# Symmetry: a fundamental resource for quantum coherence and metrology

Irénée Frérot, Tommaso Roscilde

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Irénée Frérot, Tommaso Roscilde. Symmetry: a fundamental resource for quantum coherence and metrology. Physical Review Letters, 2024, 133 (26), pp.260402. 10.1103/PhysRevLett.133.260402. hal-04642151

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Symmetry: a fundamental resource for quantum coherence and metrology

Irénée Frérot*

Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University,
Collège de France, 4 Place Jussieu, 75005 Paris, France

Tommaso Roscilde†

Univ Lyon, Ens de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

(Dated: July 2, 2024)

We introduce a new paradigm for the preparation of deeply entangled states useful for quantum metrology. We show that when the quantum state is an eigenstate of an operator A, observables G which are completely off-diagonal with respect to A have purely quantum fluctuations, as quantified by the quantum Fisher information, namely $F_Q(G) = 4\langle G^2 \rangle$. This property holds regardless of the purity of the quantum state, and it implies that off-diagonal fluctuations represent a metrological resource for phase estimation. In particular, for many-body systems such as quantum spin ensembles or bosonic gases, the presence of off-diagonal long-range order (for a spin observable, or for bosonic operators) directly translates into a metrological resource, provided that the system remains in a well-defined symmetry sector. The latter is defined e.g. by one component of the collective spin or by its parity in spin systems; and by a particle-number sector for bosons. Our results establish the optimal use for metrology of arbitrarily non-Gaussian quantum correlations in a large variety of many-body systems.

**Introduction.**— Many-body entanglement [1] is a striking feature of composite quantum systems: it is responsible for the fundamental complexity of quantum mechanics; and it represents the main resource offered by quantum devices based on the coherent control of many-body systems. Among all currently envisioned quantum technologies, quantum metrology [2, 3] represents one of the most concrete applications of entangled states. The metrological task of interest to this work is phase estimation [4], namely the reconstruction of the phase $\theta$ associated with a unitary transformation $U_{\theta} = e^{-i\theta G}$ acting on the state $\rho$ of the system, where G is a Hermitian operator. Atomic clocks, gravimeters, magnetometers etc. are all based on this functioning principle. It is well established [2] that systems of uncorrelated quantum particles do not give access to the ultimate precision on the phase $\theta$ allowed by quantum mechanics, but they are bounded by the so-called standard quantum limit of metrology. The ultimate precision (the so-called Heisenberg limit) can only be achieved in the presence of multipartite entanglement between the degrees of freedom [2, 5, 6].

It is generally believed that entanglement useful for metrology is associated with states of high purity and small fluctuations of a specific observable, whose paradigmatic example is offered by squeezed states [2, 7, 8]. Here, we unveil a different paradigm to produce arbitrarily mixed, yet highly valuable entangled states for metrology. The basic mechanism is the combination of two ingredients: the states of interest are eigenstates of a symmetry operator; and at the same time they display strong correlations in observables which are completely off-diagonal in the eigenbasis of the symmetry operator.

The sensitivity of a state to the unitary transformation $U_{\theta}$ is fundamentally related to its quantum Fisher information (QFI) $F_Q(G)$ [9] associated with the generator G

of the unitary transformation, defined as

$$F_Q(G) = 2 \sum_{nm} \frac{(p_n - p_m)^2}{p_n + p_m} |\langle n | G | m \rangle|^2 . \quad (1)$$

Here $|n\rangle, |m\rangle$ are eigenstates of $\rho$ with eigenvalues $p_n, p_m$. The uncertainty $\delta\theta$ on the estimation of the phase $\theta$ allowed by the state $\rho$ is bounded by the QFI as per the quantum Cramér-Rao bound $(\delta\theta)^2 \ge 1/F_Q(G)$. Hence a central task in quantum metrology is to prepare input states possessing a large QFI; and to identify observables O whose $\theta$ dependence $\langle O \rangle_{\theta} = \text{Tr}(U_{\theta}^{\dagger}OU_{\theta}\rho)$ can best reveal the sensitivity of the state to the unitary transformation of interest. This is expressed by the fact that the inequality $F_Q(G) \ge \xi_O^{-2}$ is tight, where $\xi_O^{-2}$ corresponds to the squared signal-to-noise ratio:

$$\xi_O^{-2} = \frac{|\langle [O, G] \rangle_{\theta}|^2}{\text{Var}(O)_{\theta}} . \quad (2)$$

