Entanglement dynamics with string measurement operators

Giulia Piccitto1,2$\star$, Angelo Russomanno3,4, and Davide Rossini1

1 Dipartimento di Fisica dell’Università di Pisa and INFN,

Largo Pontecorvo 3, I-56127 Pisa, Italy

2 Dipartimento di Matematica e Informatica, Università di Catania,

Viale Andrea Doria 6, 95125, Catania, Italy

3 Scuola Superiore Meridionale, Università di Napoli Federico II,

Largo San Marcellino 10, I-80138 Napoli, Italy

4 Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II,

Monte S. Angelo, I-80126 Napoli, Italy.

⋆<EMAIL_ADDRESS>
## Abstract

We explain how to apply a Gaussian-preserving operator to a fermionic Gaussian
state. We use this method to study the evolution of the entanglement entropy
of an Ising spin chain undergoing a quantum-jump dynamics with string
measurement operators. We find that, for finite-range string operators, the
asymptotic entanglement entropy exhibits a crossover from a logarithmic to an
area-law scaling with the system size, depending on the Hamiltonian parameters
as well as on the measurement strength. For ranges of the string which scale
extensively with the system size, the asymptotic entanglement entropy shows a
volume-law scaling, independently of the measurement strength and the
Hamiltonian dynamics. The same behavior is observed for the measurement-only
dynamics, suggesting that measurements may play a leading role in this
context.

###### Contents

  1. 1 Introduction
  2. 2 Fermionic Gaussian states
  3. 3 Evolving Gaussian states
    1. 3.1 Bogoliubov matrix evolution
    2. 3.2 QR decomposition
  4. 4 Number-preserving operators
    1. 4.1 Local operator
    2. 4.2 Two-site operator
  5. 5 Quantum jumps with string operators
    1. 5.1 Entanglement entropy
    2. 5.2 Measurement-only dynamics
    3. 5.3 Interplay with the Hamiltonian dynamics
    4. 5.4 Varying $\gamma$ and $r$
  6. 6 Conclusions
  7. A Rotation to the diagonal basis
  8. B Quantum-jump protocol
  9. C Evolution of the state: a different derivation

##  1 Introduction

Understanding the role of entanglement in quantum many-body systems is a
subject under intensive study [1, 2]. While the ground-state entanglement
properties have been widely investigated and are relatively well understood,
there are still several open questions regarding the stabilization of quantum
correlations during the non-equilibrium dynamics. By exploiting a quasi-
particle description it has been shown that in integrable systems, after a
sudden quench, the entanglement entropy linearly increases in time, and
eventually settles to a stationary value which linearly scales with the system
size (volume-law scaling) [3, 4]. A similar volume-law scaling occurs in
ergodic short-range systems [5], while in the presence of a strong disorder
the situation is more involved, due to the appearance of many-body
localization (see Ref. [6] for a review).

A natural extension of the above scenario is for a quantum system coupled to
an external environment, a setup which is crucial to applications in quantum
technologies and quantum computing, since any experimental platform is subject
to noise-induced decoherence [7, 8]. In the hypothesis of weak and Markovian
coupling with the bath, a good modeling for this setup is to assume the
evolution of the reduced density matrix of the system to be ruled by a master
equation in a Lindblad form [9]. The system-bath framework can be also
interpreted as a process in which the environment behaves as a classical
stochastic process performing random measurements on the quantum system [10,
11]. The outcome of any realization of this process is a pure-state stochastic
quantum trajectory and the density matrix obtained by averaging over such
trajectories obeys a Lindblad-type dynamic evolution [12, 13].

Here we are interested in the so-called measurement-induced phase transitions
(MIPTs), i.e., in the possibility to develop discontinuities as in the
asymptotic entanglement properties that emerge from the interplay between the
unitary dynamics (typically generating entanglement) and the measurement
processes (typically destroying it). This kind of transitions, as well as
other peculiar singularities emerging from quantum measurements, have been
first proposed and studied in unitary random circuits undergoing random
measurements (see, e.g., Refs.[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29]), which may display a transition between a phase in which the
entanglement grows with the system size and another phase where it follows an
area-law scaling (i.e., it saturates to a finite and constant value, for one
dimensional systems). Similarly, MIPTs may also arise in free-fermion random
circuits with temporal randomness [30], Majorana random circuits [31, 32],
Dirac fermions [33], non-Hermitian topological systems [34, 35, 36, 37] and
other Hamiltonian models undergoing random measurements at discrete times [38,
39, 40, 41]. When measuring string operators, MIPTs can occur in the presence
of a measurement-only dynamics, without the interplay with the Hamiltonian
[42, 43].

Recently, an increasing number of works have been focusing on MIPTs in
fermionic systems described by quadratic Hamiltonians on a lattice, undergoing
quantum trajectories under the action of random measurements of onsite
quadratic operators [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61]. One of the reasons of this interest relies in the Gaussian
structure of the states of such systems that, being entirely determined by
two-point correlation functions, are suitable for a semi-analytical treatment
up to large lattice sizes (of the order of hundreds of sites) [62, 63, 64, 65,
66, 67, 68]. In particular, it has been found that the asymptotic volume-law
scaling of the entanglement following a unitary evolution [69, 70] is unstable
under local measurements, independently of the measurement strength [44, 46],
and a MIPT may emerge. For example, the Ising chain exhibits a size-dependent
crossover towards a subextensive or an area-law entanglement regime, depending
on the system parameters, as well as on the measurement rate and the type of
unraveling [47, 48, 49, 50, 51, 71]. In a version of this model with temporal
noise also in the Hamiltonian, this crossover has been analytically proved to
be a true transition by means of the replica method [72, 73]. Similarly, it
has been found that entanglement transitions can arise, even in the absence of
a coherent evolution, from the competition between different measurement
protocols [74]. In the same context of monitored fermionic models, a couple of
recent works pointed out the importance of having long-range interactions to
stabilize larger amounts of entanglement, if the system undergoes random
onsite measurements [53, 54] (see [75, 76] for a similar phenomenon in quantum
circuits). Similar results have been found also for short-ranged Hamiltonian
in the presence of long-range continuous measurement processes [77].

Inspired by these latter findings, here we follow a complementary strategy and
address the role of non-local string measurement operators in the dynamics of
entanglement for a quadratic Hamiltonian system. The purpose of this paper is
twofold. On the one hand, we collect and review all the technical details to
implement the dynamics of Gaussian states in the presence of Gaussian-
preserving measuring operators. On the other hand, we discuss the effect of
non-local measurement operators on the dynamics of the entanglement entropy,
while keeping the Hamiltonian with short-range interactions. To this aim, we
implement a quantum-jump unraveling of the Lindblad dynamics [10, 9] for the
Ising chain (mapped to the fermionic Kitaev chain) using strings of Pauli
matrices as Lindblad operators (mapped to non-local two-point fermionic
operators). The main result is that, when the string range $r$ is comparable
with the system size $L$, contrary to what happens in the presence of local
jump operators, the measurement process can stabilize an asymptotic volume-law
for the entanglement entropy. As a consequence, we cannot exclude that a MIPT
can occur as a function of $r$. Quite interestingly, the same qualitative
conclusion applies if we switch off the Hamiltonian and consider a
measurement-only Lindblad dynamics induced by string operators, similarly to
what has been studied in Ref. [42] for measurement-only models with nonlocal
measurements of Pauli strings.

The paper is organized as follows. In Section 2 we briefly review the theory
of quadratic fermionic Hamiltonians on a lattice. In Section 3 and Appendix A
we present the technical details required to implement the dynamics of a
Gaussian state under the effect of Gaussian-preserving exponentials of
quadratic operators. We first set the general framework, and then shed light
on the dynamical equations and on the normalization of the state. In Section 4
we apply such a machinery to string operators, a particular class of operators
which generalize the one discussed in Ref. [49, 50]. Subsequently, in Section
5 we present and discuss numerical results for the Ising chain with non-local
string measurement operators. Finally, our conclusions are drawn in Section 6.
The other appendices contain some details on the quantum-jump measurement
protocol (Appendix B) and an alternative approach to determine the evolved
state with the string operator (Appendix C).

##  2 Fermionic Gaussian states

We consider a generic free-fermion system on a lattice with $L$ sites,
described by the quadratic Hamiltonian

$\hat{H}\
=\sum_{i,j=1}^{L}\left(Q_{i,j}\,\hat{c}_{i}^{\dagger}\hat{c}_{j}+P_{i,j}\,\hat{c}_{i}^{\dagger}\hat{c}_{j}^{\dagger}+\text{h.c.}\right),$
(1)

where $\hat{c}^{(\dagger)}_{j}$ denotes the anticommuting annihilation
(creation) operator on the $j$-th site. To ensure the Hermiticity of
$\hat{H}$, the coefficient matrices obey $Q^{\dagger}=Q$ and $P^{T}=-P$.

In this context, a Gaussian state has the generic form 111Hereafter we omit
extrema in all the summations running from $1$ to $L$.

$\ket{\psi}=\mathcal{N}\exp\bigg{\\{}\frac{1}{2}\sum_{p,q}Z_{p,q}\,\hat{c}_{p}^{\dagger}\,\hat{c}_{q}^{\dagger}\bigg{\\}}\ket{0}_{\hat{c}},$
(2)

with

$Z=-\big{(}u^{\dagger}\big{)}^{-1}\,v^{\dagger}\,,$ (3)

and $\mathcal{N}$ a normalization constant. Here $u$ and $v$ are $L\times L$
matrices that obey the following symplectic unitarity conditions

$u^{\dagger}\,u+v^{\dagger}\,v=I,\qquad u\,v^{\dagger}+v^{*}\,u^{T}=0\,,$ (4)

where $I$ denotes the $L\times L$ identity matrix. These conditions are
equivalent to say that the matrix

$\mathbb{U}=\begin{pmatrix}u&v^{*}\\\ v&u^{*}\end{pmatrix}$ (5)

is unitary and preserves the fermionic anticommutation relations: In fact it
defines a new set of fermionic anticommuting operators

$\hat{\gamma}_{\kappa}=\sum_{j}\left(u^{*}_{j,\kappa}\,\hat{c}_{j}+v^{*}_{j,\kappa}\,\hat{c}_{j}^{\dagger}\right)\,,$
(6)

that enjoy the property

$\hat{\gamma}_{\kappa}\ket{\psi}=0,\qquad\forall\;\kappa\in\\{1,\,\ldots,\,L\\}\,,$
(7)

meaning that $\ket{\psi}$ is the vacuum of the $\hat{\gamma}$ fermions,
$\ket{\psi}\equiv\ket{0}_{\hat{\gamma}}$ (the matrix $\mathbb{U}$ is the so-
called Bogoliubov transformation. For more details see, e.g., Refs. [66, 78]).

