ket{\pi_{\omega_{j-1},\,\omega_{1}\cdots\omega_{j-2}}^{(j-1)}}.$
(S26)

At the very end of the procedure, it leads to

$\braket{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}\big{missing}}{M_{\omega_{1}\cdots\omega_{N-1}}^{(N)}\big{missing}}{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}=0,$
(S27)

where

$M_{\omega_{1}\cdots\omega_{N-1}}^{(N)}\equiv\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{N-1},\,\cdots\omega_{N-2}}^{(N-1)}}\mathcal{M}\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\ket{\pi_{\omega_{N-1},\,\cdots\omega_{N-2}}^{(N-1)}}.$
(S28)

At this point, we have found the properties of the LMCC, i.e. Eqs. (S20),
(S23) up to (S27), are just the consequence of Eq. (S18). It is not a prior
true that they also guarantee the optimal measurement condition. However, in
this case, the optimal measurement condition can be readily seen by
substituting Eq. (S28) into Eq. (S27). The above analysis motivates the
construction recipe by Zhou at al [23], which is listed in the following:

Step 1.0: Define $M^{(1)}\equiv\text{Tr}_{(2\cdots N)}\mathcal{M}$.

Step 1.1: Hollowizing $M^{(1)}$ leads to measurement basis
$\\{\ket{\pi_{\omega_{1}}}\\}$ with

$\braket{\pi_{\omega_{1}}^{(1)}}{M^{(1)}\big{missing}}{\pi_{\omega_{1}}^{(1)}}=0.$
(S29)

Step 2.0: Use $\ket{\pi_{\omega_{1}}^{(1)}}$ as an input and define
$M_{\omega_{1}}^{(2)}\equiv\text{Tr}_{(3\cdots
N)}\braket{\pi_{\omega_{1}}^{(1)}}{\mathcal{M}\big{missing}}{\pi_{\omega_{1}}^{(1)}}$.

Step 2.1: Hollowizing $M^{(2)}$ leads to measurement basis
$\\{\ket{\pi_{\omega_{1},\,\omega_{2}}}\\}$ with

$\braket{\pi_{\omega_{1},\omega_{2}}^{(2)}}{M_{\omega_{1}}^{(2)}\big{missing}}{\pi_{\omega_{1},\,\omega_{2}}^{(2)}}=0.$
(S30)

Step ($j$+1).0: Use
$\ket{\pi_{\omega_{1}}^{(1)}},\,\cdots,\,\ket{\pi_{\omega_{1},\,\cdots\omega_{n}}^{(j)}}$as
inputs and define

$\displaystyle M_{\omega_{1},\,\cdots\omega_{j}}^{(j+1)}$
$\displaystyle\equiv\text{Tr}_{(j+2,\,\cdots
N)}\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{1},\,\cdots\omega_{j}}^{(j)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\ket{\pi_{\omega_{1},\,\cdots\omega_{j}}^{(j)}}.$
(S31)

Remark ($j$+1).0: It is straightforward to check that $M^{(j+1)}$ is
traceless.

$\displaystyle\text{Tr}_{j+1}M^{(j+1)}$
$\displaystyle=\text{Tr}_{(j+1,\,\cdots
N)}\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\ket{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}$
$\displaystyle=$
$\displaystyle\braket{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}\big{missing}}{M_{\omega_{1}\cdots\omega_{j-1}}^{(j)}\big{missing}}{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}=0.$
(S32)

###  II.2 Properties of LM

Similar with LMCC, the optimal measurement condition for LM is

$\bra{\pi_{\omega_{1}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{N}}^{(N)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\ket{\pi_{\omega_{N}}^{(N)}}=0,$
(S33)

where

$\sum_{\omega_{j}}\ket{\pi_{\omega_{j}}^{(j)}}\bra{\pi_{\omega_{j}}^{(j)}}=\mathbb{I}^{(j)},\,\forall
j\in N.$ (S34)

Again, if local measurements exist, they must satisfy following properties. We
take the summation in Eq. (S33) except for the index $\omega_{j}$ and obtain

$\braket{\pi_{\omega_{j}}^{(j)}\big{missing}}{M^{(j)}\big{missing}}{\pi_{\omega_{j}}^{(j)}}=0,\,\forall
n,$ (S35)

where

$M^{(j)}=\text{Tr}_{(1\cdots\cancel{j}\cdots N)}\mathcal{M}.$ (S36)

We note that only the first step with $j=1$ is the same as the construction
for the LMCC. We emphasize the properties of the optimal measure, i.e., Eq.
(S35) do not guarantee the optimal measurement condition (S33).

##  III The procedure of hollowization and block hollowization

As one can see from Eq. (2) in the main text, the meaning of the optimal
measurements is that the $\mathcal{M}$ operator has zero-diagonal entries in
the measurement basis. In this section, we present more mathematical details
on the “hollowization” process discussed in the main text: A traceless matrix
can be brought to zero through unitary transformations. A generalization
notion of “hollowization” for block matrices, called “block hollowization”
procedure is also discussed.

######  Proposition.

Any square traceless matrix can be transformed into a matrix with vanishing
diagonal entries through a unitary similarity transformation.

The first proof of this claim, to our best knowledge, is by Fillmore [30],
which is also discussed in the text book on matrix analysis [29]. An immediate
corollary of above proposition is that any square matrix is unitary equivalent
to a matrix with equal diagonal entries.