Here $\text{Var}(O)_{\theta} = \langle O^2 \rangle_{\theta} - \langle O \rangle_{\theta}^2$, and we have used the fact that $i\partial_{\theta}\langle O \rangle_{\theta} = \langle [O, G] \rangle_{\theta}$. O is an optimal observable for phase estimation when $\xi_O^{-2} = F_Q(G)$.

**From squeezing to symmetry projection.**— An important paradigm for the use of entangled states to improve metrological sensitivity is offered by squeezing – see Fig. 1(a) for a cartoon. Squeezing is typically formulated for observables O which are expressed as sums of local observables for a system of N degrees of freedom, $O = \sum_{i=1}^{N} O_i$. Squeezing – as associated with entanglement – occurs when the parameter $\xi_O^{-2}$ is larger than that of all separable states, i.e. of mixtures of factorized states $\otimes_{i=1}^{N} |\psi_i\rangle$. For the latter states, the parameter $\xi_O^{-2}$ is typically maximized by maximizing the “contrast” $|\langle [O, G] \rangle| -$ this is the case e.g. in ensembles of spin-1/2 particles [2]. Since the contrast is generically
<figure><img src="image_2.png" /><figcaption>FIG. 1: Different paradigms of metrological resources. (a) Squeezing of an observable O with respect to that of separable states with large contrast (see text); (b) symmetry projection, with projector P. In both panels p(A) (A = O, P) is the probability of finding a given eigenvalue of the operator A.</figcaption></figure>

not commuting with O, one has that for these states $\mathrm{Var}(O) = \sum_i \mathrm{Var}(O_i) \sim N$. On the other hand, entangled squeezed states are associated with a reduction of $\mathrm{Var}(O)$ compared to that of separable states, while preserving a finite contrast. Entangled squeezed states of spin ensembles have been obtained in a large variety of platforms [10–16], and their first uses to increase the performance of e.g. atomic clocks have been recently demonstrated [16–18].

In this work, we propose a paradigm which is alternative to squeezing of the fluctuations of local observables. Our paradigm is based on symmetry projection: the observable O of metrological interest becomes a projector P onto a subspace of the Hilbert space, which can be generically associated with the eigenspace of a further observable A. As a concrete example, for spin ensembles one can define the collective-spin operator $\mathbf{J} = \sum_{i=1}^{N} \mathbf{S}_i$, where $\mathbf{S}_i$ is the i-th spin operator. P can be thought of as the projector onto a subspace of well-defined magnetization along e.g. the z axis, in which case $A = J^z$; or of well-defined parity of the magnetization, in which case $A = (-1)^{NS-J^z}$ (for N spins of length S). The states of our interest are states $\rho$ (pure or mixed) which live entirely in the sector projected by P, namely

$$ \rho = P\rho (= \rho P = P\rho P) \qquad (3) $$

so that $\langle P \rangle = \mathrm{Tr}(\rho P) = 1$ and $\mathrm{Var}(P) = \langle P \rangle - \langle P \rangle^2 = 0$. Clearly these states are “squeezed” in terms of the fluctuations of P compared with a generic state (see Fig. 1(b)); we shall specify below under which circumstances they exhibit superior metrological properties compared to separable states. The difference with conventional squeezing is that P can be a highly non-local operator – this is clearly the case when P is the projector on a parity sector. In the following we denote such states as symmetry projected.

