# Entanglement phase transitions in non-Hermitian Kitaev chains

Longwen Zhou<EMAIL_ADDRESS>College of Physics and Optoelectronic
Engineering, Ocean University of China, Qingdao, China 266100  Key Laboratory
of Optics and Optoelectronics, Qingdao, China 266100  Engineering Research
Center of Advanced Marine Physical Instruments and Equipment of MOE, Qingdao,
China 266100

###### Abstract

The intricate interplay between unitary evolution and projective measurements
could induce entanglement phase transitions in the nonequilibrium dynamics of
quantum many-particle systems. In this work, we uncover loss-induced
entanglement transitions in non-Hermitian topological superconductors. In
prototypical Kitaev chains with local particle losses and varying hopping and
pairing ranges, the bipartite entanglement entropy of steady states is found
to scale logarithmically versus the system size in topologically nontrivial
phases and become independent of the system size in the trivial phase.
Notably, the scaling coefficients of log-law entangled phases are
distinguishable when the underlying system resides in different topological
phases. Log-law to log-law and log-law to area-law entanglement phase
transitions are further identified when the system switches between different
topological phases and goes from a topologically nontrivial to a trivial
phase, respectively. These findings not only establish the relationships among
spectral, topological and entanglement properties in a class of non-Hermitian
topological superconductors, but also provide an efficient means to
dynamically reveal their distinctive topological features.

##  I Introduction

The entanglement dynamics of open quantum many-body systems undergoing
nonunitary evolution have attracted significant attention in recent years [1,
2, 3]. An intriguing phenomena that could emerge in such contexts is the
measurement-induced entanglement transition [4, 5, 6, 7, 8, 9, 10]. It
describes a nonequilibrium phase transition of entanglement structures
following a quantum quench. In usual situations, measurement-induced
entanglement phase transitions are originated from the competition between
unitary dynamics and quantum measurements. With the increase of measurement
rates, the bipartite entanglement entropy (EE) of nonequilibrium steady states
could undergo a transition from a volume-law to an area-law scaling versus the
system size. Despite great theoretical efforts [14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], measurement-induced
entanglement transitions have also been explored experimentally in setups
including trapped ions and superconducting qubits [35, 36, 37], offering
further insights for understanding quantum information dynamics and simulating
quantum many-body systems.

Open quantum systems described by non-Hermitian Hamiltonians constitute an
important context for exploring entanglement phase transitions. Various types
of non-Hermiticity-induced entanglement transitions have been identified in
gapped or critical non-Hermitian systems made up of lattice fermions and
quantum spin chains [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. In a
non-Hermitian system with spatial nonreciprocity, the emergence of non-
Hermitian skin effects was found to accompany the transition from a volume-law
entangled to an area-law entangled phase in one spatial dimension [42]. The
development of a dissipation gap in the energy spectrum of a non-Hermitian
Hamiltonian was also found to yield a volume-to-area law entanglement phase
transition in free-fermion chains [43]. Moreover, in non-Hermitian systems
with spatially uniform or quasiperiodic randomness [45, 46, 47, 48],
entanglement phase transitions beyond the conventional volume-law to area-law
scheme could emerge due to the interplay between disorder and non-Hermitian
effects. Besides, alternated and reentrant entanglement transitions may be
engineered and controlled by time-periodic driving fields in non-Hermitian
Floquet systems [49].

In this work, we continue the study of entanglement phase transitions in non-
Hermitian systems. We focus on one-dimensional (1D) topological
superconductors with local particle losses, which have been found to possess
rich topological phases and dynamical phase transitions [50, 51, 52, 53, 54,
55, 56, 57]. In Sec. II, we introduce our model and outline the approaches of
characterizing its spectrum, topological properties and entanglement dynamics.
In Sec. III, we explore entanglement phase transitions in representative non-
Hermitian Kitaev chains with varying ranges of single-particle hopping and
superconducting pairing terms. All-round connections are established among the
spectral structures, topological transitions and entanglement phase
transitions in the considered system. Remarkably, each loss-induced transition
between topologically distinct superconducting phases is found to go hand-in-
hand with a transition in the scaling law of steady-state EE, i.e., an
entanglement phase transition. In Sec. IV, we summarize our results and
discuss potential future directions. Some further calculations details about
the entanglement dynamics are provided in the Appendix A.

##  II Model and method

We start by considering a generalized Kitaev chain [58] with local particle
losses, whose Hamiltonian can be expressed as

$\hat{H}=\frac{1}{2}\sum_{n}\left[\mu(2\hat{c}_{n}^{\dagger}\hat{c}_{n}-1)+\sum_{r}(J_{r}\hat{c}_{n}^{\dagger}\hat{c}_{n+r}+\Delta_{r}\hat{c}_{n}\hat{c}_{n+r}+{\rm
H.c.})\right].$ (1)

Here, H.c. denotes Hermitian conjugation, $n\in\mathbb{Z}$ represents the
lattice index (with the lattice constant $a=1$), and $r\in\mathbb{Z}$ controls
the ranges of hopping and pairing terms. $\hat{c}_{n}^{\dagger}$
($\hat{c}_{n}$) creates (annihilates) a spin-polarized fermion on the $n$th
lattice site. The chemical potential $\mu=u-iv$ is chosen to be complex, where
$u={\rm Re}\mu$, and $v=-{\rm Im}\mu>0$ describing the loss rate. $J_{r}$ and
$\Delta_{r}$ denote hopping and pairing amplitudes of fermions over $r$
lattice sites [$r=1$ for the nearest-neighbor (NN) coupling, $r=2$ for the
next-nearest-neighbor (NNN) coupling, etc.]. The presence of long-range
($r>1$) hopping and pairing terms allows the system to possess superconducting
phases with large topological invariants [56]. We will refer to the system
described by $\hat{H}$ in Eq. (1) with $v>0$ as a lossy Kitaev chain (LKC).

Taking the periodic boundary condition (PBC) $\hat{c}_{n}=\hat{c}_{n+L}$ and
applying the Fourier transformation
$\hat{c}_{n}=\frac{1}{\sqrt{L}}\sum_{k}e^{ikn}\hat{c}_{k}$, we can express
$\hat{H}$ in the momentum space as
$\hat{H}=\frac{1}{2}\sum_{k}\hat{\Psi}_{k}^{\dagger}H(k)\hat{\Psi}_{k}$, where
$L$ is the length of lattice, $k\in[-\pi,\pi)$ is the
quasimomentum,$\hat{\Psi}_{k}^{\dagger}=(\hat{c}_{k}^{\dagger},\hat{c}_{-k})$
is the Nambu spinor operator, and

$H(k)=h_{y}(k)\sigma_{y}+h_{z}(k)\sigma_{z},$ (2)
$h_{y}(k)=\sum_{r}\Delta_{r}\sin(kr),$ (3)
$h_{z}(k)=\mu+\sum_{r}J_{r}\cos(kr).$ (4)

