[92]. Despite this striking difference, the dynamics of both models have
important similarities in the frequent-measurement limit: emergent
classicality and a suppression of coherent dynamics, which are hallmarks of
the quantum Zeno effect for occupation measurements, and which we attribute to
a generalized Zeno effect for fermion counting. These similarities explain why
both models exhibit close to identical results in various observables in the
steady state, including density correlations, the entanglement entropy, and
the bipartite mutual information. We have found qualitative differences only
in the tripartite mutual information. Combining analytical insights from
replica Keldysh field theory with numerical results from simulations of
quantum jump trajectories, we have provided strong evidence for the absence of
a measurement-induced entanglement transition in these models. In particular,
we have shown that characteristic signatures of a critical phase can be
observed only within a well-defined and finite range of length scales.
Crucially, the upper boundary $l_{c}\sim\gamma^{-2}$ of this critical range is
only algebraically large in the measurement rate, and, therefore, observable
in numerics. Thus, our work contributes to clarifying the question of the
existence of a measurement-induced phase transition in one-dimensional free
fermions with particle number conservation.

Our work opens up several interesting directions for future research. While
there is no measurement-induced phase transition in the considered models in
one spatial dimension, such a transition does occur in higher dimensions [34,
38]. It will be interesting to study the impact of the breaking of particle
number conservation by fermion counting on dynamical critical behavior. A
classification of _dynamical_ criticality based on symmetries and conservation
laws—in analogy to the classification of dynamical criticality in
thermodynamic equilibrium [93]—is an interesting prospect. However, exploring
this possibility analytically by studying, for example, two-time correlation
functions such as
$\overline{\left\langle\hat{A}(t)\right\rangle\left\langle\hat{B}(t^{\prime})\right\rangle}$
will require a nontrivial extension of the replica Keldysh formalism.

From a more formal perspective, the stochastic Schrödinger equation (3) can be
regarded as an unravelling of a quantum master equation that is recovered by
taking the average over the stochastic increments $\mathrm{d}N_{\alpha,l}(t)$.
While here we have considered cases for which the stationary state of the
unconditional dynamics described by the master equation is trivial, this is
generically not the case, and jump operators can be chosen so as to prepare a
state of interest. For example, consider a one-dimensional fermionic chain
that is connected to particle reservoirs only at its ends and in such a way
that a finite current is driven through the system. An unravelling of the
corresponding master equation represents an interesting setup to study the
interplay of transport and entanglement dynamics on the level of individual
quantum trajectories. However, such an unravelling can, in general, not be
formulated as a random generalized measurement, and the derivation of a
replica Keldysh field theory described in Sec. V has to be modified.

Finally, it is an interesting perspective to investigate quantum jump
unravellings of quadratic fermionic Liouvillians belonging to different
symmetry classes [94, 95, 96, 97]. The resulting replica Keldysh field
theories and NLSMs will shed further light on the connections between the
dynamics of continuously monitored quantum systems in $d$ spatial dimensions
and the physics of Anderson localization in $d+1$ dimensions [98].

###### Acknowledgements.

E.S. and L.S. acknowledge support from the Austrian Science Fund (FWF) through
the projects P 33741-N and 10.55776/COE1, and from the European Union \-
NextGenerationEU. For open access purposes, the authors have applied a CC BY
public copyright license to any author accepted manuscript version arising
from this submission.

##  Appendix A Minimal physical models and their description through a
stochastic Schrödinger equation

In this appendix, we outline the derivation of the stochastic Schrödinger
equation (3) for minimal physical models of fermion counting and generalized
occupation measurements.

###  A.1 Fermion counting

We consider the setup of a quantum dot coupled to two reservoirs described in
Sec. III.1 [60]. The quantum dot corresponds to a single fermionic lattice
site, with annihilation and creation operators $\hat{\psi}$ and
$\hat{\psi}^{\dagger}$, respectively, and Hamiltonian

$\hat{H}_{0}=\omega_{0}\hat{\psi}^{\dagger}\hat{\psi}.$ (120)

We denote the reservoir annihilation and creation operators by
$\hat{b}_{\alpha,k}$ and $\hat{b}_{\alpha,k}^{\dagger}$, where $\alpha=-$ for
the reservoir that acts as the drain, $\alpha=+$ for the reservoir that acts
as the source, and $k$ is a discrete index that labels the reservoir modes.
The Hamiltonians of the reservoirs modes with energies $\omega_{\alpha,k}$
read

