# Certifying the quantum Fisher information from a given set of mean values: a
semidefinite programming approach

Guillem Müller-Rigat<EMAIL_ADDRESS>ICFO - Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Anubhav Kumar Srivastava  ICFO - Institut de
Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Stanisław Kurdziałek  Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland  Grzegorz Rajchel-
Mieldzioć  ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels (Barcelona), Spain  Maciej
Lewenstein  ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels (Barcelona), Spain  ICREA, Pg.
Lluís Companys 23, 08010 Barcelona, Spain  Irénée Frérot
<EMAIL_ADDRESS>Univ Grenoble Alpes, CNRS, Grenoble INP, Institut
Néel, 38000 Grenoble, France

###### Abstract

We introduce a semidefinite programming algorithm to find the minimal quantum
Fisher information compatible with an arbitrary dataset of mean values. This
certification task allows one to quantify the resource content of a quantum
system for metrology applications without complete knowledge of the quantum
state. We implement the algorithm to study quantum spin ensembles. We first
focus on Dicke states, where our findings challenge and complement previous
results in the literature. We then investigate states generated during the
one-axis twisting dynamics, where in particular we find that the metrological
power of the so-called multi-headed cat states can be certified using simple
collective spin observables, such as fourth-order moments for small systems,
and parity measurements for arbitrary system sizes.

##  1 Introduction

Quantum states of multipartite systems may represent useful resources for
specific quantum technology applications [12]. It is therefore a central task
for quantum information science to provide robust and reliable protocols to
estimate the resource content of a quantum state $\hat{\rho}$ prepared in a
given device. In several situations, the quantum advantage is a genuine
quantum many-body effect, such as in quantum computing [34], quantum
simulation [26, 2, 14] or quantum metrology [31]. Hence, in the context of
quantum certification – namely, the task of estimating the resource content of
a given experimental preparation –, a pervasive challenge is that of
scalability. Given that quantum state tomography quickly becomes unfeasible
for systems of increasing size, this challenge signifies that one must base
the certification protocols on partial information, typically the knowledge of
a few mean values of collective observables. A certification algorithm then
represents a conservative approach, which quantifies the minimal resource
content over all quantum states compatible with the available partial
information. A key property of all quantum resource quantifiers is that they
are convex functions $f(\hat{\rho})$ of the quantum state; namely, a quantum
resource [9], be it quantum coherence [40], thermodynamic resources [21],
quantum entanglement [20], Einstein-Podolski-Rosen steering [42], Bell non-
locality [8, 37], etc., cannot be produced (nor amplified) by mere statistical
mixing of different preparations. Convexity turns out to be a key technical
property for practical certification algorithms, as imposing compatibility of
the quantum state with a given set of mean values – namely, linear constraints
of the form ${\rm Tr}(\hat{D}_{a}\hat{\rho})=\langle\hat{D}_{a}\rangle$ for
$a=1,\dots,K$ where $\hat{D}_{a}$ are some observables forming the available
data –, maintains the convexity of $f$. Therefore, the conservative estimate
of the resource content of $\hat{\rho}$ compatible with the data can be cast
altogether as a convex optimization algorithm, hence presenting a unique and
well-defined optimum.

In this work, we focus on estimating the quantum Fisher information (QFI)
associated to unitary transformations of $\hat{\rho}$ [7, 5]. The QFI
quantifies the usefulness of the state for quantum metrology (namely, the task
of measuring small parameters, such as magnetic fields or gravitational
fields, beyond the capabilities of classical systems of the same size) [31].
While previous works have achieved significant progress in this respect [4,
15], no existing approach fully exploits the convexity of the QFI to achieve
an unbiased and certified estimate of the QFI given a set of mean values. In
the present paper, exploiting the link of the QFI with the Uhlmann fidelity,
we develop a semi-definite programming (SDP) algorithm to solve this problem.
SDP represents a well-studied special class of convex optimization problems,
for which efficient numerical routines are available [38]. Our new approach is
then applied to several problems of relevance to quantum metrology, such as
Dicke states [48, 31]. We then explore the metrological resource produced
during the out-of-equilibrium dynamics of the celebrated one-axis-twisting
dynamics (OAT) [23]. In this case, we obtain the unexpected result that few-
particle multi-headed cat states and their metrological power can be fully
assessed by low-order moments of collective spin operators. We also find that
Heisenberg-scaling metrological power can be optimally exploited by parity
measurements throughout the OAT evolution. We finally illustrate our approach
in small quantum spin chains with nearest-neighbour interactions.

##  2 Semidefinite programming solution

Quantum Fisher information.– In the context of quantum interferometry, the QFI
evaluates the sensitivity of the state $\hat{\rho}$ to unitary transformations
of the form
$\hat{\rho}(\theta)=e^{-i\theta\hat{G}}\hat{\rho}e^{+i\theta\hat{G}}$ (linear
encoding of the phase $\theta$) [31]. Here, $\hat{G}$ is the observable
generating the unitary transformation. In the specific case of magnetometry,
for instance, $\hat{G}$ is some component of the magnetic moment of the
system, and $\theta$ would incorporate both the time of evolution and the
strength of the magnetic field. Via the quantum Cramér-Rao bound [11, 35, 18],
the QFI ($F_{Q}[\hat{\rho},\hat{G}]$) quantifies the ultimate sensitivity that
can be achieved on the estimation of $\theta$ by making measurements on the
state $\hat{\rho}(\theta)$, namely [31]:

$(\Delta\theta)_{\rm
est.}^{2}\geq\frac{(\Delta\hat{O})^{2}}{(\partial_{\theta}\langle\hat{O}\rangle)^{2}}:=\Xi_{O}^{2}\geq\frac{1}{F_{Q}[\hat{\rho},\hat{G}]}\
\leavevmode\nobreak\ .$ (1)

Here, $(\Delta\theta)_{\rm est.}^{2}$ is the variance of the estimator of the
phase $\theta$, and $\hat{O}$ is some observable whose mean value
$\langle\hat{O}\rangle={\rm Tr}[\hat{O}\hat{\rho}(\theta)]$ changes with
$\theta$ as
$\partial_{\theta}\langle\hat{O}\rangle=i\langle[\hat{G},\hat{O}]\rangle$, and
$(\Delta\hat{O})^{2}=\langle\hat{O}^{2}\rangle-\langle\hat{O}\rangle^{2}$ is
the variance of $\hat{O}$. We denote by $\Xi_{O}$ the sensitivity of the
observable $\hat{O}$ in the phase estimation task. The Cramér-Rao bound [Eq.
(1)] is valid for arbitrary observables $\hat{O}$.

Statement of the certification problem.– The problem we aim at solving is to
certify the minimal QFI compatible with a set of $K$ expectation values
$\langle\hat{\mathbf{D}}\rangle=(1,\langle\hat{D}_{1}\rangle,\langle\hat{D}_{2}\rangle,...,\langle\hat{D}_{K}\rangle)$
(thereafter called data):

$\begin{array}[]{crl}\min_{\hat{\rho}=\hat{\rho}^{\dagger}}&F_{Q}[\hat{\rho},\hat{G}]&\\\
\mbox{s.t.}&\mathrm{Tr}(\hat{\mathbf{D}}\hat{\rho})&=\langle\hat{\mathbf{D}}\rangle\\\
&\hat{\rho}&\succeq 0\end{array}\\\ ,$ (2)

where $\hat{\rho}$ is a quantum state acting in a Hilbert space $\mathcal{H}$
of a given dimension (namely: a Hermitian, semidefinite positive, unit-trace
operator; note that we implement the unit-trace constraint with
$\hat{D}_{0}=\mathbb{I}$). As mentioned in the introduction, this problem is
well-posed since the QFI is a convex function of the state and the linear
constraints corresponding to the data maintain the convexity property.

QFI and Uhlmann fidelity.– The QFI can be related to a geometric notion of
distance between the infinitesimally displaced states
$\hat{\rho}_{\pm}=\hat{\rho}(\pm\delta\theta)$, namely [36, 49]:

$\mathcal{F}(\hat{\rho}_{+},\hat{\rho}_{-})=1-F_{Q}[\hat{\rho},\hat{G}](\delta\theta)^{2}+{\cal
O}[(\delta\theta)^{4}]\ ,$ (3)

with
$\sqrt{\mathcal{F}(\hat{\rho},\hat{\sigma})}=\mathrm{Tr}\left[\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right]$
the Uhlmann fidelity. Importantly, if $\hat{\rho}$ is a pure state, then the
QFI reduces to (four times) the variance of $\hat{G}$:
$F_{Q}[|\psi\rangle\langle\psi|,\hat{G}]=4[\langle\psi|\hat{G}^{2}|\psi\rangle-\langle\psi|\hat{G}|\psi\rangle^{2}]:=4(\Delta\hat{G})^{2}_{\ket{\psi}}$.
As per Eq. (3), up to fourth-order corrections in $\delta\theta$, minimizing
the QFI is equivalent to maximizing the fidelity between the states
$\hat{\rho}(\pm\delta\theta)$ given the constraints
$\mathrm{Tr}(\hat{\mathbf{D}}\hat{\rho})=\langle\hat{\mathbf{D}}\rangle$.

SDP solution to the certification problem.– To arrive at our practical
algorithm, we use the fact that the fidelity can be computed as a
semidefinite-program (SDP) [43, 38],

$\begin{array}[]{crl}\sqrt{\mathcal{F}(\hat{\rho},\hat{\sigma})}=&\max_{\hat{L}}&\Re{\mathrm{Tr}(\hat{L})}\\\
&\mbox{s.t.}&\begin{pmatrix}\hat{\rho}&\hat{L}^{\dagger}\\\
\hat{L}&\hat{\sigma}\end{pmatrix}\succeq 0\end{array}\ .$ (4)

Furthermore, it is straightforward to impose compatibility of the state with
the data in the SDP formulation, yielding the final SDP algorithm (for a given
fixed and sufficiently small $\delta\theta$):

$\begin{array}[]{crl}\sqrt{{\cal F}_{\rm
max}}=&\max_{\begin{subarray}{c}\hat{L}\\\
\hat{\rho}=\hat{\rho}^{\dagger}\end{subarray}}&\Re{\mathrm{Tr}(\hat{L})}\\\
&\mbox{s.t.}&\begin{pmatrix}\hat{\rho}_{+}&\hat{L}^{\dagger}\\\
\hat{L}&\hat{\rho}_{-}\end{pmatrix}\succeq 0\\\
&&\hat{\rho}_{\pm}=\hat{\rho}\pm i\delta\theta[\hat{G},\hat{\rho}]\\\
&&\mathrm{Tr}(\hat{\mathbf{D}}\hat{\rho})=\langle\hat{\mathbf{D}}\rangle\\\
\end{array}\ .$ (5)

The SDP algorithm delivers both the maximal fidelity and an optimal quantum
state $\hat{\rho}^{*}$, whose QFI is evaluated either as
$F_{Q}[\hat{\rho}^{*},\hat{G}]=(1-{\cal F}_{\rm max})/\delta\theta^{2}+{\cal
O}(\delta\theta^{2})$, or directly using the optimal state $\hat{\rho}^{*}$.
In Appendix A.1, we further discuss the accuracy of this certificate for small
yet finite $\delta\theta$.

Notice that the SDP algorithm can be extended to a generic (namely: non-
unitary) parameter encoding operation by defining
$\hat{\rho}_{\pm}(\theta)=\hat{\rho}(\theta)\pm\delta\theta\hat{\mathcal{L}}[\hat{\rho}(\theta)]$,
where $\hat{\mathcal{L}}$ is its corresponding Lindbladian,
$\partial_{\theta}\hat{\rho}(\theta)=\hat{\mathcal{L}}[\hat{\rho}(\theta)]$.
Such approach can be used to incorporate noise and other imperfections of the
metrological task [25]. In this situation, the QFI becomes $\theta$-dependent.

Alternative SDP algorithm.– In Appendix A.2 we introduce an alternative SDP
algorithm that avoids introducing this approximation, albeit at a higher
computational cost.