In general, quantum states $\rho$ are especially sensitive to a transformation $U_\theta$ whose generator G has a large variance in $\rho$, provided that the uncertainty is of quantum origin, namely that it stems from the fact that $[\rho, G] \neq 0$

(as captured by the QFI). For symmetry-projected states, this condition is best realized when G connects different symmetry sectors, namely when

$$ PGP = 0 \qquad PGQ + QGP = G \qquad (4) $$

where $Q = 1-P$. In this case G is completely off-diagonal on the eigenbasis of $\rho$. Our main result is that, in this case, all fluctuations of G in $\rho$ are of quantum-mechanical origin; that they are metrologically useful, regardless of the purity of $\rho$, that is: $4\mathrm{Var}(G) = F_Q(G)$; and that the sensitivity of the projector to the unitary transformation, $\xi_P^{-2}$ (defined as in Eq. (2)) equals $F_Q(G)$. Our result generalizes and strengthens a similar theorem reported in Ref. [19] in the context of spin systems, in which the classical Fisher information of the magnetization distribution in parity eigenstates is shown to coincide with the QFI.

Our results show that symmetry-projected states with strong off-diagonal correlations (namely with a large $\mathrm{Var}(G)$, ideally scaling as $\sim N^2$ if G is the sum of local quantities) are highly entangled; that this holds true regardless of their purity; and that they represent fundamental resources for quantum metrology. Below, we shall offer examples for the relevance of such states in the context of Bose-Einstein condensates and of quantum spin ensembles. Many recent experiments on quantum many-body devices have demonstrated the capability of preparing correlated states [20–24]. Symmetry preservation can then act as a very effective tool to certify which experiments prepare highly entangled states, as opposed to states with classical correlations [25].

**Symmetry-projection theorem.** – We begin the discussion of our results with the following

**Theorem.** Consider a projector P on a subspace, and a state $\rho$ having support only in that subspace – i.e. fulfilling Eq. (3). Consider then a unitary transformation $U_\theta = e^{-i\theta G}$ whose generator G sends vectors in the support of P onto the orthogonal subspace: $PGP = 0$. Then

$$ \langle P \rangle_\theta = \mathrm{Tr}(U_\theta^\dagger P U_\theta \rho) = 1 - \theta^2 \langle G^2 \rangle + \mathcal{O}(\theta^3), \quad (5) $$

where $\langle \dots \rangle = \mathrm{Tr}(\rho \dots)$. As a consequence:

$$ \xi_P^{-2} = 4\langle G^2 \rangle = F_Q(G) \qquad (6) $$

where $\xi_P^{-2}$ is defined as in Eq. (2) in the limit $\theta \to 0$.

**Proof.** We Tailor-expand $P_\theta = U_\theta^\dagger P U_\theta = P + i\theta(GP - PG) + \theta^2 GPG - (\theta^2/2)(G^2P + PG^2) + \mathcal{O}(\theta^3)$, and evaluate $\langle P \rangle_\theta = \mathrm{Tr}[P_\theta \rho]$. Since $P\rho = \rho = \rho P$, using circular invariance of the trace we have that $\mathrm{Tr}[\rho(GP - PG)] = \mathrm{Tr}[G(P\rho - \rho P)] = 0$. Similarly, we have $\mathrm{Tr}(\rho PG^2) = \mathrm{Tr}(G^2P\rho) = \langle G^2 \rangle$. Then $\mathrm{Tr}(GPG\rho) = \mathrm{Tr}(GPGP\rho) = 0$ where we used $P\rho = \rho$ and $PGP = 0$. Hence Eq. (5) follows. From this, we find $[\partial_\theta\langle P(\theta)\rangle]^2 = 4\theta^2\langle G^2 \rangle^2 + \mathcal{O}(\theta^3)$. Since $P^2 = P$, we also have that $\mathrm{Var}[P(\theta)] = \langle P \rangle - \langle P \rangle^2 = \theta^2\langle G^2 \rangle + \mathcal{O}(\theta^3)$, from which
$\xi_P^{-2} = 4\langle G^2 \rangle$. Since for an arbitrary observable $O$ we have $\xi_O^{-2} \le F_Q(G) \le 4\text{Var}(G) \le 4\langle G^2 \rangle$ [2, 4], we conclude that for $O=P$ this chain of inequalities becomes a chain of equalities, and hence Eq. (6) follows. $\square$