Here $\sigma_{x,y,z}$ are Pauli matrices in their usual representations. By
diagonalizing $H(k)$, we can find the spectrum of the system as
$E_{\pm}(k)=\pm E(k)$, where

$E(k)=\sqrt{h_{y}^{2}(k)+h_{z}^{2}(k)}.$ (5)

With the loss rate $v>0$ in the chemical potential $\mu=u-iv$, $E(k)$ becomes
complex in general. Meanwhile, $H(k)$ possesses the chiral symmetry ${\cal
S}=\sigma_{x}$, in the sense that ${\cal S}H(k){\cal S}=-H(k)$. The
topological phases of the system could then be characterized by a winding
number $w$ [56], which is defined as

$w=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\partial_{k}\phi(k),$ (6)

where $\phi(k)=\arctan[h_{y}(k)/h_{z}(k)]$. The value of $2\pi w$ is equal to
the accumulated winding angle $\phi(k)$ over the first Brillouin zone (BZ). As
the imaginary part of $\phi(k)$ has no winding [59], $w$ can only take real
values. Moreover, when the energy spectrum is gapped, $w$ will take an integer
(a half-integer) quantized value if the trajectory of vector
$[h_{y}(k),h_{z}(k)]$ encircles an even (odd) number of exceptional points
(EPs) of $H(k)$ following the change of $k$ from $-\pi$ to $\pi$ [56]. The
closing and reopening of spectrum gap will also accompany the quantized (or
half quantized) change of $w$. Therefore, we can characterize the non-
Hermitian topological phases and topological phase transitions of our system
by the winding number $w$ in Eq. (6).

In this work, we focus on the loss-induced entanglement phase transitions and
their connections with topological phases in non-Hermitian Kitaev chains. To
deal with the entanglement dynamics, we first prepare our system in a certain
initial state $|\Psi(0)\rangle$ at half-filling and evolve it over a time
duration $t$ according to the Hamiltonian $\hat{H}$ [Eq. (1)]. The resulting
state of the system then takes the form ($\hbar=1$)

$|\Psi(t)\rangle=\frac{e^{-i\hat{H}t}|\Psi(0)\rangle}{||e^{-i\hat{H}t}|\Psi(0)\rangle||},$
(7)

where the normalization factor $||e^{-i\hat{H}t}|\Psi(0)\rangle||$ arises
after taking the no-click limit of a monitored evolution [42, 43]. With the
normalized state $|\Psi(t)\rangle$, we could obtain the single-particle
correlator $C(t)$ of the system in position representation. Due to the
translational symmetry of our system, the matrix elements $C_{m,n}(t)$
($m,n=1,...,L$) of $C(t)$ in real-space can be further computed by performing
the Fourier transformation of its related generator in $k$-space, i.e.,

$C_{m,n}(t)=\frac{1}{L}\sum_{k}e^{ik(m-n)}C_{k}(t),$ (8)

where the $2\times 2$ matrix $C_{k}(t)$ can be obtained as

$C_{k}(t)=\frac{1}{2}\begin{pmatrix}1+\langle\sigma_{z}\rangle_{t}&\langle\sigma_{x}\rangle_{t}+i\langle\sigma_{y}\rangle_{t}\\\
\langle\sigma_{x}\rangle_{t}-i\langle\sigma_{y}\rangle_{t}&1-\langle\sigma_{z}\rangle_{t}\end{pmatrix}.$
(9)

The average $\langle\sigma_{j}\rangle_{t}$ ($j=x,y,z$) is taken over
normalized state $|\psi_{k}(t)\rangle$, which is evolved by $e^{-iH(k)t}$ from
the initial state vector $|\psi_{k}(0)\rangle$ at each quasimomentum $k$ (see
Appendix A for more details).

Finally, to obtain the bipartite entanglement entropy (EE), we decompose our
1D chain of length $L$ into two spatially connected subsystems A and B, whose
number of lattice sites are $l$ and $L-l$, respectively. Restricting the
lattice indices $m$ and $n$ to the portion A gives us the correlation matrix
$C_{m,n}^{{\rm A}}(t)$ ($m,n=1,...,l$) of subsystem A. Then, according to the
relation between single-particle correlation matrix and bipartite EE of a
Gaussian state [60], we can express the EE between two subsystems A and B as

$S(t)=-\sum_{j=1}^{l}[\zeta_{j}\ln\zeta_{j}+(1-\zeta_{j})\ln(1-\zeta_{j})],$
(10)

where $\\{\zeta_{j}|j=1,...,l\\}$ are the eigenvalues of the $l\times l$
correlation matrix $C^{{\rm A}}(t)$. In numerical calculations, we can obtain
$S(t)$ efficiently at any time $t$ through Eqs. (7)–(9) together with the
diagonalization of $C^{{\rm A}}(t)$. In the long-time limit, $S(t)$ will reach
a stationary value $S(L,l)\equiv\lim_{t\rightarrow\infty}S(t)$, which may
depend on the total system size, the subsystem size and other system
parameters. We can numerically find the bipartite EE $S(L,l)$ of steady states
by considering an evolution time duration $t\in[0,T]$ that is long enough,
such that any variations of $S(t)$ over time are negligible for $t\geq T$.

In the next section, we explore entanglement phase transitions in two
representative LKC models with distinct topological properties. We will see
clear changes in the scaling-law of steady-state EE when the considered system
undergo topological phase transitions. Moreover, qualitatively different
scaling behaviors of EE will be uncovered when the system resides in
topological and trivial non-Hermitian superconducting phases, respectively.

##  III Results

In this section, we investigate entanglement phase transitions in two non-
Hermitian Kitaev chains with different ranges of hopping and pairing terms. We
first reveal the spectrum and topological properties of each model. This is
followed by the demonstration of their entanglement dynamics and the related
scaling laws of steady-state EE. Finally, we establish the entanglement phase
diagrams and further unveil their connections with the topological phases and
transitions for both models.

###  III.1 LKC with nearest-neighbor hopping and pairing

We start with a “minimal” model of LKC by restricting the hopping and pairing
terms to NN sites. Referring to the Eq. (1), we choose $r=1$, let
$(J_{1},\Delta_{1})=(J,\Delta)$, and set the chemical potential $\mu=u-iv$.
The resulting system is described by the Hamiltonian

$\hat{H}_{1}=\frac{1}{2}\sum_{n}[\mu(\hat{c}_{n}^{\dagger}\hat{c}_{n}-1/2)+J\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+\Delta\hat{c}_{n}\hat{c}_{n+1}+{\rm
H.c.}].$ (11)

It has been identified that under PBC, the two energy bands $E_{\pm}(k)=\pm
E(k)$ [Eq. (5)] of this NN LKC model become gapless at $E=0$ when the system
parameters satisfy the condition $u^{2}/J^{2}+v^{2}/\Delta^{2}=1$ [56]. When
$u^{2}/J^{2}+v^{2}/\Delta^{2}<1$, there is a line gap lying along the ${\rm
Re}E$ axis between the two bands $\pm E(k)$ on the complex energy plane.
Within the gap, a pair of degenerate Majorana zero modes can be found at the
edges of the chain under the open boundary condition (OBC), and the system is
topologically nontrivial in this case [56]. When
$u^{2}/J^{2}+v^{2}/\Delta^{2}>1$, a line gap in the energy spectrum is opened
along the ${\rm Im}E$ axis, and there are no localized Majorana modes with
zero energy at system edges under the OBC. In this case, the system becomes
topologically trivial and possesses a trivial dissipation gap [56].

