# Measurement-Induced Entanglement Transitions in the Quantum Ising Chain:
From Infinite to Zero Clicks

Xhek Turkeshi<EMAIL_ADDRESS>SISSA, via Bonomea 265, 34136 Trieste, Italy
ICTP, strada Costiera 11, 34151 Trieste, Italy  JEIP, USR 3573 CNRS, Collège
de France, PSL Research University, 11 Place Marcelin Berthelot, 75321 Paris
Cedex 05, France  Alberto Biella  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
Rosario Fazio  ICTP, strada Costiera 11, 34151 Trieste, Italy  Dipartimento di
Fisica, Universitá di Napoli "Federico II", Monte S. Angelo, I-80126 Napoli,
Italy  Marcello Dalmonte  SISSA, via Bonomea 265, 34136 Trieste, Italy  ICTP,
strada Costiera 11, 34151 Trieste, Italy  Marco Schiró  JEIP, USR 3573 CNRS,
Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321
Paris Cedex 05, France

(August 29, 2024)

###### Abstract

We investigate measurement-induced phase transitions in the Quantum Ising
chain coupled to a monitoring environment. We compare two different limits of
the measurement problem, the stochastic quantum-state diffusion protocol
corresponding to infinite small jumps per unit of time and the no-click limit,
corresponding to post-selection and described by a non-Hermitian Hamiltonian.
In both cases we find a remarkably similar phenomenology as the measurement
strength $\gamma$ is increased, namely a sharp transition from a critical
phase with logarithmic scaling of the entanglement to an area-law phase, which
occurs at the same value of the measurement rate in the two protocols. An
effective central charge, extracted from the logarithmic scaling of the
entanglement, vanishes continuously at the common transition point, although
with different critical behavior possibly suggesting different universality
classes for the two protocols. We interpret the central charge mismatch near
the transition in terms of noise-induced disentanglement, as suggested by the
entanglement statistics which displays emergent bimodality upon approaching
the critical point. The non-Hermitian Hamiltonian and its associated
subradiance spectral transition provide a natural framework to understand both
the extended critical phase, emerging here for a model which lacks any
continuous symmetry, and the entanglement transition into the area law.

##  I Introduction

Recent years have seen major progress in the understanding of many-body
quantum dynamics. Two well separated limits have been discussed extensively,
the unitary dynamics of closed isolated systems polkovnikov2011nonequilibrium
; dalessio2016fromquantum  and the dissipative dynamics of open quantum
systems coupled to an external environment breuer2002the . In the former case
the system remains in a pure many-body state which, in absence of ergodicity
breaking nandkishore2015manybody ; abanin2019colloquium , acts as an efficient
bath for any of its subsystem, leading to thermalization of local observables
and volume-law entanglement entropy. In the latter, the system is
intrinsically mixed and described by a master equation for the density matrix.
The competition between unitary and dissipative couplings in this setting can
lead to non-equilibrium phase transitions but it is not expected to change the
entanglement properties of the system.

An intermediate situation which has recently attracted major attention is the
one in which the external environment represents a measurement apparatus,
which ceaselessly probe some property of the system. Here the key physics is
encoded in stochastic quantum many-body trajectories, which contain a much
richer information on the system dynamics than the average state. As a result
new dynamical phases arise, which are characterized by different entanglement
properties. For generic unitary evolution, as that encoded in random circuits
li2018quantum ; gullans2020dynamical ; gullans2020scalable ; li2019measurement
; li2019conformal ; chan2019unitary ; jian2020measurement ; zabalo2020critical
; piqueres2020mean ; skinner2019measurement ; nahum2021measurement ;
choi2020quantum ; bao2020theory ; li2020statistical ; fan2020selforganized ;
szyniszewski2019entanglement ; zhang2020nonuniversal ; snizhko2020quantum ;
shtanko2020classical ; lunt2021dimensional ; turkeshi2020measurement ;
jian2020criticality ; sang2020entanglement ; shi2020entanglement ;
ippoliti2021postselection ; lavasani2021measurement ; lavasani2020topological
; sang2020measurement ; ippoliti2021entanglement ; lang2020entanglement ;
turkeshi2021measurement , these systems show a transition between an error
correcting phase and a Zeno phase, signaled by the transition of entanglement
from a volume to an area law.

If instead the unitary dynamics is generated by an Hamiltonian, the properties
of the system depend on the nature of the latter fuji2020measurement ;
rossini2020measurement ; lunt2020measurement ; goto2020measurement ;
tang2020measurement . For instance, in the case of a free fermion Hamiltonian,
the volume law is unstable for any measurement rate to a subextensive
entanglement content cao2019entanglement ; fiskowski2021how . However, it has
been shown that the average entanglement entropy can still show a transition
between a logarithmic and area law phase at a critical measurement strength or
to display a purely logarithmic scaling, depending on the stochastic protocol
alberton2020trajectory . A logarithmic growth of the entanglement entropy in
an entire phase is particularly intriguing, given that the average state is
expected to be effectively thermal, and it is reminiscent of a critical,
conformally-invariant, phase whose origin has been so far elusive. Similar
results have been obtained for free-fermion random circuits with temporal
randomness chen2020emergent , a setting that has been recently generalized to
higher dimension tang2021quantum , or for Majorana random circuits
nahum2020entanglement ; bao2021symmetry .

A different take on measurement-induced transitions has instead focused on the
limit of post-selection, also called forced measurement phase transitions
nahum2021measurement , where only atypical trajectories with a particular
outcome of measurement are retained. In this regime several works have
discussed the relationship between measurement-induced transitions and the
spectral properties of the associated non-Hermitian Hamiltonian
biella2020manybody ; gopalakrishnan2020entanglement ; jian2021yang . More
recently a non-Hermitian Hamiltonian has been also shown to emerge from the
replica field theory associated to monitored Luttinger Liquids, leading to a
Kosterlitz-Thouless transition between a gapless and a gapped phase
buchhold2021effective .

In light of these developments it is interesting to compare the entanglement
properties of stochastic quantum dynamics and non-Hermitian Hamiltonian, in a
simple and paradigmatic setting. With such motivation, in this work we
consider the one-dimensional Quantum Ising model coupled to an environment
which continuously measures its transverse magnetization. Specifically, we
focus on two rather opposite limits of the measurement problem: the quantum
state diffusion protocol gisin1992the ; brun2002a , a very clear,
experimentally realizable measurement protocol in quantum optical systems
gardiner2000quantum , equivalent to homodyne trajectories wiseman1993quantum ;
yang2018theory  and the quantum jump protocol in the so called no-click limit,
corresponding to a purely deterministic non-Hermitian evolution, which amounts
to post-select only trajectories without jumps.

Remarkably, despite describing rather different limits, show a very similar
phenomenology in their entanglement properties. In particular, upon increasing
the ratio $\gamma$ between measurement rate and Hamiltonian coupling we find
in both protocols a transition from a critical phase to an area law phase, the
former characterized by a logarithmic scaling of the entanglement with time
and system size. Right at the transition point $\gamma_{c}$, which we find
numerically to coincide in the two protocols, the prefactor of the logarithmic
entanglement contribution, interpreted as an effective central charge,
vanishes continuously.

We note that an extended phase with logarithmic (critical) scaling of the
entanglement was found also for other free-fermionic systems under monitoring,
for example the XX chain of Ref. [alberton2020trajectory, ]. Yet this result
occurs here for a model, the Quantum Ising chain, which does not conserve
particle number but only its parity, and thus lack any obvious continuous
symmetry expected in conventional gapless systems. This suggests that the role
of microscopic symmetries in the classification of quantum phases of matter,
leading to the textbook distinction between gapped and gapless phases in
presence of discrete/continuous symmetries, needs to be revisited for quantum
many body systems under monitoring, as recently pointed out bao2021symmetry .

In this respect the no-click limit offers a natural explanation both for the
critical entanglement scaling and for its transition towards an area law:
these can be naturally understood by looking at the spectral properties of the
non-Hermitian Quantum Ising chain which undergoes, right at $\gamma_{c}$, a
subradiance transition from an extended critical phase with gapless decay
modes, to a gapped phase which is smoothly connected to the dark state of the
measurement operator.