Just like the standard diagonalization process, it is also possible discuss
simultaneous “hollowization” for multiple traceless matrices. For a $2\times
2$ Hermitian or anti-Hermitian traceless matrix, it can be parameterized by
$A=\bm{a}\cdot\bm{\sigma}$ or $A=\text{i}\bm{a}\cdot\bm{\sigma}$ , where
$\bm{a}$ is a real vector defined in $\mathbb{R}^{3}$. In this case,
simultaneous “hollowization” process have a very clear geometrical meaning:

######  Observation 1.

A set of traceless $2\times 2$ Hermitian (anti-Hermitian) matrices
$A_{j}=\bm{a}_{j}\cdot\bm{\sigma}$($A_{j}=\text{i}\bm{a}_{j}\cdot\bm{\sigma}$)
can be simultaneously hollowized iff $\\{\bm{a}_{j}\\}$ are coplanar.

###### Proof.

We first observe that a Hermitian matrix $A=\bm{a}\cdot\bm{\sigma}$ has zero
diagonal entries means they are traceless and have no $\sigma_{z}$ component,
i.e., the vector $\bm{a}$ lies on the $X-Y$ plane or

$\text{Tr}[A\sigma_{z}]=0.$ (S37)

The backward direction can be proved as follows: If all $\bm{a}_{j}$ are on
the same plane that is orthogonal to a unit normal vector $\hat{\bm{n}}$, then
a rotation from $\bm{\hat{n}}$ to $+Z$-axis suffices to make the diagonal
entries of all $A_{j}$ vanishing. The corresponding unitary reads

$U_{\theta}=\exp\left\\{-i\frac{\theta}{2}\frac{\bm{\hat{n}}\times\hat{\bm{z}}}{|\bm{\hat{n}}\times\hat{\bm{z}}|}\cdot\bm{\sigma}\right\\},$
(S38)

where $\theta$ is the angle between $\hat{\bm{n}}$ and $\hat{\bm{z}}$,i.e.,
$\cos\theta=\hat{\bm{n}}\cdot\bm{\hat{z}}$. Physically this corresponding the
basis transformation. means $U_{\theta}\ket{+\bm{n}}=\ket{\uparrow}$ and
$U_{\theta}\ket{-\bm{n}}=\ket{\downarrow}$. As consequence, we obtain

$U_{\theta}\bm{n}\cdot\bm{\sigma}U_{\theta}^{\dagger}=\sigma_{z}.$ (S39)

Using this identity, one can readily show

$\text{Tr}(U_{\theta}A_{j}U_{\theta}^{\dagger}\sigma_{z})=\text{Tr}(A_{j}U_{\theta}^{\dagger}\sigma_{z}U_{\theta})=\text{Tr}\left((\bm{a}_{j}\cdot\bm{\sigma})(\bm{n}\cdot\bm{\sigma})\right)=\bm{a}_{j}\cdot\bm{n}=0.$
(S40)

The forward direction can be easily seen by reversing the above argument. ∎

With Observation 1, it is clear that

######  Observation 2.

Two traceless $2\times 2$ Hermitian or anti-Hermitian matrices are
simultaneous hollowizable.

###### Proof.

The proof is straightforward: two vector $\bm{a}_{1}$ and $\bm{a}_{2}$ in
$\mathbb{R}^{3}$ must be coplanar. ∎

Since $\mathbb{C}^{2d}=\mathbb{C}^{2}\otimes\mathbb{C}^{d}$, one can decompose
the linear transformation on $\mathbb{C}^{d}$, i.e., a $(2d)\times(2d)$
dimensional matrix $\mathcal{A}$ in the block form

$\mathcal{A}=\left[\begin{array}[]{c|l}A_{11}&A_{12}\\\ \hline\cr
A_{21}&A_{22}\end{array}\right],$ (S41)

where $A_{ij}$ is a $d\times d$ matrix acting on the linear space
$\mathbb{C}^{d}$. Furthermore, if $A$ is Hermitian, then

$A_{11}^{\dagger}=A_{11},\,A_{22}^{\dagger}=A_{11},\,A_{12}^{\dagger}=A_{21},$
(S42)

while if $\mathcal{A}$ is anti-Hermitian,

$A_{11}^{\dagger}=-A_{11},\,A_{22}^{\dagger}=-A_{11},\,A_{12}^{\dagger}=-A_{21}.$
(S43)

Now we are in a position to state a theorem regarding the “block zero-trace”
process:

######  Observation 3.

For a traceless Hermitian (anti-Hermitian) matrix $A$ decomposed into the
block-structure form in Eq. (S41) can be brought to the “block hollowized”
form through a unitary matrix defined on the in the auxiliary space
$\mathbb{C}^{2}$, that is,$\exists$$2\times 2$ unitary matrix such that

$U^{\dagger}\mathcal{A}U=\left[\begin{array}[]{c|l}B_{11}&B_{12}\\\ \hline\cr
B_{21}&B_{22}\end{array}\right],$

with

$\text{Tr}B_{11}=\text{Tr}B_{22}=0.$ (S44)

###### Proof.

A $2\times 2$ unitary matrix can be parameterized as follows

$U=\begin{bmatrix}\ket{+\bm{n}},\,&e^{\text{i}\beta}\ket{-\bm{n}}\end{bmatrix}=\begin{bmatrix}\cos\frac{\theta}{2}&\sin\frac{\theta}{2}e^{\text{i}\beta}\\\
\sin\frac{\theta}{2}e^{i\phi}&-\cos\frac{\theta}{2}e^{\text{i}\phi+\text{i}\beta}\end{bmatrix},$
(S45)

where

$\displaystyle\ket{+\bm{n}}$
$\displaystyle=\cos\frac{\theta}{2}\ket{e_{1}}+\sin\frac{\theta}{2}e^{\text{i}\phi}\ket{e_{2}},$
(S46) $\displaystyle\ket{-\bm{n}}$
$\displaystyle=\sin\frac{\theta}{2}\ket{e_{1}}-\cos\frac{\theta}{2}e^{\text{i}\phi}\ket{e_{2}},$
(S47)

and

$\ket{e_{1}}=\begin{bmatrix}1\\\
0\end{bmatrix},\,\ket{e_{2}}=\begin{bmatrix}0\\\ 1\end{bmatrix}.$ (S48)

