††thanks: on leave from: Institut de Physique Théorique, Université Paris
Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France

# Many-body quantum Zeno effect and measurement-induced subradiance transition

Alberto Biella<EMAIL_ADDRESS>Université Paris-
Saclay, CNRS, LPTMS, 91405 Orsay, France  INO-CNR BEC Center and Dipartimento
di Fisica, Università di Trento, 38123 Povo, Italy  JEIP, USR 3573 CNRS,
Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321
Paris Cedex 05, France  Marco Schiró<EMAIL_ADDRESS>JEIP,
USR 3573 CNRS, Collège de France, PSL Research University, 11 Place Marcelin
Berthelot, 75321 Paris Cedex 05, France

###### Abstract

It is well known that by repeatedly measuring a quantum system it is possible
to completely freeze its dynamics into a well defined state, a signature of
the quantum Zeno effect. Here we show that for a many-body system evolving
under competing unitary evolution and variable-strength measurements the onset
of the Zeno effect takes the form of a sharp phase transition. Using the
Quantum Ising chain with continuous monitoring of the transverse magnetization
as paradigmatic example we show that for weak measurements the entanglement
produced by the unitary dynamics remains protected, and actually enhanced by
the monitoring, while only above a certain threshold the system is sharply
brought into an uncorrelated Zeno state. We show that this transition is
invisible to the average dynamics, but encoded in the rare fluctuations of the
stochastic measurement process, which we show to be perfectly captured by a
non-Hermitian Hamiltonian which takes the form of a Quantum Ising model in an
imaginary valued transverse field. We provide analytical results based on the
fermionization of the non-Hermitian Hamiltonian in supports of our exact
numerical calculations.

Figure 1: Cartoon of the hybrid quantum dynamics considered in this work [see
Eq. (3)] where in addition to the unitary evolution $\hat{U}$ (in our example
generated by the Quantum Ising Model) there is a stochastic measurement
process on each site $\hat{M}=\bigotimes_{i=1}^{L}\hat{M}^{(r)}_{i}$.

##  1 Introduction

Measurements lie at the heart of quantum mechanics and of its intrinsic
statistical interpretation [1]. They are known to fundamentally perturb the
state of a quantum system, leading to the famous wave-function collapse, or to
hamper its evolution in the Hilbert space, as in the celebrated Quantum Zeno
Effect (QZE), where frequent projective measurements on a system can lead to
its full localization [2, 3, 4, 5, 6, 7, 8]. In the field of quantum
information the role of measurements as a resource has been extensively
discussed and also experimentally demonstrated [9, 10, 11, 12, 13]. In the
context of quantum many-body systems the interest is much more recent.
Manifestations of many-body QZE have recently attracted interest [14, 15, 16,
17, 18, 19]. A series of theoretical works have explored the interplay between
entangling quantum many-body dynamics and frequent projective measurements,
giving rise to novel exotic entanglement phase transitions [20, 21, 22, 23,
24, 25]. A different measurement protocol is provided by a continuous
monitoring, or variable strength weak measurements, which is defined in terms
of positive operator-valued measurements (POVM) and provides a stochastic
backaction onto the system [26, 27, 28, 29]. Also in this case entanglement
phase transitions have been identified [22, 30]. A crucial feature of these
transitions is that they are invisible to the average dynamics but only
appears at the level of single quantum many-body states.

In this work we address the competition between unitary many-body dynamics and
stochastic quantum measurements in a simple and paradigmatic setting, namely
when the two processes involve degrees of freedom which do not mutually
commute, such as different components of a quantum spin. Surprisingly we find,
in the context of a one-dimensional quantum Ising model with continuous
monitoring of the local transverse magnetization, that the onset of the many-
body QZE, in which the system is freezed by measurements into a well defined
many-body state, appears sharply above a critical measurement strength, a
behavior suggestive of a novel form of quantum criticality. As for the above
mentioned entanglement phase transitions, we show here that this sharp
transition is encoded in the rare realizations of the stochastic process and
cannot be observed by only looking at the average behavior of the system. We
highlight this point by considering the no-click evolution, corresponding to a
post-selection of the rare no-click events, which can be described by a non-
Hermitian Hamiltonian [31], which in our case takes the form of a non-
Hermitian Transverse Field Ising Model (TFIM) [32]. Remarkably we show that
the sharp onset of the many body QZE can be understood in terms of a
subradiance transition [33, 34, 35, 36, 37, 38, 39] in the many-body spectrum
of the non-Hermitian TFIM, i.e. a non-analytic change in the eigenstate with
the smallest imaginary part. This result suggests an intriguing analogy with
the concept of a quantum phase transition, where the system ground-state
properties change abruptly as a result of the competition between non-
commuting operators. We provide analytical arguments based on the
fermionization of the non-Hermitian TFIM in supports of our exact numerical
calculations. We stress that the many-body nature of the problem is encoded in
the competition between unitary entangling dynamics and local measurements,
rather than in the unmeasured system itself (the quantum Ising chain) which is
integrable through free fermions techniques. We also note that to the best of
our knowledge this mapping cannot be used to exactly solve the problem for the
measurement protocol considered in this work, corresponding to a Born rule
[40], as opposed to other stochastic settings [41, 42] which also show genuine
many-body signatures such as entanglement phase transitions.