In spite of the important similarities, the two protocols differ for the
behavior of the effective central charge as a function of the measurement
rate. In particular we find this quantity to be larger in the no-click limit,
a signature of a noise-induced disentangling effect that appears clearly in
the statistics of the entanglement entropy, which shows emergent bimodality
upon approaching the critical point. Our results highlight the key role played
by the non-Hermitian Hamiltonian for the qualitative understanding of
entanglement transitions and suggest that different stochastic ensembles
provide different universality classes of a common critical phenomenon.

The paper is structured as follows. In Sec. II we introduce the model and the
measurement protocols. Then we discuss the results for the two protocols in
Sec. III, in particular the entanglement dynamics (Sec. III.1), the
entanglement scaling with system size (Sec. III.2) and the entanglement
statistics (Sec. III.3). We detail in the Appendix a derivation of the
protocol of interest, and a summary of the numerical methods used for the
simulations.

Figure 1:  Cartoon of the monitored Quantum Ising Chain. (a) Quantum state
diffusion protocol: on each lattice site a two-level system (Ising spin)
interacts with its neighbors and is subject to a weak-continuous measurement
of the up-polarized state. (b) Quantum jump protocol: the spins in the chain
interact through a non-Hermitian Hamiltonian. Occasionally a quantum jump take
place, projecting the measured degree of freedom in the up-polarized state.
The no-click limit corresponds to post-selecting only trajectories without
jumps. (See text for details).

##  II Model and Measurement Protocols

We consider a one dimensional Quantum Ising chain with Hamiltonian

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

with open boundary conditions, evolving under the competing effect of its own
unitary dynamics and the measurement of the up-polarized state
${n_{i}\equiv(\sigma^{z}_{i}+1)/2=|1\rangle\langle 1|}$ (pictorially
illustrated in Fig. 1). Here $\sigma^{\alpha}$ are the Pauli matrices. We now
introduce the two measurements protocols that will be considered in this work,
namely the quantum state diffusion and the quantum jumps with post-selection,
corresponding to the non-Hermitian Hamiltonian.

###  II.1 Stochastic Quantum Dynamics with Continuous Monitoring

Due to the monitoring, the evolution is captured by quantum trajectories
$|\psi(\boldsymbol{\xi}_{t})\rangle$ which follow the quantum state diffusion
(QSD) equation

$\displaystyle d|\psi(\boldsymbol{\xi}_{t})\rangle$
$\displaystyle=-iHdt|\psi(\boldsymbol{\xi}_{t})\rangle+\sum_{i=1}^{L}(n_{i}-\langle
n_{i}\rangle_{t})d\xi^{i}_{t}|\psi(\boldsymbol{\xi}_{t})\rangle$
$\displaystyle\quad-\frac{\gamma}{2}\sum_{i=1}^{L}(n_{i}-\langle
n_{i}\rangle_{t})^{2}dt|\psi(\boldsymbol{\xi}_{t})\rangle.$ (2)

(See Appendix A for a brief derivation of the above stochastic Schrödinger
equation). The first term in Eq. (2) represents the unitary evolution, while
the remaining encode the noise effects. The $d{\xi}^{i}_{t}$ are Îto
increments of a Wiener process
$\boldsymbol{\xi}_{t}=(\xi^{1}_{t},\xi^{2}_{t},\dots,\xi^{L}_{t})$,
responsible for the stochastic nature of the trajectories, with zero mean
$\overline{d\xi^{i}_{t}}=0$ and fulfilling the exact property
$d\xi^{i}_{t}d\xi^{j}_{t}=\gamma dt\delta^{ij}$. The last term in Eq. (2)
describes a deterministic back-action from the measuring environment.

We note the presence of a feedback mechanism in Eq. (2), as the noise
contributions couple to the fluctuations of the measured operator $\delta
n_{i}=n_{i}-\langle n_{i}\rangle_{t}$ where
$\langle\circ\rangle_{t}=\langle\psi(\boldsymbol{\xi}_{t})|\circ|\psi(\boldsymbol{\xi}_{t})\rangle_{t}$
is the average over the quantum state. The role of this feedback is to
preserve the norm of the state (and any of its cumulant) for any realization
of the noise.

It is important to stress the difference between the conditional and the mean
state cao2019entanglement . The conditional state is the quantum trajectory
itself
$\rho_{t}(\boldsymbol{\xi}_{t})=|\psi(\boldsymbol{\xi}_{t})\rangle\langle\psi(\boldsymbol{\xi})_{t}|$,
fixed by a realization of the Wiener process. Instead, the mean state is given
by

$\displaystyle\overline{\rho_{t}}=\int\mathcal{D}\boldsymbol{\xi}_{t}P(\boldsymbol{\xi}_{t})|\psi(\boldsymbol{\xi}_{t})\rangle\langle\psi(\boldsymbol{\xi}_{t})|$
(3)

with $P(\boldsymbol{\xi}_{t})$ the probability distribution of the noise.
Despite the conditional state is always pure, the mean state is mixed, and
follows the Lindblad master equation

$\displaystyle\frac{d}{dt}\overline{\rho_{t}}=-i[H,\overline{\rho_{t}}]-\frac{\gamma}{2}\sum_{i}[n_{i},[n_{i},\overline{\rho_{t}}]].$
(4)

This difference is crucial, and highlights how the two states
$\overline{\rho_{t}}$ and $\rho_{t}(\boldsymbol{\xi}_{t})$ provide rather
different information on the statistical properties of the system. For
observables which are linear in the state $O[\rho]=O\rho$, due to the
linearity of the disorder average, the mean state $\overline{\rho_{t}}$
encodes the statistical properties of $O$. Specifically, for any function
$f(O)$, we have

$\displaystyle\overline{\mathrm{tr}(\rho_{t}(\boldsymbol{\xi}_{t})f(O))}\equiv\mathrm{tr}(f(O)\overline{\rho_{t}}).$
(5)

However, the above relationship fails if one is interested in the noise-
average of quantities which depend _non-linearly_ on the density matrix. A
simple example involve the purity $\mathcal{P}(\rho)=\mathrm{tr}\rho^{2}$:
while for the conditional state
$\mathcal{P}(\rho_{t}(\boldsymbol{\xi}_{t}))=1$ (and hence its disorder
average), for the mean state we have $\mathcal{P}(\overline{\rho_{t}})<1$. In
general, similar issues arise when one is interested in more subtle
statistical correlation of the stochastic process, for example in the so
called overlaps $\overline{O_{i}(t)O_{j}(t)}=\overline{\langle
O_{i}\rangle_{t}\,\langle O_{j}\rangle_{t}}$. We emphasize that this situation
is essentially analogous to the difference between annealed and quenched
averages in disordered systems frassek2020duality .

After this parenthesis, we introduce the main observable of interest: the (von
Neumann) entanglement entropy calabrese2004entanglement ;
amico2008entanglement ; horodeski2009quantum ; laflorencie2016quantum . Given
a partition $A\cup B$, the (conditional) reduced density matrix
$\rho_{A}(\boldsymbol{\xi}_{t})=\mathrm{tr}_{B}\rho(\boldsymbol{\xi}_{t})$
encodes the bipartite entanglement. The entanglement entropy is defined as

$\displaystyle
S(\boldsymbol{\xi}_{t})=-\mathrm{tr}_{A}\left[\rho_{A}(\boldsymbol{\xi}_{t})\ln\rho_{A}(\boldsymbol{\xi}_{t})\right].$
(6)

We note that this is a well defined measure of entanglement provided that the
overall state is pure, in which case it yields the amount of Bell pairs that
can be distilled from the quantum state. This is not the case for an overall
mixed state, such as the averaged state in Eq. (3), where also classical
correlation enters into this quantity. For this reason, throughout the paper
we only consider the conditional state and the conditional average of the
entanglement entropy. To simplify the notation, we furthermore omit the
conditional specification. After evolving the state under Eq. (2), we compute
the entanglement entropy. Its average is given by

$\displaystyle\overline{S}=\int\mathcal{D}\boldsymbol{\xi}_{t}P(\boldsymbol{\xi}_{t})S(\boldsymbol{\xi}_{t}),$
(7)

as well as its full probability distribution is

$\displaystyle
P(S_{t})=\int\mathcal{D}\boldsymbol{\xi}_{t}P(\boldsymbol{\xi}_{t})\;\delta\left(S-S(\boldsymbol{\xi}_{t})\right)$
(8)

Throughout this paper we set $J=1$ and study the problem by varying the
strength of the measurement $\gamma$ and for different system sizes $L$.