Figure 1: Topological phase diagram and typical spectra of the LKC with NN
hopping and pairing. (a) shows the winding number $w$ vs the real and
imaginary parts of chemical potential $u$ and $v$. The yellow and green
regions have $w=1$ and $w=0$, respectively. The red solid dot in (a) resides
at $(u,v)=(0.8,0.1)$, with the associated spectrum of $\hat{H}_{1}$ shown in
(b). The red solid triangle of (a) is located at $(u,v)=(0.8,0.9)$, and the
associated spectrum of $\hat{H}_{1}$ is given in (c) on the complex energy
plane. Other system parameters are $J=\Delta=1$ for all panels.

In Fig. 1, we present typical energy spectra on the complex plane and the
topological phase diagram of the NN LKC model [Eq. (11)]. In Fig. 1(a), the
phase diagram is obtained by computing the topological winding number $w$ [Eq.
(6)] at different system parameters $(u,v)$. The results confirm that the NN
LKC indeed belongs to the topologically nontrivial (trivial) phase with $w=1$
($w=0$) when $u^{2}/J^{2}+v^{2}/\Delta^{2}<1$ ($>1$), with the phase boundary
$u^{2}/J^{2}+v^{2}/\Delta^{2}=1$ given by the black solid line between the
yellow and green regions in Fig. 1(a). In Figs. 1(b) and 1(c), we illustrate
the complex spectrum of the system in the topological and trivial phases,
respectively, under the OBC. A real energy gap carrying a pairs of Majorana
edge modes at $E=0$ is observed in the topological phase, while an imaginary
energy gap with no Majorana zero modes is found in the trivial phase, which
validate our theoretical predictions [56].

Figure 2: Bipartite EE vs time $t$ for the LKC with NN hopping and pairing for
the loss rate $v=0.1$ (solid lines) and $0.9$ (dotted lines). Other system
parameters are $J=\Delta=1$ and $u=0.8$. $l$ denotes the subsystem size and
the total lattice size is $L=2\times 10^{4}$.

With the above knowledge on the spectral and topological properties of the NN
LKC, it would be interesting to check whether the dynamics and scaling laws of
its EE could show distinct behaviors in different topological phases.
Following Eqs. (7)–(10), we compute the evolution of bipartite EE $S(t)$ of
the NN LKC under PBC with a large system size $L$. A set of representative
results versus different sizes $l$ of the subsystem A and at different loss
rates are shown in Fig. 2. In all the calculations, we choose the initial
state at different quasimomenta as

$|\psi_{k}(0)\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
e^{ik/2}\end{pmatrix},$ (12)

where $k=-\pi,-\pi+2\pi/L,...,\pi-2\pi/L$. Other forms of pure and non-
stationary initial states generate consistent results regarding the
entanglement dynamics. We find that when the system parameters satisfy
$u^{2}/J^{2}+v^{2}/\Delta^{2}<1$, the bipartite EE will first experience a
transient time window with a non-monotonous behavior in time, and finally
evolving to a stationary value in the late time regime. After the steady state
is reached, the EE raises monotonically with the increase of subsystem size
$l$ (solid lines in Fig. 2). On the contrary, in the parameter regime with
$u^{2}/J^{2}+v^{2}/\Delta^{2}>1$, the bipartite EE will converge to the same
steady-state value for any subsystem size $l$ after a transient time window,
which implies that $\lim_{t\rightarrow\infty}S(t)\sim l^{0}$ in this region
(dashed lines in Fig. 2). In summary, these observations suggest that in the
NN LKC, the scaling laws of steady-state EE vs the subsystem size could indeed
be qualitatively different in the topological phase with a real energy gap and
the trivial phase with an imaginary energy gap. A loss-driven entanglement
phase transition may then appear and go hand-in-hand with the topological
phase transition of the system.

Figure 3: Bipartite EE of steady states vs (a) the subsystem size $l$, and (b)
the loss rate $v$ for the LKC with NN hopping and pairing. Other system
parameters are $J=\Delta=1$ and $u=0.8$ for both panels. The lattice size of
the whole system is $L=2\times 10^{4}$. The vertical dashed line in (b)
highlights the phase transition point at $v=0.6$.

To further confirm the existence of different entangling phases in the NN LKC,
we present in Fig. 3 the steady-state EE $S(L,l)$ versus the subsystem size
$l$ and loss rate $v$. The steady-state is reached by evolving the initial
state $|\Psi(0)\rangle$ according to Eq. (7) [with $\hat{H}=\hat{H}_{1}$ in
Eq. (11)] over a long time duration, which is set to $T=2000$ in our
calculations. In Fig. 3(a), we identify the scaling-law of steady-state EE as
$S(L,l)\sim\ln[\sin(\pi l/L)]$ for $u^{2}/J^{2}+v^{2}/\Delta^{2}<1$ and
$S(L,l)\sim l^{0}$ for $u^{2}/J^{2}+v^{2}/\Delta^{2}>1$, respectively, with
$v\neq 0$. In the thermodynamic limit ($L\rightarrow\infty$), we obtain the
following scaling laws for the bipartite EE of steady states at half-filling,
i.e.,

$S(L,l)\sim\begin{cases}\ln l,&v\in(|\Delta|\sqrt{1-u^{2}/J^{2}},\infty),\\\
l^{0},&v\in(0,|\Delta|\sqrt{1-u^{2}/J^{2}}).\end{cases}$ (13)

Therefore, the EE satisfies a log-law (an area-law) vs the subsystem size in
the weak (strong) dissipation regime of the system. These two distinct
entangling phases are clearly illustrated in the regions with $v<0.6$ and
$v>0.6$ in Fig. 3(b). Remarkably, the separation point [red dashed line at
$v=0.6$ in Fig. 3(b)] between these two entangling phases is precisely
coincident with the topological transition point of the NN LKC at
$u^{2}/J^{2}+v^{2}/\Delta^{2}=1$. There should thus be a transition from a
log-law entangled topological superconducting phase to an area-law entangled
trivial phase in the NN LKC with the increase of the loss rate $v$.

Figure 4: Entanglement phase transitions in the LKC with NN hopping and
pairing. System parameters are $J=\Delta=1$ and $L=2\times 10^{4}$ for both
panels. (a) Gradient $g$ extracted from the data fitting $S(L,l)\sim
g\ln[\sin(\pi l/L)]$ of bipartite, steady-state EE vs the subsystem size $l$
at different loss rates for $u=0.8$. The vertical dashed line highlights the
phase transition point at $v=0.6$. (b) The same gradient $g$ as obtained in
(a) vs the real and imaginary parts of chemical potential $\mu=u-iv$. The
values of $g$ at different $(u,v)$ can be read out from the color bar.