Since the $U$ is also a unitary matrix on the global space
$\mathbb{C}^{2}\otimes\mathbb{C}^{d}$, we expect that the trace is preserved,
i.e.,

$\text{Tr}B_{11}+\text{Tr}B_{22}=0.$ (S49)

Therefore to satisfy Eq. (S44), it is sufficient to make the trace of one
submatrix vanishes. It is straightforward calculate

$\displaystyle B_{11}$
$\displaystyle=\cos^{2}\frac{\theta}{2}A_{11}+\cos\frac{\theta}{2}\sin\frac{\theta}{2}(e^{\text{i}\phi}A_{12}+e^{-\text{i}\phi}A_{21})+\sin^{2}\frac{\theta}{2}A_{22},$
(S50) $\displaystyle B_{22}$
$\displaystyle=\cos^{2}\frac{\theta}{2}A_{22}-\cos\frac{\theta}{2}\sin\frac{\theta}{2}(e^{\text{i}\phi}A_{12}+e^{-\text{i}\phi}A_{21})+\sin^{2}\frac{\theta}{2}A_{11},$
(S51)

which satisfies Eq. (S49) apparently. $\text{Tr}B_{11}=0$ leads to

$\displaystyle\cos\theta\text{Tr}A_{11}+\sin\theta\text{Re}\left(e^{\text{i}\phi}\text{Tr}(A_{12})\right)$
$\displaystyle=0,\>\text{if}\>\mathcal{A}=\mathcal{A}^{\dagger},$ (S52)
$\displaystyle\cos\theta\text{Tr}A_{11}+\text{i}\sin\theta\text{Im}\left(e^{\text{i}\phi}\text{Tr}(A_{12})\right)$
$\displaystyle=0,\>\text{if}\>\mathcal{A}=-\mathcal{A}^{\dagger}.$ (S53)

Thus

$\tan\theta=\begin{cases}-\frac{\text{Tr}A_{11}}{\text{Re}\left(e^{\text{i}\phi}\text{Tr}(A_{12})\right)}&\mathcal{A}=\mathcal{A}^{\dagger}\\\
\frac{\text{i}\text{Tr}A_{11}}{\text{Im}\left(e^{\text{i}\phi}\text{Tr}(A_{12})\right)}&\mathcal{A}=-\mathcal{A}^{\dagger}\end{cases},$
(S54)

which concludes the proof. ∎

##  IV Proofs and examples related to the “iterative matrix partition” (IMP)
approach

###  IV.1 Generic IMP leads to an optimal LMCC

###### Proof.

According to Eq. (8) in the main text, we know

$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}=\braket{\pi_{\omega_{1}}^{(1)}\big{missing}}{\mathcal{M}\big{missing}}{\pi_{\omega_{1}}^{(1)}}.$
(S55)

Substituting this equation into Eq. (10), we obtain

$W_{\omega_{2}\omega_{2},\,\omega_{1}}^{(\cancel{12})}=\braket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}\big{missing}}{W_{\omega_{1}\omega_{1}}^{(\cancel{1})}\big{missing}}{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}=\bra{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\bra{\pi_{\omega_{1}}^{(1)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\ket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}.$
(S56)

Iteratively, we find

$W_{\omega_{j}\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(\cancel{1\cdots
j})}=\bra{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}\cdots\otimes\bra{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\otimes\bra{\pi_{\omega_{1}}^{(1)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\ket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\otimes\cdots\ket{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}.$
(S57)

Continuing to the last step, according to Eq. (12), we obtain

$W_{\omega_{N}\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots
N})}=\braket{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}\big{missing}}{W_{\omega_{N-1}\omega_{N-1},\,\omega_{1}\cdots\omega_{N-2}}^{(\cancel{1\cdots
N-1})}\big{missing}}{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}.$
(S58)

Substituting Eq. (S57) into Eq. (S58), we arrive at

$W_{\omega_{N}\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots
N})}=\bra{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}\cdots\otimes\bra{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\otimes\bra{\pi_{\omega_{1}}^{(1)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\ket{\pi_{\omega_{2},\,\omega_{1}}^{(2)}}\otimes\cdots\ket{\pi_{\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(N)}}.$
(S59)

Upon noticing that
$W_{\omega_{N}\omega_{N},\,\omega_{1}\cdots\omega_{N-1}}^{(\cancel{1\cdots
N})}=0$ for all $(\omega_{1},\cdots,\omega_{N})$, the proof is completed ∎

###  IV.2 A degenerate IMP leads to an optimal LM

###### Proof.

The proof is straightforward based on the proof in Sec. IV.1. Since for
degenerate IMP $U_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}$ is independent
of $\omega_{1},\,\cdots,\,\omega_{j-1}$ for all $j\in[1,N]$,
$\ket{\pi_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}^{(j)}}\equiv
U_{\omega_{j},\,\omega_{1}\cdots\omega_{j-1}}\ket{e_{\omega_{j}}^{(j)}}$ is
therefore also independent of $\omega_{1},\,\cdots,\,\omega_{j-1}$. So Eq.
(S59) becomes