###  II.2 Non-Hermitian Hamiltonian

When the measurements act occasionally but abruptly on the quantum state, the
quantum trajectories are described by the following stochastic Schrödinger
equation

$\displaystyle d|\psi(\boldsymbol{N_{t}})\rangle$
$\displaystyle=-iHdt|\psi(\boldsymbol{N_{t}})\rangle-\frac{\gamma}{2}dt\sum_{i}\left(n_{i}-\langle
n_{i}\rangle_{t}\right)|\psi(\boldsymbol{N_{t}})\rangle$
$\displaystyle\quad+\sum_{i}\left(\frac{n_{i}}{\sqrt{\langle
n_{i}\rangle_{t}}}-1\right)\delta N^{i}_{t}|\psi(\boldsymbol{N_{t}})\rangle,$
(9)

where $\boldsymbol{N_{t}}$ is a Poisson process, with $\delta N^{i}_{t}=0,1$,
$(\delta N^{i}_{t})^{2}=\delta N^{i}_{t}$, and $\overline{\delta
N^{i}_{t}}=\gamma dt\langle n_{i}\rangle_{t}$. (A derivation of Eq. (9) is
given in Appendix A). The dynamics generated by Eq. (9) preserves the purity
of the state for each trajectory, and gives an alternative unravelling of the
average dynamics in Eq. (4).

When $\delta N^{i}_{t}=0$ on every site, the evolution is driven by the non-
Hermitian Hamiltonian

$\displaystyle
H_{\textup{eff}}=J\sum_{i=1}^{L-1}\sigma^{x}_{i}\sigma^{x}_{i+1}-i\frac{\gamma}{2}\sum_{i=1}^{L}n_{i},$
(10)

while when any $\delta N^{i}_{t}=1$, the last term in Eq. (9) dominates and
projects the measured degree of freedom onto the up-polarized state
$|1\rangle_{i}$. The feedback constant $\propto i\gamma\sum_{i}\langle
n_{i}\rangle_{t}$ enters as a renormalization of the wavefunction within the
non-Hermitian evolution driven by Eq. (10). In the following we consider a
post-selection of the stochastic dynamics, which arises when, among the
experimental runs, we select a portion of trajectories where measurements do
not take place (no-click limit). The evolution is therefore deterministic, and
driven by the non-Hermitian Hamiltonian in Eq. (10). In particular, the effect
of the environment enters the imaginary part of Eq. (10). Here we notice that
this limit capture different aspects of the weak measurement-induced dynamics
compared to the quantum state diffusion. In the framework of quantum jumps,
the latter arise in the limit of infinite measurement events, and hence
correspond to an infinite-click dynamics plenio1998the .

The dynamics of this non-Hermitian Hamiltonian has been considered previously
in Ref. [biella2020manybody, ], where a subradiant transition has been
observed in relationship to a quantum-Zeno effect. Here we study its
entanglement properties, namely we evolve the initial density matrix
$\rho_{0}=|00\dots 0\rangle\langle 00\dots 0|$ according to

$\displaystyle\rho(t)=\frac{e^{-iH^{\dagger}_{\rm eff}t}\rho_{0}e^{iH_{\rm
eff}t}}{\mbox{Tr}\left(\rho_{0}e^{iH_{\rm eff}t}e^{-iH^{\dagger}_{\rm
eff}t}\right)}\,,$ (11)

where the denominator take into account the term proportional to $\gamma
dt\sum_{i}\langle n_{i}\rangle_{t}$ in Eq. (9). Then, given a partition $A\cup
B$, we construct the reduced density matrix
$\rho_{A}(t)=\mathrm{tr}_{B}\rho(t)$ which encodes the bipartite entanglement.
As stressed before, the non-Hermitian Hamiltonian provide a deterministic
protocol, hence the entanglement entropy is deterministic

$\displaystyle S(t)=-\mathrm{tr}_{A}\left[\rho_{A}(t)\ln\rho_{A}(t)\right].$
(12)

To explore the stationary regime, in order to avoid residual dynamical
oscillations, we consider

${S(\infty)}\equiv\lim_{t\to\infty}\frac{1}{t-t_{\mathrm{sat}}}\int_{t_{\mathrm{sat}}}^{t}dsS(s).$
(13)

As for the monitoring case we make use of the Jordan-Wigner mapping to write
down the non-Hermitian Ising model in Eq. (10) in terms of a quadratic
fermionic model (see Appendix B).

Figure 2:  Dynamics of the entanglement entropy averaged over $N=2000$
trajectories for different values of the measurement strength $\gamma$,
different system sizes $L$ and fixing $L_{A}=L/4$. (a-c) We see a logarithmic
growth in time of the entanglement (note the log scale for the time axis) with
a saturation to a size-dependent plateau, for $\gamma<4$ and a much slower
growth to a value independent from system size, for $\gamma>4$.

##  III Results

In this section we present the results of the numerical simulations for the
quantum state diffusion and the non-Hermitian Hamiltonian. After discussion
the entanglement growth within both paradigms, we compare the entanglement
scaling at the stationary state. Finally, we investigate the entanglement
statistics of the quantum state diffusion protocol, and relate it to the non-
Hermitian results.

###  III.1 Entanglement Growth

We start considering the initial state $|\psi_{0}\rangle=|00\dots 0\rangle$
and evolve the system according to the QSD protocol defined in Eq. (2). We
make use of the Jordan-Wigner mapping to write the monitored Ising model as a
quadratic model of spinless fermions, which can be solved efficiently.
(Details on the free fermion techniques, and on the numerical implementation
are given in Appendix B). For the computation of entanglement, we choose a
subsystem $A$ of $L_{A}$ contiguous sites. The average entanglement evolution
is shown in Fig. 2. For small to intermediate values of $\gamma$ (see panels
a-c) we find a logarithmic growth in time of the entanglement,

$\displaystyle\overline{S(t)}\sim\ln t,$ (14)

which ultimately saturates at times $t_{\textup{sat}}\sim L/\gamma$. It is
worth emphasizing that already a small monitoring rate $\gamma$ is enough to
give rise to a non-trivial entanglement dynamics compared to the isolated
system oscillation (generated by $H=\sum_{i}\sigma^{x}_{i}\sigma^{x}_{i+1}$).
In other words the interplay between measurements and unitary evolution can
increase the entanglement production, at least in a certain region of the
parameters, as also noted in Ref. [biella2020manybody, ]. However, upon
further increasing the measurement rate we see a substantial change of the
entanglement growth which already for $\gamma=4$ shows a rapid saturation to a
plateau that is almost independent from the size, a trend that continues for
larger values of $\gamma$.

Figure 3:  Entanglement dynamics for the non-Hermitian Hamiltonian for
different values of the measurement rate $\gamma$ and different system sizes.
For small $\gamma$ (panel a) we see a logarithmic growth in time and a
saturation to a plateau that depends on system sizes. For large $\gamma$
(panel b) the entanglement rapidly saturates to a value that does not depend
on size. Approaching the transition point we see strong oscillations in the
dynamics.

The results of the entanglement dynamics for the non-Hermitian Ising model
(cfr. Eq. (12)) are given in Fig. 3. For small values of $\gamma$ (panels
a-c), we see that the entropy exhibits an overall logarithmic growth, which
resembles the result for the QSD protocol, but has the important difference of
exhibiting residual oscillations which dress both the growth and the
saturation regimes. Upon further increasing the measuring strength $\gamma$
the entanglement dynamics in the non-Hermitian case displays a sharp
transition towards a regime characterized by a fast approach to a stationary
value, with no oscillations. Our numerics locate the boundary between these
dynamical regimes around $\gamma\sim 4$, an estimate compatible with what
found for the QSD protocol. As we will discuss in the next section we can
understand the origin of this sharp transition by looking at the spectral
properties of the non-Hermitian Hamiltonian.

Figure 4:  Stationary-state entanglement entropy as a function of the length
of the chain. (a) Average entanglement entropy for the QSD protocol. (b) Long-
time entanglement entropy for the non-Hermitian dynamics. In both cases we see
a transition from a logarithmic _critical_ scaling, for small $\gamma$ to an
area law (constant) scaling at large $\gamma$. Different curves in the two
panels correspond to increasing values of $\gamma$, from $\gamma=0.25$ (top
curve) to $\gamma=6$ (bottom curve) with a $\Delta\gamma=0.25$.