We emphasize that in general $4\langle G^2 \rangle \ge F_Q(G)$, while the equality $4\langle G^2 \rangle = F_Q(G)$ is always realized by pure states $\rho = |\psi\rangle\langle\psi|$ with $\langle G \rangle = 0$. Finding mixed states that satisfy this equality is instead highly non-trivial. From a physical point of view, the equality implies that all fluctuations of $G$ in state $\rho$ have a quantum-coherent origin, as they stem from $G$ being completely off-diagonal in the basis of $\rho$, and this independently of the purity of $\rho$. In turn, all fluctuations of $G$ have an immediate metrological relevance, as they translate into the sensitivity of the average projector $\langle P \rangle$ to the $U_\theta$ transformation. Moreover our result has immediate implications for entanglement certification [25]. Indeed, if $G$ is a sum of local observables $G = \sum_i G_i$, the QFI serves as a witness for multipartite entanglement, excluding separability of the state when $F_Q(G) > 4 \sum_i \text{Var}(G_i)$ [26]. In the case of symmetry-projected states, if one further assumes that $PG_iP = 0$ for all $i$, this criterion simply becomes $\langle G^2 \rangle > \sum_i \langle G_i^2 \rangle$, which is automatically verified in the presence of positive correlations $\langle G_iG_j \rangle > 0$ between degrees of freedom $i \ne j$. Hence symmetry projection translates positive correlations for off-diagonal local observables into an entanglement witness.

The remaining question is then the following: where can one find symmetry-projected states which have significant off-diagonal correlations, implying strong multipartite and metrologically useful entanglement? In the next sections, we show that symmetry projection can be enforced in at least three ways: relying on super-selection rules; relying on the symmetry of Hamiltonians; or projecting the state of the system on a specific symmetry sector by non-destructive post-selection. We will illustrate these various possibilities in the case of Bose-Einstein condensates in a system of coupled bosonic modes; and in the case of interacting quantum spin (or qubits).

**Heisenberg-scaling sensitivity of tunneling rates with Bose-Einstein condensates (BEC).**— We consider the metrological task of estimating the tunneling rate $J$ between two sets of bosonic modes $A$ and $B$. The tunneling process is described by the Hamiltonian $H = JG$ where $G = -\frac{1}{2} \sum_{i=1}^N (a_i^\dagger b_i + b_i^\dagger a_i)$, with $a_i$ (resp. $b_i$) describing the bosonic modes in subsystem $A$ (resp. $B$); we imagine the modes to be pairwise coupled, as e.g. in two coupled chains or two coupled layers of localized modes. If one considers $N$ independent bosons, each localized a mode pair $(a_i, b_i)$, the quadratic error on the estimation of $J$ scales as $1/N$ (standard quantum limit). On the other hand, we can show that two BECs, one delocalized over the $\{a_i\}$ modes and the other over the $\{b_i\}$ modes, allow

for an estimation of $J$ with a quadratic error scaling as $1/N^2$ (Heisenberg scaling).

In the framework of our theorem, we consider an initial state of the form $\rho_A \otimes \rho_B$ with the two sets of modes $A$ and $B$ having a fixed boson number $N_A = \sum_{i=1}^N a_i^\dagger a_i$ (resp. $N_B = \sum_{i=1}^N b_i^\dagger b_i$): $P_{N_A} \rho_A = \rho_A$, with $P_{N_A}$ the projector onto the states with fixed boson number $N_A$ (as well as $N_B$, imagining to have a fixed total boson number). The generator $G$ is purely off-diagonal with respect to the boson number in $A$ (or $B$). Coupling the two sets of modes exposes the state to the unitary transformation $e^{-itJG}$. According to our theorem, the QFI for the estimation of $tJ$ (and hence of $J$) is given by $4\langle G^2 \rangle$, and it is saturated by measuring the time evolution of $\langle P_{N_A} \rangle$, i.e. the probability to find $N_A$ bosons in the $A$ system (or equivalently, by measuring $P_{N_B}$ in the $B$ system). One finds:

$$F_Q(G) = 4\langle G^2 \rangle = N_A + N_B + \sum_{i,j} (\langle a_i^\dagger a_j \rangle \langle b_j^\dagger b_i \rangle + \text{h.c.}) \quad (7)$$