##  2 The model and the measurement protocol

In this work we consider a stochastic quantum many-body dynamics governed by
the following protocol

$\ket{\psi(t+dt)}=\hat{M}\hat{U}\ket{\psi(t)},$ (1)

where the unitary dynamics $\hat{U}=\exp(-i\hat{\mathcal{H}}dt)$ is generated
by a Quantum Ising model,

$\hat{\mathcal{H}}=J^{x}\sum_{i=1}^{L-1}\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}$
(2)

($\hat{\sigma}^{\alpha}_{i},\alpha=x,y,z$ being the Pauli matrices acting on
the $i$-th site) and $\hat{M}$ is a measurement carried out simultaneously on
each site with probability $p_{\rm site}$. The operator $\hat{M}$ is a
(normalized) Kraus operator associated with a POVM. Specifically we have
$\hat{M}=\bigotimes_{i=1}^{L}\hat{M}_{i}^{(r)},$ with $r=0,1$ defined by

$\displaystyle\hat{M}_{i}^{(0)}$ $\displaystyle=$
$\displaystyle\hat{\Pi}^{z}_{i-}+\sqrt{1-p_{\rm meas}\ p_{\rm
site}}\,\hat{\Pi}^{z}_{i+},$ (3) $\displaystyle\hat{M}_{i}^{(1)}$
$\displaystyle=$ $\displaystyle\sqrt{p_{\rm meas}\ p_{\rm
site}}\,\hat{\Pi}^{z}_{i+},$ (5)

where $\hat{\Pi}^{z}_{i\pm}=\left(\mathbb{I}\pm\hat{\sigma}^{z}_{i}\right)/2$
are local projectors onto eigenstates of the spin $z$-component
$\hat{\sigma}^{z}_{i}\ket{\pm}_{i}=\pm\ket{\pm}_{i}$. Locally, the
probabilities of the two possible readouts are given by $p^{(1)}_{i}=p_{\rm
meas}\,p_{\rm site}\langle\Psi(t)|\hat{\Pi}^{z}_{i+}|\Psi(t)\rangle,\quad
p^{(0)}_{i}=1-p^{(1)}_{i}$.

Equations (1-3), together with the initial condition that we fix to be an
uncorrelated state with all the spin aligned along the $z-$axis i.e.
$|\Psi(0)\rangle=\bigotimes_{i=1}^{L}\ket{-}_{i}$, fully specifies the
stochastic protocol we will focus on in this work (See Figure 1 for a sketch
of the evolution). The choice of the initial state is rather natural in this
context. Starting from an uncorrelated product state we want to study how the
non local unitary evolution entangles different sites and local measurements
kill correlations. This is alternative to a scenario involving non-local
measurement processes which is expected to stabilize correlated entangled
states. While this problem is is in principle treatable theoretically it would
be very demanding to implement experimentally.

Each random realization provides a quantum many-body state encoded in the wave
function $|\Psi(t)\rangle$. We stress that Eq. (1) leads to an unnormalized
wavefunction since $\hat{M}$ is not a unitary operator. We thus normalize the
quantum state after each step of the protocol.

##  3 Results

In the following we are going to discuss the stochastic dynamics of the system
as a function of the ratio between the frequency of the measurement and the
unitary dynamics. It is natural therefore to introduce the adimensional
control parameter

$g=\frac{p_{\rm meas}}{J^{x}\ dt},$ (6)

which will play a key role in the following. Here the parameter $dt$ is finite
and defined through the unitary evolution in Eq. (1). The physics in two
limiting cases is easy to grasp. For $g\ll 1$, quantum jumps (the occurrence
of $\hat{M}_{i}^{(1)}$) take place with a very low probability ($p_{r=1}\ll
1$) regardless of the value of $p_{\rm site}$ and the projective measurement
$\hat{M}_{i}^{(0)}$ is close to the identity. The system therefore evolves
under the unitary dynamics with rare measurements. In the opposite limit $g\gg
1$ the measurements occur often with respect to the typical timescale settled
by the unitary dynamics. In this regime the stochastic evolution starting from
our initial condition stabilizes the uncorrelated state
$\ket{T}=\bigotimes_{i=1}^{L}\ket{-}_{i}$ which is also the exact fixed point
of the measurement protocol in the limiting case $p_{\rm meas}=1$. As we are
going to show below those two regimes are not smoothly connected, but a sharp
phase transition between two well defined phases occur upon increasing $g$.
Importantly, this will appear in the rare fluctuations but not in the average
dynamics that instead display a rather featureless and smooth crossover.

###  3.1 Measurement-induced entanglement

We start by studying the entanglement properties encoded in the stochastic
evolution of a single quantum state, as a function of $g$. Specifically, we
compute the entanglement entropy of a symmetric bipartition of the system

Figure 2: Top panel: Evolution of the entanglement entropy $S_{E}(t)$ of a
single quantum state for different values of $g$. Bottom panel: measurement-
induced entanglement $\Theta_{E}$ as a function of $g$. Here we set $N_{\rm
real}=100,dt=10^{-2},L=8,J^{x}=1$.