Exploiting the metrological usefulness.– After obtaning the minimal QFI for
fixed arbitrary data $\hat{\mathbf{D}}$, one may ask how to operationally
exploit the corresponding metrological resource. In other words, one aims at
finding an explicit observable $\hat{O}$ with sensitivity $\Xi_{O}^{-2}$ as
close as possible to the QFI. From the optimal state $\hat{\rho}^{*}$, such
observable can be provided with the certificate. This privileged observable
corresponds to the symmetric logarithmic derivative (SLD) [30] $S$ which is
derived from the relation
$i[\hat{G},\hat{\rho}^{*}]=\\{\hat{S},\hat{\rho}^{*}\\}/2$. For pure states,
$\hat{\rho}^{*}=\outerproduct{\Psi}{\Psi}$,
$\hat{S}=2i[\hat{G},\outerproduct{\Psi}{\Psi}]$. For this observable,
$\Xi_{S}^{-2}=\partial_{\theta}\langle S\rangle=\langle
i[\hat{G},\hat{S}]\rangle=\langle\hat{S}^{2}\rangle=F_{Q}[\hat{\rho}^{*},\hat{G}]$
is verified. Notice, however, that in practice the SLD observable might be
impractical to measure, and the practical optimal observable must be found on
a case-by-case basis [15].

Taking advantage of symmetries.– While being a convex-optimization algorithm,
the SDP of Eq. (5) involves the complete density matrix $\hat{\rho}$ as a
variable of the problem, and hence is hardly scalable beyond about $N=10$
qubits. However, in several cases of practical interest, both the generator
$\hat{G}$ and the observables $\hat{\mathbf{D}}$ respect elementary
symmetries, such as translation invariance, or even invariance under
permutation of the elementary degrees of freedom. Such symmetries can be
accommodated in the SDP by splitting the optimization into different symmetry
sectors, which yields a more scalable algorithm. As further discussed in
Appendix B, if both the observables $\hat{\mathbf{D}}$ and the generator
$\hat{G}$ obey a certain symmetry, it is sufficient to optimize over symmetric
states of the form $\hat{\rho}=\oplus_{J}\hat{\varrho}^{(J)}$, where
$\hat{\varrho}^{(J)}$ is a (non-normalized) quantum state acting on the
symmetry sector $J$ (with ${\rm Tr}[\hat{\varrho}^{(J)}]$ the probability to
find the system in the symmetry sector $J$). In essence, taking advantage of
the symmetries allows one to block-diagonalize the problem with respect to the
symmetry sectors, whose dimensions are typically much smaller than the
complete many-body Hilbert space, yielding an SDP algorithm with a more
favorable scaling (see Appendix B for a specific example on permutation
invariance).

Alternative approaches.– The central problem Eq. (2) has appeared in different
contexts throughout the literature. In Appendix C, we give a concise review of
past efforts to solve it and discuss its comparison to the algorithms proposed
here.

In the remainder of the paper, we illustrate our approach by providing the
certified metrological usefulness in several classes of states of experimental
relevance. We focus on states of an ensemble of $N$ quantum spin-1/2
particles. In Section 3, we consider Dicke states, which in contrast to spin-
squeezed states have a vanishing mean spin. In Section 4, we then consider the
family of states generated during the so-called one-axis twisting dynamics
[23] using collective spin observables of increasing orders.

##  3 Metrological usefulness of unpolarized states

In this section, we focus on unpolarized states, namely states of vanishing
mean spin: $\langle\hat{\bf J}\rangle=0$ with $\hat{\bf
J}=\sum_{i=1}^{N}\hat{\bf S}^{(i)}$ and $\hat{\bf
S}^{(i)}=(S_{x}^{(i)},S_{y}^{(i)},S_{z}^{(i)})$ individual spin-1/2 operators.
Such unpolarized states may offer a metrological advantage, the most prominent
example being balanced Dicke states, such that $\langle\hat{\bf
J}^{2}\rangle=(N/2)(N/2+1)$ (maximal total spin) and
$\langle\hat{J}_{x}^{2}\rangle=0$ (eigenstate of the collective spin along $x$
with eigenvalue $0$) [19]. The balanced Dicke state displays an exquisite
sensitivity to rotations around any axis in the $yz$ plane, with QFI given by
$F_{Q}[\hat{\rho},\hat{J}_{z}]=4\langle\hat{J}_{z}^{2}\rangle=N(N/2+1)$. To
exploit this sensitivity, a possible approach is to measure
$\langle\hat{J}_{x}^{2}\rangle(\theta)$, allowing one to reach the sensitivity
bound of the QFI [22, 27, 3, 51]. In this protocol, it is needed to estimate
the variance of $\hat{J}_{x}^{2}$, and hence to access fourth-order moments of
the collective spin. Here, we are instead interested in quantifying the
metrological usefulness of unpolarized states using only second moments of
collective spins. Our motivation for doing so is that in an experiment, it is
impossible to prepare the balanced Dicke state with unit fidelity, and it is
natural to certify the QFI using only the knowledge of
$\langle\hat{J}_{x}^{2}\rangle$, $\langle\hat{J}_{y}^{2}\rangle$,
$\langle\hat{J}_{z}^{2}\rangle$ .

Case $\langle\hat{J}_{x}^{2}\rangle=0$.– In particular, for
$\langle\hat{J}_{x}^{2}\rangle=0$, as proven in Appendix D, the QFI with
respect to the generator $\hat{J}_{{\bf n}_{\perp}}$ is equal to
$4\langle\hat{J}_{{\bf n}_{\perp}}^{2}\rangle$, where ${\bf n}_{\perp}$ is any
direction in the $yz$ plane. The constraint $\langle\hat{J}_{x}^{2}\rangle=0$
implies that any state compatible with it is an eigenstate of $\hat{J}_{x}$
and therefore it is rotationally invariant around the $x$ axis. Consequently,
$\langle\hat{J}_{y}^{2}\rangle=\langle\hat{J}_{z}^{2}\rangle=\langle\hat{\mathbf{J}}^{2}\rangle/2$.
Hence, such observation extends the property of the balanced Dicke state
(which is a pure state of maximal total spin
$\langle\hat{\mathbf{J}}^{2}\rangle=(N/2)(N/2+1)$) to general mixed states
with arbitrary total spin.

Case $\langle\hat{J}_{x}^{2}\rangle>0$.– We then investigate the effect of a
finite but small nonzero value of $\langle\hat{J}_{x}^{2}\rangle$ on the QFI
of $\hat{J}_{z}$. In Appendix E, we study the minimal $F_{Q}(\hat{J}_{z})$
compatible with a given $\langle\hat{J}_{x}^{2}\rangle$ and a given total spin
$\langle{\bf J}^{2}\rangle\sim N^{2}$. Our main conclusion is that for
$\langle\hat{J}_{x}^{2}\rangle\gtrsim 1/4$, the Heisenberg scaling
$F_{Q}(\hat{J}_{z})\sim N^{2}$ is lost. This result suggests that certifying
the metrological power of Dicke-like states requires knowledge beyond second
moments.

Questioning the Dicke squeezing parameter.– Here we question the validity of a
proposed parameter to estimate the metrological usefulness of unpolarized
states, namely the so-called Dicke squeezing parameter introduced in Ref.
[48]:

$(2\xi_{D})^{-1}=\frac{\langle\hat{J}_{y}^{2}+\hat{J}_{z}^{2}\rangle}{N(2\langle\hat{J}_{x}^{2}\rangle+1/2)}\leavevmode\nobreak\
.$ (6)

This parameter was observed numerically to quantify the metrological
usefulness of a specific class of states (namely: superpositions of Dicke
states close to the balanced Dicke state), but to our knowledge, no proof of
the general relevance of the Dicke squeezing parameter has been given in the
literature. Note that, as per the previous observation, for
$\langle\hat{J}_{x}^{2}\rangle=0$ one has that
$\mathrm{QFI}_{\hat{J}_{z}}/N=(2\xi_{D})^{-1}$. Our approach allows us to
provide a rigorous lower bound to the QFI, as opposed to $\xi_{D}$ which only
provides a conjectured lower bound based on numerical observations. In Figure
1, we compare $(2\xi_{D})^{-1}$ with the minimum QFI of $\hat{J}_{z}$
compatible with a reduced total spin of
$\langle\hat{\mathbf{J}}^{2}\rangle=0.7(N/2)(N/2+1)$ as a function of
$\langle\hat{J}_{x}^{2}\rangle$.

[width=0.5]paper/figures_paper/Dicke_param.png

Figure 1: For each value of $\langle\hat{J}_{x}^{2}\rangle$ we obtain the
minimum QFI compatible with
$\langle\hat{J}_{x}\rangle=0,\langle\hat{J}_{x}^{2}\rangle,\langle\hat{J}_{y}^{2}+\hat{J}_{z}^{2}\rangle=0.7(N/2)(N/2+1)-\langle\hat{J}_{x}^{2}\rangle$,
and $N=16$ (solid orange line). We compare it with the Dicke squeezing
parameter [Eq. (6)] (dashed blue line).

We find that the Dicke squeezing parameter [Eq. (6)] only captures the
metrological usefulness for very small values of
$\langle\hat{J}_{x}^{2}\rangle$, but in general it largely overestimates the
QFI of the state. Therefore, the main outcome of this study is that this
parameter cannot be used as a faithful estimate of the metrological usefulness
of the system without further assumptions. Notice though that in Ref. [3], a
proper squeezing parameter tailored to Dicke states has been derived, which
constitutes a valid lower bound of the QFI. However, this criterion involves
fourth moments of the collective spin, while our goal here was to include data
up to the second moments only.

##  4 Metrological usefulness of one-axis twisting-generated states

In this section we further apply our certification algorithm by considering
data produced during the one-axis twisting (OAT) dynamics starting from a
state polarized along the $x$ axis [23]:

$\ket{\Psi(t)}=e^{it\hat{J}_{z}^{2}}\ket{\theta=\pi/2,\phi=0}\ ,$ (7)

where
$\ket{\theta,\phi}=[\cos(\theta/2)\ket{\uparrow}+e^{i\phi}\sin(\theta/2)\ket{\downarrow}]^{\otimes
N}$ is a coherent spin state (CSS) of length $J=N/2$, with $(\theta,\phi)$ the
usual polar and azimuthal angles.

The OAT protocol has been widely investigated both theoretically and
experimentally to prepare metrologically useful entangled many-body states in
atomic ensembles [31]. Its realization in different regimes of structured
systems such as optical lattices has also been explored in the context of
cold-atom and Rydberg-atom quantum simulators [32, 46, 33, 10, 13, 6]. The OAT
dynamics generates spin squeezing at short times $t\lesssim 1/\sqrt{N}$ , with
strength $\xi^{-2}_{R}\sim N^{3/2}$. At longer times, the mean spin
$\langle\hat{J}_{x}\rangle$ vanishes and quantum correlations are not simply
reflected in spin squeezing. While the theoretical metrological usefulness
continues to increase with time, it becomes more demanding to certify or
exploit it, as the correlations are spread in higher moments of the collective
spin fluctuations. In particular, at times $t=\pi/q$, the state is a so-called
multi-headed cat state (MHCS), namely an equal-weight superposition of $q$ CSS
in the $xy$ plane ($\theta=\pi/2$) at angles $\phi_{m}=2\pi m/q$ for
$m\in\\{0,1,..,q-1\\}$ [1, 39]. This includes, for $t=\pi/2$, the conventional
cat state (with two heads) marking half of the total period of the dynamics.
Before discussing the metrological usefulness of MHCS, and more general states
generated during the OAT dynamics, we briefly review the case of spin-squeezed
states.

Spin-squeezed states.– Spin-squeezed states represent the most widely studied
class of many-body spin states useful for metrology, and are generated at
short time in the OAT dynamics. Those states have a nonzero mean spin
$\langle\hat{\bf J}\rangle=\langle\hat{J}_{\bf n}\rangle\neq 0$ (with ${\bf
n}$ the direction of the mean spin) and a reduced variance for a spin
component transverse to the mean spin $\langle\hat{J}_{{\bf
n}_{\perp}}^{2}\rangle<\langle\hat{J}_{\bf n}\rangle^{2}/N$ (with ${\bf
n}_{\perp}$ orthogonal to ${\bf n}$) [45]. For simplicity, let us assume that
the mean spin is along $x$ (as is the case in the OAT dynamics), and that the
minimal variance orthogonal to $x$ is along $s$ (squeezing axis). The QFI for
rotations around $as=s\times x$ (antisqueezed axis) is lower-bounded by the
spin squeezing parameter [31]:

$\xi_{R}^{-2}=\frac{\Xi_{J_{s}}^{-2}}{N}=\frac{\langle\hat{J}_{x}\rangle^{2}}{N\langle\hat{J}_{s}^{2}\rangle}\leq\frac{F_{Q}[\hat{\rho},\hat{J}_{as}]}{N}\leavevmode\nobreak\
.$ (8)

It is natural to investigate the tightness of this bound, namely, to find the
minimal $F_{Q}[\hat{\rho},\hat{J}_{as}]$ compatible with given
$\langle\hat{J}_{x}\rangle$ and $\langle\hat{J}_{s}^{2}\rangle$. This question
has been investigated previously in Ref. [4] (see Appendix C for a discussion
of the algorithm used in this paper), and the bound is known to be tight
except for data points at the boundary of the feasible set (namely, the set of
values for $\langle\hat{J}_{x}\rangle$, $\langle\hat{J}_{s}^{2}\rangle$ for
which there exists a quantum state compatible with them), where the bound is
tight within $\sim 2\%$ accuracy.