Based on the above analysis, we could identify the entanglement phase
transition and establish the entanglement phase diagram of the NN LKC, as
reported in Fig. 4. To reveal the change of scaling behaviors in the steady-
state EE, we fit $S(L,l)$ with the function $g\ln[\sin(\pi l/L)]$ in different
parameter regions and extract the coefficient $g$ as the gradient of the
associated scaling laws. In Fig. 4(a), we take $J=\Delta=1$ and the real part
of chemical potential $u=0.8$. Referring to the topological phase boundary,
$v=|\Delta|\sqrt{1-u^{2}/J^{2}}$, we find the topological transition point to
be $v=0.6$ in this case. It is clear that we have a finite $g$ when $v<0.6$
and a vanishing $g$ for $v>0.6$ in Fig. 4(a), which verifies the presence of
two different entangling phases (log-law vs area-law) and an entanglement
phase transition driven by the change of loss rate $v$ in the NN LKC.

Finally, we present the gradient $g$ extracted from the fitting $S(L,l)\sim
g\ln[\sin(\pi l/L)]$ at different $(u,v)$ in Fig. 4(b), generating the
entanglement phase diagram of the system. We observe two different regions
with distinct entanglement features ($g>0$ vs $g=0$), which are separated by
the boundary line $v=|\Delta|\sqrt{1-u^{2}/J^{2}}$ for $v\neq 0$ and
$|u|\leq|J|$. A direct comparison with the topological phase diagram in Fig.
1(a) leads to the conclusion that for the NN LKC, the bipartite EE of steady
states follows the log-law scaling vs the subsystem size in the topological
phase (with winding number $w=1$) and becomes independent of the subsystem
size in the trivial phase (with $w=0$). At $v\neq 0$, a log-law to area-law
entanglement phase transition could happen following the transition of the
system from a topologically nontrivial to a trivial phase. This entanglement
transition could thus offer a unique dynamical probe to topological phase
transitions in non-Hermitian Kitaev chains.

A possible mechanism behind the loss-driven entanglement transition in the NN
LKC is as follows. With $v>0$, the particles tend to populate the energy
levels with positive imaginary parts after being evolved over a long time
duration. In the regime with $v>|\Delta|\sqrt{1-u^{2}/J^{2}}$, such energy
levels are separated apart from those with negative imaginary parts by a
dissipation gap [see Fig. 1(c)]. The particle distribution of late-time steady
state then mimics a 1D normal band insulator at half filling, whose bipartite
EE is expected to follow an area-law scaling versus the system size. In the
regime with $v<|\Delta|\sqrt{1-u^{2}/J^{2}}$, there is no dissipation gap
between the energy levels with positive and negative imaginary parts [see Fig.
1(b)]. After a long evolution time, the steady-state population tends to form
an effective Fermi surface at $E=0$ along the ${\rm Re}E$ axis, which has two
crossing points with the spectrum. Therefore, the particle distribution of
steady state is close to a critical metallic phase in 1D systems, whose
bipartite EE is expected to scale logarithmically with respect to the system
size [61, 62, 63].

In the next subsection, we investigate entanglement transitions in an LKC with
NNN hopping and pairing terms, whose topological phases could possess larger
winding numbers. We will see that the connections between topological and
entanglement phase transitions found in this subsection could be extended to
more general situations for non-Hermitian topological superconductors.

###  III.2 LKC with next-nearest-neighbor hopping and pairing

We now consider an LKC with second neighbor hopping and pairing terms.
Referring to Eq. (1), we set the coupling range $r=2$, and the resulting
Hamiltonian of the system reads

$\hat{H}_{2}=\frac{1}{2}\sum_{n}\left[\mu(2\hat{c}_{n}^{\dagger}\hat{c}_{n}-1)+\sum_{r=1,2}(J_{r}\hat{c}_{n}^{\dagger}\hat{c}_{n+r}+\Delta_{r}\hat{c}_{n}\hat{c}_{n+r}+{\rm
H.c.})\right].$ (14)

The non-Hermitian effect is again introduced by setting the chemical potential
$\mu=u-iv$ with $v>0$. In momentum space, it can be shown that there are four
possible critical quasimomenta $\pm
k_{0}^{\pm}=\pm\arccos\\{[-J_{1}\pm\sqrt{J_{1}^{2}+8J_{2}(J_{2}-u)}]/(4J_{2})\\}$,
where the two bulk energy bands of $\hat{H}_{2}$ could touch with each other
at $E=0$ [56]. The topological phase boundaries of the system in parameter
space can then be obtained by solving the equations
$\sin(k_{0}^{\pm})(\Delta_{1}+2\Delta_{2}\cos k_{0}^{\pm})=\pm v$ [56]. In
Fig. 5(a), we plot these phase boundaries as black lines in the $u-v$ plane
and compute the topological winding number $w$ [Eq. (6)] in different
parameter regions. The results show that there are three different topological
phases with winding numbers $w=0,1,2$. The presence of NNN hopping and
coupling terms allows us to have a non-Hermitian topological superconducting
phase with a larger winding number $w=2$.

Figure 5: Topological phase diagram and typical spectra of the LKC with NNN
hopping and pairing. (a) shows the winding number $w$ vs the real and
imaginary parts of chemical potential $u$ and $v$. The yellow, green and blue
regions have $w=2$, $1$ and $0$, respectively. The red solid dot of (a)
resides at $(u,v)=(1,0.1)$, with the associated spectrum of $\hat{H}_{2}$
shown in (b). The red solid triangle of (a) is located at $(u,v)=(1,1.1)$, and
the associated spectrum of $\hat{H}_{2}$ is given in (c) on the complex energy
plane. The red solid square of (a) lies at $(u,v)=(1,2.5)$, and the associated
spectrum of $\hat{H}_{2}$ is shown in (d). Other system parameters are
$J_{1}=\Delta_{1}=1$ and $J_{2}=\Delta_{2}=1.5$ for all panels.

In the topological phase with $w=2$, the bulk energy spectrum of the system is
gapped along the ${\rm Re}E$ axis, and there are two pairs of Majorana edge
modes at $E=0$ under the OBC [see Fig. 5(b)]. With the increase of $v$, the
system could undergo a phase transition through level crossings at $E=0$,
after which it enters another topological phase with $w=1$. In this phase, the
bulk spectrum of the system holds a line energy gap along a certain angle
$\theta\in(0,\pi/2)$ with respect to the ${\rm Re}E$ axis, and a single pair
of Majorana edge modes can be found inside the gap at $E=0$ under the OBC [see
Fig. 5(c)]. Note in passing that this non-Hermitian topological
superconducting phase is different from the $w=1$ phase of NN LKC [see Fig.
1], whose spectral gap lies instead along the ${\rm Re}E$ axis. With the
further increase of $v$, the system would experience a second phase transition
and finally entering a trivial phase with $w=0$. In this phase, the spectrum
of the system develops a line gap along the ${\rm Im}E$ axis, and there are no
zero-energy Majorana edge modes within the gap under the OBC [see Fig. 5(d)].