We remark that, because of the fermionic nature of the $\hat{c}$-operators,
the matrix $Z$ in Eq (3) is antisymmetric, as can be proved by exploiting Eqs.
(4) [78]. In this context, the Nambu spinor notation is useful. Defining
$\hat{\Psi}\equiv(\hat{c}_{1},\cdots,\hat{c}_{L},\hat{c}^{\dagger}_{1},\cdots,\hat{c}_{L}^{\dagger})^{T}$,
and
$\hat{\Phi}\equiv(\hat{\gamma}_{1},\cdots,\hat{\gamma}_{L},\hat{\gamma}^{\dagger}_{1},\cdots,\hat{\gamma}_{L}^{\dagger})^{T}$,
one can rewrite Eq. (6) as

$\hat{\Phi}=\mathbb{U}^{\dagger}\hat{\Psi}\,.$ (8)

The state in Eq. (2) is Gaussian, enjoys the Wick’s theorem, and thus it is
completely determined by two-point correlation functions. Using Eq. (7) one
can show that these correlation functions have a simple expression in terms of
the $\mathbb{U}$ matrix:

$G_{i,j}\equiv\braket{\hat{c}_{i}^{\dagger}\,\hat{c}_{j}}=\big{[}v\,v^{\dagger}\big{]}_{i,j},\qquad
F_{i,j}\equiv\braket{\hat{c}_{i}\,\hat{c}_{j}}=\big{[}v\,u^{\dagger}\big{]}_{i,j}\,.$
(9)

An important case where the state has the Gaussian form in Eq. (2) is the
ground state of the Hamiltonian (1). The Hamiltonian reads

$\hat{H}=\frac{1}{2}\,\hat{\Psi}^{\dagger}\,\mathbb{H}\,\hat{\Psi}+\text{const.}\,,\
\quad\text{with}\quad\mathbb{H}=\begin{pmatrix}Q&P\\\
-\,P^{*}&-\,Q^{*}\end{pmatrix}.$ (10)

The $2L\times 2L$ Hermitian matrix $\mathbb{H}$ is called the Bogoliubov-de
Gennes Hamiltonian matrix, which can be diagonalized through the
implementation of a Bogoliubov transformation $\mathbb{U}$ of the form (5),
such that
$\mathbb{U}^{-1}\,\mathbb{H}\,\mathbb{U}=\text{diag}\,[\omega_{\kappa},-\omega_{\kappa}]_{\kappa=1,\cdots,L}$.
The above transformation defines a new set of fermionic quasi-particles
$\hat{\gamma}_{\kappa}$ [Eq. (6)], according to which the Hamiltonian of Eq.
(1) can be written as

$\hat{H}=\sum_{\kappa}\omega_{\kappa}\hat{\gamma}_{\kappa}^{\dagger}\hat{\gamma}_{\kappa}+\text{const.}\,.$
(11)

Its ground state is a Gaussian state of the form in Eq. (2), that corresponds
to the vacuum of the $\hat{\gamma}_{\kappa}$ operators [see Eq. (7)].

We note that the above formalism can be easily generalized to the case of
time-varying coefficients $Q_{i,j}(t)$ and $P_{i,j}(t)$ in Eq. (1), by
admitting a time dependence of the Bogoliubov transformation matrix
$\mathbb{U}=\mathbb{U}(t)$. Moreover, as we shall see below in Section 3, the
form of a generic Gaussian state Eq. (2) is preserved by the application of
the exponential of any operator quadratic in the $\hat{c}$ fermions.

##  3 Evolving Gaussian states

We now discuss the evolution of a generic Gaussian state $\ket{\psi}$ under
the effect of an operator of the form

$\hat{M}(s)=\operatorname{e}^{\xi s\hat{A}},$ (12a) being $s\in\mathbb{R}$,
$\xi=\\{1,i\\}$, and
$\hat{A}=\sum_{i,j}\left(D_{i,j}\,\hat{c}^{\dagger}_{i}\hat{c}_{j}+O_{i,j}\,\hat{c}^{\dagger}_{i}\hat{c}_{j}^{\dagger}+\text{h.c.}\right),$
(12b)

a generic Hermitian quadratic operator. For simplicity of notations, we assume
all coefficients in $\hat{A}$ to be real, but our analysis can be easily
generalized to the complex case.

Due to the fermionic anticommutation rules, we have $D^{T}=D$, $O^{T}=-O$.
Notice that, if $\xi=i$, the operator in Eqs. (12) describes the real-time
evolution according to a (pseudo)-Hamiltonian $\hat{A}$. On the other hand, if
$\xi=1$, it describes an imaginary-time evolution (related to the construction
of the thermal ensemble at inverse temperature $\beta=1/s$), that can be
thought of as the analytic continuation of the purely imaginary evolution. As
discussed in Section 4, in some particular cases, the action played by such
kind of operator on a given state can be intended as part of a measurement
process.

In what follows we discuss how to evaluate, for any (real) symmetric $\hat{A}$
operator, the s-evolved state

$\ket{\psi(s)}=\hat{M}(s)\ket{\psi}\,,$

taking as initial state $\ket{\psi}$ a generic Gaussian state of the form (2).
In Appendix A we show that, because of the quadratic structure of $\hat{A}$,
the state $\ket{\psi(s)}$ keeps the Gaussian form of Eq. (2):

$\ket{\psi(s)}\propto\exp\bigg{\\{}\frac{1}{2}\sum_{p,q}Z_{p,q}(s)\,\hat{c}_{p}^{\dagger}\,\hat{c}_{q}^{\dagger}\bigg{\\}}\ket{0}_{\hat{c}},$
(13)

with

$Z(s)=-\big{[}u^{\dagger}(s)\big{]}^{-1}\,v^{\dagger}(s)$ (14)

and $u(s),v(s)$ submatrices of the s-evolved Bogoliubov matrix
$\mathbb{U}(s)$. In the next Subsection we are going to show how to evaluate
the matrix $\mathbb{U}(s)$ (an alternative derivation is provided in Appendix
A).

###  3.1 Bogoliubov matrix evolution

To characterize the evolved state $\ket{\psi(s)}$, one only needs to know how
the Bogoliubov matrix $\mathbb{U}(s)$ evolves with $\hat{M}(s)$. Exploiting
the identity $\big{[}\hat{M}(s)\big{]}^{-1}=\hat{M}(-s)$, we can write

$\ket{\psi(s)}=\hat{M}(s)\ket{\psi}\propto\operatorname{e}^{\hat{\mathcal{Z}}(s)}\hat{M}(s)\ket{0}_{\hat{c}},$
(15)

where $\ket{\psi}$ is a generic Gaussian state given by Eq. (2). The operator

$\hat{\mathcal{Z}}(s)=\frac{1}{2}\sum_{p,q}Z_{p,q}\,\hat{c}_{p}^{\dagger}(s)\,\hat{c}_{q}^{\dagger}(s)$
(16)

is defined through the evolved fermions

$\hat{c}_{q}^{(\dagger)}(s)=\hat{M}(s)\,\hat{c}_{q}^{(\dagger)}\,\hat{M}(-s),$
(17)

obeying anticommutation relations
$\big{\\{}\hat{c}_{q}(s),\hat{c}_{p}^{\dagger}(s)\big{\\}}=\delta_{p,q}$. Note
that, for $\xi=1$, the transformation $\hat{M}(s)$ is not unitary and the
conjugation operation does not commute with the evolution one, hence
$\hat{c}_{q}^{\dagger}(s)\neq\big{[}\hat{M}(s)\,\hat{c}_{q}\,\hat{M}(-s)\big{]}^{\dagger}$.
Since $\hat{M}(s)\ket{0}_{\hat{c}}\propto\ket{0}_{\hat{c}}$, to determine the
evolved state, one has to calculate $\hat{\mathcal{Z}}(s)$. This can be done
by evaluating

$\partial_{s}\hat{c}_{q}^{(\dagger)}(s)=\xi\,\hat{M}(s)\,\big{[}\hat{A},\hat{c}_{q}^{(\dagger)}\big{]}\,\hat{M}(-s).$
(18)

Given the quadratic structure of the operator $\hat{A}$, the above commutators
read

$\displaystyle\big{[}\hat{A},\hat{c}^{\dagger}_{q}\big{]}$ $\displaystyle=$
$\displaystyle\phantom{-}2\sum_{i}\big{(}D_{i,q}\,\hat{c}_{i}^{\dagger}+O_{q,i}\,\hat{c}_{i}\big{)},$
(19a) $\displaystyle\big{[}\hat{A},\hat{c}_{q}\big{]}$ $\displaystyle=$
$\displaystyle-2\sum_{i}\big{(}D_{i,q}\,\hat{c}_{i}+O_{q,i}\,\hat{c}_{i}^{\dagger}\big{)}.$
(19b)

Therefore

$\displaystyle\partial_{s}\hat{c}_{q}^{\dagger}(s)$ $\displaystyle=$
$\displaystyle\phantom{-}2\xi\,\sum_{i}\,\big{(}O_{q,i}\,\hat{c}_{i}(s)+D_{i,q}\,\hat{c}_{i}^{\dagger}(s)\big{)},$
(20a) $\displaystyle\partial_{s}\hat{c}_{q}(s)$ $\displaystyle=$
$\displaystyle-2\xi\,\sum_{i}\,\big{(}D_{i,q}\,\hat{c}_{i}(s)+O_{q,i}\,\hat{c}_{i}^{\dagger}(s)\big{)}.$
(20b)

At this point, we make the Ansatz that the fermionic operators depend on $s$
only through a Bogoliubov-like matrix

$\hat{\Psi}(s)=\mathbb{U}^{\text{ev}}(s)\,\hat{\Phi},\
\quad\text{with}\quad\mathbb{U}^{\text{ev}}(s)=\begin{pmatrix}u^{\text{ev}}(s)&\bar{v}^{\text{ev}}(s)\\\
v^{\text{ev}}(s)&\bar{u}^{\text{ev}}(s)\end{pmatrix}\,.$ (21)

The expression for the submatrices of the $\mathbb{U}^{\text{ev}}(s)$ is given
in Eq. (29). Note that, while for $\xi=i$ one has
$\bar{u}^{\text{ev}}(s)=[u^{\text{ev}}(s)]^{*}$ and
$\bar{v}^{\text{ev}}(s)=[v^{\text{ev}}(s)]^{*}$ (so that
$\mathbb{U}^{\text{ev}}(s)$ is a true Bogoliubov transformation), in general
this is not true.