In the remainder of this section, we will use expectation values as produced
during the OAT dynamics [Eq. (7)] to derive, using our SDP algorithm, the
sensitivity bound to collective spin rotations. We will compare our obtained
bounds with well-known lower bounds of the QFI, such as spin squeezing [Eq.
(8)], or non-linear generalizations thereof [15].

Second moments.– We start by taking the first and second moments of collective
spin observables as input data to our algorithm, namely
$\langle\hat{J}_{a}\rangle$, and
$\mathrm{Re}\\{\langle\hat{J}_{a}\hat{J}_{b}\rangle\\}$ for
$a,b\in\\{x,y,z\\}$. Note that since $\langle\hat{\bf
J}^{2}\rangle=(N/2)(N/2+1)$ is given explicitly by the data, it is sufficient
to optimize over the totally symmetric sector whose dimension is $N+1$,
instead of the complete Hilbert space of dimension $2^{N}$. Such expectation
values contain also implicitly the spin squeezing bound of Eq. (8), where the
squeezed component $\hat{J}_{s}$ is used to sense rotations generated by
$\hat{G}=\hat{J}_{as}$. At short times, the orientations $\\{s,as\\}$ are
contained in the $yz$ plane (orthogonal to the mean spin, which is along $x$)
and depend on time. At longer times, the state wraps around the Bloch sphere
and becomes unpolarized. In this case, we use as a generator
$\hat{G}=\hat{J}_{y}$. Minimizing the QFI with respect to the generator
$\hat{G}$ allows us to compare both our SDP bound and the spin-squeezing
bound. The result is displayed in Figure 2.

[width=0.5]paper/figures_paper/OAT_linear_10.png

Figure 2: Evolution of the metrological bounds from second moments of the
collective spin during OAT dynamics for $N=10$ particles. From bottom to top:
in blue (dashed) conventional spin squeezing parameter and in orange (solid
line), the SDP bound based on the same data. We also depict horizontal lines
at $\mathrm{QFI}=0$ and $\mathrm{QFI}=N$ to represent no metrological
usefulness and the coherent state limit respectively.

The results presented in Figure 2 lead us to several important observations.
First, there is a gap between our bound and the spin squeezing parameter,
demonstrating that using the same second moments as used to build the optimal
spin-squeezing parameter, one can in fact certify more metrological usefulness
than predicted by the same squeezing parameter. As a matter of fact, adding to
the data the second moment along the mean spin
($\langle\hat{J}_{x}^{2}\rangle$) in addition to the mean spin itself
($\langle\hat{J}_{x}\rangle$) and to the squeezed second moment
($\langle\hat{J}_{s}^{2}\rangle$) is already sufficient to open this gap
between the spin squeezing parameter and the SDP bound, and gives the same QFI
lower-bound as using the full correlation matrix of the collective spin. We
note that an illustrative special case is at $t=\pi/2$. At this time, a cat
(GHZ) state is created
$\ket{\mathrm{GHZ}}=(\ket{\pi/2,0}+e^{i\Theta}\ket{\pi/2,\pi})/\sqrt{2}$. As
it has a vanishing mean spin, its spin squeezing bound is zero. It features
$\langle\hat{J}_{x}^{2}\rangle=N^{2}/4,\langle\hat{J}_{y}^{2}\rangle=\langle\hat{J}_{z}^{2}\rangle=N/4$,
which implies that it exhibits maximal QFI with respect to rotations generated
by $\hat{J}_{x}$,
$F_{Q}[\ket{\mathrm{GHZ}},\hat{J}_{x}]=4(\Delta\hat{J}_{x})^{2}=N^{2}$.
However, using only second moments it is not possible to certify this
metrological usefulness, as they are always compatible with a state invariant
under rotations around $x$ such as a statistical mixture of two coherent spin
states
$[\mathrm{GHZ}]=(\outerproduct{\pi/2,0}{\pi/2,0}+\outerproduct{\pi/2,\pi}{\pi/2,\pi})/2$.
Nonetheless, one can certify the metrological usefulness of rotations around
$y$ (or $z$), for which $F_{Q}[[\mathrm{GHZ}],\hat{J}_{y}]=N$, which is
equivalent to considering rotations of a coherent spin state polarized along
$\pm x$. Finally, we notice that more generally, at all times at which the
mean spin is zero, our SDP algorithm certifies metrological usefulness for
rotations around $y$ with a QFI lower bound $\sim N$ (see Appendix F for
details).

In order to further reveal the metrological usefulness of non-spin-squeezed
states beyond the small QFI lower-bound reported in Figure 2, it is necessary
to consider data involving higher moments of the collective spin.

Fourth moments.– We then add to our SDP algorithm the information on third and
fourth moments:
$\mathrm{Re}\\{\langle\hat{J}_{a}\hat{J}_{b}\hat{J}_{c}\rangle\\}$,
$\mathrm{Re}\\{\langle\hat{J}_{a}\hat{J}_{b}\hat{J}_{c}\hat{J}_{d}\rangle\\}$
for $a,b,c,d\in\\{x,y,z\\}$, in addition to the data considered in Figure 2.
We emphasize that in Ref. [15], a method is proposed to incorporate
systematically such quantities in generalized squeezing bounds, quantifying
the optimal sensitivity to collective spin rotations for all observables in
the span of the data operators. Namely, if the data include mean values of the
form $\\{\langle\hat{O}_{a}\rangle\\}_{a}=\langle\hat{\mathbf{O}}\rangle$ and
second moments of it
$\\{\langle\hat{O}_{a}\hat{O}_{b}\rangle\\}_{a,b}=\langle\hat{\mathbf{O}}\hat{\mathbf{O}}^{T}\rangle\\}$,
then Ref. [15] allows one to build the optimal observable of the form
$\hat{O}=\mathbf{v}\cdot\hat{\mathbf{O}}$ such that $\Xi_{O}^{-2}$ is maximal,
as well as the optimal direction for collective spin rotations. Here, if one
considers $\hat{\mathbf{O}}$ up to second moments, using fourth-order moments
in $\hat{\mathbf{D}}$ is sufficient to derive the corresponding bound. The
details of the approach of Ref. [15] can be found in Appendix C. Using the
said approach, we find that at short times, the optimal generator is
$\hat{G}=\hat{J}_{as}$ (as was already the case using only up to second
moments of the collective spin); then until $t\lesssim 0.15\pi$ the optimal
generator is $\hat{G}=\hat{J}_{x}$; then for $t\gtrsim 0.27\pi$ the optimal
generator is $\hat{G}=\hat{J}_{y}$. We compare the non-linear spin squeezing
parameter obtained using the method of Ref. [15] to the QFI lower-bound
obtained by our SDP, using up to fourth-order moments, and for the same
optimal generator. Notice that in general the optimal generator as given by
Ref. [15] need not be the same as the one for our approach. The result of the
comparison of both sensitivity bounds for $N=10$ particles is shown in Figure
3. We also plot the QFI of the OAT state $\ket{\Psi(t)}$ with respect to
$\hat{G}$. (Note that in Figure 3, we have added infinitesimal noise to the
quantum state which modifies the data as
$\langle\hat{\mathbf{D}}\rangle\mapsto(1-p)\langle\hat{\mathbf{D}}\rangle+p\mathrm{Tr}(\hat{\mathbf{D}})/(N+1)$
with $p=10^{-4}$; this has a negligible effect on the QFI of the state or the
generalized squeezing bound, and avoids singular results in the SDP bound.)

[width=0.5]paper/figures_paper/OAT_nonlinear_10.png

Figure 3: Evolution of the metrological bounds from fourth moments of the
collective spin during OAT dynamics for $N=10$ particles. From bottom to top:
in blue (dashed) nonlinear spin squeezing parameter as introduced in Ref.
[15], in orange (solid line), the SDP bound, based on the same data. The green
dotted line is $4(\langle\hat{G}^{2}\rangle-\langle\hat{G}\rangle^{2})$ and
corresponds to the QFI of the OAT-evolved state $\ket{\Psi(t)}$ of Eq. (7)
with respect to the same generator $\hat{G}$ as used to compute the other
bounds.

We observe consistently that our bound systematically improves over the non-
linear squeezing parameter of Ref. [15] as soon as the non-gaussianity of the
state becomes significant (that is, at times $t/\pi\gtrsim 0.1$). However, we
notice that in contrast to the previous case considering only up to second-
order observables, here the gap appears to close when increasing $N$ (see
Appendix F for a detailed discussion on the scaling). Furthermore, a striking
fact is that close to times $t=\pi/4,\pi/3$, where MHCS are realized, our
bound nearly saturates the full QFI of the state. This is remarkable, for MHCS
are highly non-gaussian states and are rather characterized by high-order
interference fringes stemming from the coherent superpositions of several
well-separated coherent spin states, while our bound requires using spin
fluctuations only up to order four. We observe that for the 4-MHCS at time
$t=\pi/4$ and for $N=2,4,6,8,10$, our bound actually reaches the maximal QFI,
namely $N(N+1)/2$. This result stems from the unexpected fact that by fixing
the collective spin fluctuations up to order four, one actually fully
specifies the state: there exists an operator $\hat{W}$, built as a linear
combination of the data, such that the state corresponds to its extremal
eigenvalue: $\min_{\hat{\rho}}{\rm
Tr}(\hat{\rho}\hat{W})={\bra{4\mathrm{MHCS}}}\hat{W}{\ket{4\mathrm{MHCS}}}=\lambda_{\min}(\hat{W})$,
where $\lambda_{\min}$ denotes the minimum eigenvalue of $\hat{W}$ with
corresponding eigenvector $\ket{4\mathrm{MHCS}}$. In Appendix G we outline a
systematic procedure to find such operator (if it exists). However, we observe
that this result is a finite-size effect, only valid in few-particle systems:
indeed, as the number of spins is increased, the peaks for our lower-bound to
QFI at times $t=\pi/4,\pi/3$ drop. For the 4-MHCS, the certified QFI drops
below $N$ (coherent-state limit) for $N=24$. Regarding the 3-MHCS, the
certification drops below $N$ already for $N=10$ (see Appendix F for further
details). For larger particle numbers, higher order moments then become
necessary to certify the metrological resource of highly non-gaussian states
such as MHCS.

Parity measurements.– As a last illustration of our algorithm, we consider the
use of parity operators $\hat{\Pi}_{a}=(-1)^{\hat{J}_{a}+N/2}$, which are
paradigmatic operators to exploit the maximal metrological usefulness of the
cat (GHZ) state generated at time $t=\pi/2$ in the OAT dynamics [24]. In
analogy with Eq. (8), one can construct the parity squeezing parameter:

$\Xi^{-2}_{\Pi}=\frac{|\langle[\hat{J}_{a},\hat{\Pi}_{b}]\rangle|^{2}}{1-\langle\hat{\Pi}_{b}\rangle^{2}}\
.$ (9)

For the GHZ state,
$\ket{\mathrm{GHZ}}=(\ket{\pi/2,0}+e^{i\Theta}\ket{\pi/2,\pi})/\sqrt{2}$, the
optimal axes are $a=x$ and $b$ contained in the $yz$ plane, yielding
$\Xi^{-2}_{\Pi}=N^{2}$, which is the maximal sensitivity allowed by quantum
mechanics (the so-called Heisenberg limit). Such a result can be also obtained
with the approach of Ref. [15] using as operators
$\hat{\mathbf{O}}=\\{\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{\Pi}_{x},\hat{\Pi}_{y},\hat{\Pi}_{z}\\}$,
and including their expectation values and the correlation matrix
$\langle\hat{\mathbf{O}}\hat{\mathbf{O}}^{T}\rangle$. Using this approach, the
optimal generator throughout the OAT dynamics is in the anti-squeezed
direction $\hat{J}_{as}$ for times $t\lesssim 0.27\pi$, and then $\hat{J}_{x}$
for longer times. As in the previous illustrations, our proposed SDP-based
algorithm is used to find a stronger metrological sensitivity based on the
same data and with respect to such optimal generator. In Figure 4 the
corresponding bounds are compared. There, we depict the maximal QFI associated
with collective spin rotations, that is, (four times) the largest eigenvalue
of the covariance matrix
$C_{ab}=\mathrm{Re}\\{\langle\hat{J}_{a}\hat{J_{b}}\rangle\\}-\langle\hat{J}_{a}\rangle\langle\hat{J}_{b}\rangle$
for $a,b\in\\{x,y,z\\}$.