$W_{\omega_{N}\omega_{N}}^{(\cancel{1\cdots
N})}=\bra{\pi_{\omega_{1}}^{(1)}}\otimes\bra{\pi_{\omega_{2}}^{(2)}}\otimes\cdots\bra{\pi_{\omega_{N}}^{(N)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\ket{\pi_{\omega_{2}}^{(2)}}\otimes\cdots\ket{\pi_{\omega_{N}}^{(N)}},$
(S60)

that is $\mathcal{M}$ is hollowized with projective LM basis. ∎

###  IV.3 An optimal LM allows a degenerate IMP

###### Proof.

Given the optimal measurement condition for LM, Eq. (S33), we construct

$U^{(j)}=\sum_{\omega_{j}}\ket{\pi_{\omega_{j}}^{(j)}}\bra{e_{\omega_{j}}^{(j)}}.$
(S61)

It is then straightforward to check that the application of $U^{(1)}$ on the
first qubit leads to

$W_{\omega_{1}\omega_{1}}^{(\cancel{1})}=\braket{\pi_{\omega_{1}}^{(1)}\big{missing}}{\mathcal{M}\big{missing}}{\pi_{\omega_{1}}^{(1)}}.$
(S62)

According to Eq. (S33), we find

$\text{Tr}_{(2\cdots
N)}W_{\omega_{1}\omega_{1}}^{(\cancel{1})}=\sum_{\omega_{2},\,\cdots\omega_{N}}\bra{\pi_{\omega_{2}}^{(1)}}\otimes\cdots\otimes\bra{\pi_{\omega_{N}}^{(N)}}W_{\omega_{1}\omega_{1}}^{(\cancel{1})}\ket{\pi_{\omega_{2}}^{(1)}}\otimes\ket{\pi_{\omega_{N}}^{(N)}}=0.$
(S63)

Next, we apply $U^{(2)}$on the second qubit, leading to

$W_{\omega_{2}\omega_{2},\,\omega_{1}}^{(\cancel{12})}=\braket{\pi_{\omega_{2}}^{(2)}\big{missing}}{W_{\omega_{1}\omega_{1}}^{(\cancel{1})}\big{missing}}{\pi_{\omega_{2}}^{(2)}}.$
(S64)

Similarly with Eq. (S63), one can easily show

$\text{Tr}_{(3\cdots
N)}W_{\omega_{2}\omega_{2},\,\omega_{1}}^{(\cancel{12})}=0.$ (S65)

With mathematical induction, one can prove that

$W_{\omega_{j}\omega_{j},\,\omega_{j-1}\cdots\omega_{1}}^{(\cancel{1\cdots
j})}=\bra{\pi_{\omega_{j}}^{(j)}}\cdots\otimes\bra{\pi_{\omega_{2}}^{(2)}}\otimes\bra{\pi_{\omega_{1}}^{(1)}}\mathcal{M}\ket{\pi_{\omega_{1}}^{(1)}}\otimes\ket{\pi_{\omega_{2}}^{(2)}}\otimes\cdots\ket{\pi_{\omega_{j}}^{(j)}},$
(S66)

with

$\text{Tr}W_{\omega_{j}\omega_{j},\,,\,\omega_{j-1}\cdots\omega_{1}}^{(\cancel{1\cdots
j})}=0,\,\forall j.$ (S67)

∎

###  IV.4 A two-qubit tutorial example for the “iterative matrix partition”
approach

Let us now illustrate IMP approach with the two-qubit pure state

$\left|\psi_{\lambda}\right\rangle=\cos\beta|01\rangle+\sin\beta
e^{i\lambda}|10\rangle.$ (S68)

where $\beta$ is a parameter independent of $\lambda$. It can be readily
calculated that

$\mathcal{M}=\text{i}\sin 2\beta(|01\rangle\langle 01|-|10\rangle\langle
10|)-\text{i}\cos 2\beta(e^{i\lambda}|10\rangle\langle
01|+e^{-i\lambda}|01\rangle\langle 10|).$ (S69)

In the first step of the IMP, according to Sec. III in the Supplemental
Material [27], we choose

$U^{(1)}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&1\\\
e^{i\phi}&-e^{i\phi}\end{array}\right]\text{, }$ (S70)

where the phase $e^{i\phi}$ is arbitrary , leading to

$\begin{array}[]{l}W_{11}^{(\cancel{1})}=\frac{i}{2}\left[\begin{array}[]{cc}-\sin
2\beta&-\cos 2\beta e^{i(\lambda-\phi)}\\\ -\cos 2\beta
e^{i(\phi-\lambda)}&\sin 2\beta\end{array}\right],\\\
W_{22}^{(\cancel{1})}=\frac{i}{2}\left[\begin{array}[]{cc}-\sin 2\beta&\cos
2\beta e^{i(\lambda-\phi)}\\\ \cos 2\beta e^{i(\phi-\lambda)}&\sin
2\beta\end{array}\right].\end{array}$ (S71)

Next, a unitary matrix $U^{(2)}$ which is independent of $\omega_{1}$

$U^{(2)}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&-e^{i\left(\lambda-\phi-\frac{\pi}{2}\right)}\\\
e^{-i\left(\lambda-\phi-\frac{\pi}{2}\right)}&1\end{array}\right]\text{. }$
(S72)

The comparison between QFI and CFI corresponding to the LM basis forming the
local unitary transformations Eqs. (S70, S72) are depicted in Figure S1, which
confirms universality of the saturation of the qCRB for 2-qubit pure states.

Figure S1:  The ratio between CFI and QFI as functions of parameter $\lambda$
for the example of two-qubit state given by Eq. (S68). The CFI is computed
from the LM basis found by the IMP approach while the QFI is computed
according to Eq. (1).