###  III.2 Entanglement Scaling and Effective Central Charge

As we have seen, at long enough times the entanglement reaches a stationary
value in both protocols. We first discuss the scaling of this stationary
entanglement with system size. This usually provides key insights on the
properties of the system at long-times. To this extent we chose a subsystem of
$L_{A}=L/4$ contiguous spins and plot in Fig. 4 the long-time entanglement as
a function of $L$ and different values of $\gamma$.

For the quantum state diffusion protocol, we can clearly distinguish a regime
of logarithmic entanglement growth with system size for small $\gamma$, which
eventually evolves into a much weaker dependence, an almost constant plateau
for large measurement rates (see Fig. 4 (a)). As we are going to discuss
further below those two regimes are separated by a sharp entanglement
transition.

We note that a logarithmic scaling of the average entanglement in an extended
region of parameters is a particularly intriguing result which was obtained
before in free-fermion models with a continuous $U(1)$ symmetry
alberton2020trajectory  and more recently in connection with a BKT transition
bao2021symmetry ; buchhold2021effective . Here we find this result for a model
which only features a discrete $Z_{2}$ symmetry and whose ground-state would
show logarithmic scaling of the entanglement only at the critical point. This
suggests that a different mechanism, not directly related to the microscopic
symmetries of the problem, could be behind the origin of this seemingly
_critical_ phase.

Interestingly, the finite-size scaling of the entanglement entropy for the
non-Hermitian Hamiltonian shows a behavior which is qualitatively similar to
the QSD protocol, as we show in Fig. 4 (b), including a logarithmic scaling
for small $\gamma$ and an area-law (constant) scaling for large measurement
rates, separated by a sharp transition. We also note that the two scaling
regimes are connected by a finite-size crossover, absent in the stochastic QSD
protocol, whose characteristic scale we found to diverge at the transition
(not shown).

For equilibrium quantum critical systems the prefactor of the logarithmic
scaling of the entanglement entropy is controlled by the central charge of the
associated Conformal Field Theory(CFT) calabrese2004entanglement , a universal
quantity which roughly measures the number of degrees of freedom in the CFT.
Following this analogy Ref. [alberton2020trajectory, ] introduced an
_effective central-charge_ for an XX spin-chain under continuous monitoring,
which turned out to be a non-universal function of the measurement rate. Here
we follow this route and define an effective central charge for both the QSD
protocol and the non-Hermitian Hamiltonian by fitting the data in Fig. 4
according to the ansatz

$\begin{split}\mathrm{QSD}&\qquad\qquad\overline{S}=\frac{1}{3}c_{\mathrm{QSD}}(\gamma)\ln
L+a\,,\\\ H_{\mathrm{eff}}&\qquad\qquad
S=\frac{1}{3}c_{H_{\mathrm{eff}}}(\gamma)\ln L+b\,.\end{split}$ (15)

Here $a$ and $b$ are two system-size independent quantities.

We plot the result in Fig. 5 and find that $c(\gamma)$ decreases with $\gamma$
and, quite remarkably, vanishes for both protocols around the same value of
measurement strength $\gamma_{c}\simeq 4$, that we therefore establish as
critical value for the entanglement transitions between logarithmic and area
law regimes. Despite this similarity we notice that the effective central
charge shows a rather different dependence from $\gamma$ in the QSD and in the
non-Hermitian Hamiltonian protocols. In particular, a direct comparison in
Fig. 5 shows that the effective central charges coincide at small and large
values of $\gamma$, while deviate in a region before the critical point. Here
the non-Hermitian effective central charge turns out to be larger than the one
associated to the QSD protocol, a point on which we will come back in the next
section.

Figure 5:  Effective central charge as a function of the measurement rate
$\gamma$, for the QSD protocol and the non-Hermitian dynamics. In both cases
we find that $c(\gamma)$ decreases monotonously and vanishes around
$\gamma_{c}\simeq 4$, which locates the entanglement transition between
logarithmic scaling and area law. Interestingly the two central charges
coincide at small and large values of $\gamma$, while approaching the critical
point we find the non-Hermitian one to be larger, possibly vanishing at
$\gamma_{c}$ in a non-analytic way.

Before closing this section we note that many of the properties found for the
entanglement dynamics of the non-Hermitian Hamiltonian can be naturally
understood in terms of the spectral properties of the non-Hermitian Ising
model in Eq. (10). In fact this can be easily diagonalized once written in
fermionic language hickey2013time ; lee2014heralded ; biella2020manybody , and
takes the form

$\displaystyle H_{\rm
eff}=\sum_{k>0}\Lambda_{k}\left(\bar{\eta}_{k}\eta_{k}+\eta_{-k}\bar{\eta}_{-k}\right)+E_{0},$
(16)

where $\bar{\eta}_{k},\eta_{k}$ are fermionic operators obtained from a non-
Hermitian Bogolyubov rotation, $E_{0}$ is an additive constant fixing the
energy zero, and $\Lambda_{k}$ are the quasiparticle energies. We note that
these are in general complex, as the model lacks any PT symmetry and read

$\displaystyle\Lambda_{k}=2\sqrt{1-\frac{\gamma^{2}}{16}+i\frac{\gamma}{2}\cos
k}$ (17)

Figure 6:  (a-f) Probability distribution of the entanglement entropy for
different values of $\gamma$ and different system sizes. The realization
collected for these functions are $N=64000$. We see that upon increasing
$\gamma$, the statistic evolves from a simple Gaussian (panel a-b) to a skewed
distribution (panel c-e), with emergent bimodality. At large $\gamma$, the
distribution is peaked at $S_{\infty}\simeq 0$ (panel f). The solid line show
the average (blue line) over the trajectories for the quantum state diffusion,
and the stationary value of the non-Hermitian Hamiltonian (orange line). (g-h)
Different statistical estimators $\langle\langle S\rangle\rangle$ versus
system size for different values of $\gamma$. Note the logarithmic scaling in
the $L$-axis.

The non-Hermitian Ising model features a subradiance transition at a critical
value of the measurement rate, given exactly by $\gamma_{c}=4$, at which the
state with smallest imaginary part changes in a non-analytic way Biella2020.
For small $\gamma<\gamma_{c}$ the imaginary part of the spectrum in Eq. (17)
is _gapless_ and, consistently, the correlation functions in this phase decay
as a power-law lee2014heralded . We emphasize that for a non-Hermitian problem
the usual Goldstone theorem relating a gapless spectrum with a continuous
broken symmetry does not hold. From this point of view the logarithmic scaling
of the entanglement entropy can be naturally understood, since the entire
phase $\gamma<\gamma_{c}$ has soft-modes with small quasi-degenerate lifetime,
which are responsible for the slow growth of the entanglement. On the other
hand the real-part of the spectrum is gapped for $\gamma<\gamma_{c}$, leading
to residual oscillations in the entanglement dynamics.

Above the subradiance transition for $\gamma>\gamma_{c}$ a gap opens up in the
imaginary part of the spectrum $\Lambda_{k}$, which is consistent with the
fast entanglement dynamics found in this regime. The correlation functions in
this phase are expected to decay exponentially over a length diverging as
$\gamma\rightarrow\gamma_{c}^{+}$, a fact that can naturally explain the
emergence of a crossover in the finite-size entanglement scaling. We note that
in the subradiant phase the long-time dynamics is controlled by a pair of
states with the smallest non-zero imaginary part. These in the strong
measurement limit coincide with the dark states of the measurement operator,
which are factorized (uncorrelated) eigenstates of the transverse
magnetization, therefore justifying the small, system-size independent
entanglement entropy.

The insights on the non-Hermitian Hamiltonian could suggest that an effective
field theory description could be possible for $\gamma<\gamma_{c}$ and that
the effective central charge could be computed using field theory replica
techniques calabrese2009entanglement , which however we do not attempt here.
We also note that, in this perspective, an effective central charge which
depends from a parameter in a non-universal way seems very natural, and in
fact several non-unitary field theories have been shown to enjoy this property
bianchini2014entanglement ; couvreur2017entanglement ; chen2020emergent .