[width=0.5]paper/figures_paper/OAT_linpar_N10.png

Figure 4: Evolution of the metrological bounds from second moments of spin and
parity measurements during OAT dynamics for $N=10$ particles. From bottom to
top: in blue (dashed) non-linear squeezing parameter derived from Ref. [15]
using
$\hat{\mathbf{O}}=\\{\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{\Pi}_{x},\hat{\Pi}_{y},\hat{\Pi}_{z}\\}$
as measurement operators; orange curve (solid line) – the corresponding SDP
bound using the same data. In green (dotted) parity squeezing parameter of Eq.
(9) with $a=y$ (generator direction) and $b=x$ (measurement direction). In red
(dashed-dotted) is plotted the maximal QFI of the OAT-generated states with
respect to collective spin rotations.

We observe that at the times where the generator is $\hat{J}_{as}$, our bound
captures the full QFI of the state, a strong improvement with respect to the
prediction of Ref. [15]. In particular, we verify that it is sufficient in our
algorithm to input the data
$\langle\hat{\Pi}_{x}\rangle,\mathrm{Im}\\{\langle\hat{\Pi}_{x}\hat{J}_{as}\rangle\\}$
to saturate the full QFI of the state. In addition, we have verified that this
QFI can be saturated using simply the parity squeezing parameter introduced in
Eq. (9), with directions $a=as,b=x$. This simple observation highlights the
possibility to use parity operators to exploit the maximal sensitivity during
the whole OAT dynamics before the generation of the GHZ state.

At longer times ($t\gtrsim 0.27\pi$), when the optimal generator as predicted
by Ref. [15] is $\hat{J}_{x}$, our bound only yields a modest improvement with
respect to the non-linear squeezing parameter of Ref. [15]. However, we notice
our bound can be drastically improved up to times $t\lesssim 0.35\pi$, again
saturating the full QFI of the state, by just changing the generator from
$\hat{J}_{x}$ to $\hat{J}_{y}$. Also in this case, the bound is saturated by
the parity squeezing parameter of Eq. (9), taking as orientations $a=y,b=x$
(see the green dotted curve of Figure 4). Related findings on the relevance of
parity measurements to saturate the QFI throughout the evolution were reported
in previous works [16, 29], pointing out the crucial role played by parity
conservation during the dynamics.

##  5 Conclusions

Certifying the resource content of a multipartite quantum system for metrology
applications is an important task for quantum technologies. In this work, we
introduced semidefinite programming algorithms to find the minimal sensitivity
to unitary transformation, as quantified by the quantum Fisher information,
compatible with an arbitrary set of mean values. These algorithms are general
and improve over previous approaches by fully exploiting the convexity of the
problem. We have shown that one can take advantage of symmetries to
drastically reduce the numerical cost of the implementation. As illustrations,
we have focused on relevant families of states of spin-$1/2$ ensembles. In the
case of Dicke states, we have shown that a so-called Dicke squeezing parameter
introduced in the past [48] overestimates the certifiable metrological
usefulness. We have then focused on states generated in the one-axis-twisting
dynamics [23], where we obtained several significant results. First, we showed
that using only the first and second moments of the collective spin, one can
certify more metrological usefulness than given by the Wineland spin squeezing
parameter [45]. Including up to fourth-order moments, we made the surprising
observation that, for up to a dozen of spins, the metrological power of so-
called multi-headed cat states can be almost optimally certified. Finally, we
found that using the sensitivity of parity operators (which are $N$-body
correlation functions), close to the maximal QFI of the state can be certified
throughout the whole dynamical evolution. Our results are directly relevant to
many experiments where collective spin fluctuations are sampled and could be
used to certify the metrological resource of such systems. Future applications
of our approach could include spatially-structured systems, such as those
studied in quantum simulators, and Appendix H proposes a first numerical study
in this direction.

Code and data availability.– The code and data to construct the plots are
available at the following link,
https://github.com/anubhavks/SDP_QFI_partialinfo.git, which includes a brief
tutorial on the numerical implementation of the method introduced in this
paper.

We thank Rafał Demkowicz-Dobrzański, Janek Kołodyński and Géza Tóth for useful
discussions. IF acknowledges support from the European Union’s Horizon 2020
research and innovation programme under the Marie-Skłodowska-Curie grant
agreement No 101031549 (QuoMoDys). GMR, AKS, GRM, and ML acknowledge the
support from: ERC AdG NOQIA; Ministerio de Ciencia y Innovation Agencia
Estatal de Investigaciones (PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I “Retos Colaboración” QUSPIN
RTC2019-007196-7); MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya; Fundació
Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund
FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU (PASQuanS2.1,
101113690); EU Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); EU Horizon
Europe Program (Grant Agreement 101080086 — NeQST), National Science Centre,
Poland (Symfonia Grant No. 2016/20/W/ST4/00314); ICFO Internal “QuantumGaudi”
project; European Union’s Horizon 2020 research and innovation program under
the Marie-Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No
847648 (“La Caixa” Junior Leaders fellowships ID100010434:
LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012,
LCF/BQ/PR21/11840013). AKS acknowledges support from the European Union’s
Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-
Curie Grant Agreement No. 847517. SK acknowledges support from the National
Science Center (Poland) grant No.2020/37/B/ST2/02134. Views and opinions
expressed are, however, those of the author(s) only and do not necessarily
reflect those of the European Union, European Commission, European Climate,
Infrastructure and Environment Executive Agency (CINEA), nor any other
granting authority. Neither the European Union nor any granting authority can
be held responsible for them.

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##  Appendix A Two SDP algorithms

In this Appendix, we detail the main technical considerations of two SDPs
solving the central problem as formalized in Eq. (2). Firstly, for the
algorithm based on maximizing fidelity as presented in the main text, we
discuss the effect of the small parameter $\delta\theta$. Secondly, we provide
a comprehensive derivation of the second SDP, which solves exactly the problem
of Eq. (2) by maximizing the variance of a locally unbiased estimator.
Finally, we compare the performance of both algorithms in terms of accuracy
and running computational times.

###  A.1 Finite $\delta\theta$ considerations of the main-text SDP

In order to illustrate the error introduced by a finite $\delta\theta$ in the
SDP algorithm, we consider the following example: a system of $N=4$ qubits, in
which we bound the QFI with respect to the generator $\hat{G}=\hat{J}_{z}$,
imposing compatibility with the arbitrarily-chosen data:
$\langle\hat{J}_{x}+0.2\hat{J}_{z}\rangle=0.6078,\langle\hat{J}_{y}^{2}\rangle=0.5795$.
As in the main text, we denote by
$\hat{J}_{a}=\sum_{i=1}^{4}\hat{\sigma}_{i}^{a}/2$ the collective spin along
orientation $a$, with $\hat{\sigma}_{i}^{a}$ the Pauli matrices
($a\in\\{x,y,z\\}$). The SDP is solved for a range of values for
$\delta\theta$. For each value of $\delta\theta$, we compute the QFI via two
methods:

  1. 1.

By inverting the relation
$\mathcal{F}_{\max}=1-\mathcal{F}_{Q}[\hat{\rho}^{*}_{\delta\theta},\hat{G}](\delta\theta)^{2}+{\cal
O}(\delta\theta^{4})$, the minimal QFI can be estimated from the fidelity
resulting from the SDP,
$\mathcal{F}_{Q}[\hat{\rho}^{*}_{\delta\theta},\hat{G}]=(1-\mathcal{F}_{\max})/(\delta\theta)^{2}$
(up to second-order corrections in $\delta\theta$).

  2. 2.

By computing the QFI directly from the optimal $\hat{\rho}^{*}_{\delta\theta}$
which is itself obtained as an output of the SDP algorithm. With the spectral
decomposition of the state $\hat{\rho}=\sum_{p}p\hat{\Pi}^{(p)}$, where $p$
are the eigenvalues and $\hat{\Pi}_{p}$ the projectors to the respective
eigenspaces, the QFI with respect to $\hat{G}$ can be obtained with the closed
formula,

$F_{Q}[\hat{\rho},\hat{G}]=2\sum_{p+p^{\prime}>0}\frac{(p-p^{\prime})^{2}}{(p+p^{\prime})}\mathrm{Tr}(\hat{G}\hat{\Pi}^{(p)}\hat{G}\hat{\Pi}^{(p^{\prime})})\
\ .$ (10)

By construction, this method always gives a QFI realizable by a physical
quantum state.

We compare such ways to lower-bound the QFI in Figure 5, where we plot both
fidelity and QFI as a function of $\delta\theta$.

[width=0.45]paper/figures_paper/finite_dtheta_fidelity.png

(a)

[width=0.45]paper/figures_paper/finite_dtheta_QFI.png

(b)

Figure 5: Fidelity (left) and QFI (right) bounds as a function of
$\delta\theta$. In blue (solid line), the value computed from the fidelity
$\mathcal{F}_{\max}$ (SDP output). The orange dashed line corresponds to the
final inferred QFI,
$F_{Q}[\hat{\rho}^{*}_{\delta\theta=0.02},\hat{G}]=0.2401$. The green dotted
curve is deduced from the QFI of the optimal state
$F_{Q}[\hat{\rho}^{*}_{\delta\theta},\hat{G}]$ for each $\delta\theta$ using
Eq. (10).

From Figure 5 we observe that for smaller values of $\delta\theta$ than a
certain finite $\delta\theta_{\rm critical}$, the fidelity is a concave
function of $\delta\theta$. After convergence of the SDP, this leads by means
of method 1 to underestimation of the QFI bound. On the other hand, using
method 2, a QFI value is given that it is exactly realizable by
$\hat{\rho}^{*}_{\delta\theta}$. Thus, this method provides an overestimation,
generally an upper bound to the method 1. In the limit
$\delta\theta\rightarrow 0$, in practice $\delta\theta\sim 10^{-2}$, the two
methods converge yielding a tight lower bound. Hereinafter, we will use method
2, because as shown in Figure 5b it is the tightest in the limit of small
$\delta\theta$.