##  V Proof of Theorem 2

###### Proof.

The projectors of a projective measurement on the $j$-th qubit is
parameterized by

$\Pi_{\omega_{j}}^{(j)}=\frac{\mathbb{I}^{(j)}+(-1)^{\omega_{j}}\bm{n}\cdot\bm{\sigma}}{2},\;\omega_{j}=\pm
1.$ (S73)

It is straightforward to calculate

$\displaystyle\prod_{j}\Pi_{\omega_{j}}^{(j)}$
$\displaystyle=\frac{1}{2^{N}}\left(\mathbb{I}^{(1)}+(-1)^{\omega_{1}}\bm{n}^{(1)}\cdot\bm{\sigma}^{(1)}\right)\otimes\left(\mathbb{I}^{(2)}+(-1)^{\omega_{2}}\bm{n}^{(2)}\cdot\bm{\sigma}^{(2)}\right)\otimes\cdots\left(\mathbb{I}^{(N)}+(-1)^{\omega_{N}}\bm{n}^{(N)}\cdot\bm{\sigma}^{(N)}\right)$
$\displaystyle=\frac{1}{2^{N}}\left(\mathbb{I}+\sum_{\alpha\subseteq\mathscr{X},\,\alpha\neq\emptyset}(-1)^{\sum_{j\in\alpha}\omega_{j}}\otimes_{j\in\alpha}n^{(j)}\cdot\bm{\sigma}^{(j)}\right).$
(S74)

Upon defining

$\mathcal{N}_{\alpha}=\otimes_{j\in\alpha}n^{(j)}\cdot\bm{\sigma}^{(j)},$
(S75)

the optimal measurement condition (S33) is equivalent to

$\sum_{\alpha\subseteq\mathscr{X}_{N},\,|\alpha|\geq
1}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0,$
(S76)

where $|\alpha|$ denotes the cardinality of the set $\alpha$ and we have used
the fact that $\text{Tr}\mathcal{M}=0$. Apparently, Eq. (17) in Theorem 2 is
sufficient for Eq. (S76) to hold. We prove the converse in several steps as
follows:

Step 1: Let us first observe, upon setting $\omega_{N}=1$ and $\omega_{N}=-1$,
respectively, while keeping all the remaining $\omega_{j}$’s fixed, we find

$\displaystyle\sum_{\alpha\subseteq\mathscr{X}_{N}\backslash\\{N\\},\,|\alpha|\geq
1}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})+\sum_{\beta\cup\\{N\\},\,\beta\subseteq\mathscr{X}_{N}\backslash\\{N\\},\,|\beta|\geq
1}(-1)^{\sum_{j\in\beta}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\beta\cup\\{N\\}})$
$\displaystyle=0,\,\omega_{N}=1,$ (S77)
$\displaystyle\sum_{\alpha\subseteq\mathscr{X}_{N}\backslash\\{N\\},\,|\alpha|\geq
1}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})-\sum_{\beta\cup\\{N\\},\,\beta\subseteq\mathscr{X}_{N}\backslash\\{N\\},\,|\beta|\geq
1}(-1)^{\sum_{j\in\beta}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\beta\cup\\{N\\}})$
$\displaystyle=0,\,\omega_{N}=-1.$ (S78)

Summing over these two equations, we find

$\sum_{\alpha\subseteq\mathscr{X}_{N}\backslash\\{N\\},\,|\alpha|\geq
1}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0.$
(S79)

Iterating above procedure, we find

$\sum_{\alpha\subseteq\mathscr{X}_{N}\backslash\\{N,\,N-1,\,\cdots
2\\},\,|\alpha|\geq
0}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0,$
(S80)

i.e.,

$\text{Tr}(\mathcal{M}\mathcal{N}_{1})=0.$ (S81)

In a similar manner, one can show

$\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0,\,|\alpha|=1.$ (S82)

Step 2: Substituting Eq. (S82) into Eq. (S76), we find

$\sum_{\alpha\subseteq\mathscr{X}_{N},\,|\alpha|\geq
2}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0.$
(S83)

Following similar manipulation in Step 1, one can first choose some
$\omega_{j}$ and set it to be $+1$ and $-1$, respectively whiling keeping the
remaining ones fixed. Then summing over the two equation corresponding to
$\omega_{j}=\pm 1$, the index $j$ is then removing from equation, leading to

$\sum_{\alpha\subseteq\mathscr{X}_{N}\backslash\\{j\\},\,|\alpha|\geq
2}(-1)^{\sum_{j\in\alpha}\omega_{j}}\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0.$
(S84)

Iterating this process, we find

$\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0,\,|\alpha|=2.$ (S85)

Now one can clearly tell, when we arrive at Step $N$, we will arrive at

$\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0,\;|\alpha|=N,$ (S86)

which concludes the proof. ∎

##  VI Proof of Theorem 3

###### Proof.

For the $j$-th qubit, the most general rank-$1$ measurement can be
parameterized as follows:

$\tilde{\Pi}_{\omega_{j}}^{(j)}=x_{\omega_{j}}\ket{+\bm{n}_{\omega_{j}}^{(j)}}\bra{+\bm{n}_{\omega_{j}}^{(j)}}=\frac{x_{\omega_{j}}}{2}\left(\mathbb{I}^{(j)}+\bm{n}_{\omega_{j}}^{(j)}\cdot\bm{\sigma}^{(j)}\right),$
(S87)

where $x_{\omega_{j}}\in(0,\,1]$ and the normalization condition

$\sum_{\omega_{j}}\tilde{\Pi}_{\omega_{j}}^{(j)}=\mathbb{I}^{(j)},$ (S88)

leads to

$\sum_{\omega_{j}}x_{\omega_{j}}^{(j)}=2,\quad\sum_{\omega_{j}}x_{\omega_{j}}\bm{n}_{\omega_{j}}^{(j)}=0.$
(S89)