$S_{E}(t)=-{\rm Tr}\left[\hat{\rho}_{A}(t)\log_{2}\hat{\rho}_{A}(t)\right],$
(7)

where $\hat{\rho}_{A}(t)={\rm Tr}_{B}\left[\ket{\psi(t)}\bra{\psi(t)}\right]$
is the reduced density matrix of the left halve (labelled as $A$) of the
system. In the purely unitary (and deterministic) case, $g=0$, this quantity
displays coherent Rabi oscillations in the range $0\leq S_{E}\leq 1$ (see Fig.
2, top panel). Remarkably, we find that for small finite $g$ measurements can
produce more entanglement between the two halves of the chain with respect to
the unitary case, as shown by the entanglement entropy of a single stochastic
realization, going above $S_{E}=1$. For large $g$ instead, as expected, the
entanglement production is strongly suppressed. In order to quantify this
entanglement generation with respect to the unitary case ($g=0$) we consider
the probability $P(S_{E})$ that a certain value of $S_{E}$ occurs during the
stochastic evolution and then we evaluate the measurement-induced entanglement
$\Theta_{E}$

$\Theta_{E}=\int_{S_{E}>1}S_{E}P(S_{E})\ dS_{E},$ (8)

which is a measure of the entanglement generated on top of the unitary
dynamics averaged over $N_{\rm real}$ stochastic realizations. In Fig.2
(bottom panel) we show the behavior of $\Theta_{E}$ as a function of $g$ and
for $p_{\rm site}=1$. We find a surprising non-monotonic behavior of the
excess entanglement as the strength of the measurements is increased, with a
maximum around $g\sim 1$ and a rapid decrease for larger value of $g$,
ultimately expected as the QZE kicks in and the system is more and more
projected towards the product state $|T\rangle$. A similar behavior is found
for $p_{\rm site}\neq 1$. In fact we find evidence for a data collapse
suggesting the excess entanglement $\Theta_{E}$ only depends on the product
$g\ p_{\rm site}$ (see App. B) This result shows that before the onset of the
many body QZE the system at the single trajectory level is highly entangled,
due to the interplay between unitary dynamics and measurements. As we are
going to discuss below this result is the consequence of a transition between
two phases, very different in nature, controlled by $g$.

Figure 3: Top Panel: Probability distribution of the overlap
$O_{\psi,T}=\left|\braket{\psi}{T}\right|$ between the long-time state of the
stochastic dynamics in Eq. (1) and the Zeno state $|T\rangle$. Increasing the
strength of the measurements $g$ the distribution develops a secondary peak,
which becomes dominant for large $g$. Here we set $N_{\rm
real}=100,dt=10^{-2},L=8,J^{x}=1$. Bottom Panel: Overlap between the long-time
state of the non-Hermitian evolution and the Zeno state $|T\rangle$ as a
function of $g$ and for different sizes $L$. For $L=8$ we compare this overlap
with the result obtained in the full dynamics by tracking the secondary peak
(triangles) and find excellent agreement for $g>g_{c}$.

###  3.2 Onset of many-body quantum Zeno effect

As already discussed, for $g\gg 1$ the measurements drive the system towards
the uncorrelated product state $|T\rangle$. This is a signature of the QZE
which appears in the stochastic dynamics of local observables, such as the
transverse magnetization $m_{z}$ whose probability distribution features a
rich evolution (see App. D).

Here we discuss the onset of QZE in the probability distribution of the
overlap $O_{\psi,T}=\left|\braket{\psi}{T}\right|$ at long times, that we plot
in Fig. 3 (top panel) for different values of $g$. For $g\lesssim 1$ the
function $P(O_{\psi,T})$ is peaked at $O_{\psi,T}=0$, signaling the correlated
nature of $\ket{\psi(t)}$ in the regime where the unitary quantum dynamics
dominates, but display a skewed distribution with fat tails slowly decaying
for larger value of $O_{\psi,T}$. The broad statistics of $O_{\psi,T}$
suggests that taking the average overlap as measure of the onset of the QZE
misses important aspects of the physics, and that a-typical values of the
overlap might play an important role in the stochastic dynamics.
Interestingly, for $g\gtrsim 1$ a second peak emerges at $O_{\psi,T}>0$ and
the distribution acquires a strong bi-modal character. As $g$ is increased
further into the deep quantum Zeno regime this secondary peak (i) shifts
towards values of order one, $O_{\psi,T}\sim 1$, indicating that the long-time
limit state resembles more and more the uncorrelated product state
$|T\rangle$, and (ii) acquires larger weight eventually becoming the dominant
peak of the distribution. Next, we show that the sharp onset of this secondary
peak in the overlap distribution can be understood by considering the no-click
evolution, corresponding to a post-selection of the rare dynamics with outcome
$r=0$ for all the lattice sites.

Figure 4:  Top Panel: Ratio between the probability that the subradiant state
is inside the Zeno subspace and its complement [$f=O_{{\rm
sub},T}/\left(1-O_{{\rm sub},T}\right)$] as a function of $g$, showing two
clearly distinct behaviors. At small $g$ this ratio is constant and small,
$f<1$ decreasing with $L$, while for large $g$ we find $f\sim g^{2}$. Bottom
panel: Scaling with system size $L$ of the overlap $O_{{\rm sub},T}$ between
the subradiant state and the Zeno state $|T\rangle$, for two values of $g$.
While at large $g$ the overlap depends weakly on $L$, we find indication for a
power-law scaling at small $g$. Here we set $dt=10^{-2},J^{x}=1$.