Figure 6: Bipartite EE vs time $t$ for the LKC with NNN hopping and pairing
for the loss rate $v=0.1$ (solid lines), $0.8$ (dash-dotted lines) and $2.5$
(dotted lines). Other system parameters are $J_{1}=\Delta_{1}=1$,
$J_{2}=\Delta_{2}=1.5$ and $u=1$. $l$ denotes the subsystem size and the total
lattice size is $L=2\times 10^{4}$.

After unveiling the rich spectral and topological features of the NNN LKC, we
are ready to explore its entanglement dynamics. Referring again to the Eqs.
(7)–(10), we can find the evolution of bipartite EE $S(t)$ of the NNN LKC with
a large system size $L$ under PBC. Exemplary results vs different sizes $l$ of
the subsystem A and different loss rates are presented in Fig. 6. We have also
considered a half-filled system and used Eq. (12) as the initial state at
different quasimomenta throughout the calculations. The results show that in
the case with a large loss rate ($v=2.5$ in Fig. 6), the EE does not change
with the increase of subsystem size $l$ after a long evolution time (dotted
lines in Fig. 6), which implies an area-law scaling of the steady-state EE vs
$l$. This case corresponds to the spectrum of NNN LKC in Fig. 5(d), where
there exists a dissipation gap and the system resides in a topologically
trivial phase. In the situation with a small loss rate ($v=0.1$ in Fig. 6),
the EE grows monotonically with the increase of the subsystem size $l$ after a
long-time evolution (solid lines in Fig. 6). It corresponds to the case with a
line energy gap along the ${\rm Re}E$ axis in the spectrum [Fig. 5(b)], and
the system resides in a topological phase with winding number $w=2$. When the
loss rate takes an intermediate value ($v=0.8$ in Fig. 6), the EE again rises
with the increase of the subsystem size $l$ in the late time regime.
Nevertheless, its growth rate vs $l$ is smaller compared to the case of
$v=0.1$. The system also belongs to a topological phase with a smaller winding
number $w=1$ and possessing a gapped spectrum as shown in Fig. 5(c). These
observations suggest that for the NNN LKC, the bipartite EE in steady states
may possess different scaling properties when the system belongs to distinct
topological superconducting phases with distinguishable spectral features.

Figure 7: Bipartite EE of steady states vs (a) the subsystem size $l$, and (b)
the loss rate $v$ for the LKC with NNN hopping and pairing. Other system
parameters are $J_{1}=\Delta_{1}=1$, $J_{2}=\Delta_{2}=1.5$ and $u=1$ for both
panels. The lattice size of the whole system is $L=2\times 10^{4}$. Vertical
dashed lines in (b) denote two topological transition points of the system.

To further decode the scaling laws of bipartite EE in steady states, we
consider an evolution over a long time duration of $T=2000$, and obtain the
final values of EE $S(L,l)$ at different subsystem sizes $l$ and loss rates,
as shown in Fig. 7. The methodology of computing $S(L,l)$ here is in parallel
with that employed in Sec. III.1. We find that in the weak and intermediate
dissipation regions [$v=0.1,0.5$ and $v=1.1,1.5$ in Fig. 7(a)], the EE
$S(L,l)$ is proportional to $\ln[\sin(\pi l/L)]$ for a fixed system size $L$.
In the limit of large $L$, this relation reduces to the log-law scaling
$S\sim\ln l$. In the strong dissipation region [$v=2.1,2.5$ in Fig. 7(a)], the
steady-state EE becomes independent of $l$, leading to an area-law entangled
phase with $S\sim l^{0}$. More precisely, under the condition $v>0$, we
numerically find the following three parameter regions in which the bipartite
EE of steady states show distinguishable scaling behaviors in the limit
$L\rightarrow\infty$, i.e.,

$S(L,l)\sim\begin{cases}g\ln l,&v<\min|h_{y}(k_{0}^{\pm})|,\\\ g^{\prime}\ln
l,&\min|h_{y}(k_{0}^{\pm})|<v<\max|h_{y}(k_{0}^{\pm})|,\\\
l^{0},&v>\max|h_{y}(k_{0}^{\pm})|.\end{cases}$ (15)

Here, the coefficients $g$ and $g^{\prime}$ both depend on the system
parameters, and we always have $g>g^{\prime}$. The expression of $h_{y}(k)$ is
given by Eq. (3) after setting $r=1,2$. $k_{0}^{\pm}\in[-\pi,\pi)$ are the
critical quasimomenta where the energy bands could touch when topological
phase transition happens [56]. The three distinct scaling regions of $S(L,l)$
are clearly visible from the EE vs loss rate $v$ in Fig. 7(b). Notably, the
separation points between these entangling phases tend out to be identical to
the topological transition points of the NNN LKC [56]. Despite a transition
from a log-law entangled topological phase to an area-law entangled trivial
phase, there should also be a possible entanglement transition between two
non-Hermitian topological superconducting phases with winding numbers $w=2$
and $w=1$ following the variation of the loss rate $v$.

Figure 8: Entanglement phase transitions in the LKC with NNN hopping and
pairing. System parameters are $J_{1}=\Delta_{1}=1$, $J_{2}=\Delta_{2}=1.5$
and $L=2\times 10^{4}$ for both panels. (a) Gradient $g$ extracted from the
data fitting $S(L,l)\sim g\ln[\sin(\pi l/L)]$ of bipartite, steady-state EE vs
the subsystem size $l$ at different loss rate $v$ for $u=1$. Vertical dashed
lines highlight topological transition points of the system. (b) The same
gradient $g$ as obtained in (a) vs the real and imaginary parts of chemical
potential $\mu=u-iv$. The values of $g$ at different $(u,v)$ can be figured
out from the color bar.

To confirm the presence of entanglement transitions in the NNN LKC and build
their connections with topological phase transitions, we perform numerical
fitting for the bipartite EE of steady states as $S(L,l)\sim g\ln[\sin(\pi
l/L)]$, and extract the coefficients $g$ at different system parameters. The
results are shown in Fig. 8. The two red dashed lines in Fig. 8(a) correspond
to the loss rates where topological phase transitions happen in the system
[56]. We see that they separate the configuration of $g$ into three
distinguishable regions. In the left and middle regions (with weak and
intermediate dissipation), the values of $g$ are finite and decrease gradually
with the increase of $v$. The system then belongs to log-law entangled phases
in these domains. In the right region, the gradient $g$ becomes pinned to
zero, which implies that the system has entered an area-law entangled phase
with $S(L,l)\sim l^{0}$. It is noteworthy that the derivatives of $g$ with
respect to the loss rate $v$ undergo two discontinuous changes at the
topological phase transition points. Therefore, the loss-induced topological
transitions in the NNN LKC also accompany entanglement phase transitions
characterized by different scaling laws in the bipartite EE of steady states.