According to the qCRB-saturating condition (S7), we find

$\text{Tr}\left(\mathcal{M}\otimes_{j}\tilde{\Pi}_{\omega_{j}}^{(j)}\right)=0,$
(S90)

which can be simplified as

$\text{Tr}\left(\mathcal{M}\otimes_{j}(\ket{+\bm{n}_{\omega_{j}}^{(j)}}\bra{+\bm{n}_{\omega_{j}}^{(j)}})\right)=\text{Tr}\left(\mathcal{M}\otimes_{j}\frac{\mathbb{I}^{(j)}+\bm{n}_{\omega_{j}}^{(j)}\cdot\bm{\sigma}^{(j)}}{2}\right)=0.$
(S91)

Without loss of generality, one can assuming
$\omega_{j}=1,\,2,\cdots,\,d_{j}$. For $j$-th qubit we set $\omega_{j}=1$ and
by summing over all the remaining $\omega$’s find,

$\text{Tr}\left(\mathcal{M}\ket{+\bm{n}_{1}^{(j)}}\bra{+\bm{n}_{1}^{(j)}}\right)=\frac{1}{2}\text{Tr}\left(\mathcal{M}(\mathbb{I}^{(j)}+\bm{n}_{1}^{(j)}\cdot\bm{\sigma}^{(j)})\right)=0,$
(S92)

which reduces to

$\text{Tr}\left(\mathcal{M}\mathcal{N}_{\alpha}\right)=0,$ (S93)

where $\alpha=\\{j\\}$ and $\mathcal{N}_{\alpha}$ is defined as in Eq. (S75)
with $\bm{n}^{(j)}$ identified as $\bm{n}_{1}^{(j)}$. Similarly, by
considering two qubits $j,k$, we obtain

$\text{Tr}\left(\mathcal{M}\frac{(\mathbb{I}^{(j)}+\bm{n}_{1}^{(j)}\cdot\bm{\sigma}^{(j)})(\mathbb{I}^{(k)}+\bm{n}_{1}^{(k)}\cdot\bm{\sigma}^{(k)})}{2}\right)=0,$
(S94)

which leads to Eq. (S93) with $\alpha=\\{j,\,k\\}$. In a similar manner, by
induction,

$\text{Tr}(\mathcal{M}\mathcal{N}_{\alpha})=0,\quad\forall\alpha\subseteq\mathscr{X}_{N}.$
(S95)

∎

##  VII Proof of Theorem 4

###### Proof.

For simplicity, we shall suppress the time-dependence of $U_{\lambda}$ on $t$.
For unitary channels and pure initial states, i.e.,

$\rho_{\lambda}=U_{\lambda}\rho_{0}U_{\lambda}^{\dagger},$ (S96)

with $\rho_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$, it is straightforward to compute

$\partial_{\lambda}\rho_{\lambda}=\partial_{\lambda}U_{\lambda}\rho_{0}U_{\lambda}^{\dagger}+U_{\lambda}\rho_{0}\partial_{\lambda}U_{\lambda}^{\dagger}=-\text{i}U_{\lambda}[G_{\lambda},\,\rho_{0}]U_{\lambda}^{\dagger}.$
(S97)

Thus we find

$\mathcal{M}=2[\rho_{\lambda},\,\partial_{\lambda}\rho_{\lambda}]=-2\text{i}U_{\lambda}[\rho_{0},\,[G_{\lambda},\,\rho_{0}]]U_{\lambda}^{\dagger}.$
(S98)

In the special case of $U_{\lambda}=e^{-\text{i}H_{0}t\lambda}$,
$G_{\lambda}=tH_{0}$, $M$ takes a simple form

$\mathcal{M}=-2\text{i}tU_{\lambda}[\rho_{0},\,[H_{0},\,\rho_{0}]]U_{\lambda}^{\dagger}.$
(S99)

Alternatively, in terms of $\rho_{\lambda}$

$\mathcal{M}=2\text{i}t\left(\left\\{H_{0}\text{,}\,\rho_{\lambda}\right\\}-2\langle\psi_{0}|H_{0}|\psi_{0}\rangle\rho_{\lambda}\right),$
(S100)

Substituting Eq. (S98) into Eq. (17), we arrive at

$\text{Tr}\left([\rho_{0},\,[G_{\lambda},\,\rho_{0}]]\mathcal{N}_{\alpha}^{(\text{H})}\right)=0,$
(S101)

Eq. (S101) can be rewritten as

$\text{Tr}\left([G_{\lambda},\,\rho_{0}][\mathcal{N}_{\alpha}^{(\text{H})},\,\rho_{0}]\right)=0.$
(S102)

leads to Eq. (22) in the main text.  ∎

##  VIII Details on the three-qubit counter-example

Let us first note the symmetries in the initial states. We denote the
computational basis

$\ket{e_{1}}=\ket{001},\,\ket{e_{2}}=\ket{010},\,\ket{e_{3}}=\ket{001},$
(S103) $\mathcal{V}_{o}=\text{span}\\{\ket{e_{\alpha}}\\}_{\alpha=1}^{3},$
(S104)

and the orthogonal complement of $\mathcal{V}_{o}$ as
$\mathcal{V}_{o}^{\perp}$. Clearly, $\mathcal{V}_{o}$ is the odd parity
subspace with one ”$\ket{1}$” state. We observe that when acting on
computational basis, $\sigma_{x}^{(j)}$ and $\sigma_{y}^{(j)}$ flip the
computational basis for the $j$-th qubit while $\sigma_{z}^{(j)}$ left the
computational basis unchanged, that is