###  3.3 No-click dynamics and subradiance transition

In absence of a click the POVM reduces to
$\hat{M}=\bigotimes_{i=1}^{L}\hat{M}_{i}^{(0)}$ regardless of the value of the
system parameters. Fixing $p_{\rm site}=1$ and parametrizing the measurement
rate as $p_{\rm meas}=\gamma dt\rightarrow 0$, we obtain that the dynamics can
be described by an effective non-Hermitian evolution (see App. C for a
discussion of its validity). We thus get $\ket{\psi(t+dt)}\simeq
e^{-i\hat{\mathcal{H}}_{\rm eff}dt}\ket{\psi(t)}$, where we have introduced
the non-hermitian Hamiltonian

$\hat{\mathcal{H}}_{\rm
eff}=\hat{\mathcal{H}}-i\hat{\Gamma}=J^{x}\sum_{i=1}^{L-1}\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}-i\frac{\gamma}{2}\sum_{i=1}^{L}\frac{\hat{\sigma}^{z}_{i}+\mathbb{I}}{2}.$
(9)

The continuous monitoring generates a purely imaginary transverse field,
proportional to the strength of the measurement $\gamma$, which does not
commute with the Hamiltonian $\hat{\mathcal{H}}$. The no-click dynamics is
therefore described by a TFIM with a purely imaginary transverse field.

Figure 5: Quasiparticle spectrum $\Lambda_{k}$ [given in Eq. (12)] of the non-
Hermitian TFIM for different values of $g=\gamma/J^{x}$. For convenience, we
choose the sign convention for each $k$ such that the imaginary part of
$\Lambda_{k}$ is always negative [32]. The real part (left panel), ${\rm
Re}\left[\Lambda_{k}\right]=\Delta_{k}$, is gapped at $k=\pm\pi/2$ for
$g<g_{c}$ while becomes gapless for $g>g_{c}$. The imaginary part (right
panel), ${\rm Im}\left[\Lambda_{k}\right]=\Sigma_{k}$, always displays two
maxima located at $k=\pm\pi/2$. For $g<g_{c}$ we have $\Sigma_{k=\pm\pi/2}=0$
while for $g>g_{c}$ the imaginary part of the quasiparticle spectrum is such
that $\Sigma_{k}<0,\ \forall k$.

We notice that non-Hermitian extensions of classical Ising models have a long
tradition in statistical physics [43, 44], while the non-Hermitian TFIM has
been much less explored (see however Ref. [45] for a realization in the
context of full-counting statistics transitions). Since this model is not PT
symmetric [46], its many-body spectrum is in general complex.

The non-Hermitian evolution at long times drive the state to the eigenstate of
$\hat{\mathcal{H}}_{\rm eff}$ with smaller imaginary part, the so called
subradiant state, since
$\ket{\psi(t)}=\sum_{\alpha=1}^{N}\braket{\tilde{\alpha}}{\psi(0)}e^{-itE_{\alpha}}\ket{\alpha}$
where $\bra{\tilde{\alpha}}$/$\ket{\alpha}$ are the left/right eigenvectors of
$\hat{\mathcal{H}}_{\rm eff}$, $E_{\alpha}=\omega_{\alpha}-i\Gamma_{\alpha}$
are the corresponding complex eigenvalues and $N$ is the dimension of the
Hilbert space. Remarkably, $\hat{\mathcal{H}}_{\rm eff}$ undergoes a
subradiance transition [33, 34, 35, 36, 37, 38, 39, 31] at a critical value of
$g$ which is signaled by a non-analytic behavior of the eigenvalue with
smaller imaginary part, $\Gamma_{\rm sub}=\min\left(\Gamma_{\alpha}\right)$
(see App. E). While for a non-Hermitian Hamiltonian such non-analyticity can
occur sharply also in a finite size system in the following we will focus on
those properties which are genuine many-body effects depending on the system
size. The structure of the subradiant state when $g\rightarrow\infty$ can be
immediately read from Eq. (9) and gives $\ket{{\rm sub}}=\ket{T}$. In Figure 3
(bottom panel) we plot the overlap $O_{{\rm sub},T}$ between the subradiant
state and the initial uncorrelated state $\ket{T}$ as a function of $g$,
showing a monotonous decrease from $O_{{\rm sub},T}\simeq 1$ (as $g$ is
decreased) and a rapid drop around $g\simeq g_{c}$. Remarkably the behavior
for $g>g_{c}$ matches with the one obtained from the full stochastic evolution
by tracking the secondary peak in the probability distribution (see Fig.3). A
more clear indication of a critical behavior in the overlap can be seen by
plotting the ratio between the probability of the subradiant state being
inside or outside the Zeno subspace, as we do in Figure 4 (top panel)

$f=\frac{O_{{\rm sub},T}}{1-O_{{\rm sub},T}}.$ (10)

We see that $f<1$ for $g<g_{c}$, while for $g>g_{c}$ we have a large value of
$f$, namely the subradiant state is essentially in the Zeno subspace. We
stress that this sharp localization in the Hilbert space is highly nontrivial
since the Zeno subspace dimension does not increase with the size of the
system while the dimension of the full Hilbert space increases exponentially
with $L$. Numerically we find $f\sim g^{2}$ for large $g$, from which we get
$O_{{\rm sub},T}=1-A/g^{2}$ at large value of $g$. Interestingly the overlap
scales differently with system sizes depending on the value of $g$, as we show
in Figure 4 (bottom panel), where an almost constant overlap is found for
large $g$ while a power-law scaling is found for $g<g_{c}$ ($O_{{\rm
sub},T}\sim L^{-\beta}$, with $\beta=0.049\pm 0.003$).