In the limit of ideal BECs with perfect off-diagonal long-range order, one has $\langle a_i^\dagger a_j \rangle = N_A/N := n_A$ and $\langle b_i^\dagger b_j \rangle = N_B/N := n_B$, hence $4\langle G^2 \rangle = N(n_A + n_B) + 2N^2 n_A n_B$. The Heisenberg scaling $F_Q(G) \sim N^2$ is fundamentally related to the entanglement between the $N$ modes of $A$ (resp. $B$) associated with the existence of a BEC. Indeed, for a fully separable state $\rho_A = \sum_\lambda p_\lambda \otimes_\lambda^N \rho_i(\lambda)$ with $p_\lambda$ an arbitrary probability distribution and $\rho_i(\lambda)$ a density matrix for mode $a_i$ (and similarly for the $B$ system), one has the inequality [26] $F_Q(G) \le \sum_i (\langle a_i^\dagger b_i + b_i^\dagger a_i \rangle)^2 \sim N$. More explicitly, taking uniform densities $\langle a_i^\dagger a_j \rangle = N_A/N = n_A$ (and similarly for $B$), one finds the separable bound $F_Q(G) \le F_{\text{sep}} = N(n_A + n_B + 2n_A n_B)$, leading to a quadratic error on the estimation of $J$ scaling no faster than $1/N$. A uniform ideal BEC on the $B$ system (namely $\langle b_i^\dagger b_j \rangle = N_B/N$) leads to a QFI in the form $F_Q(G) = F_{\text{sep}} + (N_B/N) \sum_{i \ne j} (\langle a_i^\dagger a_j \rangle + \text{c.c.})$. Therefore, the condition $\sum_{i \ne j} \text{Re}\langle a_i^\dagger a_j \rangle > 0$ witnesses mode entanglement, since it violates the bound for separable states. This illustrates the fact that entanglement generated solely by the indistinguishability of quantum particles represents a consistent resource for metrology [27]. In the case of atomic BECs, the symmetry of the initial state $\rho$ corresponds to a super-selection rule enforced by atom-number conservation. Our metrological protocol hence requires only post-selection on specific initial numbers $N_A, N_B$ of atoms in the two modes.

**Quantum spin ensembles.**— We now consider systems of $N$ quantum $S = 1/2$ spins [44], prepared initially in a coherent spin state $|\psi_0\rangle = |\to_x\rangle \otimes^N$ fully polarized along the $x$ axis, and where the parity projector $P_x = (1 + \prod_{i=1}^N \sigma_i^x)/2$ is both fixed by the initial state, $P_x|\psi_0\rangle = |\psi_0\rangle$, and preserved during the dynamics. In particular, $P_x$ is conserved by all interaction Hamiltonians of the form $H_{\text{int}} = - \sum_{i,j} \sum_{a \in \{x,y,z\}} J_{ij}^a \sigma_i^a \sigma_j^a,$
<figure><img src="image_3.png" /><figcaption>FIG. 2: Preparation of long-range correlated quantum states in a fixed parity sector. (a,b) Preparing an initial state polarized along the transverse field $\Omega$, and then ramping down $\Omega$ towards low-energy states of the parity-preserving Hamiltonian, allows one to prepare resource-states for metrology even at finite energy density and extensive entropy. The critical energy density difference $\Delta e_c$ with respect to the ground-state energy density $e_{gs}$ marks the loss of long-range order. (c) Quantum-circuit projection of an arbitrary qubit state $\rho$ onto a well-defined parity sector, depending on the measurement $M_Z$ of $\sigma^z$ for the ancilla qubit.</figcaption></figure>