Finally, we present the scaling coefficients $g$ of $S(L,l)$ vs the real and
imaginary parts of chemical potential $\mu$ in Fig. 8(b), which forms the
entanglement phase diagram of the system. In comparison with the topological
phase diagram in Fig. 5(a), we conclude that the topological phases of NNN LKC
with different winding numbers $w=2$, $1$ and $0$ indeed exhibit different
scaling laws in the EE vs system sizes. Both the two topologically nontrivial
phases are log-law entangled, whereas the topologically trivial phase is area-
law entangled. Each topological phase transition further goes hand-in-hand
with an entanglement phase transition.

The physical mechanism behind these loss-driven entanglement transitions is
similar to that discussed for the NN LKC in Sec. III.1. Starting with an
initial state at half-filling, the final population of the steady state tends
to fill the energy levels with positive imaginary parts after a long evolution
time. The interface between filled and empty states then possesses four, two
and no crossing points with the bulk energy spectrum of $\hat{H}_{2}$ in the
topological phases with winding numbers $w=2$, $1$ and $0$, respectively (see
Fig. 5), yielding two critical-type, log-law entangled phases and a gapped,
area-law entangled phase. The entanglement phase transitions identified here
may also provide us with an alternative strategy to dynamically probe and
distinguish non-Hermitian topological superconducting phases with large
winding numbers.

##  IV Conclusion

In this study, we unveiled entanglement phase transitions within 1D non-
Hermitian topological superconductors. Our investigation focused on Kitaev
chains exhibiting local particle loss, revealing distinct scaling laws of EE
corresponding to different non-Hermitian superconducting phases. Specifically,
the bipartite EE of steady states demonstrated an area-law scaling in the
topologically trivial phase and a log-law scaling in the nontrivial phase.
Notably, the scaling coefficients of the log-law become distinguishable when
the system is situated in superconducting phases with distinct topological
winding numbers. Furthermore, we identified entanglement phase transitions
coinciding with topological phase transitions in non-Hermitian Kitaev chains
at identical system parameters. Our study established generic connections
among spectral, topological, and entanglement phase transitions in two
distinct lossy Kitaev chains with varying hopping and pairing ranges. These
findings not only uncovered the richness and unique characteristics of
entanglement transitions in a class of non-Hermitian systems, but also
introduced an efficient dynamical probe for detecting and distinguishing
different non-Hermitian superconducting phases with diverse spectral and
topological properties.

In future research, exploring entanglement phase transitions in non-Hermitian
topological superconductors with disorder, subject to time-periodic driving,
and extending the investigation to higher spatial dimensions holds
considerable interest. The critical properties of entanglement transitions in
non-Hermitian systems and their responses to many-body interactions deserve
more thorough explorations. Additionally, the experimental realization of non-
Hermitian Kitaev chains and the detection of entanglement and topological
phase transitions within these systems offer intriguing directions for future
studies.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12275260 and 11905211), the Fundamental Research Funds for the
Central Universities (Grant No. 202364008), and the Young Talents Project of
Ocean University of China.

##  Appendix A Calculation details of the EE dynamics

In this Appendix, we discuss how to obtain the $k$-space generator [Eq. (9)]
of single-particle correlation matrix in real space. We first transform the
Hamiltonian in Eq. (1) to another real-space fermionic basis through the
Bogoliubov-de Gennes (BdG) transformation [64]. Let us introduce creation and
annihilation operators of normal fermions $\hat{f}_{j}^{\dagger}$ and
$\hat{f}_{j}$, which satisfy the anticommutation relations

$\\{\hat{f}_{i},\hat{f}_{j}\\}=\\{\hat{f}_{i}^{\dagger},\hat{f}_{j}^{\dagger}\\}=0,\qquad\\{\hat{f}_{i},\hat{f}_{j}^{\dagger}\\}=\delta_{ij}.$
(16)

With these operators, we can expand the annihilation operator $\hat{c}_{n}$ in
Eq. (9) as

$\hat{c}_{n}=\sum_{j=1}^{L}(\varphi_{jn}\hat{f}_{j}+\phi_{jn}\hat{f}_{j}^{\dagger}).$
(17)

Here the real coefficients $\varphi_{jn}$ and $\phi_{jn}$ satisfy the
normalization condition $\sum_{j=1}^{L}(\varphi_{jn}^{2}+\phi_{jn}^{2})=1$.
Assuming that the Hamiltonian $\hat{H}$ in Eq. (1) is diagonalized in the
basis
$\\{\hat{f}_{1},...,\hat{f}_{L},\hat{f}_{1}^{\dagger},...,\hat{f}_{L}^{\dagger}\\}$,
we can express $\hat{H}$ as

$\hat{H}=\sum_{\ell}E_{\ell}\hat{f}_{\ell}^{\dagger}\hat{f}_{\ell}.$ (18)

Using the commutator formula $[AB,C]=A\\{B,C\\}-\\{A,C\\}B$ together with Eqs.
(17) and (18), we find that

$[\hat{H},\hat{f}_{j}]=-E_{j}\hat{f}_{j},\qquad[\hat{H},\hat{f}_{j}^{\dagger}]=E_{j}\hat{f}_{j}^{\dagger},$
(19)
$[\hat{H},\hat{c}_{n}]=-\sum_{j}E_{j}(\varphi_{jn}\hat{f}_{j}-\phi_{jn}\hat{f}_{j}^{\dagger}).$
(20)

Meanwhile, from Eqs. (1) and (17), we can equivalently express the commutator
$[\hat{H},\hat{c}_{n}]$ as

$\displaystyle[\hat{H},\hat{c}_{n}]=$
$\displaystyle-\sum_{j}\mu(\varphi_{jn}\hat{f}_{j}+\phi_{jn}\hat{f}_{j}^{\dagger})$
$\displaystyle-\sum_{j}\sum_{r}\frac{J_{r}}{2}(\varphi_{jn+r}+\varphi_{jn-r})\hat{f}_{j}$
$\displaystyle-\sum_{j}\sum_{r}\frac{J_{r}}{2}(\phi_{jn+r}+\phi_{jn-r})\hat{f}_{j}^{\dagger}$
$\displaystyle+\sum_{j}\sum_{r}\frac{\Delta_{r}}{2}(\phi_{jn+r}-\phi_{jn-r})\hat{f}_{j}$
$\displaystyle+\sum_{j}\sum_{r}\frac{\Delta_{r}}{2}(\varphi_{jn+r}-\varphi_{jn-r})\hat{f}_{j}^{\dagger}.$
(21)

Equating the right hand sides of Eqs. (20), (21) and dropping the redundant
index $j$, we arrive at the following BdG self-consistent equations

$+\mu\varphi_{n}+\sum_{r}\frac{J_{r}}{2}(\varphi_{n-r}+\varphi_{n+r})+\sum_{r}\frac{\Delta_{r}}{2}(\phi_{n-r}-\phi_{n+r})=E\varphi_{n},$
(22)
$-\mu\phi_{n}-\frac{J_{r}}{2}\sum_{r}(\phi_{n-r}+\phi_{n+r})-\sum_{r}\frac{\Delta_{r}}{2}(\varphi_{n-r}-\varphi_{n+r})=E\phi_{n}.$
(23)