The above Ansatz is equivalent to assume that the $s$-evolved state
$\ket{\psi(s)}$ is a Bogoliubov vacuum $\hat{\gamma}_{\mu}(s)\ket{\psi(s)}=0,\
\forall\mu$. By substituting this Ansatz in Eq. (20), we obtain a set of
differential equations for the elements of the evolved matrix:

$\displaystyle\partial_{s}u^{\text{ev}}_{q,\mu}(s)$ $\displaystyle=$
$\displaystyle-2\xi\,\sum_{i}\big{(}D_{q,i}\,u^{\text{ev}}_{i,\mu}(s)+O_{q,i}\,v^{\text{ev}}_{i,\mu}(s)\big{)},$
(22a) $\displaystyle\partial_{s}\bar{v}^{\text{ev}}_{q,\mu}(s)$
$\displaystyle=$
$\displaystyle-2\xi\,\sum_{i}\big{(}D_{q,i}\,\bar{v}^{\text{ev}}_{i,\mu}(s)+O_{q,i}\,\bar{u}^{\text{ev}}_{i,\mu}(s)\big{)},$
(22b) $\displaystyle\partial_{s}v^{\text{ev}}_{q,\mu}(s)$ $\displaystyle=$
$\displaystyle\phantom{-}2\xi\,\sum_{i}\big{(}D_{q,i}\,v^{\text{ev}}_{i,\mu}(s)+O_{q,i}\,u^{\text{ev}}_{i,\mu}(s)\big{)},$
(22c) $\displaystyle\partial_{s}\bar{u}^{\text{ev}}_{q,\mu}(s)$
$\displaystyle=$
$\displaystyle\phantom{-}2\xi\,\sum_{i}\big{(}D_{q,i}\,\bar{u}^{\text{ev}}_{i,\mu}(s)+O_{q,i}\,\bar{v}^{\text{ev}}_{i,\mu}(s)\big{)},$
(22d)

with the initial condition:
$u^{\text{ev}}_{q,\mu}(0)=[\bar{u}^{\text{ev}}_{q,\mu}(0)]^{*}=u_{q,\mu}$, and
$v^{\text{ev}}_{q,\mu}(0)=[\bar{v}^{\text{ev}}_{q,\mu}(0)]^{*}=v_{q,\mu}$. In
a compact form, these equations read

$\partial_{s}\mathbb{U}^{\text{ev}}(s)=-2\,\xi\,\mathbb{A}\,\mathbb{U}^{\text{ev}}(s),\quad\text{with
}\ \mathbb{A}=\begin{pmatrix}\phantom{-}D&\phantom{-}O\\\ -O&-D\end{pmatrix},$
(23)

being $D^{T}=D$, $O^{T}=-O$. This leads to the formal solution

$\mathbb{U}^{\text{ev}}(s)=\operatorname{e}^{-2\xi\mathbb{A}s}\mathbb{U}.~{}$
(24)

The matrix $\mathbb{U}^{\text{ev}}(s)$ is not a priori characterizing a
fermionic state in that it should be unitary, to guarantee the normalization
of the state, and should obey Eqs. (4), to impose the anticommutation
relations for the $\hat{\Psi}(s)$. While the former requirement is
automatically fulfilled for $\xi=i$ (because of the unitarity of the evolution
operator), in the case $\xi=1$ it must be forced by performing a QR-
decomposition, as discussed in details in the next subsection. Once the
unitarity of the matrix is imposed, Eqs. (4) are satisfied if the Bogoliubov
matrix preserves the symplectic structure of Eq. (6), and this happens for
$\hat{A}$ being Hermitian.

By substituting the normalized s-evolved Bogoliubov matrix in $Z(s)$ [cf. Eq.
(14)], we obtain the expression for $\ket{\psi(s)}$. If the above conditions
are satisfied, this is a Gaussian fermionic state, as the one in Eq. (13). A
more detailed discussion about this point is provided in Appendix A.

###  3.2 QR decomposition

We now discuss how to restore the unitarity of the s-evolved Bogoliubov matrix
$\mathbb{U}^{\text{ev}}(s)$, when considering a norm-non preserving evolution
with $\xi=1$, that is in general obtained by QR-decomposing the s-evolved
Bogoliubov matrix $\mathbb{U}^{\text{ev}}(s)$. This procedure keeps the $Z(s)$
matrix in Eq. (14) – and then the Gaussian state – invariant, as we are going
to see below. Nevertheless, it is necessary to apply the known formulae [78]
for evaluating the entanglement entropy and the local observables.

The QR-decomposition is a procedure that allows one to decompose a generic
matrix as $K=U_{Q}R$, with $U_{Q}$ a unitary matrix and $R$ an upper
triangular one. In what follows, we provide an argument to show that, if
$\hat{A}=\hat{A}^{\dagger}$, the unitary matrix obtained by QR-decomposing the
s-evolved Bogoliubov matrix $\mathbb{U}^{\text{ev}}(s)$ describes a fermionic
state, i.e., it has the symplectic form of Eq. (6). The straightforward way
would be to check it by construction, a procedure that requires some involved
calculations. Here we choose a different strategy. Since $R$ is positive
definite (we have checked it numerically), the QR decomposition must be
unique. Therefore, we assume $U_{Q}$ to have the right structure and then we
show that it exists a decomposition that is compatible with this assumption.

Let us consider a symmetric operator $\hat{A}$ defined in Eq. (12b) and the
associated coefficient matrix $\mathbb{A}$ defined in Eq. (23). For simplicity
let us assume $\hat{A}$ to have nearest-neighbors interactions, i.e.
$D_{i,j}=O_{i,j}=0$ if $j\neq i+1$, and let as assume $D_{i,i+1}=O_{i,i+1}$
(e.g. the Kitaev Hamiltonian). The most general case does not have an
analytical solution, but the main argument of this section should hold
independently. Because of the symmetries of $D$ and $O$, we have that
$\\{D,O\\}=0$. 333This simply follows from the fact that
$(\\{D,O\\})^{\dagger}=-\\{D,O\\}$. Being $D$ and $O$ real matrices, it
implies that the anticommutator vanishes. We notice that

$\mathbb{A}^{2n}=\begin{pmatrix}X^{2n}&0\\\
0&X^{2n}\end{pmatrix},\qquad\mbox{with }\,\,X^{2}=D^{2}-O^{2}.$ (25)

By exploiting the block diagonal form of Eq. (25) and the identities

$\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}=\cosh(x),\qquad\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!}=\frac{\sinh(x)}{x},$
(26)

we can write

$\operatorname{e}^{-2\xi\mathbb{A}s}=\sum_{n=0}^{\infty}\left[\frac{(-2\xi\mathbb{A}s)^{2n}}{(2n)!}+\frac{-2\xi\mathbb{A}s\
(-2\xi\mathbb{A}s)^{2n}}{(2n+1)!}\right]=\begin{pmatrix}\text{ch}^{+}(s)&\text{sh}(s)\\\
-\text{sh}(s)&\text{ch}^{-}(s)\end{pmatrix},$ (27)

where, for shortness of notation, we have posed

$\text{ch}^{\pm}(s)=I\,\cosh(-2\xi Xs)\pm D\,\sinh(-2\xi
Xs)\,X^{-1},\quad\quad\text{sh}(s)=O\,\sinh(-2\xi Xs)\,X^{-1}.$ (28)

Notice that $[\text{ch}^{\pm}(s)]^{\dagger}=\text{ch}^{\pm}(s)$, while
$\text{sh}^{\dagger}(s)=-\text{sh}(s)$. Moreover, since $\\{D,O\\}=0$, we have
$[D,\text{ch}^{\pm}(s)]=[O,\text{ch}^{\pm}(s)]=0$ and
$[D,\text{sh}(s)]=[O,\text{sh}(s)]=0$. This means that

$\begin{pmatrix}u^{\text{ev}}(s)&\bar{v}^{\text{ev}}(s)\\\
v^{\text{ev}}(s)&\bar{u}^{\text{ev}}(s)\end{pmatrix}=\begin{pmatrix}\text{ch}^{+}(s)\,u+\text{sh}(s)\,v&\text{ch}^{+}(s)v^{*}+\text{sh}(s)\,u^{*}\\\
-\text{sh}(s)\,u+\text{ch}^{-}(s)\,v&-\text{sh}(s)\,v^{*}+\text{ch}^{-}(s)\,u^{*}\end{pmatrix}.$
(29)

We can QR-decompose $\mathbb{U}^{\text{ev}}(s)$

$\mathbb{U}^{\text{ev}}(s)=\mathbb{U}(s)\,\text{R}(s),$ (30)

making the Ansatz that the $\mathbb{U}(s)$ preserves the symplectic form of
Eq. (6), so that (30) can be rewritten as

$\begin{pmatrix}u^{\text{ev}}(s)&\bar{v}^{\text{ev}}(s)\\\
v^{\text{ev}}(s)&\bar{u}^{\text{ev}}(s)\end{pmatrix}=\begin{pmatrix}u(s)&v^{*}(s)\\\
v(s)&u^{*}(s)\end{pmatrix}\begin{pmatrix}r_{1}(s)&r_{3}(s)\\\
0&r_{2}(s)\end{pmatrix}.$ (31)

In what follows, we show that there exists a decomposition compatible with
this Ansatz, by imposing Eq. (4) and that, because of the unicity, it must be
the only possible. Let us assume to have derived $r_{1}(s)$ by construction.
We can explicitly write $\mathbb{U}(s)=\mathbb{U}^{\text{ev}}(s)R(s)^{-1}$,
exploiting the fact that the inverse of an upper block triangular matrix
remains an upper block triangular matrix:

$\begin{pmatrix}u(s)&v^{*}(s)\\\
v(s)&u^{*}(s)\end{pmatrix}=\begin{pmatrix}u^{\text{ev}}(s)\,r_{1}^{-1}(s)&u^{\text{ev}}(s)\,r_{3}^{-1}(s)+\bar{v}^{\text{ev}}(s)\,r_{2}^{-1}(s)\\\
v^{\text{ev}}(s)\,r_{1}^{-1}(s)&v^{\text{ev}}(s)\,r_{3}^{-1}(s)+\bar{v}^{\text{ev}}(s)\,r_{2}^{-1}(s)\end{pmatrix}.$
(32)

We can substitute these expressions in Eq. (4) to obtain

$u^{\dagger}u+v^{\dagger}v=\big{[}r_{2}^{T}(s)\big{]}^{-1}\,\Big{(}[\bar{u}^{\text{ev}}]^{T}(s)\,u^{\text{ev}}(s)+[\bar{v}^{\text{ev}}]^{T}(s)\,v^{\text{ev}}(s)\Big{)}\,\big{[}r_{1}(s)\big{]}^{-1}=I.$
(33)

Since
$[\bar{u}^{\text{ev}}]^{T}(s)\,u^{\text{ev}}(s)+[\bar{v}^{\text{ev}}]^{T}(s)\,v^{\text{ev}}(s)=I$,
the above equation provides a relation between two of the blocks of the matrix
$R(s)$:

$r_{2}(s)=\big{[}r_{1}^{T}(s)\big{]}^{-1}.$ (34)

By imposing the second constraint in Eq. (4), we find the relation for
$r_{3}(s)$

$r_{3}(s)=-r_{1}^{T}\Big{(}[u^{\text{ev}}]^{\dagger}(s)\,\bar{v}^{\text{ev}}(s)+[v^{\text{ev}}]^{\dagger}(s)\,\bar{u}^{\text{ev}}(s)\Big{)}^{-1}\Big{(}[u^{\text{ev}}]^{\dagger}(s)\,u^{\text{ev}}(s)+[v^{\text{ev}}]^{\dagger}(s)\,v^{\text{ev}}(s)\Big{)}.$
(35)

As a last comment we notice that the matrix $Z(s)$ in Eq. (14) – and then the
Gaussian state – are left invariant by the application of the QR
decomposition:

$Z(s)=\big{[}u^{\dagger}(s)\big{]}^{-1}\,v(s)^{\dagger}=\big{\\{}\big{[}u^{\text{ev}}(s)\,r_{1}^{-1}(s)\big{]}^{\dagger}\big{\\}}^{-1}\
\big{[}v^{\text{ev}}(s)\,r_{1}^{-1}(s)\big{]}^{\dagger}=\big{\\{}[u^{\text{ev}}]^{\dagger}(s)\big{\\}}^{-1}\,[v^{\text{ev}}]^{\dagger}(s).$
(36)