##  4 Analytical approach

The non-hermitian TFIM in Eq. (9) can be diagonalised exactly using the
Jordan-Wigner representation of the spin operators into fermionic
creation/annihilation operators and a generalised Bogoliubov transformation,
as we discuss more in detail in App. A. The final result, in terms of new
fermionic operators $\hat{\bar{\gamma}}_{k},\hat{\gamma}_{k}$ reads (fixing
$J^{x}=1$)

$\hat{\mathcal{H}}_{\rm
eff}=\sum_{k>0}\Lambda_{k}\left(\hat{\bar{\gamma}}_{k}\hat{\gamma}_{k}+\hat{\bar{\gamma}}_{-k}\hat{\gamma}_{-k}-1\right)-i\frac{g}{g_{c}}L,$
(11)

where the quasiparticle spectrum is given by

$\Lambda_{k}=\pm 2\sqrt{1-(g/g_{c})^{2}+2ig/g_{c}\,\cos
k}\equiv\Delta_{k}+i\Sigma_{k}$ (12)

and, following Ref. [32] we choose $\Sigma_{k}<0$ for all $k$, and we have
introduced the critical coupling $g_{c}=4$. The behavior of this spectrum,
plotted in Figure 5, is particularly interesting around $k=\pi/2$: for
$g<g_{c}$ the real-part is gapped, with a gap closing (in the thermodynamic
limit $L\to\infty$) as $\Delta=\Delta_{k=\pi/2}\sim\sqrt{1-(g/g_{c})^{2}}$
(with $\Sigma_{k=\pi/2}=0$); for $g>g_{c}$ the real-part of the energy
spectrum remains pinned to zero ($\Delta_{k=\pi/2}=0$) while the imaginary
part become non-vanishing as
$\Sigma=\Sigma_{k=\pi/2}\sim-\sqrt{(g/g_{c})^{2}-1}$. Right at the critical
point, $g=g_{c}$, the spectrum around $k=\pm\pi/2$ acquires a very unusual
power-law scaling

$\Lambda_{k}\sim|k\mp\pi/2|^{1/2}\left[\pm\mbox{sign}(k\mp\pi/2)-i\right].$
(13)

This result has direct consequences in the structure of the two subradiant
states merging at $g_{c}$, whose structure factor is strongly peaked at
$k\simeq\pi/2$ before the transition (see App. F). Finally, from the
quasiparticle spectrum (12) we can obtain the many-body eigenstates (see
"Methods"), in particular the emerging subradiant state which has an
(intensive) complex energy

$\varepsilon_{\rm sub}=\frac{1}{2\pi}\int_{-\pi}^{0}\Lambda_{k}\
dk-i\frac{g}{g_{c}}.$ (14)

At weak coupling, $g/g_{c}\ll 1$, the subradiant eigenvalue has a real-part
which does not depend on $g$ and a linearly growing imaginary part,
$\varepsilon_{\rm sub}\sim 1-ig/g_{c}$. At strong coupling instead,
$g/g_{c}\gg 1$ we find from Eq. (14) a purely imaginary eigenvalue, which goes
to zero as $\varepsilon_{\rm sub}\sim-2ig_{c}/g$, in agreement with our exact
numerical calculations (see App. A).

We conclude pointing out that the physics highlighted in this work would be
completely invisible to the dynamics of the density matrix representing the
statistical mixture of $N$ quantum states (here labelled by $n$)
$\rho(t)=\frac{1}{N}\sum_{n=1}^{N}\ket{\Psi_{n}(t)}\bra{\Psi_{n}(t)}$. Given
the Kraus operators definedin Eq. (3), the density matrix $\rho(t)$ obeys the
following Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation [47]

$\partial_{t}\rho(t)=-\frac{i}{\hbar}\left[\hat{\mathcal{H}}_{\rm
eff},\rho(t)\right]+\sum_{i=1}^{L}\hat{L}_{i}\rho(t)\hat{L}_{i}^{\dagger},$
(15)

where $\hat{L}_{i}=\sqrt{\gamma}\ \hat{\Pi}^{z}_{i+}$. The master equation
(15) models a Quantum Ising model in presence of local dephasing processes
taking place at a rate $\gamma$. It leads to the a fully depolarized steady-
state density matrix ($\lim_{t\to\infty}\rho(t)=\rho_{\rm SS}$)

$\rho_{\rm SS}=\frac{1}{2^{L}}\bigotimes_{i=1}^{L}\mathbb{I}_{i}.$ (16)

Therefore, the steady-state density matrix retains no information about the
statistical features of the observables studied in this work. However, the
analysis of the spectrum of the GKLS equation could reveal useful information
associated to the typical time scale associated to the onset of the Zeno
regime [8].

##  5 Conclusion

In this work we have shown that the many-body QZE in a quantum Ising chain,
where strong continuous monitoring of the transverse magnetization leads to a
complete freezing of the system, appears sharply upon increasing the ratio
between measurement strength and coherent (entangling) interaction. Crucially,
this transition is not encoded in the average dynamics, but it is rather
controlled by rare fluctuations which we show are perfectly captured by
looking at the no-click evolution. In the limit of continuous measurements
this reduces to a non-hermitian TFIM which we show, using exact numerics and a
mapping to fermions, to undergo a sharp subradiance transition characterized
by a non-analytic behavior of the slowest decay mode and by a gap closing in
the quasiparticle spectrum. This work establish a rather general important
connection between many body QZE, subradiance transition in non-hermitian
systems and rare fluctuation of the stochastic dynamics of a measured quantum
many-body system. Future directions include the investigation of the role of
spatially inhomogeneous measurements ($p_{\rm site}<1$) leading to dilute
disorder [48, 49, 50, 51], different initial conditions and higher
dimensionality. The study of a model where the unmeasured dynamics is governed
by an interacting model represents a further intriguing research direction.