$\sigma_{a}^{(j)}|e_{\alpha}\rangle\in\mathcal{V}_{o}^{\perp},\>\forall\alpha=1,\,2,\,3,\,a=x,\,y.$
(S105)
$\sigma_{a}^{(j)}\sigma_{b}^{(k)}\sigma_{c}^{(l)}|e_{\alpha}\rangle\in\mathcal{V}_{o}^{\perp},\,\forall\alpha=1,\,2,\,3,\,a,\,b,\,c=x,\,y.$
(S106)

More generally, for an operator $\mathcal{O}$ of odd parity, i.e., a
combination of Pauli operators, flips the states at odd number of times,

$\mathcal{O}|e_{\alpha}\rangle\in\mathcal{V}_{o}^{\perp},\>\forall\alpha=1,\,2,\,3,$
(S107)

Furthermore, even for operators with even parity, we note that

$\sigma_{a}^{(1)}\sigma_{b}^{(2)}\ket{001}\in\mathcal{V}_{o}^{\perp},\,\sigma_{a}^{(2)}\sigma_{b}^{(3)}\ket{100}\in\mathcal{V}_{o}^{\perp},\,\text{etc},\,a=x,\,y.$
(S108)

Next, we note that initial states $\ket{\psi_{0}}$ also possesses permutation
symmetry of spins, as well as $U(1)$ symmetry, i.e.,

$\sum_{j}\sigma_{z}^{(j)}\ket{\psi_{0}}=\ket{\psi_{0}}.$ (S109)

The parity, permutation and $U(1)$ symmetries will simplify our subsequent
calculations dramatically.

Now we are in a position to list the all the equations in Eq. (22). When
$\lambda=0$, $G_{\lambda}(t)=tH_{0}$, where

$H_{0}=\sigma_{x}^{(1)}\sigma_{x}^{(2)}+\sigma_{x}^{(2)}\sigma_{x}^{(3)}+\sigma_{y}^{(1)}\sigma_{y}^{(2)}+\sigma_{y}^{(2)}\sigma_{y}^{(3)},$
(S110)

and

$\mathcal{N}_{\alpha}^{(\text{H})}(t)=\mathcal{N}_{\alpha}(0).$ (S111)

So Eq. (22) becomes

$\text{Cov}\left(\mathcal{N}_{\alpha}(0)H_{0}\right)_{\ket{\psi_{0}}}=0.$
(S112)

First, taking $\alpha=\\{j\\}$ with $j\in\\{1,2,3\\}$, we find

$\braket{\psi_{0}\big{missing}}{\left(\bm{n}^{(j)}\cdot\bm{\sigma}^{(j)}\right)H_{0}\big{missing}}{\psi_{0}}-\langle\psi_{0}|H_{0}|\psi_{0}\rangle
n_{z}^{(j)}=0.$ (S113)

With Eq. (S107), the above equation reduces to

$n_{z}^{(j)}\left(\braket{\psi_{0}\big{missing}}{\sigma_{z}^{(j)}H_{0}\big{missing}}{\psi_{0}}-\langle\psi_{0}|H_{0}|\psi_{0}\rangle\right)=0.$
(S114)

Using Eq. (S108) and the permutation symmetry of $\ket{\psi_{0}}$,

$\displaystyle\braket{\psi_{0}\big{missing}}{\sigma_{a}^{(j)}\sigma_{a}^{(k)}\big{missing}}{\psi_{0}}$
$\displaystyle=\braket{\psi_{0}\big{missing}}{\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{\psi_{0}}$
$\displaystyle=\frac{1}{3}\left(\braket{010\big{missing}}{\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{100}+\braket{100\big{missing}}{\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{010}\right)$
$\displaystyle=\frac{2}{3},\;a=x,\,y.$ (S115)

Since $\sigma_{z}^{(3)}\ket{e_{\alpha}}\in\mathcal{V}_{o}$ with
$\alpha=1,\,2,\,3$, we conclude

$\braket{\psi_{0}\big{missing}}{\sigma_{z}^{(3)}\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{\psi_{0}}=\braket{\psi_{0}\big{missing}}{\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{\psi_{0}}=\frac{2}{3}.$
(S116)

Then according to the permutation symmetry and $U(1)$ symmetry (S109), we
conclude

$\braket{\psi_{0}\big{missing}}{\sigma_{z}^{(1)}\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{\psi_{0}}=\braket{\psi_{0}\big{missing}}{\sigma_{z}^{(2)}\sigma_{a}^{(1)}\sigma_{a}^{(2)}\big{missing}}{\psi_{0}}=0.$
(S117)

In summary,

$\braket{\psi_{0}\big{missing}}{\sigma_{z}^{(l)}\sigma_{a}^{(j)}\sigma_{a}^{(k)}\big{missing}}{\psi_{0}}=\begin{cases}\frac{2}{3}&j\neq
k\neq l\neq j\\\ 0&l=j\>\text{or}\>l=k\end{cases},$ (S118)

which also implies

$\braket{\psi_{0}\big{missing}}{\sigma_{x}^{(j)}\sigma_{y}^{(k)}\big{missing}}{\psi_{0}}=0,\,j\neq
k.$ (S119)

According to Eq. (S115), we conclude

$\langle\psi_{0}|H_{0}|\psi_{0}\rangle=\frac{8}{3},$ (S120)

and according to Eq. (S118)