###  A.2 Alternative SDP algorithm

In this section, we derive an alternative SDP algorithm computing the minimal
QFI compatible with a set of mean values. This second algorithm does not
involve the fidelity, and has the advantage that it is then no longer
necessary to introduce the small parameter $\delta\theta$ as discussed in the
previous section. Let us start with the following optimization problem whose
solution is the inverse of the QFI [17, 50] :

$\begin{array}[]{crl}F_{Q}[\hat{\rho},\hat{G}]^{-1}=&\min_{\hat{O}=\hat{O}^{\dagger}}&\mathrm{Tr}(\hat{\rho}\hat{O}^{2})\\\
&\mbox{s.t.}&\mathrm{Tr}\left(i[\hat{\rho},\hat{G}]\hat{O}\right)=1\\\
&&\mathrm{Tr}\left(\hat{\rho}\hat{O}\right)=0\\\ \end{array}\ .$ (11)

The operator $\hat{O}$ can be interpreted as an observable representing the
estimator for the unknown phase of the unitary transformation
$\hat{U}(\theta)=e^{-i\theta\hat{G}}$. It can be decomposed as
$\hat{O}=\sum_{i}\tilde{\theta}(i)\ket{i}\bra{i}$, where
$\left\\{\ket{i}\right\\}$ is the measurement basis and $\tilde{\theta}(i)$ is
the value of the estimator of $\theta$ associated with the $i$-th measurement
outcome. The objective function of the problem is the variance of the
observable $\hat{O}$, while the constraints ensure local unbiasedness. The
last constraint, $\mathrm{Tr}(\hat{\rho}\hat{O})=0$, can be dropped without
affecting the result of the minimization. Indeed, any $\hat{O}$ can be
decomposed as $\hat{O}=\alpha\mathbb{1}+\hat{O}_{0}$, where
$\mathrm{Tr}(\hat{\rho}\hat{O}_{0})=0$, changing $\alpha$ does not affect the
first constraint, and
$\mathrm{Tr}(\hat{\rho}\hat{O}^{2})=\mathrm{Tr}(\hat{\rho}\hat{O}_{0}^{2})+\alpha^{2}\mathrm{Tr}(\mathbb{1})$
is clearly minimal when $\alpha=0$, which implies that
$\mathrm{Tr}(\hat{\rho}\hat{O})=0$. Let us reformulate as an SDP the former
problem:

$\begin{array}[]{crl}F_{Q}[\hat{\rho},\hat{G}]^{-1}=&\min_{\hat{O},\hat{O}_{2}}&\mathrm{Tr}(\hat{\rho}\hat{O}_{2})\\\
&\mbox{s.t.}&\mathrm{Tr}\left(i[\hat{\rho},\hat{G}]\hat{O}\right)=1\\\
&&\begin{pmatrix}\hat{O}_{2}&\hat{O}\\\ \hat{O}&\mathbb{1}\end{pmatrix}\succeq
0\\\ \end{array}\ .$ (12)

As a consequence of the last constraint, matrices $\hat{O}$ and $\hat{O}_{2}$
are Hermitian, and $\hat{O}_{2}\succeq\hat{O}^{2}$ due to Schur’s complement
condition. Therefore, the minimum of the objective function is achieved when
$\hat{O}_{2}=\hat{O}^{2}$ (because $\hat{\rho}$ is positive-semidefinite),
which means that problems Eq. (11) and Eq. (12) are equivalent. Our task is to
minimize the QFI over $\hat{\rho}$ satisfying certain constraints, so the
inverse of the QFI must be maximized and the problem we want to solve is:

$\begin{array}[]{crl}\max{\hat{\rho}}&\min_{\hat{O},\hat{O}_{2}}&\mathrm{Tr}(\hat{\rho}\hat{O}_{2})\\\
&\mbox{s.t.}&\mathrm{Tr}\left(i[\hat{\rho},\hat{G}]\hat{O}\right)=1\\\
&&\begin{pmatrix}\hat{O}_{2}&\hat{O}\\\ \hat{O}&\mathbb{1}\end{pmatrix}\succeq
0\\\
&&\mathrm{Tr}(\hat{\mathbf{D}}\hat{\rho})=\langle\hat{\mathbf{D}}\rangle\\\
&&\hat{\rho}\succeq 0\end{array}\ .$ (13)

To simplify the problem, we are going to change it from “max-min” to “max-max”
– to achieve this, we will use a strong duality theorem for SDPs, and change
the minimization over $\hat{O},\hat{O}_{2}$ to a maximization.

Let us start with rewriting the minimization problem of Eq. (12) into a
canonical form:

$\begin{array}[]{crl}&\min_{\hat{X}}&\mathrm{Tr}(\hat{C}\hat{X})\\\
&\mbox{s.t.}&\Phi(\hat{X})=\hat{B}\\\ &&\hat{X}\succeq 0\\\ \end{array}\ ,$
(14)

where $\Phi$ is a linear, Hermitian-preserving map (notice that $\hat{X}$ and
$\hat{B}$ are not necessarily of the same dimension). The dual maximization
problem, whose solution is the same as of the primal one, reads [44]

$\begin{array}[]{crl}&\max_{\hat{Y}=\hat{Y}^{\dagger}}&\mathrm{Tr}(\hat{B}\hat{Y})\\\
&\mbox{s.t.}&\Phi^{*}(\hat{Y})\preceq\hat{C}\\\ \end{array}\ ,$ (15)

where $\Phi^{*}$ is the adjoint map, satisfying

$\mathrm{Tr}\left(\Phi(\hat{X})\hat{Y}\right)=\mathrm{Tr}\left(\hat{X}\Phi^{*}(\hat{Y})\right)$
(16)

for all Hermitian $\hat{X},\hat{Y}$. Let $d$ be the dimension of $\hat{\rho}$,
and $\Phi$ be a map from $2d\times 2d$ to $(2d+1)\times(2d+1)$ Hermitian
matrices, defined as

$\Phi\left(\hat{X}=\begin{pmatrix}\hat{X}_{00}&\hat{X}_{01}\\\
\hat{X}_{01}^{\dagger}&\hat{X}_{11}\end{pmatrix}\right)=\begin{pmatrix}\frac{1}{2}\mathrm{Tr}\left(i[\hat{\rho},\hat{G}](\hat{X}_{01}+\hat{X}_{01}^{\dagger})\right)&0&0\\\
0&i(\hat{X}_{01}-\hat{X}_{01}^{\dagger})&0_{d\times d}\\\ 0&0_{d\times
d}&\hat{X}_{11}\end{pmatrix},$ (17)

$0_{d\times d}$ is a $d\times d$ zero matrix, $\hat{X}_{ij}$ are $d\times d$
blocks of $\hat{X}$. The problem Eq. (12) is equivalent to Eq. (14) for such
$\Phi$ if

$\hat{C}=\begin{pmatrix}\hat{\rho}&0_{d\times d}\\\ 0_{d\times d}&0_{d\times
d}\end{pmatrix},$ (18)

and

$\hat{B}=\begin{pmatrix}1&0&0\\\ 0&0_{d\times d}&0_{d\times d}\\\ 0&0_{d\times
d}&\mathbb{1}_{d\times d}\end{pmatrix}.$ (19)

Since $\hat{X}$ is constrained to be positive-semidefinite, it must also be
Hermitian, which, together with equalities
$\hat{X}_{01}=\hat{X}_{01}^{\dagger}$ and $\hat{X}_{11}=\mathbb{1}$ (which
follow from condition $\Phi(\hat{X})=\hat{B}$) implies that $\hat{X}$ has
structure $\hat{X}=\begin{pmatrix}\hat{O}_{2}&\hat{O}\\\
\hat{O}&\mathbb{1}\end{pmatrix}$, where $\hat{O}$, $\hat{O}_{2}$ are both
Hermitian. Moreover,
$\mathrm{Tr}\left(\hat{X}\hat{C}\right)=\mathrm{Tr}\left(\hat{\rho}\hat{O}_{2}\right)$,
and from $\Phi(\hat{X})=\hat{B}$ follows that
$\mathrm{Tr}\left(i[\hat{\rho},\hat{G}]\hat{O}\right)=1$, which shows that the
problem Eq. (12) is indeed equivalent to the canonical form Eq. (14) with
parameters given by Eq. (17), Eq. (18), Eq. (19). Let us write down the
Hermitian matrix $\hat{Y}$ as

$\hat{Y}=\begin{pmatrix}y_{00}&y_{01}&y_{02}\\\
y_{01}^{*}&\hat{Y}_{11}&\hat{Y}_{12}\\\
y_{02}^{*}&\hat{Y}_{12}^{\dagger}&\hat{Y}_{22}\end{pmatrix},$ (20)

where $\hat{Y}_{ij}$ are $d\times d$ blocks, and plug in this representation
of $\hat{Y}$ into Eq. (16) with $\Phi$ defined by Eq. (17). After expanding
both sides of Eq. (16) and comparing them, we obtain

$\Phi^{*}(Y)=\begin{pmatrix}0_{d\times
d}&i\hat{Y}_{11}+\frac{i}{2}y_{00}[\hat{\rho},\hat{G}]\\\
-i\hat{Y}_{11}+\frac{i}{2}y_{00}[\hat{\rho},\hat{G}]&\hat{Y}_{22}\end{pmatrix}.$
(21)

Moreover, $\mathrm{Tr}(\hat{B}\hat{Y})=y_{00}+\mathrm{Tr}(\hat{Y}_{22})$, so
the SDP problem dual to Eq. (12) is

$\begin{array}[]{crl}F_{Q}[\hat{\rho},\hat{G}]^{-1}=&\max_{y_{00},\hat{Y}_{11},\hat{Y}_{22}}&y_{00}+\mathrm{Tr}(\hat{Y_{22}})\\\
&\mbox{s.t.}&\begin{pmatrix}\hat{\rho}&-i\hat{Y}_{11}-\frac{i}{2}y_{00}[\hat{\rho},\hat{G}]\\\
i\hat{Y}_{11}-\frac{i}{2}y_{00}[\hat{\rho},\hat{G}]&-\hat{Y}_{22}\end{pmatrix}\succeq
0\\\ \end{array},\ $ (22)

$y_{00}$ is real, and $\hat{Y}_{11},\hat{Y}_{22}$ are Hermitian because
$\hat{Y}$ is Hermitian. We are now almost ready to express problem Eq. (13) as
a single SDP, in which the maximization is performed also over $\hat{\rho}$.
The only problem is that variables $y_{00}$ and $\hat{\rho}$ are multiplied in
the constraint of Eq. (22). To deal with this issue, let us make the following
observation: when block matrix satisfies $\begin{pmatrix}A_{00}&A_{01}\\\
A_{10}&A_{11}\end{pmatrix}\succeq 0$, then
$\begin{pmatrix}x^{2}A_{00}&xA_{01}\\\ xA_{10}&A_{11}\end{pmatrix}\succeq 0$
for any real $x$. This is a direct consequence of Schur’s complement condition
for matrix positivity. This observation allows us to change the constraint
from Eq. (22) to

$\begin{pmatrix}y_{00}^{2}\hat{\rho}&-i\hat{M}-\frac{i}{2}[y_{00}^{2}\hat{\rho},\hat{G}]\\\
i\hat{M}-\frac{i}{2}[y_{00}^{2}\hat{\rho},\hat{G}]&-\hat{Y}_{22}\end{pmatrix}\succeq
0,$ (23)

where $\hat{M}=y_{00}\hat{Y}_{11}$ is an arbitrary Hermitian matrix, which
will be our new independent variable (notice that neither $\hat{Y}_{11}$ nor
$\hat{M}$ enter into the problem’s objective). Let us introduce a new matrix
variable $\hat{\rho}^{\prime}=y_{00}^{2}\hat{\rho}$ satisfying
$\mathrm{Tr}(\hat{\rho}^{\prime})=y_{00}^{2}$, and impose a condition

$\begin{pmatrix}\mathrm{Tr}(\hat{\rho}^{\prime})&y_{00}\\\
y_{00}&1\end{pmatrix}\succeq 0,$ (24)

which is equivalent to $\mathrm{Tr}(\hat{\rho}^{\prime})\geq y_{00}^{2}$. When
the objective function $y_{00}^{2}+\mathrm{Tr}(\hat{Y}_{22})$ is maximized,
then the above inequality becomes equality. This allows us to treat $y_{00}$
and $\hat{\rho}^{\prime}$ as independent variables. Importantly, constraint
Eq. (23) is linear with $\hat{\rho}^{\prime}$. Finally, the constraints
$\mathrm{Tr}(\hat{\bm{D}}\hat{\rho})=\langle\hat{\bm{D}}\rangle$ are
equivalent to
$\mathrm{Tr}(\hat{\bm{D}}\hat{\rho}^{\prime})=\langle\hat{\bm{D}}\rangle\mathrm{Tr}(\hat{\rho}^{\prime})$.
All these observations allow us to write the expression for the inverse of
minimal QFI,

$\begin{array}[]{crl}[F_{Q}(\hat{G})]^{-1}=&\max_{\hat{\rho}}&F_{Q}^{-1}[\hat{\rho},\hat{G}]\\\
&\mbox{s.t.}&\mathrm{Tr}(\hat{\bm{D}}\hat{\rho})=\langle\hat{\bm{D}}\rangle\\\
&&\rho\succeq 0\\\ \end{array}\ ,$ (25)

as a single SDP:

$\begin{array}[]{crl}[F_{Q}(\hat{G})]^{-1}=&\max_{\hat{\rho}^{\prime},y_{00},\hat{M},\hat{Y}_{22}}&y_{00}+\mathrm{Tr}(\hat{Y}_{22})\\\
&\mbox{s.t.}&\begin{pmatrix}\hat{\rho}^{\prime}&-i\hat{M}-\frac{i}{2}[\hat{\rho}^{\prime},\hat{G}]\\\
i\hat{M}-\frac{i}{2}[\hat{\rho}^{\prime},\hat{G}]&-\hat{Y}_{22}\end{pmatrix}\succeq
0\\\ &&\begin{pmatrix}\mathrm{Tr}(\rho^{\prime})&y_{00}\\\
y_{00}&1\end{pmatrix}\succeq 0\\\
&&\mathrm{Tr}(\hat{\bm{D}}\hat{\rho}^{\prime})=\langle\hat{\bm{D}}\rangle\mathrm{Tr}(\hat{\rho}^{\prime})\\\
&&\hat{Y}_{22}=\hat{Y}_{22}^{\dagger},\hat{M}=\hat{M}^{\dagger}\end{array}.\ $
(26)

###  A.3 Comparison

In this section, we compare the efficiency and accuracy of the two previously-
introduced SDP algorithms. As an example, we consider a system of $N$ qubits
in a totally symmetric state, i.e.
$\langle\hat{\mathbf{J}}^{2}\rangle=(N/2)(N/2+1)$, where
$\hat{\mathbf{J}}^{2}$ is the total spin. In the data, we include a mean spin
$\langle\hat{J}_{x}\rangle=0.8$ and the fluctuations in an orthogonal
direction $\langle\hat{J}_{y}^{2}\rangle=0.4$. We find the minimal QFI with
respect to the generator $\hat{J}_{z}$ compatible with such expectation values
via the two methods, varying $N$ between $N=4$ and $N=46$ (for even values).
For each $N$ we record both the computing time and corresponding QFI lower-
bound. The number of particles $N$ is related to the dimension of the totally
symmetric subspace as $D=N+1$. The corresponding results are shown in Figure
6.