namely XYZ models with arbitrary two-body interac-
tions, as well as by Hamiltonians describing an external
magnetic field oriented along x: $H_{\text{field}} = -\sum_{i} \Omega_{i} \sigma_{i}^{x}$. Our
theorem then guarantees that, at all times, spin fluctua-
tions for components in the yz plane are fully quantum-
coherent; and that the sensitivity of the state to spin
rotations around an axis in the same plane is optimally
exploited by simply measuring the parity projector $P_{x}$
itself. A simple symmetry constraint entails therefore
very strong consequences on the nature of fluctuations in
the system, and their potential application for quantum
metrology. In particular, our result applies to widely
different model Hamiltonians for quantum simulators
whose interactions possess different symmetries but all
preserve the parity $P_{x}$, such as transverse-field Ising mod-
els $H_{\text{Ising}} = -\sum_{i,j} J_{ij} \sigma_{i}^{z} \sigma_{j}^{z} - \Omega \sum_{i} \sigma_{i}^{x}$ (with $Z_{2}$ symme-
try) [28–31]; XY models $H_{\text{XY}} = -\sum_{i,j} J_{ij}(\sigma_{i}^{x} \sigma_{j}^{x} + \sigma_{i}^{y} \sigma_{j}^{y})$
(with $U(1)$ symmetry) [28, 32]; and XXZ (Heisenberg)
models $H_{\text{XXZ}} = -\sum_{i,j} J_{ij}(\sigma_{i}^{x} \sigma_{j}^{x} + \sigma_{i}^{y} \sigma_{j}^{y} - \sigma_{i}^{z} \sigma_{j}^{z})$ (with
$SU(2)$ symmetry) [33–35]. Notice that the one-axis-
twisting Hamiltonian (OAT), whose dynamics is imple-
mented in many recent experiments [11, 13–15, 36], may
be viewed as a special instance of XY models, by writing
$H_{\text{OAT}} = \chi J_{z}^{2}/N = \frac{\chi}{N} [\mathbf{J}^{2} - (1/4) \sum_{i,j} (\sigma_{i}^{x} \sigma_{j}^{x} + \sigma_{i}^{y} \sigma_{j}^{y})]$,
where the conserved total spin $\mathbf{J}^{2}$ is just a constant off-
set. Therefore our results provide the optimal strategy
to exploit metrologically all the non-Gaussian spin states
[2, 37, 38] produced by the OAT dynamics after the ap-
pearance of maximal spin squeezing (for a time $t \sim N^{1/3}$)
and before the appearance of the Schrödinger cat state
(for a time $t = \pi N/(2\chi)$), by measuring the sensitivity
of the parity $P_{x}$ to rotations around the y axis.

The initial state is the ground state of $H_{\text{field}}$. A gen-
eral strategy to prepare metrologically useful states is
then to evolve $|\psi_{0}\rangle$ with any of the parity-preserving

Hamiltonians above $H = H_{\text{int}} + H_{\text{field}}$, including pos-
sibly time-dependent parameters, towards a state dis-
playing long-range correlations for spin components in
yz plane. Such long-range correlations typically emerge
when the system displays an ordered phase at low energy,
spontaneously breaking the $Z_2$, $U(1)$ or $SU(2)$ symme-
try of the Hamiltonian $H_{\text{int}}$ in the thermodynamic limit.
Maximal correlations are expected in the ground state of
$H_{\text{int}}$, which requires to implement an adiabatic evolution
starting from the initial state, namely the ground state
of $H_{\text{field}}$. In a realistic (quasi-adiabatic) evolution im-
plemented in experiments in which e.g. $H_{\text{field}}$ is slowly
ramped down (see Fig. 2(a-b)), the final state has fi-
nite energy density above the ground state. One may
then naïvely conclude that the resulting state is akin
to a thermal mixed state, implying that quantum cor-
relations have acquired a short-ranged natureeven when
total correlations are long-ranged [39–42]. Yet, at vari-
ance with this thermal picture of correlations, our theo-
rem establishes that the preparation of correlated state
corresponds to the onset of fully quantum correlations
for off-diagonal observables, regardless of the purity of
the final state, whenever parity is preserved. Typical
phase diagrams in the external-field/energy-density plane
are illustrated in Fig. 2(a,b): whenever parity-preserving
quasi-adiabatic evolutions end up in a long-range ordered
phase, they give rise to macroscopic quantum Fisher in-
formation (i.e scaling as $N^2$) for a completely off-diagonal
collective-spin component, such as the y or z compo-
nent of the collective spin (uniform or staggered). For
instance, ferromagnetic transverse-field Ising models de-
velop $F_Q(J^z) \sim O(N^2)$ at finite energy density when the
connectivity of their interactions is sufficiently high (e.g.
in dimensions $d \ge 2$ with nearest-neighbor interactions).
Similarly, $F_Q(J^y) \sim O(N^2)$ at finite energy density in
ferromagnetic XY models when $J_{ij} \sim 1/r_{ij}^{\alpha}$ with $r_{ij}$ the distance between sites, and $\alpha < d+2$ in $d \le 2$ dimensions [43]; and for arbitrarily short-ranged interactions in $d=3$ – the latter results holds as well for the SU(2) symmetric Heisenberg model.