###  III.3 Entanglement Statistics

Additional insights are given by the statistical fluctuations of the
entanglement in the quantum state diffusion protocol, a feature that goes
beyond the simple average so far discussed and that is encoded in the full
probability distribution, $P(S_{t})$, defined in Eq. (8). Concretely, we
investigate this distribution at stationarity and evaluate
$P(S_{\infty})=\lim_{t\to\infty}P(S_{t})$.

As we see in Fig. 6, the statistics of entanglement shows a rather rich
evolution with $\gamma$. In particular we see that in the logarithmic phase
($\gamma<\gamma_{c}$), the statistics changes dramatically, much more than
what one could have guessed from the average value, whose dependence from
$\gamma$ can be reabsorded in a renormalization of the central charge. For
small values of the measurement rate, $\gamma\lesssim 1$ (panel a), the
entanglement distribution is a normal Gaussian which broadens asymmetrically
for $\gamma\sim 2$ (panel b) and develops an emergent bimodality as the
critical point $\gamma_{c}$ is approached (panels c-e). Specifically we see
that already for $\gamma=2.75$ a secondary peak, for lower values of the
entanglement, starts to emerge. Upon further increasing $\gamma$ we see a
transfer of weight between the high-entanglement and low-entanglement peaks,
with the latter ultimately becoming the dominant one as the system enters the
area-law phase. Here, for $\gamma>\gamma_{c}$ the entanglement distribution is
strongly skewed and peaked around small values of the entropy.

It is interesting to discuss the size dependence of the histograms, which is
rather different depending on the value of $\gamma$. In particular, as we show
in Fig. 6 (g), where we plot the average, median and typical entanglement for
different $L$ and $\gamma$, we find that for $\gamma\lesssim 2.5$ all
indicators scale logarithmically with $L$. Instead, Fig. 6 (h) shows that
approaching the critical point ($\gamma=3.75$) and above it ($\gamma=8$) the
statistical estimators deviate from each others, a signature of the non-
gaussianity of the entanglement distribution. In the area-law phase for
$\gamma>\gamma_{c}$ the scaling of all the statistical proxies considered
becomes independent on the system size.

The emergence of bimodality in the entanglement statistics is an interesting
result, which suggests that within the log-phase the entanglement is not a
self-averaging quantity, and that limiting the discussion to the average value
could hide important features of the problem. A similar result was found in
the monitored XX model under QSD protocol alberton2020trajectory  and, in Ref
[biella2020manybody, ] for the statistics of the return probability and the
local transverse magnetization under a quantum jump stochastic protocol. In
this latter case it was shown that the non-Hermitian Hamiltonian captured a
secondary peak in the distribution of the relevant observables.

It is therefore interesting to compare the statistical properties of the
entanglement in the QSD protocol with the non-Hermitian Hamiltonian. This
comparison is proposed in Fig. 6, where we indicate with a dashed line the
long-time entanglement in the non-Hermitian case and with a blue line the
average entanglement of the QSD protocol. We see that indeed the non-Hermitian
Hamiltonian provides a good proxy for the average entanglement both for small
$\gamma$, when entanglement fluctuations are gaussian, as well as for large
$\gamma$ when the statistics is dominated by small values of the entanglement
entropy. For intermediate values of $\gamma$ instead, when the statistics
becomes strongly bimodal and fluctuations due to the noise are more relevant,
the entanglement content of the non-Hermitian Hamiltonian is systematically
larger than the average, reflecting the atypical nature of the no-click
process. This is not surprising in retrospect, as the QSD protocol contains
randomness which is expected to decrease the entanglement content of the
system as opposed to the translational-invariant non-Hermitian protocol. We
note also that in addition to the leading scaling behavior of the
entanglement, captured by the effective central charge, there is also a non-
universal correction to the entanglement that depends in principle on $\gamma$
and could play a role in the precise comparison between the two protocols.

##  IV Conclusion

In this work we have discussed measurement-induced criticality in a Quantum
Ising chain coupled to a monitoring environment. We have focused on two
different limits of the measurement problem, corresponding respectively to the
quantum state diffusion protocol and the no-click dynamics, the latter
described by a non-Hermitian Ising model.

In both cases we found a sharp phase transition in the entanglement properties
of the system, as a function of the measurement rate $\gamma$. The
entanglement dynamics out of an initial product state evolves from a slow
logarithmic growth at small measurement rates to a fast approach to a
stationary value at large $\gamma$. The stationary state entanglement shows a
_critical_ logarithmic scaling with respect to system size, with an effective
central charge which changes continuously with $\gamma$ and vanishes at a
critical strength $\gamma_{c}$, indicating a transition towards an area-law
scaling which we found numerically to coincide in the two protocols. The
existence of an extended critical phase with logarithmic scaling of the
average entanglement is particularly surprising for our Ising model which,
differently from others stochastic measurement problems considered recently in
the literature, lacks a continuous symmetry usually associated with gapless
critical behavior. The no-click limit and associated non-Hermitian evolution
provides a natural mechanism for this phenomenology, in terms of a spectral
(subradiant) phase transition from a critical phase with gapless decay modes
at small measurements to a gapped area-law phase. We also find important
differences between the two protocols, in particular in the behavior of the
effective central charge as a function of the measurement strength which turns
out to be larger in the non-Hermitian case. This effect, which seems to
suggest different universality classes depending on the specific measurement
ensemble, can be understood in terms of a disentangling effect of the noise.

###### Acknowledgements.

XT thanks S. Pappalardi, V. Vitale for discussions on related topics. We
acknowledge computational resources on the Collége de France IPH cluster. XT
acknowledges computing resources at Cineca Supercomputing center through the
Italian SuperComputing Resource Allocation via the ISCRA grant EntHybDy. The
work of MD and XT is partly supported by the ERC under grant number 758329
(AGEnTh), by the MIUR Programme FARE (MEPH), and has received funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 817482. R.F. acknowledges partial financial support from the
Google Quantum Research Award. This work has been carried out within the
activities of TQT. AB acknowledges funding by LabEx PALM
(ANR-10-LABX-0039-PALM). MS acknowledges support from the ANR grant
"NonEQuMat" (ANR-19-CE47-0001).

##  Appendix A Continuous measurements and stochastic Schrödinger equations

This section is an overview of stochastic quantum dynamics in the Hilbert
space (see Ref. [plenio1998the, ; brun2002a, ; jacobs2006a, ;
wiseman2009quantum, ] for general review). We first introduce the positive-
operator value measurements (POVM), which arise naturally when system and
environment interact for a finite time. In the limit of weak measurement, the
system evolve through a stochastic Schrödinger equation, the details of which
depend on the environment specifics. We consider two possible choices of
system-environment setups which naturally translate, respectively, in the
quantum state diffusion Eq. (2) and in the quantum jump equation Eq. (9). From
the latter, the non-Hermitian quantum Hamiltonian Eq. (10) is obtained through
post-selection. These equations correspond to different unravelling of the
same master equation, highlighting the independence of the mean state Eq. (3)
from the stochastic protocols.

###  A.1 Weak measurements and quantum trajectories

Measurement lies at the core of quantum mechanics. Given a state
$|\psi\rangle$, and an observable $O$, the von Neumann postulate states that,
when a measurement takes place, the wavefunction collapses in the eigenvector
$|n\rangle$ of $O$ corresponding to the measurement outcome $o_{n}$. This can
be seen as the transformation

$\displaystyle|\psi\rangle\mapsto|\psi^{\prime}\rangle=\frac{P_{n}|\psi\rangle}{||P_{n}|\psi\rangle||}.$
(18)

Here $P_{n}=|n\rangle\langle n|$ is the projector to the eigenspace of $o_{n}$
(the observable is resolved as $O=\sum_{n}o_{n}P_{n}$), and
$p_{n}\equiv\langle\psi|P_{n}|\psi\rangle$ is the measurement probability.

These projective measurements are an idealization: they are implicitly
instantaneous, and they collapse a pure state into a pure state. In contrast,
real world experiments are imperfect: they require a finite time of
interaction with the system, and in general the outcome state is mixed.
Positive operator-value measurements represent a generalization of projective
measurements, which are suitable to treat situations where the interaction
time between system and ancilla is comparable to the energy-scale of the
system.