##  4 Number-preserving operators

We now consider a generic number-preserving operator

$\hat{A}_{\mathcal{I},\mathcal{J}}=\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\hat{c}_{i}^{\dagger}\hat{c}_{j},$
(37)

where $\mathcal{I},\mathcal{J}$ are two sets of indices labeling certain sites
on the system, and apply

$\hat{M}_{\mathcal{I},\mathcal{J}}=\operatorname{e}^{\beta\hat{A}_{\mathcal{I},\mathcal{J}}}$
(38)

to a Gaussian state. We focus on this particular example of the theory
presented in Section 3.1, since it generalizes the construction discussed in
Ref. [49, 50]. By exploiting the anticommutation relations, it is possible to
show that
$\hat{A}_{\mathcal{I},\mathcal{J}}^{n}=\alpha^{n-1}\hat{A}_{\mathcal{I},\mathcal{J}}$,
with $\alpha=|\mathcal{I}\cap\mathcal{J}|$. In fact

$\displaystyle\hat{A}_{\mathcal{I},\mathcal{J}}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i,l\in\mathcal{I}}\,\sum_{j,m\in\mathcal{J}}\hat{c}^{\dagger}_{i}\hat{c}_{j}\hat{c}^{\dagger}_{l}\hat{c}_{m}=\sum_{i,l\in\mathcal{I}}\,\sum_{j,m\in\mathcal{J}}\big{(}\hat{c}^{\dagger}_{i}\hat{c}_{j}\delta_{l,m}+\hat{c}_{i}^{\dagger}\hat{c}_{j}\hat{c}_{m}\hat{c}^{\dagger}_{l}\big{)}$
(39) $\displaystyle=$
$\displaystyle\sum_{i,l\in\mathcal{I}}\,\sum_{j,m\in\mathcal{J}}\Big{[}\hat{c}^{\dagger}_{i}\hat{c}_{j}\delta_{l,m}+\tfrac{1}{2}\hat{c}_{i}^{\dagger}(\hat{c}_{j}\hat{c}_{m}+\hat{c}_{m}\hat{c}_{j})\hat{c}^{\dagger}_{l}\Big{]}=|\mathcal{I}\cap\mathcal{J}|\,\sum_{i\in\mathcal{I}}\,\sum_{j\in\mathcal{J}}\hat{c}^{\dagger}_{i}\hat{c}_{j},\quad$

where the last equality holds, since $j$ and $m$ run in the same set.
Therefore, if $\mathcal{I}\cap\mathcal{J}=\varnothing$, we have
$\hat{A}_{\mathcal{I},\mathcal{J}}^{2}=0$ and
$\hat{M}_{\mathcal{I},\mathcal{J}}=\hat{1}+\beta\hat{A}_{\mathcal{I},\mathcal{J}}$,
where $\hat{1}$ denotes the identity operator. Otherwise

$\hat{M}_{\mathcal{I},\mathcal{J}}=\sum_{n=0}^{\infty}\frac{(\beta\hat{A}_{\mathcal{I},\mathcal{J}})^{n}}{n!}=\hat{1}+\frac{(\operatorname{e}^{\alpha\beta}-1)\hat{A}_{\mathcal{I},\mathcal{J}}}{\alpha}.$
(40)

In what follows, without loss of generality, we set
$\beta\equiv\log(\alpha+1)/\alpha$, so that
$\hat{M}_{\mathcal{I},\mathcal{J}}=\hat{1}+\hat{A}_{\mathcal{I},\mathcal{J}}$
reduces to that studied in Ref. [49, 50] for
$\mathcal{I}=\mathcal{J}=\\{i\\}$. As discussed in Appendix B, the application
of such a $\hat{M}_{\mathcal{I},\mathcal{J}}$ can be thought as the
measurement step of a quantum-jump dynamics.

We want to evaluate the effect of this operator on a Gaussian state

$\ket{\psi}_{\hat{M}_{\mathcal{I},\mathcal{J}}}\equiv\hat{M}_{\mathcal{I},\mathcal{J}}\ket{\psi}=(\hat{1}+\hat{A}_{\mathcal{I},\mathcal{J}})\ket{\psi}.$
(41)

Following the procedure discussed in Section 3.1, we write the evolved state
as

$\ket{\psi}_{\hat{M}_{\mathcal{I},\mathcal{J}}}\propto\exp\bigg{\\{}\frac{1}{2}\sum_{p,q}Z_{p,q}\,\hat{k}^{\dagger}_{p}\,\hat{k}^{\dagger}_{q}\bigg{\\}}\ket{0},$
(42)

where we introduced the transformed fermionic operators
$\hat{k}_{q}^{(\dagger)}=\hat{M}_{\mathcal{I},\mathcal{J}}\,\hat{c}_{q}^{(\dagger)}\,\hat{M}_{\mathcal{I},\mathcal{J}}^{-1}$.
By exploiting the fact that
$\hat{M}_{\mathcal{I},\mathcal{J}}^{-1}=\hat{1}-(\alpha+1)^{-1}\hat{A}_{\mathcal{I},\mathcal{J}}$,
we derive the exact expression of $\hat{k}^{(\dagger)}$

$\displaystyle\hat{k}_{q}^{\dagger}$ $\displaystyle=$
$\displaystyle\hat{c}^{\dagger}_{q}+\sum_{j\in\mathcal{J}}\delta_{q,j}\sum_{i\in\mathcal{I}}\hat{c}^{\dagger}_{i},$
(43a) $\displaystyle\hat{k}_{q}$ $\displaystyle=$
$\displaystyle\hat{c}_{q}-(\alpha+1)^{-1}\sum_{j\in\mathcal{J}}\delta_{q,j}\sum_{i\in\mathcal{I}}\hat{c}_{i}.$
(43b)

This leads to the following expressions for the evolved Bogoliubov matrix:

$\displaystyle
u_{q,\mu}^{\text{ev}}=u_{q,\mu}-(\alpha+1)^{-1}\sum_{j\in\mathcal{J}}\delta_{q,j}\sum_{i\in\mathcal{I}}u_{i,\mu},$
$\displaystyle\bar{u}_{q,\mu}^{\text{ev}}=u^{*}_{q,\mu}+\sum_{j\in\mathcal{J}}\delta_{q,j}\sum_{i\in\mathcal{I}}u^{*}_{i,\mu},$
(44a)
$\displaystyle\bar{v}_{q,\mu}^{\text{ev}}=v^{*}_{q,\mu}-(\alpha+1)^{-1}\sum_{j\in\mathcal{J}}\delta_{q,j}\sum_{i\in\mathcal{I}}v^{*}_{i,\mu},$
$\displaystyle
v_{q,\mu}^{\text{ev}}=v_{q,\mu}+\sum_{j\in\mathcal{J}}\delta_{q,j}\sum_{i\in\mathcal{I}}v_{i,\mu}.$
(44b)

As discussed in Appendix C, an alternative derivation can be obtained from the
evolved antisymmetric matrix $Z$.

###  4.1 Local operator

The above results can be applied straightforwardly to local (l) operators, as
for the one used in Ref. [49, 50],
$\hat{A}_{\mathcal{I},\mathcal{J}}^{\text{l}}\equiv\hat{n}_{i}=\hat{c}_{i}^{\dagger}\hat{c}_{i}$,
such that
$\hat{M}_{\mathcal{I},\mathcal{J}}^{\text{l}}\equiv\operatorname{e}^{\log(2)\hat{A}^{\text{l}}}=\hat{1}+\hat{c}_{i}^{\dagger}\hat{c}_{i}$.
The Bogoliubov matrix, evolves according to Eqs. (44), which read

$\displaystyle u_{q,\mu}^{\text{ev},\,\text{l}}=(1-\delta_{q,i}/2)u_{q,\mu},$
$\displaystyle\quad\bar{u}^{\text{ev},\,\text{l}}_{q,\mu}=(1+\delta_{q,i})u^{*}_{q,\mu},$
(45a)
$\displaystyle\bar{v}^{\text{ev},\,\text{l}}_{q,\mu}=(1-\delta_{q,i}/2)v^{*}_{q,\mu},$
$\displaystyle\quad v_{q,\mu}^{\text{ev},\,\text{l}}\
=(1+\delta_{q,i})v_{q,\mu}.$ (45b)

This evolution is implemented by applying to the $\mathbb{U}$ a block-diagonal
matrix T with entries $\text{T}_{i+L,i+L}=\text{T}_{i,i}^{-1}=2$ (see Eq. (1)
of Ref. [50]).

###  4.2 Two-site operator

We now consider a non-local (nl) operator with support on two sites
($\mathcal{I}=\mathcal{J}=\\{i,j\\}$):

$\hat{A}_{\mathcal{I},\mathcal{J}}^{\text{nl}}=\big{(}\hat{c}_{i}^{\dagger}+\hat{c}_{j}^{\dagger}\big{)}\big{(}\hat{c}_{i}+\hat{c}_{j}\big{)},$
(46)

Since $\alpha=|\mathcal{I}\cap\mathcal{J}|=2$, we have
$\hat{M}_{\mathcal{I},\mathcal{J}}^{\text{nl}}\equiv\operatorname{e}^{\frac{\log(3)}{2}\hat{A}^{\text{nl}}}=\hat{1}+\hat{A}_{\mathcal{I},\mathcal{J}}^{\text{nl}}$.
Without losing in generality, in the following we will focus on one
dimensional systems and set $j=i+r$ (with $r>0$). As we shall see in the next
Section, a measurement operator as the one in Eq. (46) may unveil a richer
physics than local ones. In fact, in the spin-$1/2$ language, it corresponds
to a string operator. The evolved Bogoliubov matrix can be derived from Eqs.
(44) which, in this case, read

$\displaystyle u_{q,\mu}^{\text{ev},\,\text{nl}}$ $\displaystyle=$
$\displaystyle
u_{q,\mu}-\tfrac{1}{3}(\delta_{q,i}+\delta_{q,i+r})(u_{i,\mu}+u_{i+r,\mu}),$
(47a) $\displaystyle\bar{v}^{\text{ev},\,\text{nl}}_{q,\mu}$ $\displaystyle=$
$\displaystyle{v}^{*}_{q,\mu}-\tfrac{1}{3}(\delta_{q,i}+\delta_{q,i+r})(v^{*}_{i,\mu}+v^{*}_{i+r,\mu}),$
(47b) $\displaystyle\bar{u}^{\text{ev},\,\text{nl}}_{q,\mu}$ $\displaystyle=$
$\displaystyle{u}^{*}_{q,\mu}+(\delta_{q,i}+\delta_{q,i+r})(u^{*}_{i,\mu}+u^{*}_{i+r,\mu}),$
(47c) $\displaystyle v^{\text{ev},\,\text{nl}}_{q,\mu}$ $\displaystyle=$
$\displaystyle
v_{q,\mu}+(\delta_{q,i}+\delta_{q,i+r})(v_{i,\mu}+v_{i+r,\mu}).$ (47d)

##  5 Quantum jumps with string operators

In this section we use the machinery presented above to describe the evolution
of a quantum Ising chain under the effect of a quantum-jump dynamics generated
by $\hat{M}_{\mathcal{I},\mathcal{J}}^{\text{nl}}$ [cf. Eq. (46) and below],
with emphasis on the evolution of the entanglement entropy.