Next, it can be verified that the Eqs. (22) and (23) can be generated by a
fermionic Hamiltonian $\hat{{\cal H}}$. It owns two internal degrees of
freedom $(\alpha,\beta)$ in each unit cell but has no superconducting pairing
terms, i.e.,

$\hat{{\cal H}}=\frac{1}{2}\sum_{n}\left[\mu\hat{{\bf
c}}_{n}^{\dagger}\sigma_{z}\hat{{\bf c}}_{n}+\sum_{r}\hat{{\bf
c}}_{n}^{\dagger}(J_{r}\sigma_{z}-i\Delta_{r}\sigma_{y})\hat{{\bf
c}}_{n+r}+{\rm H.c.}\right],$ (24)

where $\hat{{\bf
c}}_{n}^{\dagger}=(\hat{c}_{n,\alpha}^{\dagger},\hat{c}_{n,\beta}^{\dagger})$.
Taking the PBC and applying the Fourier transformation $\hat{{\bf
c}}_{n}^{\dagger}=\frac{1}{\sqrt{L}}\sum_{k}e^{-ikn}\hat{{\bf
c}}_{k}^{\dagger}$ with the momentum-space creation operator $\hat{{\bf
c}}_{k}^{\dagger}=(\hat{c}_{k,\alpha}^{\dagger},\hat{c}_{k,\beta}^{\dagger})$,
we arrive at $\hat{{\cal H}}=\sum_{k}\hat{{\bf c}}_{k}^{\dagger}{\cal
H}(k)\hat{{\bf c}}_{k}$. Here the Bloch Hamiltonian

${\cal
H}(k)=\sum_{r}\Delta_{r}\sin(kr)\sigma_{y}+[\mu+\sum_{r}J_{r}\cos(kr)]\sigma_{z},$
(25)

which is formally identical to the $H(k)$ in Eq. (2). In the basis of
$\hat{{\bf c}}_{k}$ and $\hat{{\bf c}}_{k}^{\dagger}$, we can now compute the
$k$-space generator $C_{k}(t)$ of correlation matrix in Eq. (8) as [43]

$C_{k}(t)=\begin{pmatrix}\langle\hat{c}_{k,\alpha}^{\dagger}\hat{c}_{k,\alpha}\rangle_{t}&\langle\hat{c}_{k,\alpha}^{\dagger}\hat{c}_{k,\beta}\rangle_{t}\\\
\langle\hat{c}_{k,\beta}^{\dagger}\hat{c}_{k,\alpha}\rangle_{t}&\langle\hat{c}_{k,\beta}^{\dagger}\hat{c}_{k,\beta}\rangle_{t}\end{pmatrix}.$
(26)

Since the system is noninteracting, we can express the single particle
operators in the first quantized formalism as

$\hat{c}_{k,\alpha}^{\dagger}\hat{c}_{k,\alpha}\rightarrow|\alpha\rangle\langle\alpha|,\qquad\hat{c}_{k,\beta}^{\dagger}\hat{c}_{k,\beta}\rightarrow|\beta\rangle\langle\beta|,$
(27)
$\hat{c}_{k,\alpha}^{\dagger}\hat{c}_{k,\beta}\rightarrow|\alpha\rangle\langle\beta|,\qquad\hat{c}_{k,\beta}^{\dagger}\hat{c}_{k,\alpha}\rightarrow|\beta\rangle\langle\alpha|,$
(28)

where $\\{|\alpha\rangle,|\beta\rangle\\}$ forms the two by two internal
subspace of the BdG Hamiltonian ${\cal H}(k)$ with the identity matrix
$\sigma_{0}\equiv|\alpha\rangle\langle\alpha|+|\beta\rangle\langle\beta|$.
Finally, with the help of Pauli matrices, we can identify

$\sigma_{x}=|\alpha\rangle\langle\beta|+|\beta\rangle\langle\alpha|,$ (29)
$\sigma_{y}=-i(|\alpha\rangle\langle\beta|-|\beta\rangle\langle\alpha|),$ (30)
$\sigma_{z}=|\alpha\rangle\langle\alpha|-|\beta\rangle\langle\beta|.$ (31)

Therefore, the matrix elements of $C_{k}(t)$ in Eq. (26) can be equivalently
expressed as
$\langle\hat{c}_{k,\alpha}^{\dagger}\hat{c}_{k,\alpha}\rangle_{t}=\langle\sigma_{0}+\sigma_{z}\rangle_{t}/2$,
$\langle\hat{c}_{k,\beta}^{\dagger}\hat{c}_{k,\beta}\rangle_{t}=\langle\sigma_{0}-\sigma_{z}\rangle_{t}/2$,
$\langle\hat{c}_{k,\alpha}^{\dagger}\hat{c}_{k,\beta}\rangle_{t}=\langle\sigma_{x}+i\sigma_{y}\rangle_{t}/2$
and
$\langle\hat{c}_{k,\beta}^{\dagger}\hat{c}_{k,\alpha}\rangle_{t}=\langle\sigma_{x}-i\sigma_{y}\rangle_{t}/2$,
yielding the expression of $C_{k}(t)$ in Eq. (9).