Given a set of positive operators $E_{n}$ which sum to the identity
$\sum_{n}E_{n}=1$, it is possible to define the probabilities
$p_{n}=\mathrm{tr}(\rho E_{n})$. Differently from the projective case, these
operators are not orthogonal, hence the knowledge of $E_{n}$ does not fix the
state after the measurement. It is necessary to know a set of operators
$A_{n,k}$ such that $E_{n}=\sum_{k}A^{\dagger}_{nk}A_{nk}$ (dubbed Kraus
operators). Then, the state after the measurement outcome corresponding to
$E_{n}$ is

$\displaystyle\rho\mapsto\rho^{\prime}=\frac{\sum_{k}A_{nk}\rho
A_{nk}^{\dagger}}{\mathrm{tr}(\rho E_{n})}.$ (19)

It is clear that in general, given a pure density matrix $\rho$, the outcome
measurement in Eq. (19) is mixed. The condition to preserve purity is that
$E_{n}=A^{\dagger}_{n}A_{n}$, i.e., there is only one Kraus operator per
measurement. In this case Eq. (19) can be recast in a wave-function
perspective

$\displaystyle|\psi\rangle\mapsto|\psi^{\prime}\rangle=\frac{A_{n}|\psi\rangle}{\sqrt{\langle\psi|E_{n}|\psi\rangle}}.$
(20)

POVM can be implemented by allowing the system to interact with an ancilla for
a finite time, and apply afterward a projective measurement to the ancilla.
The limit when this interaction is small is the weak measurement, when
typically the state of the system is mildly perturbed.

As an example, we consider the toy model of a two-level system
$|\psi\rangle=\sum_{n=0,1}c^{S}_{n}|n\rangle_{S}$ which interacts with a two-
level ancilla initialized in a state
$|a\rangle=\sum_{k=0,1}c^{A}_{k}|k\rangle_{A}$. If their interaction is
infinitesimal, the unitary evolution of the combined system initialized in
$|\Psi\rangle=|\psi\rangle\otimes|a\rangle$ is given by

$\displaystyle U(\epsilon)=\exp\left[-i\epsilon\sum_{j}O_{j}^{S}\otimes
O_{j}^{A}\right].$ (21)

After the unitary transformation, entanglement is generated, and the state up
to first order in $\epsilon$ is given by

$\displaystyle|\Psi^{\prime}\rangle(\epsilon)$
$\displaystyle=\sum_{m,n}(c^{S}_{n}c^{A}_{m}-i\epsilon
c^{\mathrm{ent}}_{nm})|n\rangle_{S}\otimes|m\rangle_{A}$ $\displaystyle
c_{nm}^{\textup{ent}}$
$\displaystyle=\sum_{j}\sum_{n^{\prime},m^{\prime}=0,1}O^{S}_{j,n^{\prime},n}O^{A}_{j,m^{\prime},m}c^{S}_{n^{\prime}}c^{A}_{m^{\prime}}$
$\displaystyle O^{\mu}_{j,k,k^{\prime}}$ $\displaystyle\equiv\langle
k|O^{\mu}|k^{\prime}\rangle_{\mu},\quad\mu=S,A.$ (22)

Rearranging the above equation, the system is given by

$\displaystyle|\Psi^{\prime}\rangle(\epsilon)=\alpha(\epsilon)|\phi_{0}\rangle_{S}\otimes|0\rangle_{A}+\beta(\epsilon)|\phi_{1}\rangle_{S}\otimes|1\rangle_{A},$
(23)

where the constants $\alpha,\beta$ and $\phi_{k}$ are read out matching Eq.
(22) and Eq. (23). If a measurement is now performed on the ancilla along the
$z$-direction, one obtains

$\displaystyle|\psi\rangle=\begin{cases}|\phi_{0}\rangle,&\text{with
probability }\quad p_{0}=|\alpha(\epsilon)|^{2},\\\
|\phi_{1}\rangle,&\text{with probability }\quad
p_{1}=|\beta(\epsilon)|^{2}.\end{cases}$ (24)

Hence, POVM can be implemented by carefully designing the system-ancilla
interaction, the ancilla initial state, and the projective measurement on the
ancilla. To conclude, let us notice that, in the case of two-level ancilla,
the Kraus operator are one for each measurement type. Hence, the measurements
are generated by $E_{k}=A^{\dagger}_{k}A_{k}$ $k=0,1$. This is a convenient,
but not unique, setting to guarantee the purity-preservation of the state.
Furthermore, as stressed, the precise form of the operators $A_{k}$ depends on
the microphysics of system and ancilla. As we shall see, to derive a
stochastic Schrödinger equation it is usually necessary to expand up to second
order in the parameter $\epsilon$. (Nonetheless, the above argument is still
valid. ) In the following we shall assume that the Kraus operators (and hence
the POVM) are given by the problem, thus simplifying the discussion.

###  A.2 Quantum state diffusion

We consider the following Kraus operators

$\displaystyle A_{0}$ $\displaystyle=\sqrt{\frac{1}{2}}|0\rangle\langle
0|+\sqrt{\frac{1}{2}-\epsilon}|1\rangle\langle 1|$ (25) $\displaystyle A_{1}$
$\displaystyle=\sqrt{\frac{1}{2}}|0\rangle\langle
0|+\sqrt{\frac{1}{2}+\epsilon}|1\rangle\langle 1|.$ (26)

Starting from the state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, after
the measurement the state is in one of the following two states

$\displaystyle|\phi_{0}\rangle$
$\displaystyle=\frac{A_{0}|\psi\rangle}{\sqrt{p_{0}}}=\alpha\left(|\beta|^{2}\epsilon+\frac{3}{2}|\beta|^{4}\epsilon^{2}\right)|0\rangle$
$\displaystyle+\beta\left(\epsilon(1-|\beta|^{2})+\frac{\epsilon^{2}}{2}(-1-2|\beta|^{2}+3|\beta|^{4})\right)|1\rangle$
(27) $\displaystyle|\phi_{1}\rangle$
$\displaystyle=\frac{A_{1}|\psi\rangle}{\sqrt{p_{1}}}=\alpha\left(-|\beta|^{2}\epsilon+\frac{3}{2}|\beta|^{4}\epsilon^{2}\right)|0\rangle$
$\displaystyle+\beta\left(-\epsilon(1-|\beta|^{2})+\frac{\epsilon^{2}}{2}(-1-2|\beta|^{2}+3|\beta|^{4})\right)|1\rangle,$
(28)

with probability respectively $p_{0}=1/2-|\beta|^{2}\epsilon$ and
$p_{1}=1/2+|\beta|^{2}\epsilon$. The above post-measurement state can be
collected into a compact differential form, by introducing the random binomial
variable $dW=\mp\epsilon$, and the operator $n=|1\rangle\langle 1|$. We have

$\displaystyle d|\psi\rangle$
$\displaystyle=|\psi^{\prime}\rangle-|\psi\rangle=2\epsilon^{2}(\langle
n\rangle^{2}-2\langle n\rangle n)|\psi\rangle$ $\displaystyle+dW(n-\langle
n\rangle)|\psi\rangle-\frac{\epsilon^{2}}{2}(n-\langle
n\rangle)^{2}|\psi\rangle,$ (29)

where $\langle\circ\rangle=\langle\psi|\circ|\psi\rangle$ is the state
average. The expression can be simplified by centering the random variable
$dW\mapsto d\xi=dW-\overline{dW}$, since the average
$\overline{dW}=2\beta^{2}\epsilon^{2}=2\langle L\rangle\epsilon^{2}$ due to
the $\mathcal{O}(\epsilon)$ unbalance in the outcome probability. The variance
is preserved at leading order $d\xi^{2}=\epsilon=dW^{2}$, and the equation
reads

$\displaystyle d|\psi\rangle$ $\displaystyle=d\xi(n-\langle
n\rangle)|\psi\rangle-\frac{\epsilon^{2}}{2}(n-\langle
n\rangle)^{2}|\psi\rangle.$ (30)

Eq. (30) gives one interaction with a single ancilla qubit. In order to derive
a stochastic Schrödinger equation we subsequently couple a series of
identically prepared ancillas with the same paradigm. Hence, Eq. (30) is
iterated multiple time. Taking the scaling limit, with infinitesimal time step
we have

$\displaystyle\frac{d|\psi(\xi_{t})\rangle}{dt}$
$\displaystyle=\frac{d\xi_{t}}{dt}(n-\langle
n\rangle_{t})|\psi(\xi_{t})\rangle$
$\displaystyle\qquad-\frac{\epsilon^{2}}{2dt}(n-\langle
n\rangle_{t})^{2}|\psi(\xi_{t})\rangle.$ (31)

Fixing the scaling $\epsilon^{2}=\gamma dt$ reduce this equation to

$\displaystyle d|\psi(\xi_{t})\rangle$ $\displaystyle=d\xi_{t}(n-\langle
n\rangle_{t})|\psi(\xi_{t})\rangle$
$\displaystyle\qquad-\frac{\gamma}{2}dt(n-\langle
n\rangle_{t})^{2}|\psi(\xi_{t})\rangle.$ (32)

This is the quantum state diffusion equation for a two-level qubit.