## Acknowledgements

We thanks useful discussions with L. Mazza and R. Fazio. The computations were
performed on the Collége de France IPH computer cluster. We acknowledge
support from the ANR grant “NonEQuMat”(ANR-19-CE47-0001). AB acknowledges
funding by LabEx PALM (ANR-10-LABX-0039-PALM).

## APPENDICES

##  Appendix A Diagonalization of the non-Hermitian quantum Ising model

We consider the non-Hermitian Transverse Field Ising Model (TFIM) introduced
in the main text, whose Hamiltonian reads

$\hat{\mathcal{H}}_{\rm
eff}=\hat{\mathcal{H}}-i\hat{\Gamma}=J^{x}\sum_{i=1}^{L-1}\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}-i\frac{\gamma}{2}\sum_{i=1}^{L}\frac{\hat{\sigma}^{z}_{i}+1}{2}$
(17)

and we take periodic boundary conditions for the spin
$\sigma^{x}_{L+1}=\sigma^{x}_{1}$. Using the Jordan-Wigner transformation,
which maps spin-$1/2$ operators to fermions through the identities

$\displaystyle\sigma^{x}_{j}$ $\displaystyle=$
$\displaystyle\hat{K}_{j}\left(\hat{c}_{j}+\hat{c}^{\dagger}_{j}\right),$ (18)
$\displaystyle\sigma^{z}_{j}$ $\displaystyle=$ $\displaystyle
1-2\hat{c}^{\dagger}_{j}\hat{c}_{j},$ (19)

where the so called string operator $\hat{K}_{j}$, defined as

$\hat{K}_{j}=\left[\prod_{l<j}\left(1-2\hat{c}^{\dagger}_{l}\hat{c}_{l}\right)\right]\,,$
(20)

ensures the proper fermionic algebra. We can rewrite Eq. (17) as

$\displaystyle\hat{\mathcal{H}}_{\rm eff}$
$\displaystyle=J^{x}\sum_{i=1}^{L-1}\left(\hat{c}^{\dagger}_{i}\hat{c}_{i+1}+\hat{c}^{\dagger}_{i}\,\hat{c}^{\dagger}_{i+1}+{\rm
H.c.}\right)$ (22)
$\displaystyle-i\frac{\gamma}{2}\sum_{i=1}^{L}\,\left(1-\hat{c}^{\dagger}_{i}\hat{c}_{i}\right)$
$\displaystyle+(-1)^{N_{F}}J^{x}\left(\hat{c}^{\dagger}_{L}\hat{c}_{1}+\hat{c}^{\dagger}_{L}\,\hat{c}^{\dagger}_{1}+{\rm
H.c.}\right).$

In the last expression $N_{F}=\sum_{i=1}^{L}\hat{c}^{\dagger}_{i}\hat{c}_{i}$
is the total number of fermions and $(-1)^{N_{F}}$ the associated parity. We
notice that in the Hamiltonian case (purely real transverse field) the
fermionic parity is conserved and as such the last term is just a phase that
can be handled by proper choice of boundary conditions, i.e. by posing
$\hat{c}_{L+1}=(-1)^{N_{F}+1}\hat{c}_{1}$. In the non-Hermitian case this is
in general not the case, however one can show that starting from an initial
state which is eigenstate of the fermionic parity (as it is our case here) the
latter will remain conserved throughout the evolution. Therefore we fixed the
parity to be even (as we start from the $N_{F}=0$ sector), which corresponds
to anti-periodic boundary conditions (ABC). Going to Fourier space, and fixing
the values of $k$ appropriate for ABC, we obtain

$\displaystyle\hat{\mathcal{H}}_{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{k}\,\varepsilon_{k}\,\hat{c}^{\dagger}_{k}\hat{c}_{k}-i\frac{\gamma}{2}L$
(24) $\displaystyle+J\sum_{k}\,i\sin
k\left(\hat{c}^{\dagger}_{k}\,\hat{c}^{\dagger}_{-k}-\hat{c}_{-k}\hat{c}_{k}\right)$

with $\varepsilon_{k}=i\gamma/2+2J\cos k$. We notice that last term on the
right hand side, deriving from Eq. (22) which is crucial to ensure the non-
Hermitian Hamiltonian has a positive definite imaginary part. Restricting the
sum to positive values of $k$ and introducing Nambu-Gorkov spinor notation
$\Psi_{k}=\left(\hat{c}_{k},\hat{c}^{\dagger}_{-k}\right)^{\rm T}$ we get
$\hat{\mathcal{H}}_{\rm
eff}=\sum_{k>0}\bar{\Psi}_{k}h_{k}\Psi_{k}-i\frac{\gamma}{4}L$ where we
defined the generalized Bogoliubov matrix

$h_{k}=\begin{pmatrix}\varepsilon_{k}&-i\eta_{k}\\\
i\eta_{k}&-\varepsilon_{k}\end{pmatrix},$ (25)

where $\eta_{k}=-2J^{x}\sin k$. We can then diagonalise the matrix $h_{k}$ and
introduce new fermionic operators $\hat{\bar{\gamma}}_{k},\hat{\gamma}_{k}$ to
obtain