[width=0.45]paper/figures_paper/comparison_time.png

(a)

[width=0.45]paper/figures_paper/comparison_preci.png

(b)

Figure 6: a) Computing time for both approaches for different values of $N$
and $\delta\theta=0.01$ for the SDP used in the main text. b) Relative
difference of the two bounds ($F$ stands for fidelity and $V$ for variance)
for the same values of $N$.

In Figure 6, we observe how the fidelity-based SDP is marginally more
efficient than the variance-based SDP. This fact stems from the extra
parameter and constraint existing in the latter approach. In panel b) we study
the discrepancy between the two bounds. We find that due to the finite
$\delta\theta$ the fidelity-based method slightly overestimates the variance
bound, within a $10^{-4}$ relative accuracy.

##  Appendix B Permutation invariance

In this section, we discuss the implementation of symmetries in order to
reduce the computational complexity of our SDP algorithm. For the sake of
concreteness, we consider a system of $N$ qubits. A generic density matrix
describing the system is fully specified by exponentially-many parameters. We
will show that if the observables and the generator are invariant under
permutations of the qubits (PI), it is sufficient to optimize over
$\mathcal{D}_{N}=(1+\lfloor N/2\rfloor)(3(1+N)^{2}-2(2+3N)\lfloor
N/2\rfloor+4\lfloor N/2\rfloor^{2})/3-1$ (that is: polynomially-many) real
parameters representing a state which is block-diagonal with respect to the
symmetry sectors. We hence obtain a drastic scaling reduction.

We first define the local basis. The state of each spin $i\in\\{1,2,..,N\\}$
is fully described by the Pauli vector
$\hat{\boldsymbol{\sigma}}^{(i)}=(\hat{\sigma}^{(i)}_{x},\hat{\sigma}^{(i)}_{y},\hat{\sigma}^{(i)}_{z})$,
where $\hat{\sigma}^{(i)}_{a}$ applies Pauli matrix $a$ in party $i$ with
trivial action in the remaining qubits. From these operators, typical PI
observables that we can form are collective spin
$\hat{\mathbf{J}}=\sum_{i=1}^{N}\mathbf{S}^{(i)}$ (elementary), where
$\mathbf{S}=\hat{\boldsymbol{\sigma}}/2$, and moments, i.e., polynomials
thereof. All those operators have in common that they commute with a unitary
representation of any permutation of $N$ parties, $\pi\in\mathcal{S}_{N}$,
$\hat{U}_{\pi}$. Such unitary is described by its action in the computational
basis
$\hat{U}_{\pi}\ket{a_{1}b_{2}c_{2}...}=\ket{a_{\pi(1)}b_{\pi(2)}c_{\pi(3)}...}$.

Sufficiency of optimizing over PI states.– In this paragraph we show that it
is sufficient to optimize over PI states if both the observables in
$\hat{\mathbf{D}}$ and the generator $\hat{G}$ are PI. The proof consists in
two steps. First, we notice that if there is a state compatible with the data,
then there always exists also a PI state compatible with the data. Indeed, for
any state $\hat{\rho}$ we can generate the PI state by applying the averaging
map $\hat{\rho}_{\rm
PI}=(N!)^{-1}\sum_{\pi\in\mathcal{S}_{N}}\hat{U}_{\pi}^{\dagger}\hat{\rho}\hat{U}_{\pi}$.
Both states $\hat{\rho}$ and $\hat{\rho}_{\rm PI}$ will yield the same
expectation values for PI observables. The second step is to establish a
relation between the QFI of $\hat{\rho}$ and that of $\hat{\rho}_{\rm PI}$.
Since $\hat{U}_{\rm{PI}}$ commute with the generator $\hat{G}$,
$F_{Q}[\hat{\rho},\hat{G}]=F_{Q}[\hat{U}_{\pi}^{\dagger}\hat{\rho}\hat{U}_{\pi},\hat{G}]$
for any $\pi\in\hat{S}_{N}$. Finally, by the convexity of QFI from the
averaging map, we see that $F_{Q}[\hat{\rho}_{\rm PI},\hat{G}]\leq
F_{Q}[\hat{\rho},\hat{G}]$, concluding that in our optimization for the search
of the minimum QFI, PI states are sufficient.

Block-diagonal representation of PI states.– Here, we derive an efficient
representation of PI states. First, one can check that any elementary PI
operator $\hat{A}_{\rm PI}$ commutes with the total spin
$\hat{\mathbf{J}}^{2}$. Hence, it can be decomposed in the block-diagonal
form, $\hat{A}_{\rm PI}=\oplus_{J=J_{\min}}^{J_{\max}}\hat{A}^{(J)}$, where
$J$ label the degenerate eigenspaces of $\hat{\mathbf{J}}^{2}$ with eigenvalue
$J(J+1)$, where $J_{\min}=0,1/2$, for even, odd $N$ and $J_{\max}=N/2$. All
sectors $J$ have coarse-grained dimension $2J+1$, which is the number of PI
inequivalent ways to combine $N$ spins-$1/2$ to form a collective state of
total spin $J$. Now, each vector represents a set of highly degenerated
orthogonal states that cannot be distinguished by any PI observable. With
previous considerations, we can identify each sector $J$ as a magnified spin,
of length $J$. Accordingly, the irreducible representations are
$\hat{\mathcal{J}}=\oplus_{J=J_{\mathrm{min}}}^{J_{\max}}\hat{\mathbf{j}}^{(J)}$
with $\hat{\mathbf{j}}^{(J)}$ the vector of spin-$J$ matrices. We can
establish the correspondence
$\mathbf{v}\cdot\hat{\mathcal{J}}\sim\mathbf{v}\cdot\hat{\mathbf{J}}$ as they
fulfill the same algebra. Moreover, beyond linear combination, for any
polynomial function $f$, $f(\hat{\mathcal{J}})\sim f(\hat{\mathbf{J}})$.

We can finally count the number of real parameters of our representation. Each
block $J$ is of dimension $2J+1=D_{J}$, representing generally an $D_{J}\times
D_{J}$ Hermitian matrix. An Hermitian matrix of size $D_{J}$ is described by
$D_{J}^{2}$ real parameters. Adding all blocks yields
$\sum_{J=J_{\min}}^{N/2}D_{J}^{2}=\mathcal{D}_{N}+1$. Note that we can remove
a parameter due to the unit trace constraint.

With the above block structure, the SDP based on maximizing fidelity, Eq. (5)
of the main text, can be expressed in the following way:

$\begin{array}[]{crl}\sqrt{{\cal F}_{\rm
max}}=&\max_{\left\\{\begin{subarray}{c}\hat{\varrho}^{(J)}=[\hat{\varrho}^{(J)}]^{\dagger}\\\
\hat{L}^{(J)}\end{subarray}\right\\}_{J=J_{\min}}^{J_{\max}}}&\sum_{J=J_{\min}}^{J_{\max}}\Re{\mathrm{Tr}(\hat{L}^{(J)})}\\\
&\mbox{s.t.}&\forall J\in\\{J_{\min},J_{\min}+1,...,J_{\max}\\}\\\
&&\begin{pmatrix}\hat{\rho}_{+}^{(J)}&[\hat{L}^{(J)}]^{\dagger}\\\
\hat{L}^{(J)}&\hat{\rho}_{-}^{(J)}\end{pmatrix}\succeq 0\\\
&&\hat{\rho}_{\pm}^{(J)}=\hat{\varrho}^{(J)}\pm
i\delta\theta[\hat{G}^{(J)},\hat{\varrho}^{(J)}]\ ,\\\
&&\sum_{J=J_{\min}}^{J_{\max}}\mathrm{Tr}(\hat{\mathbf{D}}^{(J)}\hat{\varrho}^{(J)})=\langle\hat{\mathbf{D}}\rangle\end{array}\
.$ (27)

Our second approach, as introduced in Appendix A.2 Eq. (26), also admits a
similar block decomposition:

$\begin{array}[]{crl}[F_{Q}(\hat{G})]^{-1}=&\max_{\left\\{\begin{subarray}{c}y_{00}^{(J)}\in\mathbb{R}\\\
\hat{\rho}^{\prime(J)}=[\hat{\rho}^{\prime(J)}]^{\dagger}\\\
\hat{Y}_{22}^{(J)}=[\hat{Y}_{22}^{(J)}]^{\dagger}\\\
\hat{M}^{(J)}=[\hat{M}^{(J)}]^{\dagger}\end{subarray}\right\\}_{J=J_{\min}}^{J_{\max}}}&\sum_{J=J_{\min}}^{J_{\max}}\left[y_{00}^{(J)}+\mathrm{Tr}(\hat{Y}_{22}^{(J)})\right]\\\
&\mbox{s.t.}&\forall J\in\\{J_{\min},J_{\min}+1,...,J_{\max}\\}\\\
&&\begin{pmatrix}\hat{\rho^{\prime}}^{(J)}&-i\hat{M}^{(J)}-i[\hat{G}^{(J)},\hat{\rho}^{\prime(J)}]/2\\\
i\hat{M}^{(J)}-i[\hat{G}^{(J)},\hat{\rho}^{\prime(J)}]/2&-\hat{Y}_{22}^{(J)}\end{pmatrix}\succeq
0\\\ &&\begin{pmatrix}\mathrm{Tr}(\hat{\rho}^{\prime(J)})&y_{00}^{(J)}\\\
y_{00}^{(J)}&1\end{pmatrix}\succeq 0\ ,\\\
&&\sum_{J=J_{\min}}^{J_{\max}}\mathrm{Tr}(\hat{\mathbf{D}}^{(J)}\hat{\rho}^{\prime(J)})=\langle\hat{\mathbf{D}}\rangle\sum_{J=J_{\min}}^{J_{\max}}\mathrm{Tr}(\hat{\rho}^{\prime(J)})\end{array}\
.$ (28)

Note that all the sectors must be taken into account for the optimization.
Minimization over a single sector will lead generally to an upper bound of the
true minimum. Besides, more than one sector may be required to meet the data
constraints. However, if we set the total spin to its extremal value,
$\langle\hat{\mathbf{J}}^{2}\rangle=J_{\max}(J_{\max}+1)$, optimizing over the
total symmetric subspace $J=J_{\max}=N/2$ is sufficient.

##  Appendix C Comparison with alternative approaches

The central problem of the paper, Eq. (2), has appeared in several contexts
throughout the literature. In this Appendix, for completeness, we provide a
concise overview of some of its manifestations and how the corresponding
solutions compare with the present approach.

###  C.1 Relaxations

Here, we describe the methods that provide a valid lower bound for our
problem, but however, there is no guarantee of its optimality, nor of the
existence of a quantum realization for it. Note that this is not the case for
our SDP-based approach, as by construction, there is a quantum state
compatible with the data with minimal QFI.