We consider the Hamiltonian

$\hat{H}=-J\sum_{j}\hat{\sigma}_{j}^{x}\hat{\sigma}_{j+1}^{x}-h\sum_{j}\hat{\sigma}_{j}^{z},$
(48)

where $\hat{\sigma}_{j}^{\alpha}$ ($\alpha=x,y,z$) are the usual spin-$1/2$
Pauli matrices acting on the $j$-th site, while $J>0$ denotes the
ferromagnetic spin-spin coupling and $h$ the transverse magnetic field. In the
thermodynamic limit, the ground state exhibits a zero-temperature continuous
transition at $|h/J|=1$, separating a paramagnetic (symmetric) phase, for
$|h/J|>1$, from a ferromagnetic (symmetry-broken) phase, for $|h/J|<1$ [79,
80]. Through a Jordan-Wigner transformation [81, 82],

$\hat{\sigma}_{j}^{z}=1-2\hat{n}_{j},\quad\hat{\sigma}_{j}^{+}=\hat{K}_{j}\,\hat{c}_{j},\
\
\text{with}\quad\hat{K}_{j}=\Pi_{j^{\prime}}^{j-1}(1-2\,\hat{n}_{j^{\prime}}),$
(49)

the Ising model (48) can be mapped into the so-called Kitaev chain

$\hat{H}\
=-J\sum_{j}\left(\hat{c}_{j}^{\dagger}\hat{c}_{j+1}+\hat{c}_{j}^{\dagger}\hat{c}_{j+1}^{\dagger}+\text{h.c.}\right)-2h\sum_{j}\hat{c}^{\dagger}_{j}\hat{c}_{j}\,.$
(50)

This Hamiltonian is of the form (1), with

$Q_{j,j}=h,\ \ Q_{j,j+1}=Q_{j+1,j}=-J/2,\qquad P_{j,j+1}=-P_{j+1,j}=-J/2\,.$
(51)

Hereafter we work in units of $\hbar=1$, set $J=1$ as the energy scale, and
assume periodic boundary conditions for fermions.

We implement the quantum-jump protocol described in Appendix B on the
fermionic model above, by initializing the system in the ground state of the
Hamiltonian (50) with a given value of $h$ and then Trotterizing the time
evolution in steps $\delta t\propto(4L\gamma)^{-1}$, to ensure convergence,
being $\gamma$ the strength of the measurements. The emerging dynamics is one
of the possible unravelings [10] of the Lindblad master equation [9]

$\dot{\rho}(t)=-i\big{[}\hat{H},\rho(t)\big{]}-\gamma\sum_{j}\Big{(}\hat{A}_{j}(r)\,\rho(t)\,\hat{A}_{j}(r)-\tfrac{1}{2}\\{\hat{A}_{j}^{2}(r),\rho(t)\\}\Big{)}\,,$
(52)

being
$\hat{A}_{j}(r)=(\hat{c}_{j}^{\dagger}+\hat{c}_{j+r}^{\dagger})(\hat{c}_{j}+\hat{c}_{j+r})\,$
the same operator as in Eq. (46), with $i\equiv j+r$. 444Note that, with our
convention for the boundary conditions, we have
$\hat{c}^{(\dagger)}_{j+r}\equiv\hat{c}^{(\dagger)}_{j+r-L}$ for any $j+r>L$.
In the spin language, this operator is a sum of strings of Pauli matrices, of
length $r$:

$\hat{A}_{j}(r)=\hat{1}+\hat{\sigma}_{j}^{-}(\hat{K}_{j}\hat{K}_{j+r})\,\hat{\sigma}_{j+r}^{+}+\hat{\sigma}_{j}^{+}(\hat{K}_{j}\hat{K}_{j+r})\,\hat{\sigma}_{j+r}^{-}\,-(\hat{\sigma}_{j}^{z}+\hat{\sigma}_{j+r}^{z})/2\,,$
(53)

with $\hat{\sigma}_{j}^{\pm}=(\hat{\sigma}_{j}^{x}\pm
i\hat{\sigma}_{j}^{y})/2$. Hereafter, we refer to it as the string operator.

We point out that our framework is similar to the one presented in Ref. [42],
although with some important qualitative differences. The former is described
by a random sequence of stabilizer-formalism measurements of Pauli string
operators. These operators have a fixed range $r$ and are taken from a random
distribution of strings, located around different sites, and made of different
sets of onsite Pauli operators.

In this work, in contrast, we are following an unraveling of the quantum
master equation Eq. (52). As detailed in Appendix B, we apply random quantum-
jump measurements of string operators $\hat{1}+\hat{A}_{j}(r)$, located at
different sites, but with the structure fixed as the one in Eq. (53). This is
sufficient to achieve a measurement-induced nontrivial dynamics, with
important consequences on the entanglement generation.

It is important to emphasize that the operators $\hat{1}+\hat{A}_{j}(r)$ have
the exponential form given by Eqs. (38) and (40) (see also Sec. 4). In this
way, the state keeps its Gaussian form under the application of the quantum
jumps, and a numerical analysis of quite large system sizes is possible.

###  5.1 Entanglement entropy

Let us focus our attention on the entanglement entropy. Given a pure state
$\ket{\psi}$, there is an unambiguous way to quantify its quantum
correlations. We divide the system in two subchains, one with length $\ell$
and the other with length $L-\ell$. After evaluating the density matrix
$\rho_{\ell}=\text{Tr}_{L-\ell}\big{[}\\!\ket{\psi}\bra{\psi}\\!\big{]}$
reduced to the subsystem $\ell$, the entanglement entropy is quantified by its
von Neumann entropy [7, 8]:

$S_{\ell}[\rho_{\ell}]=-\rho_{\ell}\log\rho_{\ell}\,.$ (54)

For the Ising model we are discussing herewith, it is known that after, a
sudden quench in a purely Hamiltonian dynamics, the entanglement entropy
approaches a stationary value that increases linearly with $\ell$, thus
obeying a volume-law behavior [3, 4]. As discussed in Refs. [44, 45, 46, 47,
48, 49, 50, 51], this dynamical volume law is generally destroyed by any
arbitrary small local weak measurement process. In this scenario, the
asymptotic entanglement entropy is conjectured to exhibit either a logarithmic
scaling or an area-law behavior, depending on the strength of measurements and
of the Hamiltonian parameters. We are going to show that the situation is
quite different for the case of the nonlocal operators defined in Eq. (53).

In practice, to avoid spurious contributions from classical correlations, we
evaluate Eq. (54) on each stochastic quantum trajectory $\ket{\psi_{t}}$
(defined in Appendix B), and then average over different stochastic
realizations. 555For details on how to evaluate the von Neumann entropy of the
reduced density matrix of Gaussian states see, e.g., Ref. [78]. For simplicity
of notation, we denote the ensemble-averaged entanglement entropy as

$S_{\ell}(t)\equiv\big{\langle}\\!\\!\big{\langle}S_{\ell}\big{[}\rho_{\ell}^{\text{stoch}}(t)\big{]}\big{\rangle}\\!\\!\big{\rangle},\qquad\mbox{where}\quad\rho_{\ell}^{\text{stoch}}(t)=\text{Tr}_{L-\ell}\big{[}\\!\ket{\psi_{t}}\bra{\psi_{t}}\\!\big{]},$
(55)

and where the symbol
$\big{\langle}\\!\\!\big{\langle}\cdots\big{\rangle}\\!\\!\big{\rangle}$
denotes the average over different stochastic realizations. We also define the
asymptotic entanglement entropy as

$\overline{S}_{\ell}=\lim_{T\to\infty}\int_{t^{\star}}^{T}dt^{\prime}S_{\ell}(t^{\prime}).~{}$
(56)

We remark that, in all the simulations shown hereafter, we keep the size of
the partition fixed to $\ell=L/4$ and study the scaling with $L$.

Below we present two different scenarios for fixed $\gamma=0.5$ and for $r=1,\
4,\ L/2$: the measurement-only dynamics (Section 5.2) and the full quantum-
jump dynamics (Section 5.3). Finally, in Section 5.4, we discuss how these
results may change when varying the measurement rate $\gamma$, as well as the
range $r$ and the partition size $\ell$.

###  5.2 Measurement-only dynamics

Figure 1: Top panels: the ensemble-averaged entanglement entropy $S_{\ell}(t)$
versus the rescaled time $\gamma t$, along the quantum-jump process with
$\hat{H}=0$. The different panels refer to $r=1,\,4,\,L/2$ [resp. (a), (b),
(c)]. The various curves are for different sizes $L$ [see color code in (a)].
Bottom panels: scaling of the long-time asymptotic value $\overline{S}_{\ell}$
versus $L$, for $r=1,\,4$, and $\overline{S}_{\ell}/L$ versus $1/L$, for
$r=L/2$ [resp. (d), (e), and (f)]. The dashed line in (f) extrapolates to the
asymptotic value for $L\to\infty$.

We first neglect the Hamiltonian contribution, i.e., we implement the quantum-
jump protocol described in Appendix B with $\hat{H}=0$. The Hamiltonian is
only used to set the initial state as its corresponding ground state with
$h=0.5$. However, we have carefully checked that the choice of the initial
state does not affect the asymptotic value reached by $S_{\ell}(t)$, but only
its transient dynamics. 777Note that the perfectly paramagnetic state with
spins pointing along the $z$ direction ($h=\infty$) is a dark state of the
dissipation, therefore, initializing the system close to such state leads to
slower convergence. The advantage of studying a measurement-only dynamics is
to isolate the effects of measuring the string operators, without considering
the interplay with the unitary dynamics.

The three upper panels of Figure 1 report the ensemble-averaged entanglement
entropy $S_{\ell}(t)$ versus the rescaled time $\gamma t$, for three ranges of
the string measurement operator: $r=1$ [panel (a)], $r=4$ [panel (b)], and
$r=L/2$ [panel (c)], and for different system sizes $L$. We note that, for any
finite value of $r$ (i.e., which does not depend on $L$), the entanglement
entropy displays metastable plateaus, whose lifetime generally increases with
some power of $L$ (see the discussion below, following the rescaled data of
Fig. 3). This may cause some difficulties in estimating the long-time
asymptotic behavior, especially for large system size. However we checked that
the qualitative scaling of the entanglement entropy with $L$ does not change
if one considers the value reached in one of such steady values. Moreover,
since their lifetime grows with the size, in the thermodynamic limit we can
treat the first emerging plateau as the effective stationary state.

Going into the details, for $r=1$ we observe that the measurement-only
dynamics generally destroys entanglement in time and the system stabilizes on
a low-entangled asymptotic state [Fig. 1(a)]. This is confirmed by the data
reported in Fig. 1(d), which shows the behavior of the asymptotic entanglement
entropy $\overline{S}_{\ell}$ with the system size, as extrapolated by the
second plateau stabilizing around the rescaled time $\gamma t\gtrsim
10^{-2}L^{2}$ [see also Fig. 3(a)]. A decreasing behavior in $L$ is clearly
visible: $\overline{S}_{\ell}$ should saturate to a constant value for small
enough ratio $r/L$, i.e., in the limit of local measurements, thus obeying an
area-law behavior, as expected by a comparison with the case where onsite
operators are measured [49, 50].