$\braket{\psi_{0}\big{missing}}{\sigma_{z}^{(j)}H_{0}\big{missing}}{\psi_{0}}=\begin{cases}\frac{4}{3}&j=1,\,3\\\
0&j=2\end{cases}.$ (S121)

Therefore we obtain $n_{z}^{(j)}=0$. Consequently, we can parameterize as

$\bm{n}^{(j)}\cdot\bm{\sigma}^{(j)}=\cos\beta^{(j)}\sigma_{x}^{(j)}+\sin\beta^{(j)}\sigma_{y}^{(j)}.$
(S122)

Combing this parameterization with parity symmetry, we find for
$\alpha=\\{1,\,2,3\\}$, Eq. (S112) is satisfied automatically thanks to the
parity symmetry. Next we take $\alpha=\\{k,\,l\\}$ in Eq. (S112) with $k\neq
l,\,k,\,l\in\\{1,2,3\\}$, we find

$\braket{\psi_{0}\big{missing}}{\left(\bm{n}^{(k)}\cdot\bm{\sigma}^{(k)}\otimes\bm{n}^{(l)}\cdot\bm{\sigma}^{(l)}\right)H_{0}\big{missing}}{\psi_{0}}-\braket{\psi_{0}\big{missing}}{\bm{n}^{(k)}\cdot\bm{\sigma}^{(k)}\otimes\bm{n}^{(l)}\cdot\bm{\sigma}^{(l)}\big{missing}}{\psi_{0}}\langle\psi_{0}|H_{0}|\psi_{0}\rangle=0,$
(S123)

which implies,

$\cos\left(\beta^{(k)}-\beta^{(l)}\right)=0.$ (S124)

for any pair of $(k,\,l)$, which is impossible.

##  IX Analytical calculations of Eq. (27)

Define a $N$-qubit highest-weight state $|j,\,j\rangle\equiv|0^{\otimes
N}\rangle$ with $j=N/2$ such that $S_{z}|j,\,j\rangle=j|j,\,j\rangle$ and
$S_{+}|j,\,j\rangle=0$. Then

$\left(S_{-}\right)^{p}|j,\,j\rangle=\sqrt{\frac{N!p!}{(N-p)!}}|j,\,j-p\rangle,\quad
p=0,1,\ldots,N$ (S125)

where $|j,\,j-p\rangle$ is the eigenstate of $S_{z}$ such that
$S_{z}|j,\,j-p\rangle=(\frac{N}{2}-p)|j,\,j-p\rangle$.

The spin coherent state Eq. (26) can be equivalently expressed as [40]

$|\psi_{0}\rangle=\frac{\exp\left(\mu
S_{-}\right)}{\left(1+|\mu|^{2}\right)^{\frac{N}{2}}}|j,\,j\rangle=\frac{1}{\left(1+|\mu|^{2}\right)^{\frac{N}{2}}}\sum_{p=0}^{N}\sqrt{\frac{N!}{p!(N-p)!}}\mu^{p}|j,\,j-p\rangle,\quad\mu=\tan\left(\frac{\theta}{2}\right)e^{i\phi},\>\theta\in[0,\pi),\phi\in[0,2\pi).$
(S126)

Define the generating function

$\displaystyle F(\beta)$
$\displaystyle\equiv\langle\psi_{0}|e^{\text{i}S_{z}\beta}|\psi_{0}\rangle=\left(1+|\mu|^{2}\right)^{-N}e^{\text{i}\beta
N/2}\sum_{p=0}^{N}\left(\frac{(N)!}{p!(N-p)!}\right)\left(\left|\mu\right|^{2}e^{-\text{i}\beta}\right)^{p}$
$\displaystyle=\frac{\left(e^{\text{i}\beta/2}+\left|\mu\right|^{2}e^{-\text{i}\beta/2}\right)^{N}}{\left(1+\left|\mu\right|^{2}\right)^{N}},$
(S127)

where in the last step we have used
$\sum_{p=0}^{N}\frac{(N)!}{p!(N-p)!}\left|\mu\right|^{2p}=\left(1+\left|\mu\right|^{2}\right)^{N}$.
We note

$\text{Var}[S_{z}^{2}]_{\ket{\psi_{0}}}=\left\langle\psi_{0}\left|S_{z}^{4}\right|\psi_{0}\right\rangle-\left(\left\langle\psi_{0}\left|S_{z}^{2}\right|\psi_{0}\right\rangle\right)^{2}=\left[\frac{\partial^{4}F(\beta)}{\partial\beta^{4}}-\left(\frac{\partial^{2}F(\beta)}{\partial\beta^{2}}\right)^{2}\right]\bigg{|}_{\beta=0}.$
(S128)

Carrying out the derivatives, we obtain

$\displaystyle\text{Var}[S_{z}^{2}]_{\ket{\psi_{0}}}=$
$\displaystyle\left(-\frac{3}{8}(1-\cos\theta)^{4}+\frac{3}{2}(1-\cos\theta)^{3}-\frac{7}{4}(1-\cos\theta)^{2}+\frac{1}{2}(1-\cos\theta)\right)N$
(S129)
$\displaystyle+\left(\frac{5}{8}(1-\cos\theta)^{4}-\frac{5}{2}(1-\cos\theta)^{3}+3(1-\cos\theta)^{2}+\cos\theta-1\right)N^{2}$
$\displaystyle+\left(-\frac{1}{4}(1-\cos\theta)^{4}+(1-\cos\theta)^{3}-\frac{5}{4}(1-\cos\theta)^{2}+\frac{1}{2}(1-\cos\theta)\right)N^{3}.$

One can immediate read off the expression for $f_{k}(\cos\theta)$ from above
equation.