**Parity projection.**– An alternative route towards preparing highly entangled states is offered by projecting an arbitrary qubit state $\rho$ onto a given parity sector. Active parity projection has the power of turning even a classically correlated state into an entangled one. Parity projection can be achieved with a conceptually simple quantum circuit, namely a sequence of C-NOT gates with an auxiliary qubit, followed by the measurement of the latter. This is illustrated in Fig. 2(c) for parity projection in the $z$ basis: starting from an arbitrary basis state $|\tilde{z}\rangle = |z_1, \dots, z_N\rangle$ with $z_i \in \{0,1\}$, the sequence of C-NOT gates realizes the linear transformation $|\tilde{z}\rangle \otimes |0\rangle \to |\tilde{z}\rangle \otimes |(1+P(\tilde{z}))/2\rangle$ with $P(\tilde{z})$ the parity of the magnetization of $|\tilde{z}\rangle$. With an arbitrary input state $\rho \otimes |0\rangle\langle 0|$, measuring the auxiliary qubit then allows one to implement the parity projection $\rho \to \rho_{\text{even}} = P\rho P$ (resp. $\rho_{\text{odd}} = Q\rho Q$) by post-selecting the $0$ (resp. $1$) measurement outcome. Most importantly, the transformation $\rho \to P\rho P + Q\rho Q$ preserves two-body correlations for the $x$ spin components, namely: $\text{Tr}(\rho\sigma_i^x\sigma_j^x) = \text{Tr}[(P\rho P + Q\rho Q)\sigma_i^x\sigma_j^x]$ for all $i,j$. Indeed $\sigma_i^x\sigma_j^x = P\sigma_i^x\sigma_j^x P + Q\sigma_i^x\sigma_j^x Q$, since $\sigma_i^x\sigma_j^x$ is parity preserving. Hence post-selecting the state with the strongest correlations between $P\rho P$ or $Q\rho Q$ allows one to turn the arbitrary correlations of $\rho$ into quantum ones. The simplest example of this construction is obtained by taking $\rho$ as the coherent spin state $|\to_x\rangle^{\otimes N}$: the post-selected even / odd parity states are cat states ($|\to_x\rangle^{\otimes N} \pm |\leftarrow_x\rangle^{\otimes N}/\sqrt{2}$, which have maximal QFI for rotations around the $x$ axis, $F_Q(J_x) = N^2$.

**Conclusions.**– We have shown that whenever the quantum dynamics of many-body systems remains constrained to a given symmetry sector, all correlations of off-diagonal observables maintain a fully quantum nature, regardless of the purity of the quantum state; and that they directly translate into a metrological resource. Our result opens the path to the systematic certification and metrological use of a broad class of entangled states, such as e.g. non-Gaussian entangled spin states which are naturally prepared in quantum spin ensembles, going beyond the case of spin-squeezed states.

**Acknowledgements.**– TR acknowledges support of PEPR-Q “QubitAF”.

<footnote>* Electronic address: irenee.frerot@lkb.upmc.fr</footnote>

<footnote>† Electronic address: tommaso.roscilde@ens-lyon.fr</footnote>

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