Let us make some final comments. (i) The scaling $\epsilon=\sqrt{\gamma dt}$
is typical when dealing with Kraus operators (for instance, see Ref.
[plenio1998the, ; brun2002a, ; breuer2002the, ] for an indepth discussion).
This is due to a Markovian approximation, which is assumed to hold for the
systems of interest. (ii) The generalization to multiple qubits is trivial.
Recovering the noise terms in Eq. (2) require the introduction of $L$
uncorrelated Îto terms $d\xi_{i}$ with $\overline{d\xi}=0$ and
$d\xi_{i}d\xi_{j}=\gamma dt\delta_{ij}$. (iii) For the time evolution in the
limit $dt\to 0$, the binomial variable can be approximated by a Gaussian
increment, in the same spirit as obtaining a Wiener process from a random
walk.

###  A.3 Quantum jumps and non-Hermitian Hamiltonian

We consider the following Kraus operators

$\displaystyle A_{0}$ $\displaystyle=|0\rangle\langle
0|+\cos\epsilon|1\rangle\langle 1|$ (33) $\displaystyle A_{1}$
$\displaystyle=\sin{\epsilon}|1\rangle\langle 1|.$ (34)

In this situation, given the state
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, the measurements $E_{0}$ and
$E_{1}$ are unbalanced, with probability $p_{0}=1-|\beta|^{2}\epsilon^{2}$ and
$p_{1}=|\beta|^{2}\epsilon^{2}$, respectively. As in the previous section, we
introduce the operator $n=|1\rangle\langle 1|$. The state after the
measurement is in either of the following states

$\displaystyle|\phi_{0}\rangle$
$\displaystyle=|\psi\rangle+\frac{1}{2}\alpha\langle
n\rangle\epsilon^{2}|0\rangle+\frac{1}{2}\beta(\langle
n\rangle-1)\epsilon^{2}|1\rangle$
$\displaystyle=|\psi\rangle-\frac{1}{2}\left(n-\langle
n\rangle\right)\epsilon^{2}|\psi$ (35) $\displaystyle|\phi_{1}\rangle$
$\displaystyle=|1\rangle.$ (36)

In differential form, we have

$\displaystyle d|\psi\rangle=-\frac{1}{2}\left(n-\langle
n\rangle\right)\epsilon^{2}|\psi\rangle+\left(\frac{n}{\sqrt{\langle
n\rangle}}-1\right)\delta N|\psi\rangle.$ (37)

Here $\delta N$ is a Poisson process: $\delta N=0(1)$ with probability
$p_{0}(p_{1})$. Since $\overline{\delta N}=|\beta|^{2}\epsilon^{2}$, the state
is unlikely to experience a measurement and usually is affected only by the
Kraus operator $A_{0}$. However, when $\delta N=1$, the second term in Eq.
(37) dominate the infinitesimal one, and project the state onto $|1\rangle$.
Notice that in this case $\delta N^{2}=\delta N$.

Suppose that the system interacts with $n$ ancilla qubits. As previously
discussed, the stochastic Schrödinger equation is given once the scaling
$\epsilon^{2}=\gamma dt$ is set

$\displaystyle d|\psi(N_{t})\rangle$
$\displaystyle=-\frac{1}{2}\left(n-\langle n\rangle_{t}\right)\gamma
dt|\psi(N_{t})\rangle$ $\displaystyle\qquad+\left(\frac{n}{\sqrt{\langle
n\rangle_{t}}}-1\right)\delta N_{t}|\psi(N_{t})\rangle.$ (38)

This is the so-called quantum jump equation. The first term is a non-Hermitian
Hamiltonian, while the second term is the quantum jump. When the event
happens, the state is projected to $|1\rangle\langle 1|$. It is interesting to
ask how much is the probability that the state does never jump? The
probability of this no-click dynamics for one-qubit is given by

$\displaystyle P_{\textup{no-
click}}=\Pi_{n=0}^{\infty}p_{0}\simeq|\alpha|^{2}.$ (39)

Thus, the number of trajectories with no-jump effect is substantial.

We conclude with few remarks. (i) The generalization to many-body is trivial,
once $L$ uncorrelated variables are included, and the Hamiltonian $H$ of the
system is added. The final equation for the quantum jumps is Eq. (9). The non-
Hermitian Hamiltonian in Eq. (10) is recasted when post-selecting the
trajectories without jumps. Notice that the contribution $\propto\gamma
dt\sum_{i}\langle n_{i}\rangle_{t}$ factorize as an overall rescaling of the
wavefunction, and cancels out in the average. (ii) In the situation of a spin-
chain, the probability of no-click dynamics (hence non-Hermitian generated
dynamics) depends on the system size. In practice, one chooses among all the
trajectories the portion where no jump has occurred. This post-selection
process is key in investigating non-Hermitian physics in experimental setups.
(ii) Both the quantum jump and the quantum state diffusion equations induce
the same Lindblad evolution Eq. (4) for the mean state. This difference
highlights how quantum trajectories contain substantial more information than
the average state. Importantly, the non-Hermitian Hamiltonian coincide with
the non-coherent term of the Lindblad equation (4).

##  Appendix B Free fermion techniques and numerical implementation

In this section we present a summary of the simulation techniques used in this
paper. (For a general reference on the ideas used here, see Ref.
[mbeng2020the, ]). These methods are based on the fermionic representation of
the Ising model, which can be applied also to Eq. (2) (Eq. (10)), as the noise
(non-Hermitian) term preserve the quadratic structure. We first review the
Jordan-Wigner transformation, and explain how the time-evolution reads in
fermionic variables. Due to the Gaussian preservation of both quantum state
diffusion and non-Hermitian Hamiltonian, the state is fully characterized by
its two point function. Afterwards, we show how the entanglement entropy is
derived from the correlation matrix. Finally, we give some detail of the
numerical implementation.

###  B.1 Jordan-Wigner mapping

The Jordan-Wigner transformation is defined by

$\displaystyle\sigma^{x}_{i}$
$\displaystyle=K_{i}(c_{i}+c^{\dagger}_{i}),\quad K_{i}=\prod_{j<i}(2n_{i}-1)$
(40) $\displaystyle n_{i}$ $\displaystyle=c_{i}^{\dagger}c_{i}.$ (41)

Here, the string operator $K_{i}$ ensure the fermionic algebra. For the
quantum state diffusion Eq. (2), the dynamics maps to

$\displaystyle d|\psi(\boldsymbol{\xi}_{t})\rangle$
$\displaystyle=-i\sum_{i=1}^{L-1}(c^{\dagger}_{i}c_{i+1}+c^{\dagger}_{i}c^{\dagger}_{i+1}+\mathrm{h.c.})dt|\psi(\boldsymbol{\xi}_{t})\rangle$
$\displaystyle\quad+\sum_{i=1}^{L}(n_{i}-\langle
n_{i}\rangle_{t})d\xi^{i}_{t}|\psi(\boldsymbol{\xi}_{t})\rangle$
$\displaystyle\qquad-\frac{\gamma}{2}dt\sum_{i=1}^{L}(n_{i}-\langle
n_{i}\rangle_{t})^{2}|\psi(\boldsymbol{\xi}_{t})\rangle.$ (42)

Instead for the non-Hermitian Hamiltonian Eq. (10), we have

$\displaystyle d|\psi\rangle$
$\displaystyle=-i\sum_{i=1}^{L-1}(c^{\dagger}_{i}c_{i+1}+c^{\dagger}_{i}c^{\dagger}_{i+1}+\mathrm{h.c.})dt|\psi\rangle$
$\displaystyle\qquad-\frac{\gamma}{2}dt\sum_{i=1}^{L}n_{i}|\psi\rangle.$ (43)

Within this formalism, if the initial state of the system is Gaussian, then
the state is Gaussian at each time step. In particular, all the observables
are encoded in the correlation matrix

$\displaystyle G_{ij}=\begin{pmatrix}\langle
c^{\dagger}_{i}c_{j}\rangle&\langle c^{\dagger}_{i}c_{j}^{\dagger}\rangle\\\
\langle c_{i}c_{j}\rangle&\langle c_{i}c^{\dagger}_{j}\rangle.\end{pmatrix}$
(44)

In the next subsection, we show how the dynamics can be reduced to the time-
evolution of a $2L\times 2L$ matrix.