Loong Len et al.– In Ref. [25] a method was proposed to lower bound the
classical Fisher information (FI), $F$, of an observable $\hat{O}$ from
moments of it. Up to K-th moments, they defined:

$F^{(K)}=\mathbf{b}^{T}A^{-1}\mathbf{b}\leq F\ ,$ (29)

where the vector $\mathbf{b}$ has components
$b_{p}=\partial_{\theta}\langle\hat{O}^{p}\rangle=\langle
i[\hat{G},\hat{O}^{p}]\rangle$, and the matrix $A$,
$A_{pq}=\langle\hat{O}^{p+q}\rangle$ for $p,q\in\\{0,1,..,K\\}$. The
sensitivity $\Xi_{O}^{-2}$ is recovered in the first order ($K=1$). By
construction, $F^{(K)}\leq F^{(K+1)}$, and in the limit $K\rightarrow\infty$,
it converges to $F$. The FI is a lower bound of the QFI. Thus, $F^{(K)}$
constitutes also a proper lower bound of QFI. Such values can be compared in
our approach with the minimal QFI compatible with the data
$\langle\hat{\mathbf{D}}\rangle=\\{\langle
i[\hat{G},\hat{O}^{p}]\rangle,\langle\hat{O}^{p+q}\rangle\\}_{p,q\in\\{0,1,..,K\\}}$.

Gessner et al.– In Ref. [15] they ask which generator
$\hat{G}\in\mathrm{Span}(\hat{\mathbf{G}})$ and observable
$\hat{O}\in\mathrm{Span}(\hat{\mathbf{G}}\oplus\hat{\mathbf{O}})$ capture the
maximal sensitivity $\Xi^{-2}_{O}$, which reads:

$\Xi_{O}^{-2}=\max_{\mathbf{n}\
\mathrm{s.t.}||\mathbf{n}||=1}\mathbf{n}^{T}\underbrace{P_{\hat{\mathbf{G}}}\Im{\mathcal{C}}\Re{\mathcal{C}}^{-1}\Im{\mathcal{C}}P_{\hat{\mathbf{G}}}}_{:=M}\mathbf{n}=\lambda_{\max}(M)\
,$ (30)

where $P_{\hat{\mathbf{G}}}$ projects to the subspace generated by
${\hat{\mathbf{G}}}$ and
$\mathcal{C}=\langle\hat{\mathbf{O}}\hat{\mathbf{O}}^{T}\rangle-\langle\hat{\mathbf{O}}\rangle\langle\hat{\mathbf{O}}\rangle^{T}$
is the complex covariance matrix. The optimal generator is
$G=\mathbf{n}_{G}\cdot\hat{\mathbf{G}}$ where $\mathbf{n}_{G}$ is the maximal
eigenvector of $M$ and the observable is
$\hat{O}=\mathbf{n}_{O}\cdot(\hat{\mathbf{O}}\oplus\hat{\mathbf{G}})$, with
$\mathbf{n}_{O}=\Re{\mathcal{C}}^{-1}\Im{\mathcal{C}}\mathbf{n}_{G}/||\mathbf{n}_{O}||$
yielding $\Xi_{O}^{-2}=\langle
i[\hat{G},\hat{O}]\rangle^{2}/(\Delta\hat{O})^{2}=\lambda_{\max}(M)$.

This approach is used in the main text to find the optimal generator for a set
of expectation values during OAT dynamics. For example, if spin and its
fluctuations are given, the method recovers the squeezing and anti-squeezing
axes for $\mathbf{n}_{O}$ and $\mathbf{n}_{G}$ respectively. By taking
$\hat{\mathbf{O}}$ up to second moments, we can derive a nonlinear squeezing
parameter beyond spin. Such bound is also compared with our SDP value in the
main text with the same data, i.e. considering data
$\langle\hat{\mathbf{D}}\rangle$ up to the fourth moments of the collective
spin. In general, this approach yields a valid but not optimal lower bound to
the QFI given the data.

Tóth et al.– In Ref. [41] an SDP-based approach was proposed to find lower
bounds of the QFI given few arbitrary expectation values. The starting point
is to notice that, as proven in Ref. [47], the QFI is the convex roof of the
variance:

$F_{Q}[\hat{\rho},\hat{G}]=4\min_{\hat{\rho}=\sum_{k}p_{k}\outerproduct{\psi_{k}}{\psi_{k}}}\sum_{k}p_{k}(\Delta\hat{G})^{2}_{\ket{\psi_{k}}}\
.$ (31)

Then, the variance of a pure state $\ket{\psi}\in\mathcal{H}$ can be
linearized in the two-copy space,
$(\Delta\hat{G})^{2}_{\ket{\psi}}=\mathrm{Tr}[(\hat{G}^{2}\otimes\mathbb{I}-\hat{G}\otimes\hat{G})(\outerproduct{\psi}{\psi}\otimes\outerproduct{\psi}{\psi})]$.
In this embedding, Eq. (31) is rewritten as a minimization of separable
symmetric states $\hat{\sigma}\in\mathrm{SEP}(\mathcal{H}\otimes\mathcal{H})$.
The separability condition in general cannot be incorporated as a positive-
semidefinite constraint. However, one can relax the previous restriction to
states with positive partial trace (PPT) and formalize the following SDP:

$\begin{array}[]{crl}\mathrm{QFI}_{\hat{G}}^{\rm
PPT}=&4\min_{\hat{\sigma}=\hat{\sigma}^{\dagger},\hat{\sigma}\succeq
0,\mathrm{Tr}(\hat{\sigma})=1}&\mathrm{Tr}[(\hat{G}^{2}\otimes\mathbb{I}-\hat{G}\otimes\hat{G})\hat{\sigma}]\\\
&\mbox{s.t.}&\hat{S}\hat{\sigma}\hat{S}=\hat{\sigma},\ \hat{\sigma}^{t}\succeq
0\\\
&&\mathrm{Tr}[\hat{\mathbf{D}}\mathrm{tr}(\hat{\sigma})]=\langle\hat{\mathbf{D}}\rangle\end{array}\
,$ (32)

where $\hat{S}$ is the exchange operator (swapping the two subsystems) with
action $\hat{S}\ket{\psi}\otimes\ket{\phi}=\ket{\phi}\otimes\ket{\psi}$, $t$
the partial transpose and $\mathrm{tr}$ the partial trace with respect to a
subsystem. Note that this method requires the optimization over a matrix of
size $D^{2}$ (with $D$ the dimension of the Hilbert space $\mathcal{E}$),
while in our proposed algorithm, the largest matrices are of size $2D$. This
fact compromises the scalability of the method. For example, with $D=11$ and
the data used in Figure 6, we verify that the present algorithm is $\sim 1300$
times slower than our proposed solutions. As for generic Hilbert spaces
$\mathcal{H}$, $\mathrm{SEP}\subsetneq\mathrm{PPT}$, the solution of the
problem Eq. (32) will generally underestimate the physical bound. However, for
most practical purposes, e.g. the setting used in Figure 6, the bound obtained
here is compatible with the bound given by our methods. Furthermore, the
approach explained here is tailored to linear encoding. Its extension to
arbitrary maps is not direct as the algorithm is based on the convex-roof
construction Eq. (31) which was only proven valid for unitary transformations
(see in particular the last remark in Ref. [47]).

###  C.2 Exact methods

Here, we describe the approaches leading to a tight bound, i.e., exactly
reproducible by a quantum state.

Suggestion by Kołodyński.–Another possibility is to use the definition of QFI
from SLD, $\hat{S}$.

$\begin{array}[]{crl}\mathrm{QFI}_{\hat{G}}=&\min_{\hat{S},\hat{\rho}=\hat{\rho}^{\dagger}}&\mathrm{Tr}(i[\hat{G},\hat{S}]\hat{\rho})\\\
&\mbox{s.t.}&i[\hat{G},\hat{\rho}]=\\{\hat{S},\hat{\rho}\\}\\\
&&\mathrm{Tr}(\hat{\mathbf{D}}\hat{\rho})=\langle\hat{\mathbf{D}}\rangle\\\
&&\hat{\rho}\succeq 0\end{array}\ .$ (33)

From Eq. (33) we realize that the optimization over the state $\hat{\rho}$ for
a given $\hat{S}$ is an SDP. Conversely, for a fixed state $\hat{\rho}$,
finding $\hat{S}$ is a linear problem. Consequently, one could alternate
between these two problems in a see-saw strategy to find the optimal global
solution. In our work, we incorporate implicitly both problems as a single
SDP.

For completeness we shall introduce yet another way to compute the QFI which
was derived in Ref. [28] from the error propagation formula (c.f. Eq. (1) main
text).

$F_{Q}[\hat{\rho},\hat{G}]=\max_{\hat{X}=\hat{X}^{\dagger}}-\langle
X^{2}\rangle_{\hat{\rho}}+2\langle i[\hat{G},\hat{X}]\rangle_{\hat{\rho}}$
(34)

The RHS of Eq. (34) is thus a valid lower bound of the QFI given the data
$\hat{\mathbf{D}}=\\{\hat{X}^{2},i[\hat{G},\hat{X}]\\}$. Such approach has
been exploited to compute, also with a see-saw strategy, the so-called QFI of
a channel $\Lambda$, defined as the optimization problem
$\max_{\hat{\rho}}F_{Q}[\Lambda(\hat{\rho}),\hat{G}]$ over all proper density
matrices $\hat{\rho}$.

Apellaniz et al.– We compare our approach with the method introduced in Ref.
[4]. The solution of the central problem of the main text, Eq. (2), is there
recovered with the Legendre transform formalism as a maximization over lower
bounds, $\inf_{\mu}\mathcal{W}_{\mu}(\mathbf{r}):=\beta(\mathbf{r})$, where

$\mathcal{W}_{\mu}(\mathbf{r})=\mathbf{r}\cdot\langle\mathbf{D}\rangle+\lambda_{\rm
min}\left[(\hat{G}-\mu\mathbb{I})^{2}-\mathbf{r}\cdot\hat{\mathbf{D}}\right]\
,$ (35)

with $\lambda_{\rm min}$ denoting the smallest eigenvalue. For any value of
$\mathbf{r}$, $\beta(\mathbf{r})$ will lower bound the minimum QFI with
respect to $\hat{G}$ over states $\hat{\rho}$ compatible with
$\mathrm{Tr}(\hat{\mathbf{D}}\hat{\rho})=\langle\hat{\mathbf{D}}\rangle$,
namely, the solution of the central problem Eq. (2) (main text). In other
words, $\mathbf{r}$ parameterizes a family of valid lower bounds, the stronger
of which will be the
$\max_{\mathbf{r}}\beta(\mathbf{r})=\mathrm{QFI}_{\hat{G}}$. As verified in
the same Ref. [4] one can prove that by construction $\beta$ is tightly
coinciding with the solution of the problem.

While the maximization over $\mathbf{r}$ is convex, this is not the case for
$\mu$, which turns the problem nonconvex. Now, for each update of $\mathbf{r}$
in the maximization algorithm, one must search the global minimum on $\mu$
over the large spectral range on $\hat{G}$,
$\mu\in[\lambda_{\min}(\hat{G}),\lambda_{\max}(\hat{G})]$. In Figure 7, we
show for the same example described in the Appendix A the landscape which is
needed to explore to find the minimum QFI.

[width=0.6]paper/figures_paper/Apellaniz.png

Figure 7: Blue solid curve $\mathcal{W}_{\mu}(\mathbf{r})$ of Eq. (35), for
the optimized $\mathbf{r}$ maximizing its global minimum (orange dashed line),
$\mathbf{r}^{*}=\arg\max_{\mathbf{r}}\beta(\mathbf{r})$. For the present
example, we have $\mathbf{r}^{*}=(1.31,-0.95)$. We check that the minimum,
$\min_{\mu}\mathcal{W}_{\mu}(\mathbf{r}^{*})=0.2401$, is compatible with our
previous value introduced in Appendix A. In green (dotted), the same for a
random (not optimal) setting $\mathbf{r}=(0.89,0.08)$ whose global minimum
constitutes generally a lower bound of the QFI.

We verify several local minima appearing forcing to sweep over all the range
of $\mu$. This fact contrasts with our proposal, which just requires
considering an infinitesimal parameter $\delta\theta$. Once more, such method
is based on the QFI as a convex roof Eq. (31), and it is especially suited to
evaluate the metrological usefulness in phase estimation tasks such as
interferometry. Its generalization to arbitrary parameter-encoding protocols
is not addressed in Ref. [4].