## References

  * [1] A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, in Quantum Science and Technology, edited by A. Bayat, S. Bose, and H. Johannesson (Springer International, Cham, Switzerland, 2022).
  * [2] M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  * [3] B. Skinner, Introduction to random unitary circuits and the measurement-induced entanglement phase transition, arXiv:2307.02986.
  * [4] Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  * [5] X. Cao, A. Tilloy, and A. D. Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys. 7, 024 (2019).
  * [6] B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  * [7] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019).
  * [8] Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  * [9] M. Szyniszewski, A. Romito and H. Schomerus, Entanglement transition from variable-strength weak measurements, Phys. Rev. B 100, 064204 (2019).
  * [10] R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Ludwig, Entanglement transitions from holographic random tensor networks, Phys. Rev. B 100, 134203 (2019).
  * [11] A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B 101, 060301 (2020).
  * [12] Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020).
  * [13] M. J. Gullans and D. A. Huse, Dynamical purification phase transition induced by quantum measurements, Phys. Rev. X 10, 041020 (2020).
  * [14] S. Goto and I. Danshita, Measurement-induced transitions of the entanglement scaling law in ultracold gases with controllable dissipation, Phys. Rev. A 102, 033316 (2020).
  * [15] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum error correction in scrambling dynamics and measurement-induced phase transition, Phys. Rev. Lett. 125, 030505 (2020).
  * [16] Q. Tang and W. Zhu, Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations, Phys. Rev. Res. 2, 013022 (2020).
  * [17] C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum circuits, Phys. Rev. B 101, 104302 (2020).
  * [18] D. Rossini and E. Vicari, Measurement-induced dynamics of many-body systems at quantum criticality, Phys. Rev. B 102, 035119 (2020).
  * [19] Y. Fuji and Y. Ashida, Measurement-induced quantum criticality under continuous monitoring, Phys. Rev. B 102, 054302 (2020).
  * [20] J. Iaconis, A. Lucas, and X. Chen, Measurement-induced phase transitions in quantum automaton circuits, Phys. Rev. B 102, 224311 (2020).
  * [21] M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, Entanglement phase transitions in measurement-only dynamics, Phys. Rev. X 11, 011030 (2021).
  * [22] A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory, PRX Quantum 2, 010352 (2021).
  * [23] O. Lunt, M. Szyniszewski, and A. Pal, Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits, Phys. Rev. B 104, 155111 (2021).
  * [24] A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transitions in symmetric random quantum circuits, Nat. Phys. 17, 342 (2021).
  * [25] S.-K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang, Measurement-induced phase transition in the monitored Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 127, 140601 (2021).
  * [26] J. Willsher, S.-W. Liu, R. Moessner, and J. Knolle, Measurement-induced phase transition in a chaotic classical many-body system, Phys. Rev. B 106, 024305 (2022).
  * [27] S. Sharma, X. Turkeshi, R. Fazio, and M. Dalmonte, Measurement-induced criticality in extended and long-range unitary circuits, SciPost Phys. Core 5, 023 (2022).
  * [28] P. Sierant, M. Schirò, M. Lewenstein, and X. Turkeshi, Measurement-induced phase transitions in ($d+1$)- dimensional stabilizer circuits, Phys. Rev. B 106, 214316 (2022).
  * [29] M. Block, Y. Bao, S. Choi, E. Altman and N. Y. Yao, Measurement-induced transition in long-range interacting quantum circuits, Phys. Rev. Lett. 128, 010604 (2022).
  * [30] B. J. J. Khor, M. Wampler, G. Refael, and I. Klich, Measurement-induced chirality: Diffusion and disorder, Phys. Rev. B 108, 214305 (2023).
  * [31] M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear Sigma Models for Monitored Dynamics of Free Fermions, Phys. Rev. X 13, 041045 (2023).
  * [32] I. Poboiko, P. Pöpperl, I. V. Gornyi, and A. D. Mirlin, Theory of Free Fermions under Random Projective Measurements, Phys. Rev. X 13, 041046 (2023).
  * [33] R. Morral-Yepes, A. Smith, S. L. Sondhi, and F. Pollmann, Entanglement Transitions in Unitary Circuit Games, PRX Quantum 5, 010309 (2024).
  * [34] G. Lee, T. Jin, Y. Wang, A. McDonald, and A. Clerk, Entanglement Phase Transition Due to Reciprocity Breaking without Measurement or Postselection, PRX Quantum 5, 010313 (2024).
  * [35] C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Measurement-induced quantum phases realized in a trapped-ion quantum computer, Nat. Phys. 18, 760 (2022).
  * [36] J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nat. Phys. 19, 1314 (2023).
  * [37] Google Quantum AI and Collaborators, Measurement-induced entanglement and teleportation on a noisy quantum processor, Nature (London) 622, 481 (2023).
  * [38] Á. Bácsi and B. Dóra, Dynamics of entanglement after exceptional quantum quench, Phys. Rev. B 103, 085137 (2021).
  * [39] S. Gopalakrishnan and M. J. Gullans, Entanglement and purification transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 126, 170503 (2021).
  * [40] X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schirò, Entanglement transitions from stochastic resetting of non-Hermitian quasiparticles, Phys. Rev. B 105, L241114 (2022).
  * [41] X. Turkeshi and M. Schirò, Entanglement and correlation spreading in non-Hermitian spin chains, Phys. Rev. B 107, L020403 (2023).
  * [42] K. Kawabata, T. Numasawa, and S. Ryu, Entanglement phase transition induced by the non-Hermitian skin effect, Phys. Rev. X 13, 021007 (2023).
  * [43] Y. L. Gal, X. Turkeshi and M. Schirò, Volume-to-area law entanglement transition in a non-Hermitian free fermionic chain, SciPost Phys. 14, 138 (2023).
  * [44] E. Granet, C. Zhang, and H. Dreyer, Volume-law to area-law entanglement transition in a nonunitary periodic Gaussian circuit, Phys. Rev. Lett. 130, 230401 (2023).
  * [45] K. Li, Z.-C. Liu, and Y. Xu, Disorder-Induced Entanglement Phase Transitions in Non-Hermitian Systems with Skin Effects, arXiv:2305.12342.
  * [46] T. Qian and L. Zhou, Correlation-induced phase transitions and mobility edges in a non-Hermitian quasicrystal, arXiv:2310.01275.
  * [47] L. Zhou, Entanglement phase transitions in non-Hermitian quasicrystals, Phys. Rev. B 109, 024204 (2024).
  * [48] S. Li, X. Yu, and Z. Li, Emergent entanglement phase transitions in non-Hermitian Aubry-André-Harper chains, Phys. Rev. B 109, 024306 (2024).
  * [49] L. Zhou, Entanglement phase transitions in non-Hermitian Floquet systems, arXiv:2310.11351.
  * [50] K. Kawabata, Y. Ashida, H. Katsura, and M. Ueda, Parity-time-symmetric topological superconductor, Phys. Rev. B 98, 085116 (2018).
  * [51] A. McDonald, T. Pereg-Barnea, and A. A. Clerk, Phase-Dependent Chiral Transport and Effective Non-Hermitian Dynamics in a Bosonic Kitaev-Majorana Chain, Phys. Rev. X 8, 041031 (2018).
  * [52] M. T. van Caspel, S. E. T. Arze, and I. P. Castillo, Dynamical signatures of topological order in the driven-dissipative Kitaev chain, SciPost Phys. 6, 026 (2019).
  * [53] S. Lieu, Non-Hermitian Majorana modes protect degenerate steady states, Phys. Rev. B 100, 085110 (2019).
  * [54] N. Okuma and M. Sato, Topological Phase Transition Driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics, Phys. Rev. Lett. 123, 097701 (2019).
  * [55] L. Zhou, Non-Hermitian Floquet topological superconductors with multiple Majorana edge modes, Phys. Rev. B 101, 014306 (2020).
  * [56] L. Zhou and Q. Du, Non-Hermitian topological phases and dynamical quantum phase transitions: a generic connection, New J. Phys. 23, 063041 (2021).
  * [57] L. Zhou and D. Zhang, Non-Hermitian Floquet Topological Matter—A Review, Entropy 25, 1401 (2023).
  * [58] A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Sov. Phys. Usp. 44, 131 (2001).
  * [59] B. Zhu, Y. Ke, H. Zhong, and C. Lee, Dynamic winding number for exploring band topology, Phys. Rev. Res. 2, 023043 (2020).
  * [60] I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A: Math. Theor. 42, 504003 (2009).
  * [61] C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424, 44 (1994).
  * [62] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 2004, P06002 (2004).
  * [63] P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A: Math. Theor. 42, 504005 (2009).
  * [64] P. G. de Gennes, _Superconductivity of Metals and Alloys_ (CRC, New York, 1999).