$\hat{\mathcal{H}}_{\rm
eff}=\sum_{k>0}\Lambda_{k}\left(\hat{\bar{\gamma}}_{k}\hat{\gamma}_{k}+\hat{\bar{\gamma}}_{-k}\hat{\gamma}_{-k}-1\right)-i\frac{\gamma}{4}L,$
(26)

where the quasiparticle spectrum reads

$\displaystyle\Lambda_{k}$ $\displaystyle=$ $\displaystyle\pm
2J^{x}\sqrt{1-(g/g_{c})^{2}+2ig/g_{c}\,\cos k}$ (27) $\displaystyle\equiv$
$\displaystyle\Delta_{k}+i\Sigma_{k}$ (29)

and we have introduced the critical coupling $g_{c}=4$. Once the Hamiltonian
is diagonalised we can write down the vacuum state of $\hat{\mathcal{H}}_{\rm
eff}$

$|\rm{vac}\rangle=\prod_{k>0}\hat{\gamma}_{-k}\hat{\gamma}_{k}|0\rangle$ (30)

with intensive energy $\varepsilon_{\rm vac}=E_{\rm
vac}/L=-\frac{1}{L}\sum_{k>0}\Lambda_{k}-i\frac{\gamma}{4}$ which reads (in
the thermodynamic limit)

$\displaystyle\varepsilon_{\rm vac}/2J^{x}$ $\displaystyle=$
$\displaystyle-\int_{0}^{\pi}\frac{dk}{2\pi}\sqrt{1-(g/g_{c})^{2}+2ig/g_{c}\,\cos
k}$ (33) $\displaystyle-i\frac{g}{2g_{c}}.$

Similarly, any excited state of the form

$|\left\\{n_{k}\right\\}\rangle=\prod_{k}\left(\hat{\bar{\gamma}}_{k}\right)^{n_{k}}|\rm
vac\rangle$ (34)

with $n_{k}=0,1$ such that $\sum_{k}n_{k}=N_{F}$, given the even parity
sector, has eigenvalue

$E(\left\\{n_{k}\right\\})=E_{\rm vac}+\sum_{k}n_{k}\Lambda_{k}.$ (35)

For example, the state $\ket{T}=\bigotimes_{i=1}^{L}\ket{-}_{i}$ (that is the
exact subradiant state of $\hat{\mathcal{H}}_{\rm eff}$ in the
$g/g_{c}\to\infty$ limit) can be written in this fermionic language as the
maximally occupied state, with $n_{k}=1$ for all $k$. It has eigenvalue

$\displaystyle\varepsilon_{\rm sub}/2J^{x}$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{0}\frac{dk}{2\pi}\sqrt{1-(g/g_{c})^{2}+2ig/g_{c}\,\cos
k}$ (38) $\displaystyle-i\frac{g}{2g_{c}}.$

##  Appendix B Excess entanglement

As discussed in the main text, we quantify the excess entanglement via the
quantity $\Theta_{E}$ defined in Eq. (8). In this appendix we show how
$\Theta_{E}$ depends on $p_{\rm site}$. In Fig.6 we show $\Theta_{E}$ for
several different values of $p_{\rm site}$. Curves collapse when plotted
against $g\ p_{\rm site}$ [with $g$ defined in Eq. (6)].

Figure 6: Measurement-induced entanglement $\Theta_{E}$ as a function of $g\
p_{\rm site}$ for several values of $p_{\rm site}$. Here we set $N_{\rm
real}=100,dt=10^{-2},L=8,J^{x}=1$.

This result shows that that for this specific quantity the control parameter
is given by

$g\ p_{\rm site}=\frac{p_{\rm tot}}{J^{x}\ dt},$ (39)

where $p_{\rm tot}=p_{\rm site}\ p_{\rm meas}$, indicating that the spatial
density and the strength of the measurements play an equivalent role in the
generation of the excess entanglement.

##  Appendix C Validity of the non-Hermitian effective Hamiltonian

In this appendix we test the validity of the non-Hermitian effective
Hamiltonian $\hat{\mathcal{H}}_{\rm eff}$. To this purpose we compare the full
no-click evolution with the one generated by the $\hat{\mathcal{H}}_{\rm eff}$
[Eq. (9)] for different values of $g$.

As we can see from Fig. 7 we dynamics induced by $\hat{\mathcal{H}}_{\rm eff}$
reproduce exactly the no-click evolution of the entanglement entropy $S_{E}$
for $g\lesssim 1$. After this threshold the results for the entanglement
entropy slightly differ quantitatively but behaves qualitatively in the same
way.

Figure 7: Comparison between the full no-click evolution with the one
generated by the $\hat{\mathcal{H}}_{\rm eff}$ [Eq. (9)] for different values
of $p_{\rm meas}=\gamma dt$. Here we set $dt=10^{-2},L=8,J^{x}=1$.

##  Appendix D Onset of the Zeno effect

In this section we show appearance of the Zeno dynamics as $g$ is increased.
The onset of the SZ phase can be inspected monitoring the probability
distribution of the local magnetization $P(m^{z})$ which represents the
probability of having a certain value of the local magnetization during the
dynamics $m^{z}=\bra{\psi(t)}\hat{\sigma}^{z}_{i}\ket{\psi(t)},\forall i$. For
$g\lesssim 1$ the distribution $P(m^{z})$ explores almost uniformly all the
possible values of the magnetization with two peaks progressively emerging at
$m^{z}_{-}$ and $m^{z}_{+}$, located respectively at $m_{z}<0$ and
$m_{z}\simeq 1$. For $g\gtrsim 1$ the structure of the probability
distribution change dramatically. Beside the usual peak at $m^{z}\approx 1$,
the second peak at $m^{z}_{-}$ sharper and moves toward $m^{z}=-1$ as $g$ is
increased. Finally, for $g\gg 1$ the distribution is a delta function peaked
at $m^{z}=-1$. In the top left panel we also show the behaviour of $m_{z}^{-}$
as a function of $g$.