##  Appendix D Bound for Dicke states of arbitrary spin

In this Appendix, we prove the observation in the main text regarding the QFI
of the manifold of states with $\langle\hat{J}_{x}^{2}\rangle=0$. First, we
notice that such condition implies that the state is an eigenstate of
$\hat{J}_{x}$ with eigenvalue $0$. That is,

$\hat{\rho}=\sum_{J}p_{J}\hat{\Pi}^{(J)}_{J_{x}=0}\ ,$ (36)

where $\hat{\Pi}^{(J)}_{J_{x}=0}$ are projectors properly orthonormalized
projecting onto subspaces generated by states with well-defined zero spin
projection along $x$, and total spin $J$.

Next, we use Eq. (10) to compute the QFI of $\hat{\rho}$ with respect
$\hat{J}_{z}$ with $p=(p_{J},0)$ and
$\hat{\Pi}^{(p)}=(\hat{\Pi}^{(J)}_{J_{x}=0},\hat{\Pi}_{J_{x}\neq 0})$, where
$\hat{\Pi}_{J_{x}\neq 0}=\mathbb{I}-\sum_{J}\hat{\Pi}^{(J)}_{J_{x}=0}$ is the
completion of the basis. We can split the double sum as:

$F_{Q}[\hat{\rho},\hat{J}_{z}]=2\sum_{p_{J}+p_{J^{\prime}}>0}\frac{(p_{J}-p_{J^{\prime}})^{2}}{p_{J}+p_{J^{\prime}}}\mathrm{Tr}(\hat{J}_{z}\hat{\Pi}^{(J)}_{J_{x}=0}\hat{J}_{z}\hat{\Pi}^{(J^{\prime})}_{J_{x}=0})+4\sum_{p_{J}>0}p_{J}\mathrm{Tr}(\hat{J}_{z}\hat{\Pi}^{(J)}_{J_{x}=0}\hat{J}_{z}\hat{\Pi}_{J_{x}\neq
0})+0\ .$ (37)

We observe that $\hat{J}_{z}$ does not connect sectors of different total spin
$J$, and therefore
$\mathrm{Tr}(\hat{J}_{z}\hat{\Pi}^{(J)}_{J_{x}=0}\hat{J}_{z}\hat{\Pi}^{(J^{\prime})}_{J_{x}=0})=0$
for $J\neq J^{\prime}$. This implies that the first sum in Eq. (37) is zero;
and the second one is simply
$4\langle\hat{J}_{z}^{2}\rangle=F_{Q}[\hat{\rho},\hat{J}_{z}]$.

##  Appendix E Scaling of the QFI bound with $N$ close to Dicke states

As detailed in the main text, the condition $\langle\hat{J}_{x}^{2}\rangle=0$
is quite stringent. Here, we use our approach to find the minimal QFI
compatible with some finite $\langle\hat{J}_{x}^{2}\rangle$ and
$\langle\hat{\mathbf{J}}^{2}\rangle$. In particular, we fix the total spin to
its maximal value, $\langle\hat{\mathbf{J}}^{2}\rangle=J_{\max}(J_{\max}+1)$
with $J_{\max}=N/2$, so that we can restrict the optimization to the totally
symmetric subspace and reach easily $N=70$ on a standard computer. This
enables us to study the scaling of the QFI with $N$. For
$\langle\hat{J}_{x}^{2}\rangle=0$, we verified the Heisenberg scaling as the
QFI is $\sim\order{J_{\max}^{2}}=\order{N^{2}}$. In Figure 8, we apply a small
perturbation $\langle\hat{J}_{x}^{2}\rangle=aN/4$ with $a\ll 1$ and study how
the QFI bound is modified.

[width=0.45]paper/figures_paper/Dicke_N.png

(a)

[width=0.45]paper/figures_paper/Dicke_Jx.png

(b)

Figure 8: a) Minimum QFI compatible with
$\langle\hat{\mathbf{J}}^{2}\rangle=(N/2)(N/2+1)$ and
$\langle\hat{J}_{x}^{2}\rangle=aN/4$ for different values of $a$ and number of
particles $N$. In panel b) the same plot for nonzero values of $a$ with the
horizontal axis rescaled to $\langle\hat{J}_{x}^{2}\rangle$. The dashed
vertical line indicates $\langle\hat{J}_{x}^{2}\rangle=0.25$.

Figure 8 suggests that the Heisenberg scaling is valid only when $\langle
J_{x}^{2}\rangle\lesssim 1/4$, where the QFI bound shows a maximum. Beyond
this point, the Heisenberg scaling is lost and the metrological usefulness is
reduced.

##  Appendix F Scaling of the QFI bound with N with respect to squeezing
bounds

In this Appendix, we analyze the scaling of the advantage offered by our
method with respect to the analytical lower bound given by the squeezing
parameter ensuring the OAT dynamics. In the main text, for $N=10$ particles,
we show a gap existing between such two bounds when up to second moments and
fourth moments are considered. Here, we investigate its robustness as the
number of parties is increased. Specifically, in Figure 9 we display the gap
per spin between the two bounds for both cases for $N$ up to 30.

[width=0.45]paper/figures_paper/OAT_lin_gaps.png

(a)

[width=0.45]paper/figures_paper/OAT_quad_gaps.png

(b)

Figure 9: Gap between our SDP-based bound and the analytical squeezing lower
bound with the same data is considered: a) up to second moments where we
depict also the $\mathrm{QFI}=0.32N$ horizontal line, b) up to fourth moments,
during the OAT dynamics and the line $\mathrm{QFI}=0$. The squeezing lower
bound for second moments corresponds to the conventional spin squeezing
parameter. If, in addition, up to fourth moments are taken into account, such
bound corresponds to the generalized spin squeezing parameter introduced in
Ref. [15].

In Figure 9a, we observe that at intermediate times, the gap is proportional
to the particle number, leading to a robust improvement with respect to the
usual spin squeezing parameter. At times $t\gtrsim 0.15\pi$, when the state
becomes unpolarized, spin squeezing vanishes while our method certifies
metrological usefulness of $\mathrm{QFI}/N\approx 0.32$, which is finite but
smaller than the coherent-state limit $\mathrm{QFI}/N=1$.

The gap when up to fourth-order moments are considered with respect to the
quadratic spin squeezing parameter, introduced in Ref. [15], is shown in
Figure 9b. Unlike the linear case, we observe at intermediate times the gap
saturating to zero as $N$ is increased, signaling no metrological advantage
with respect to the squeezing bound, except in the vicinity of the MHCS at
$t=\pi/3,\pi/4$. It allows us to assess the robustness of the MHCS detection
as reported in the main text. On the one hand, the usefulness of the 3-MHCS is
already lost for $N=14$. On the other, the detection of the 4-MHCS is stronger
and does not drop below $N$ (coherent limit) until $N=24$. For larger system
sizes, higher moments become necessary to detect the metrological usefulness
of such states.

##  Appendix G Derivation of the data-driven witness $\hat{W}$

As noticed in the main text, using up to fourth-order moments in the 4-MHCS
for $N\leq 10$ (even), the very same state is recovered with unit fidelity as
the output of our SDP approach. This suggests that there is only a single
state, namely the 4-MHCS, compatible with the data.

Here, in order to verify this fact, we outline a generic procedure to
construct a witness operator $\hat{W}$ which is based on a certain data
$\hat{\mathbf{D}}$ (e.g. up to fourth order moments) and whose unique ground
state corresponds to a given state $\ket{\Psi}$ (e.g. the 4-MHCS). Without
loss of generality, we can set $\lambda_{\min}(\hat{W})=0$ as the identity
operator is in the data list. Then, we need to find an operator
$\hat{W}\in\mathrm{Span}(\hat{\mathbf{D}})$ such that it features $\ket{\Psi}$
as its unique minimal eigenstate. Such problem can be cast as an SDP:

$\begin{array}[]{crl}&\max_{\mathbf{n},\mathrm{cutoff}\geq\Delta\geq
0}&\Delta\\\ &\mbox{s.t.}&\hat{W}:=\mathbf{n}\cdot\hat{\mathbf{D}}\\\
&&\hat{W}\ket{\Psi}=0\\\
&&\ker(\outerproduct{\Psi}{\Psi})^{\dagger}\hat{W}\ker(\outerproduct{\Psi}{\Psi})\succeq\Delta\mathbb{I}\end{array}\
,$ (38)

where $\ker(\outerproduct{\Psi}{\Psi})$ projects to the space orthogonal to
$\outerproduct{\Psi}{\Psi}$. Intuitively, $\Delta$ is the gap between the
ground energy $\lambda_{\min}(\hat{W})$ and the first excitation (with
multiplicity), which is to be maximized. If the optimal $\Delta$ is zero, it
means that the ground state is degenerate, and a witness $\hat{W}$ cannot be
found. Conversely, $\Delta>0$ implies that we succeed in finding a witness
$\hat{W}$ based on $\hat{\mathbf{D}}$ such that $\ket{\Psi}$ is its unique
extremal operator. In this case, we add the constraint
$\Delta\leq\mathrm{cutoff}$ to avoid unbounded solutions.

With the above method, we find the existence of such witness for $N\leq 10$
(even). Note that for $N\leq 4$, up to fourth moments data are tomographically
complete, so a witness could be simply
$\hat{W}=\mathbb{I}-\outerproduct{\mathrm{4MHCS}}{\mathrm{4MHCS}}$. Here, we
check that unexpectedly for $N=6,8,10$ fourth moments are also sufficient to
determine uniquely the 4-MHCS.

##  Appendix H Spin chain

In this last appendix, we go beyond permutationally-invariant states such as
those generated in the OAT dynamics and turn to spatially-structured models.
In particular, we consider the nearest-neighbour version of the OAT
Hamiltonian studied in the main text, namely the Ising model:

$\hat{H}=\sum_{l=0}^{N-2}\hat{\sigma}_{z}^{(l)}\hat{\sigma}_{z}^{(l+1)}\ ,$
(39)

where the $N$ spins form a one-dimensional chain with open boundary
conditions. Because of the reduced interaction range, the evolution from the
initial CSS along $x$ ($\ket{\pi/2,0}=\ket{\rightarrow}^{\otimes N}$) will
produce squeezing on a longer timescale than in the OAT model. The spatial
structure of the model motivates the use of spatially-structured observables,
beyond the collective observables considered in the rest of the paper. Hence,
we use structure factor defined as:

$\mathcal{C}^{(k)}_{ab}=\sum_{l\neq
l^{\prime}}\langle\hat{\sigma}_{a}^{(l)}\hat{\sigma}_{b}^{(l^{\prime})}\rangle
e^{2\pi ik(l-l^{\prime})/N}\ ,$ (40)

where $a,b\in\\{x,y,z\\}$. The structure factor captures the spin fluctuations
at momentum $k\in\\{0,1,..,N-1\\}$. Indeed, it can be expressed as the second
moment of the collective spin at momentum $k$,
$\hat{J}^{(k)}_{a}=(1/2)\sum_{l=0}^{N-1}\hat{\sigma}_{a}^{(l)}e^{2\pi ikl/N}$,
as
$\mathcal{C}^{(k)}_{ab}=\langle\hat{J}^{(k)}_{a}\hat{J}^{(-k)}_{b}\rangle-N\delta_{ab}/4$.
Such operators generalize to arbitrary momentum $k$ the uniform collective
spin used throughout this work, $\hat{\mathbf{J}}=\hat{\mathbf{J}}^{(k=0)}$.

Here, we will consider data of second moments
$\\{\mathcal{C}_{ab}^{(k)}\\}_{k\leq K}$ only up to momentum $K\leq N-1$ as
well as the mean spin $\langle\hat{\mathbf{J}}\rangle$. There is no squeezing
parameter based on such data as it is not algebraically closed. Indeed, two
momenta $k,k^{\prime}$ couple to the sum $k+k^{\prime}\mod N$. Nevertheless,
our algorithm is flexible in the data and we are able to verify that nonzero-
momentum observables can improve the metrological bound. In Figure 10 we show
the results for $N=4$ taking as generator $\hat{J}_{as}$. We observe that the
inclusion of momentum $K=1$ indeed improves over the QFI lower bound. This
illustrates the potentiality of our approach to certify the metrological
usefulness in generic multipartite systems, using spatially-structured data.
Note however that in this general case, our algorithm requires computing
memory resources scaling exponentially with $N$.

[width=0.5]paper/figures_paper/K.png

Figure 10: For $N=4$. Orange (dashed) bound for $K=0$. Green (dotted) bound
for $K=1$ (i.e. including $k=0$ and $k=1$). Blue (solid), QFI of the coherent
state evolved with $\hat{H}$ Eq. (39).