###  B.2 Dynamics in the fermionic formalism

The equation of motion for both the protocol can be written as
$d|\psi(\boldsymbol{\xi}_{t})\rangle=dZ_{t}|\psi(\boldsymbol{\xi}_{t})\rangle$,
where $dZ_{t}$ is quadratic in the creation/destruction fields. For the
quantum state diffusion

$\displaystyle dZ_{t}$
$\displaystyle=dt\left(-iH-\frac{\gamma}{2}\sum_{i}(n_{i}-\langle
n_{i}\rangle_{t})^{2}\right)$
$\displaystyle\qquad+\sum_{i}d\xi^{i}_{t}(n_{i}-\langle n_{i}\rangle_{t}),$
(45)

while for the non-Hermitian Hamiltonian we have

$\displaystyle
dZ_{t}=dt\left(-iH-\frac{\gamma}{2}\sum_{i}n_{i}\right)=-idtH_{\mathrm{eff}}.$
(46)

Both these instances can be treated on the same foot, once the operator
$dZ_{t}$ is specified. From the operator $dZ_{t}$, it is possible to write
down an equation of motion for the correlation matrix Eq. (44). (This is
obtained by taking the differential of Eq. (44), simplifying the fermionic
algebra, and expanding up to order $\mathcal{O}(dt)$).

However, in practical implementation, this method is not the most efficient. A
better strategy, which we follow in this paper, is the route pioneered in Ref.
[cao2019entanglement, ]. Since the system is Gaussian, for any $t$, there
exist a unitary matrix

$\displaystyle\mathcal{U}_{t}=\begin{pmatrix}U_{t}&V^{\dagger}_{t}\\\
V_{t}&U^{\dagger}_{t},\end{pmatrix}$ (47)

such that

$\displaystyle|\psi\rangle_{t}=\frac{1}{\sqrt{|\det
U_{t}|}}\exp\left(-\frac{1}{2}\sum_{i,j,k}(U_{t})_{i,k}(V_{t}^{\dagger})_{k,j}c^{\dagger}_{i}c^{\dagger}_{j}\right)|0\rangle.$
(48)

This condition is equivalent to finding a unitary transformation that maps the
original fermionic fields $(c_{i},c_{j}^{\dagger})$ to a new set
$(\chi_{i},\chi_{j}^{\dagger})$ such that $\chi_{i}|\psi\rangle_{t}=0$ for all
$i$. Explicitly, labelling a vector of fermionic field as
$\mathbf{c}=(c_{1},c_{2},\dots,c_{L})$,

$\displaystyle\begin{pmatrix}\mathbf{\chi}\\\
\mathbf{\chi}^{\dagger}\end{pmatrix}=\mathcal{U}_{t}\begin{pmatrix}\mathbf{c}\\\
\mathbf{c}^{\dagger}.\end{pmatrix}$ (49)

From the definition of the correlation matrix and Eq. (48), and after few
manipulation in fermionic algebra, we have the relationship

$\displaystyle G(t)=\mathcal{U}_{t}\begin{pmatrix}\mathbf{1}_{L\times
L}&\mathbf{0}_{L\times L}\\\ \mathbf{0}_{L\times L}&\mathbf{0}_{L\times
L}\end{pmatrix}\mathcal{U}^{\dagger}_{t}.$ (50)

Hence, the full evolution is encoded in $\mathcal{U}_{t}$. The equation of
motion for this object are easily derived from Eq. (49) and using the
annihilation condition $\chi_{i}|\psi\rangle_{t}=0$. In practice it can be
implemented into two steps. (i) Integrate the Heisenberg equation of motion
between $t$ and $t+dt$ (with initial condition
$\mathcal{U}_{t=0}=\mathcal{U}_{0}$ given)

$\displaystyle d\mathcal{U}_{t}=2d\mathcal{Z}_{t}\mathcal{U}_{t}.$ (51)

(ii) Apply a renormalization to guarantee that $U_{t+dt}$ is unitary (for
instance a QR decomposition). This guarantees that $|\psi\rangle_{t+dt}$ is in
the form Eq. (48) with the operators given in $U_{t+dt}$. For concrete details
on the numerical implementation, see Sec. B.4.

###  B.3 Majorana fermions and entanglement entropy

For a Gaussian state, Wick’s theorem reduce the computation of entanglement
from diagonalizing a $2^{L}\times 2^{L}$ matrix to a $2L\times 2L$ linear
problem. For this goals, it is convenient to introduce two species of Majorana
fermions

$\displaystyle a_{j,1}=c_{j}+c_{j}^{\dagger},\qquad
a_{j,2}=i(c^{\dagger}_{j}-c_{j}).$ (52)

Then the matrix

$\displaystyle V=\begin{pmatrix}\mathbf{1}_{L\times L}&\mathbf{1}_{L\times
L}\\\ -i\mathbf{1}_{L\times L}&i\mathbf{1}_{L\times L}\end{pmatrix},$ (53)

transforms the correlation function from Dirac fermions to Majorana
$W=VGV^{\dagger}$. Due to the Majorana algebra
$\\{a_{i,p},a_{j,q}\\}=\delta_{i,j}\delta_{p,q}$, the Majorana correlation can
be decomposed into $W=\mathbf{1}+i\tilde{W}$. The matrix $\tilde{W}$ is real
and anti-symmetric, and is the key actor in computing any observable. In
particular, entanglement entropy on a bipartition $A\cup B$ is obtained by
diagonalizing $\tilde{W}|_{A(B)}$, that is restricting the indices of the
Majorana fermions within either partition. If the eigenvalues of
$\tilde{W}|_{A}$ are $\lambda_{k}$, then the entanglement entropy is given by

$\displaystyle S$
$\displaystyle=-\sum_{k}\left(\frac{1+\lambda_{k}}{2}\ln\frac{1+\lambda_{k}}{2}+\frac{1-\lambda_{k}}{2}\ln\frac{1-\lambda_{k}}{2}\right).$
(54)

###  B.4 Numerical implementation

The numerical implementation can be adapted to the protocol considered. For
the quantum state diffusion, it is practical to divide the unitary step from
the noise term $d\mathcal{Z}_{t}=-i\mathcal{H}dt+dT$. The Hamiltonian $H$ can
be stored and exponentiated only once, while the noise term $dT$ is diagonal,
with elements $dT_{ij}=\delta_{ij}(d\xi^{i}+\gamma dt(2\langle
n_{i}\rangle-1))/2$ that needs to be computed repeatedly at each time step.

The resulting evolution then is

$\displaystyle\tilde{\mathcal{U}}_{t+dt}=\mathcal{N}e^{2dT}e^{2i\mathcal{H}dt}\mathcal{U}_{t},$
(55)

where we have neglected higher order terms $\mathcal{O}(d\xi dt)$ (Trotter
approximation). The normalization $\mathcal{N}$ is unimportant, as Eq. (55) is
followed by a QR decomposition
$\tilde{\mathcal{U}}_{t+dt}=\mathcal{Q}_{t+dt}\mathcal{R}_{t+dt}$, and by
setting $\mathcal{U}_{t+dt}=\mathcal{Q}_{t+dt}$. As previously discussed, this
condition guarantees the state is in the form Eq. (48) and the correlation
function is obtained through Eq. (50).

For the non-Hermitian quantum evolution, it is sufficient to exponentiate once
the operator $d\mathcal{Z}_{t}=-i\mathcal{H}_{\mathrm{eff}}dt$ and apply it
subsequently to generate the evolution

$\displaystyle\tilde{\mathcal{U}}_{t+dt}=\mathcal{N}e^{2i\mathcal{H}_{\mathrm{eff}}dt}\mathcal{U}_{t}.$
(56)

After each time step, the QR decomposition is implemented to normalize the
state. In both situations, the entanglement entropy is computed as described
in Sec. B.3.

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