Figure 8: Probability distribution of the local magnetization $P(m^{z})$ for
different values of $g$. In the top left panel we show the position of
$m_{z}^{-}$, the leftmost peak of $P(m^{z})$. We also compare it with
magnetization of the subradiant state of $\hat{\mathcal{H}}_{\rm eff}$. Here
we set $p_{\rm site}=1,dt=10^{-2},L=8,J^{x}=1,N_{\rm real}=100$.

We stress that the value of $m^{z}$ unambiguously determine the quantum state
since the local single-site density matrix can be parametrized as
$\hat{\rho}_{\rm
loc}=\frac{1}{2}\left(\mathbb{I}+m^{z}\hat{\sigma}^{z}\right).$ It is clear
that the only free parameter is $m^{z}$, therefore the structure of $P(m^{z})$
directly gives information about the onset of the Zeno effect (i.e. the system
state is mostly frozen next to one of the measurement eigenstates, yet rarely
performs quantum jumps between them). In Fig. 8 we show the probability
distribution of the local magnetization $P(m^{z})$ which represents the
probability of having a certain value of the local magnetization during the
dynamics $m^{z}=\bra{\psi(t)}\hat{\sigma}^{z}_{i}\ket{\psi(t)},\forall i$. In
the top left panel we compare the position of the peak at $m^{z}_{-}$ with the
magnetization of the subradiant state $\bra{\rm sub}\hat{\Sigma}^{z}\ket{\rm
sub}/L$. The two curves show a perfect agreement for $g>4$.

##  Appendix E Spectral signatures of the superradiance transition

While in the main text we showed the emergence of the subradiance transition
computing the overlap with the Zeno subspace, here we compute the spectral
properties of the non-Hermitian Hamiltonian $\hat{\mathcal{H}}_{\rm eff}$ [Eq.
(9)]. In particular, In Fig. 9 (left panel) we show the behavior of the width
of the subradiand state as a function of the dimensionless coupling $g$. The
real part of the subradiant eigenvalue vanishes for $g=g_{c}^{ST}$, where it
merges with an anti-subradiant partner, and remains pinned there for
$g>g^{ST}_{c}$ (left panel, inset). The transition to the subradiant regime is
sharp also at finite size and the critical value of $g$ slowly shifts with
size. A finite size scaling analysis of the critical point $g_{c}^{ST}$
(bottom panel, inset) gives $g_{c}^{ST}=\mathcal{A}+\mathcal{B}\ L^{-1}$ with
$\mathcal{A}=3.61\pm 0.02$ and $\mathcal{B}=4.5\pm 0.1$. We can thus infer the
behavior in the thermodynamic limit
$(\lim_{L\to\infty}g_{c}^{ST}=\mathcal{A}$).

Figure 9: Subradiant Transition in the Non-Hermitian Hamiltonian. Left panel:
Subradiant width $\Gamma_{\rm sub}$ as a function of $g$. (Inset: Real and
Imaginary part of the subradiand and its anti-subradiant partner as a function
of $g$). Right panel: finite size scaling of the critical point $g_{c}^{ST}$
computed from the behavior of $\Gamma_{\rm sub}$ in the left panel. Here we
set $dt=10^{-2},J^{x}=1$.

##  Appendix F $k$-mode occupancy

Here we show the $k$-mode occupancy of the subradiant and anti-subradiant
state (the many-body state that becomes degenerate with the subradiant at the
transition) defined as

Figure 10: Population in $k$-space [see Eq.(F)] of the subradiant (top panel)
and anti-subradiant (bottom panel) state. Here we set $J^{x}=1$ and vary
$\gamma$ in order to change the dimensionless coupling $g$.

$\displaystyle\braket{\hat{n}_{k}}$ $\displaystyle=$
$\displaystyle\braket{\hat{\sigma}^{+}_{k}\hat{\sigma}^{-}_{k}}$
$\displaystyle=$
$\displaystyle\frac{2}{L+1}\sum_{i,j=1}^{L}\sin(ki)\sin(kj)\braket{\hat{\sigma}^{+}_{i}\hat{\sigma}^{-}_{j}},$

with $k=n\pi/(L+1)$, $n=1,\dots,L$ as imposed by open boundary conditions and
where we used the Fourier transform of the spin operators

$\hat{\sigma}^{+}_{k}=\sqrt{\frac{2}{L+1}}\sum_{i=1}^{L}\sin(ki)\
\hat{\sigma}^{+}_{i}.$ (42)

The result is shown in Fig. 10. The anti-subradiant state (bottom panel)
remains mainly populates the $k=\pi/2$ mode of the chain as $g$ is varied
across the transition. The superradiant state (top panel) is peaked around
$k=\pi/2$ for $g/g_{c}<1$. After the transition the $k$-mode distribution is
sharply depleted since $\ket{\rm sub}\to\ket{T}$ as $g/g_{c}\to\infty$.

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