When the correlation range of the dissipative string operator is increased, as
for $r=4$ [Fig. 1(b)], the measurements still produce a low entangled state.
In fact, the asymptotic value of the entanglement entropy approaches a
constant value with $L$, with superimposed staggered oscillations, probably
due to finite-size commensurability effects between $r$, $\ell$, $L$ [Fig.
1(e)]. It is worth noticing that, also in this case, we extrapolated the
asymptotic value $\overline{S}_{\ell}$ over the first observable metastable
plateau itself. When going to even later times, we find that the subsequent
plateau, corresponding to a lower value of the entropy, is still obeying an
area-law scaling (at least for $L\leq 48$, so that we were able to reach such
plateau).

The above scenario dramatically changes when $r$ is chosen in such a way to be
a thermodynamic fraction of $L$ [see Fig. 1(c), for $r=L/2$]. In that case,
the measurement-only dynamics is projecting over Bell pairs at arbitrary
distances, thus quickly leading the bipartite entanglement to an asymptotic
stationary value which undergoes a volume-law behavior. This is clarified by
Fig. 1(f), which shows $\overline{S}_{\ell}/L$ versus $L^{-1}$: the saturation
of $\overline{S}_{\ell}/L$ to a finite value for $L^{-1}\to 0$ (see the dashed
line, which extrapolates to the asymptotic value in the thermodynamic limit)
clearly evidences a linear growth $\overline{S}_{\ell}\propto L$ for large
system sizes.

###  5.3 Interplay with the Hamiltonian dynamics

Figure 2: Same plots of Figure 1, but in the simultaneous presence of the
Kitaev Hamiltonian $\hat{H}$ and of the string measurement process. We observe
qualitatively similar behaviors, although with different time scales. Here we
fix $\gamma=0.5$, $J=1$, $h=0.1$, and start from the ground state of $\hat{H}$
with $h_{i}=0.5$.

We now switch on the Kitaev Hamiltonian $\hat{H}$ and follow the full quantum-
jump dynamics for $h=0.1$, starting from the ground state with $h_{i}=0.5$. As
for the case with $\hat{H}=0$, we have checked that the asymptotic values
reached by the entanglement entropy are independent of $h_{i}$, while they may
depend on $h$ (although the qualitative behaviors with $L$ are unaffected by
this choice).

Figure 2 displays $S_{\ell}(t)$ versus $\gamma t$ (upper panels) and the
asymptotic value $\overline{S}_{\ell}$ versus $L$, for the three ranges of
$r=1$ [panels (a),(d)], $r=4$ [panels (b),(e)], and $r=L/2$ [panels (c),(f)].
The data shown have been obtained for $\gamma=0.5$. In Fig. 2(a) we focus on
the local case $r=1$. The long-time averaged entanglement entropy attains a
stationary value that, as for the measurement-only case, slightly decreases
with the system size, thus reflecting an area-law behavior [see Fig. 2(d),
which shows $\overline{S}_{\ell}$ corresponding to the plateau reached after a
transient time $\gamma t\gtrsim 10^{-2}L^{2}$, as visible in Fig. 3(c)]. The
value of the asymptotic entanglement entropy is slightly larger than the one
obtained for the measurement-only dynamics, without the entangling action of
$\hat{H}$ [compare with Fig. 1(a)].

Figure 2(b) reports the case $r=4$, where we observe convergence in time to a
single plateau, up to the times we were able to achieve numerically. Comparing
the time behavior of the various curves with the corresponding ones for
$\hat{H}=0$ [see Fig. 1(b)], it is reasonable to associate this only plateau
with the late-time second plateau emerging with $\hat{H}=0$. In the presence
of the Hamiltonian dynamics, the transient time appears to be reduced, with
respect to that for the measurement-only dynamics. The asymptotic data shown
in Fig. 2(e) ($x$-axis in logarithmic scale) suggest that
$\overline{S}_{\ell}$ may grow sub-extensively with $L$, for small system
sizes, while we expect it to eventually attain an area-law regime for large
enough $L$. Physically, this can be understood by the fact that, for $L$ much
larger than $r$, the dissipation becomes effectively local and the
entanglement growth is thus eventually suppressed.

Figure 3: Same plots as in Fig. 1(a) and Fig. 2(a) for $r=1$ (i.e., without
and with the Hamiltonian $\hat{H}$, respectively), such that the time ($x$
axis) has been rescaled as $t\to\gamma t/L^{2}$. The two insets show the time
$t^{\star}$ at which the entanglement entropy $S_{\ell}(t^{\star})$ reaches
the threshold value $0.75$ (dashed line in the two main frames), as a function
of $L^{2}$.

In contrast with that and analogously as for the case without the Hamiltonian,
if the range $r$ of the jump operator is comparable with the total size $L$,
the measurements can stabilize a volume-law scaling of the entanglement, as we
can see in Fig. 2(c), referring to $r=L/2$. Here $\overline{S}_{\ell}$ quickly
attains, in time, an asymptotic value that follows a volume-law scaling with
$L$ [see the data in Fig. 2(f) for $\overline{S}_{\ell}/L$ versus $L^{-1}$,
which display convergence to a finite value in the limit $L^{-1}\to 0$, and
compare them with Fig. 1(f)].

It is finally worth commenting on the metastable plateaus displayed by the
entanglement entropy, when considering ranges not extensively scaling with the
system size. The curves reported in Fig. 3 are for $r=1$ and correspond to
those of Fig. 1(a) and Fig. 2(a), without and with the Hamiltonian,
respectively. Here we changed the time axis according to $t\to\gamma t/L^{2}$
and noticed a fairly good convergence to an asymptotic scaling behavior with
$L$. In fact, as highlighted in the two insets, the time $t^{\star}$ at which
the entanglement entropy $S_{\ell}(t^{\star})$ reaches a given threshold value
scales as $t^{*}\sim L^{2}$.

A less clear situation emerges at larger values of $r$. For example, when
$r=4$, with the numerical data at our disposal we could not find a clear
signature of rescaling with $L$. Nonetheless, it emerges that the metastable
plateau for $\hat{H}=0$ has a lifetime that grows superlinearly with $L$.

###  5.4 Varying $\gamma$ and $r$

In the numerical analysis presented so far, the measurement rate has been
always kept fixed. It is however quite natural that the parameter $\gamma$
should have an impact on the stabilization of the entanglement, when the
measurement-induced dynamics competes with the (unitary) Hamiltonian dynamics.
In fact, for fixed and finite values of $r$, we observe a change of behavior
in the entanglement scaling with $L$, from a logarithmic growth to an area-law
scaling, when varying $\gamma$ and in the presence of $\hat{H}$. An example of
this is reported in Fig. 4(a) where we show, for $r=1$, the asymptotic
entanglement entropy $\overline{S}_{\ell}$ versus the system size $L$ for
different measurement rates. In this case, a clear distinction between the two
above mentioned regimes is observable for $\gamma_{c}\approx 0.15$. This
behavior agrees with what has been already observed for a dynamics with onsite
measurement operators [49, 50]. To support this result in Fig. 4(b) we plot
the rescaled entanglement entropy $\overline{S}/\log(L)$ vs $\gamma$ for
different $L$. As expected, within the error bars, the curves collapse for
$\gamma\lesssim 0.15$, indicating a logarithmic scaling. For larger $\gamma$,
instead, the various curves drop, suggesting that the asymptotic entanglement
entropy grows slower than $\log(L)$. On the other hand, for $r$ growing
extensively with $L$, we expect that the volume-law scaling of the asymptotic
entanglement entropy is robust.

Coming to the dependence of the entanglement dynamics on the range $r$ of the
string measurement operators, differently from the onsite case, this should
crucially depend on the system size $L$, since, as discussed above, the
entanglement scaling is affected by the ratio $L/r$. Unfortunately, the
numerical effort required to derive the results does not allow us to reach
quantitative conclusions, because of the strong finite-size effects we have to
deal with. However, our numerics suggests the existence of a critical value of
$n=L/r$ ($n>1$), separating a regime in which the asymptotic entanglement
entropy does not change with the system size (i.e., $n\gg 1$), to a regime
where it exhibits a linear growth with $L$ (i.e., $n\sim 1$).

Figure 4: (a): Asymptotic entanglement entropy $\overline{S}_{\ell}$ versus
the system size $L$, for $r=1$ and different measurement rates (color scale).
Here we fix $J=1$ and quench the field from $h_{i}=0.5$ to $h=0.1$, as in Fig.
2. For $\gamma\lesssim 0.15$ we observe a logarithmic scaling, while for
$\gamma\gtrsim 0.15$ an area-law behavior sets in. Note the logarithmic scale
on the $x$ axis. (b): Rescaled asymptotic entanglement $\overline{S}/\log(L)$
vs $\gamma$. For $\gamma\lesssim 0.15$ the rescaled curves (are overlapped
within the error bars), while they separate for larger measurement rates.

From our simulations, we cannot rule out the possibility that, in the
thermodynamic limit, a volume-law scaling emerges for any non-diverging value
of $n$. A qualitative argument supporting this possibility is the following.
Assuming that two subsequent measurements with overlapping operators of range
$r$ (which denotes the length of the corresponding string in spin-$1/2$
language) occur, these may strongly influence each other and thus generate
large entanglement. Roughly speaking, the ratio between the probability
$p_{\rm ov}$ that the operators in subsequent measurements overlap and the
probability $p_{\rm nov}$ that they do not is $p_{\rm ov}/p_{\rm nov}\sim
2r/(L-2r)$. In the thermodynamic limit, this ratio remains nonvanishing only
when $r=O(L)$ (i.e., $n\sim 1$). Thus it is reasonable to assume a volume-law
scaling of the entanglement entropy whenever $L/r$ is not diverging.

We finally point out that, although the numerical analysis described above has
been performed for a partition size fixed to $\ell=L/4$, we have done further
checks to test the impact of this choice on our results. We found that the
general scenario is qualitatively robust, at least for small and for large
ranges $r$ of the string measurement operator, compared with $\ell$. Different
behaviors, however, may appear when $r$ becomes of the order of the partition
size $\ell$, suggesting that the important ratio to distinguish between the
different entanglement scalings is $r/\ell$, rather than $r/L$.

##  6 Conclusions

We have unveiled some aspects of the entanglement-entropy dynamics of
fermionic many-body Gaussian states, in the presence of a Gaussian-preserving
evolution, with a special focus on the possible emergence of measurement-
induced phase transitions in such systems. The first part of the paper
presents a detailed technical discussion on how to maintain and treat
Gaussianity, when exponentials of quadratic fermion operators are applied to a
Gaussian state. Our purpose is to provide the reader with a comprehensive
guide to implement fermionic Gaussian evolutions and study the dynamics (not
necessarily unitary) of such systems. Among all the possible treatable
evolutions, we focused on the quantum-jump unraveling of a Lindblad dynamics
induced by a particular class of quadratic fermion operators, having a simple
definition in the fermionic language, but a non-trivial string structure in
terms of the Pauli matrices. We called them string operators.

In the second part of the paper, after having derived the equation of motions
for a quantum-jump dynamics induced by the string operators, we have solved it
for the quantum Ising chain, focusing on the time behavior of the entanglement
entropy. Our main result is that the scaling of the entanglement entropy with
the system size $L$ strongly depends on the range $r$ of the string operators.
Because of numerical limitations we cannot investigate system sizes large
enough to claim that this is an entanglement transition rather than a simple
crossover. For short strings ($r$ finite), we see that the asymptotic
entanglement entropy obeys a non-extensive scaling behavior. In this case, the
measurement operator is effectively local in space and thus it asymptotically
destroys quantum correlations that may be generated by the unitary dynamics.
For string ranges comparable with the system size ($r\sim L$), the measurement
process may become highly entangling and is able to stabilize a volume-law
scaling in the asymptotic entanglement entropy.

Remarkably, the picture above is valid both for a measurement-only dynamics
and for a dynamics where also the effect of a short-range Hamiltonian is
considered. So, the measurement dynamics of the string operators is strong
enough to induce different entanglement scalings by itself.

It would be tempting to address dynamical behaviors induced by more than a
single type of string measurement operators (e.g., with different $r$), in
such a way to explore the possibility of having frustration of the
measurements [42, 74]. A further natural extension of our work could be to
understand how the MIPT scenario would change in the presence of long-range
measurement operators (e.g., by considering some power-law decay of
dissipators with the distance). Finally, novel dynamical mechanisms may emerge
in systems beyond the free-fermion paradigm, due to the role of interactions
in the entanglement dynamics under variable-range monitoring.

## Acknowledgements

We acknowledge fruitful discussions with V. Alba, M. Collura, J. De Nardis, R.
Fazio, M. Schirò, and X. Turkeshi.

#### Funding information

This work has been partly funded by the Italian MIUR through PRIN Project No.
2017E44HRF and by PNRR MUR Project PE0000023-NQSTI.

##  Appendix A Rotation to the diagonal basis

Here we discuss a different derivation of the $s$-evolved state
$\ket{\psi(s)}=\hat{M}(s)\ket{\psi}$, where $\ket{\psi}$ is a generic Gaussian
state [Eq. (2)], and the $\hat{M}(s)=\operatorname{e}^{\xi s\hat{A}}$ is
defined in Eqs. (12).

The operator $\hat{A}$ [Eq. (12b)] is quadratic in the $\hat{c}$ fermions and
can be written compactly as
$\hat{A}=\hat{\Psi}^{\dagger}\mathbb{A}\hat{\Psi}$, where
$\hat{\Psi}=(\hat{c}_{1},\cdots,\hat{c}_{L},\hat{c}^{\dagger}_{1},\cdots,\hat{c}_{L}^{\dagger})^{T}$
is the Nambu spinor associated to the $\hat{c}$ fermions and $\mathbb{A}$ the
Bogoliubov-de Gennes matrix associated to $\hat{A}$, as defined in Eq. (23).
This matrix can be diagonalized by a Bogoliubov transformation $\mathbb{W}$,
such that

$\mathbb{W}^{\dagger}\mathbb{A}\mathbb{W}=\text{diag}\,[\epsilon_{\nu},-\epsilon_{\nu}]_{\nu=1,\cdots,L}\,,\quad{\rm
and}\quad\hat{A}=\sum_{\nu}\epsilon_{\nu}\left(\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}-\hat{\eta}_{\nu}\hat{\eta}_{\nu}^{\dagger}\right)\,.$
(57)

The unitary matrix $\mathbb{W}$ defines a new set of $\hat{\eta}$ fermions,
whose associated Nambu spinor is
$\hat{\Theta}\equiv(\hat{\eta}_{1},\cdots,\hat{\eta}_{L},\hat{\eta}^{\dagger}_{1},\cdots,\hat{\eta}_{L}^{\dagger})^{T}=\mathbb{W}^{\dagger}\hat{\Psi}$.

Let us now evaluate $\ket{\psi(s)}$. We remind that the initial state
$\ket{\psi}$ is is the vacuum of the $\hat{\gamma}$ fermions, defined through
the Bogoliubov transformation in Eq. (6):
$\hat{\Phi}=\mathbb{U}^{\dagger}\hat{\Psi}$, where
$\hat{\Phi}=(\hat{\gamma}_{1},\cdots,\hat{\gamma}_{L},\hat{\gamma}^{\dagger}_{1},\cdots,\hat{\gamma}_{L}^{\dagger})^{T}$.
These fermions can be transformed into those diagonalizing $\mathbb{A}$ by
means of a rotation

$\hat{\Phi}=\mathbb{U}^{\dagger}\,\mathbb{W}\,\hat{\Theta}=\mathbb{Q}^{\dagger}\,\hat{\Theta},\
\ \text{being}\ \ \mathbb{Q}=\begin{pmatrix}u_{\eta}&v_{\eta}^{*}\\\
v_{\eta}&u_{\eta}^{*}\end{pmatrix}\,.$ (58)

Therefore we have the relation

$\mathbb{U}=\mathbb{W}\,\mathbb{Q}\,,$ (59)

with $\mathbb{U}$ given in Eq. (5). In the $\hat{\eta}$ representation, the
state (2) takes the form

$\ket{\psi}\propto\exp\bigg{\\{}\frac{1}{2}\sum_{\mu,\nu}Z_{\mu,\nu}^{(\eta)}\,\hat{\eta}_{\mu}^{\dagger}\,\hat{\eta}_{\nu}^{\dagger}\bigg{\\}}\ket{0}_{\hat{\eta}},$
(60)

with $Z^{(\eta)}=-\big{[}u_{\eta}^{\dagger}\big{]}^{-1}\,v_{\eta}^{\dagger}$.

Going on analogously as in Sec. 3.1, we can write the $s$-evolved state as

$\ket{\psi^{\prime}}\propto\exp\bigg{\\{}\frac{1}{2}\sum_{\mu,\nu}Z_{\mu,\nu}^{(\eta)}\,\hat{\eta}^{\dagger}_{\mu}(s)\,\hat{\eta}^{\dagger}_{\nu}(s)\bigg{\\}}\hat{M}(s)\ket{0}_{\hat{\eta}}\,,$
(61)

with
$\hat{\eta}^{\dagger}_{\nu}(s)\equiv\hat{M}(s)\,\hat{\eta}^{\dagger}_{\nu}\,\hat{M}(-s)$.
The operator $\hat{M}(s)$ is number-preserving, therefore we can exploit the
methods of Section 4. We first notice that
$\hat{M}(s)\ket{0}_{\eta}\propto\ket{0}_{\eta}$, in fact

$\operatorname{e}^{\xi
s\sum_{\nu}\epsilon_{\nu}(\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}-\hat{\eta}_{\nu}\hat{\eta}_{\nu}^{\dagger})}\ket{0}_{\hat{\eta}}=\operatorname{e}^{-\xi
s\sum_{\nu}\epsilon_{\nu}}\ket{0}_{\hat{\eta}}\,,$ (62)

because $\hat{\eta}_{\nu}\ket{0}_{\hat{\eta}}=0$,
$\forall\,\nu\in\\{1,\,\ldots,\,L\\}$. Then we notice that, since
$(\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu})^{n}=\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}$
(cf. Section 4), we have

$\displaystyle\hat{M}(\pm s)=\operatorname{e}^{\pm\xi s\hat{A}}=$
$\displaystyle\operatorname{e}^{\mp\xi
s\sum_{\nu}\epsilon_{\nu}}\prod_{\nu}\left[1+\sum_{n=1}^{\infty}\frac{(\pm
2\xi
s\epsilon_{\nu})^{n}}{n!}\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}\right]$
$\displaystyle=$ $\displaystyle\operatorname{e}^{\mp\xi
s\sum_{\nu}\epsilon_{\nu}}\prod_{\nu}\left[1+(\operatorname{e}^{\pm 2\xi
s\epsilon_{\nu}}-1)\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}\right]\,,$ (63)

where we have applied the diagonal form of $\hat{A}$, in Eq. (57). Therefore,
according to the fact that
$\big{[}\hat{\eta}_{\mu}^{\dagger}\hat{\eta}_{\mu},\hat{\eta}_{\nu}^{\dagger}\big{]}=\delta_{\mu,\nu}\hat{\eta}_{\mu}^{\dagger}$,
we can write

$\hat{\eta}^{\dagger}_{\nu}(s)=\big{[}1+(\operatorname{e}^{2\xi
s\epsilon_{\nu}}-1)\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}\big{]}\
\hat{\eta}_{\nu}^{\dagger}\ \big{[}1+(\operatorname{e}^{-2\xi
s\epsilon_{\nu}}-1)\hat{\eta}_{\nu}^{\dagger}\hat{\eta}_{\nu}\big{]}=\operatorname{e}^{2\xi
s\epsilon_{\nu}}\hat{\eta}_{\nu}^{\dagger}.$ (64)

Substituting into Eq. (61) we get

$\ket{\psi(s)}\propto\exp\bigg{\\{}\frac{1}{2}\sum_{\mu,\nu}\tilde{Z}_{\mu,\nu}^{(\eta)}\,\hat{\eta}^{\dagger}_{\mu}\,\hat{\eta}^{\dagger}_{\nu}\bigg{\\}}\ket{0}_{\hat{\eta}}\,,\
\ \text{with}\ \ \tilde{Z}^{(\eta)}_{\mu,\nu}\equiv
Z_{\mu,\nu}^{(\eta)}\,\operatorname{e}^{2\xi
s(\epsilon_{\mu}+\epsilon_{\nu})}.$ (65)

We can also write
$\tilde{Z}^{(\eta)}=-\big{[}{u_{\eta}^{\text{ev}}}^{\dagger}(s)\big{]}^{-1}{v_{\eta}^{\text{ev}}}^{\dagger}(s)$,
with

$\begin{pmatrix}u_{\eta}^{\text{ev}}(s)\\\
v_{\eta}^{\text{ev}}(s)\end{pmatrix}=\begin{pmatrix}\operatorname{e}^{-2\xi
s\boldsymbol{\epsilon}}&0\\\ 0&\operatorname{e}^{2\xi
s\boldsymbol{\epsilon}}\end{pmatrix}\begin{pmatrix}u_{\eta}\\\
v_{\eta}\end{pmatrix},\ \ \text{being}\ \
\boldsymbol{\epsilon}=\text{diag}(\epsilon_{\nu}).$ (66)

At this point, if $\xi=1$, we apply the QR decomposition to
$\begin{pmatrix}u_{\eta}^{\text{ev}}(s)\\\
v_{\eta}^{\text{ev}}(s)\end{pmatrix}$, as explained in Sec. 3.2. In this way,
the matrix $\tilde{Z}^{(\eta)}$ remains unchanged, but can be written as
$\tilde{Z}^{(\eta)}=-\big{[}{u_{\eta}}^{\dagger}(s)\big{]}^{-1}{v_{\eta}}^{\dagger}(s)$,
with ${u_{\eta}}(s)$, ${v_{\eta}}(s)$ obeying the symplectic unitarity
conditions Eq. (4) and written as

$\begin{pmatrix}u_{\eta}^{\text{ev}}(s)\\\
v_{\eta}^{\text{ev}}(s)\end{pmatrix}=\begin{pmatrix}u_{\eta}(s)\\\
v_{\eta}(s)\end{pmatrix}r_{1}\,,$ (67)

with $r_{1}$ some upper triangular matrix $L\times L$.

We have therefore reduced to the standard Gaussian form for the evolved state
$\ket{\psi(s)}$ in the $\hat{\eta}$ representation, with the $Z$ matrix given
by
$\tilde{Z}^{(\eta)}=-\big{[}{u_{\eta}}^{\dagger}(s)\big{]}^{-1}{v_{\eta}}^{\dagger}(s)$,
and the symplectic unitary matrix

$\mathbb{Q}(s)=\begin{pmatrix}u_{\eta}(s)&v_{\eta}^{*}(s)\\\
v_{\eta}(s)&u^{*}_{\eta}(s)\end{pmatrix}$ (68)

providing the set of fermionic operators $\hat{\gamma}_{\nu}(s)$ annihilating
$\ket{\psi(s)}$ [we have
$\hat{\Phi}(s)=\mathbb{Q}^{\dagger}(s)\,\hat{\Theta}$, with
$\hat{\Phi}(s)\equiv(\hat{\gamma}_{1}(s),\cdots,\hat{\gamma}_{L}(s),\hat{\gamma}^{\dagger}_{1}(s),\cdots,\hat{\gamma}_{L}^{\dagger}(s))^{T}$].
At this point it is possible to come back to the $\hat{c}$ representation. We
can use Eq. (59), and obtain the Bogoliubov matrix $\mathbb{U}(s)$ [the one
such that $\hat{\Phi}(s)=\mathbb{U}^{\dagger}(s)\hat{\Psi}$] as
$\mathbb{U}(s)=\mathbb{W}\,\mathbb{Q}(s)$. Applying $\mathbb{W}$ on the left
of both sides of Eq. (66) and writing

$\mathbb{U}(s)=\begin{pmatrix}u(s)&v^{*}(s)\\\ v(s)&u^{*}(s)\end{pmatrix}\,,$

we obtain

$\begin{pmatrix}u(s)\\\ v(s)\end{pmatrix}r_{1}=\operatorname{e}^{-2\xi
s\mathbb{A}}\begin{pmatrix}u\\\ v\end{pmatrix}\,.$ (69)

[To get this result, we used the relation
$\mathbb{U}(s)=\mathbb{W}\,\mathbb{Q}(s)$, as well as Eqs. (57), (59), and
(67)]. The $Z$ matrix of the evolved state in this representation is thus
$Z(s)=-\big{[}{u}^{\dagger}(s)\big{]}^{-1}\,{v}^{\dagger}(s)$. Notice that Eq.
(69) is the same result of Eq. (24), when the QR decomposition discussed in
Sec. 3.2 is applied.

As a last comment we point out that Eq. (69) refers to the first two blocks of
the Bogoliubov matrix $\mathbb{U}$ only. This is consistent with the fact
that, in the Nambu spinor notation, the degrees of freedom are doubled and
there is some redundant information that can be neglected.

##  Appendix B Quantum-jump protocol

We start from the following master equation in the Lindblad form:

$\dot{\rho}(t)={\mathcal{L}}\big{[}\rho(t)\big{]}\equiv-i\big{[}\hat{H},\rho(t)\big{]}-\gamma\sum_{j}\Big{(}\hat{m}_{j}\rho(t)\,\hat{m}_{j}^{\dagger}-\tfrac{1}{2}\\{\hat{m}_{j}^{\dagger}\,\hat{m}_{j},\rho(t)\\}\Big{)},$
(70)

where $\rho(t)$ is the density matrix describing the state of the system at
time $t$ and $\hat{m}_{j}^{(\dagger)}$ are the Lindblad jump operators, while
${\mathcal{L}}$ denotes the Lindbladian superoperator acting on $\rho$. The
solution of Eq. (70) is, in general, a mixed state that can be thought of as
an average over many pure-state quantum trajectories of a suitable stochastic
process. Given a Lindbladian, there are many stochastic processes (or
unravelings) that produce the same averaged density matrix [10]. Among them,
in this paper we focus on the so-called quantum-jump unraveling [83], whose
implementation is described below.

We define the non-Hermitian Hamiltonian

$\hat{H}_{\text{eff}}=\hat{H}-\frac{i\gamma}{2}\sum_{j}\hat{m}_{j}^{\dagger}\,\hat{m}_{j},$
(71)

then we rewrite Eq. (70)

$\dot{\rho}(t)=-i\big{[}\hat{H}_{\text{eff}},\rho(t)\big{]}-\gamma\sum_{j}\hat{m}_{j}\rho(t)\,\hat{m}_{j}^{\dagger}.$
(72)

In this form, the Lindblad dynamics ruled by Eq. (70) can be interpreted as a
deterministic non-Hermitian time evolution of a pure quantum state
$\ket{\psi_{t}}$ generated by $\hat{H}_{\text{eff}}$, plus a stochastic part
given by the possibility of applying $\hat{m}_{j}$ to the state
$\ket{\psi_{t}}$ [cf. the second term in the right-hand-side of Eq. (72)]. For
$\hat{m}_{j}$ Hermitian, this dynamics can be thought of as an occasional, yet
abrupt, measurement process in which, at any time, the system has a chance to
be POVM-measured [7, 8], i.e., to undergo the action of one of the operators
$\hat{m}_{j}$, taken from a distribution (see more details below).

In this framework, Eq. (70) can be solved by implementing the following
stochastic process. The time evolution is discretized in time slices of step
$dt$. Then, at each step:

  1. 1.

With probability
$\pi_{j}=\gamma\braket{\hat{m}^{\dagger}_{j}\,\hat{m}_{j}}dt$, the jump
operator is applied:

$\ket{\psi_{t+dt}}\mapsto\frac{\hat{m}_{j}\ket{\psi_{t}}}{||\hat{m}_{j}\ket{\psi_{t}}||};$
(73)

  2. 2.

With probability $\pi=1-\sum_{j}\pi_{j}$ the non-Hermitian evolution occurs:

$\ket{\psi_{t+dt}}\mapsto\frac{\operatorname{e}^{-i\hat{H}_{\text{eff}}}\,dt\ket{\psi_{t}}}{||\operatorname{e}^{-i\hat{H}_{\text{eff}}\,dt}\ket{\psi_{t}}||}.$
(74)

Note that the Lindblad master equation in Eq. (70) is invariant under a shift
of the jump operators $\hat{m}_{j}\mapsto\hat{1}+\hat{m}_{j}$. In practice,
one can implement the protocol described above with the Lindblad operators
$\hat{1}+\hat{m}_{j}$ (as we actually do in Sec. 5), instead of $\hat{m}_{j}$,
and obtain the same averaged density matrix, although the single trajectories
$\ket{\psi_{t}}$ may behave differently.

We remark that, since the von Neumann entropy of Eq. (54) quantifies the
amount of genuine quantum correlations of a pure state, we first evaluate it
over the single pure-state quantum trajectory and then perform the ensemble
averaging. It is important to fix the order of the two operations of measuring
and ensemble averaging, since they are commuting only for linear functions of
the state $\rho$ (as for expectation values of observables), but not for
quantities as the one in Eq. (54). For a more detailed discussion see, e.g.,
Refs. [44, 49, 50].

##  Appendix C Evolution of the state: a different derivation

In this appendix we provide a different derivation of the analytic expression
for the evolved state $\ket{\psi}_{\hat{M}_{\mathcal{I},\mathcal{J}}}$, by
evaluating the operator
$\hat{\mathcal{Z}}(1)=\frac{1}{2}\sum_{p,q}Z_{p,q}\,\hat{c}_{p}^{\dagger}(1)\,\hat{c}_{q}^{\dagger}(1)$,
in Eq. (16). To this purpose we notice that

$\big{[}\hat{A}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}\big{]}=\frac{1}{2}\sum_{p,q}\sum_{i\in\mathcal{I}}\big{(}Z_{p,\mathcal{J}}\delta_{q,i}+Z_{q,\mathcal{J}}\delta_{p,i}\big{)}\,\hat{c}^{\dagger}_{p}\,\hat{c}^{\dagger}_{q},$
(75)

where we have defined
$\hat{\mathcal{Z}}=\frac{1}{2}\sum_{p,q}Z_{p,q}\,\hat{c}_{p}^{\dagger}\,\hat{c}_{q}^{\dagger}$
and $Z_{p,\mathcal{J}}=\sum_{j\in\mathcal{J}}Z_{p,j}$. Consequently
$\big{[}\mathcal{\hat{Z}},[\hat{M}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}]\big{]}=0$.
This allows to write

$\hat{M}_{\mathcal{I},\mathcal{J}}\,\operatorname{e}^{\mathcal{\hat{Z}}}=\operatorname{e}^{\mathcal{\hat{Z}}}\left(\hat{M}_{\mathcal{I},\mathcal{J}}+\big{[}\hat{M}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}\big{]}\right),$
(76)

that, together with $\hat{M}_{\mathcal{I},\mathcal{J}}\ket{0}=\ket{0}$, leads
to

$\hat{M}_{\mathcal{I},\mathcal{J}}\ket{\psi}_{\hat{M}_{\mathcal{I},\mathcal{J}}}=\mathcal{N}\operatorname{e}^{\mathcal{\hat{Z}}}\left(\hat{1}+\big{[}\hat{A}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}\big{]}\right)\ket{0}.$
(77)

Since
$\big{[}\hat{A}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}\big{]}^{2}=0$, we
can write
$\exp\big{\\{}[\hat{A}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}]\big{\\}}=\hat{1}+\big{[}\hat{A}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}\big{]}$
and, therefore,

$\hat{M}_{\mathcal{I},\mathcal{J}}\ket{\psi}_{\hat{M}_{\mathcal{I},\mathcal{J}}}=\mathcal{N}\operatorname{e}^{\mathcal{\hat{Z}}}\operatorname{e}^{[\hat{A}_{\mathcal{I},\mathcal{J}},\mathcal{\hat{Z}}]}\ket{0}\equiv\mathcal{N}\operatorname{e}^{\mathcal{\hat{Z}}^{\prime}}\ket{0},$
(78)

being

$\mathcal{\hat{Z}}^{\prime}=\frac{1}{2}\sum_{i\in\mathcal{I}}\sum_{p,q}\Big{[}Z_{p,q}+\big{(}Z_{p,\mathcal{J}}\delta_{q,i}+Z_{q,\mathcal{J}}\delta_{p,i}\big{)}\Big{]}\hat{c}^{\dagger}_{p}\,\hat{c}^{\dagger}_{q}.$
(79)

For the local case $\hat{M}_{\mathcal{I},\mathcal{J}}^{\text{l}}$ in Subsec
4.1, the jumped state
$\ket{\psi}_{\hat{M}_{\mathcal{I},\mathcal{J}}^{\text{l}}}$ is described by
the antisymmetric matrix $\big{[}Z^{\text{l}}\big{]}^{\prime}$, with

$\big{[}Z^{\text{l}}\big{]}^{\prime}_{p,q}=(1+\delta_{q,i}+\delta_{p,i})Z_{p,q}.$
(80)

On the other hand, for the non-local case
$\hat{M}_{\mathcal{I},\mathcal{J}}^{\text{nl}}$ in Subsec. 4.2, the
antisymmetric matrix $\big{[}Z^{\text{nl}}\big{]}^{\prime}$ reads

$\big{[}Z^{\text{nl}}\big{]}^{\prime}_{p,q}=Z_{p,q}+(\delta_{p,i}+\delta_{p,i+r})(Z_{i,q}+Z_{i+r,q}).$
(81)

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