$\hat{H}_{\alpha}=\sum_{k}\omega_{\alpha,k}\hat{b}_{\alpha,k}^{\dagger}\hat{b}_{\alpha,k},$
(121)

and the reservoirs are coupled to the quantum dot by

$\hat{H}_{\mathrm{coup}}=\mathrm{i}\sum_{\alpha,k}\left(T_{\alpha,k}\hat{\psi}^{\dagger}\hat{b}_{\alpha,k}-T_{\alpha,k}^{\ast}\hat{b}_{\alpha,k}^{\dagger}\hat{\psi}\right),$
(122)

with tunneling coefficients $T_{\alpha,k}$. Both reservoirs are assumed to be
in thermodynamic equilibrium at a low temperature, with chemical potentials
chosen such that the reservoir modes at the energy $\omega_{0}$ are empty in
the drain and occupied in the source. We model this scenario by assuming the
dot and reservoirs to be in the pure state
$\lvert\Psi(0)\rangle=\lvert\psi(0)\rangle\otimes\lvert
0_{-}\rangle\otimes\lvert 1_{+}\rangle$ at time $t=0$, with $\lvert
0_{-}\rangle$ and $\lvert 1_{+}\rangle$ denoting empty and occupied states of
the drain and source, respectively.

Our goal is to derive a stochastic Schrödinger equation describing the
dynamics of the quantum dot. As a first step, we move to an interaction frame
with respect to the bath Hamiltonians, Eq. (121), and integrate the
Schrödinger equation for the state $\lvert\Psi(t)\rangle$ of the dot and
reservoirs in small time steps $\Delta t$. In Born and Markov approximations,
the increment of the wave function,
$\Delta\lvert\Psi(t)\rangle=\lvert\Psi(t+\Delta
t)\rangle-\lvert\Psi(t)\rangle$, is given by [99]

$\Delta\lvert\Psi(t)\rangle=\left[-\mathrm{i}\hat{H}_{\mathrm{eff}}\Delta
t+\sqrt{2}\hat{L}_{-}\Delta\hat{B}_{-}^{\dagger}(t)+\sqrt{2}\hat{L}_{+}\Delta\hat{B}_{+}(t)\right]\lvert\Psi(t)\rangle.$
(123)

The effective Hamiltonian reads

$\hat{H}_{\mathrm{eff}}=\hat{H}_{0}-\mathrm{i}\sum_{\alpha=\pm}\hat{L}_{\alpha}^{\dagger}\hat{L}_{\alpha},$
(124)

where $\hat{L}_{-}=\sqrt{\gamma_{-}/2}\,\hat{\psi}$ and
$\hat{L}_{+}=\sqrt{\gamma_{+}/2}\,\hat{\psi}^{\dagger}$ are the jump operators
defined in Eq. (4) for a single lattice site. Further, in Eq. (123), we have
introduced quantum Itô increments,

$\Delta\hat{B}_{\alpha}(t)=\sqrt{\frac{2}{\gamma_{\alpha}}}\int_{t}^{t+\Delta
t}\mathrm{d}t^{\prime}\sum_{k}T_{\alpha,k}\hat{b}_{\alpha,k}\mathrm{e}^{-\mathrm{i}\omega_{\alpha,k}t^{\prime}},$
(125)

which obey the anticommutation relations

$\left\\{\Delta\hat{B}_{\alpha}^{\vphantom{\dagger}}(t),\Delta\hat{B}_{\alpha}^{\dagger}(t^{\prime})\right\\}=\Delta
t\,\delta_{t,t^{\prime}}^{\vphantom{\dagger}},$ (126)

where $\delta_{t,t^{\prime}}$ is the Kronecker delta for discrete times $t$
and $t^{\prime}$. Finally, the loss and gain rates appearing in Eq. (123) are
defined as $\gamma_{\alpha}=2\pi\rho_{\pm}(\omega_{0})\left\lvert
T_{\pm}(\omega_{0})\right\rvert$, where $\rho_{\alpha}(\omega_{0})$ and
$T_{\alpha}(\omega_{0})$ are the density of states and tunneling amplitudes at
the energy of the dot, which are introduced upon taking the continuum limit
such that
$\sum_{k}\to\int_{0}^{\infty}\mathrm{d}\omega\,\rho_{\alpha}(\omega)$ and
$T_{\alpha,k}\to T_{\alpha}(\omega)$ [60].

The operators $\hat{L}_{-}\Delta\hat{B}_{-}^{\dagger}(t)$ and
$\hat{L}_{+}\Delta\hat{B}_{+}(t)$ in Eq. (123) describe the transfer of a
fermion from the quantum dot to the drain and from the source to the dot,
respectively. Since the Itô increments anticommute at different times, a
fermion that has been transferred from the dot to the drain at time $t$ cannot
be transferred back at a later time $t^{\prime}$. Formally, the state
$\Delta\hat{B}_{-}^{\dagger}(t)\lvert 0_{-}\rangle$ is annihilated by
$\Delta\hat{B}_{-}(t^{\prime})$. An analogous argument applies to the transfer
of fermions from the source to the dot. Therefore, in Eq. (123), the operators
$\Delta\hat{B}_{-}(t)$ and $\Delta\hat{B}_{+}^{\dagger}(t)$ do not occur.

In analogy to photon counting in quantum optics, fermion counting is
implemented by measuring the number operators,

$\hat{N}_{\alpha}(t)=\frac{\Delta\hat{B}_{\alpha}^{\dagger}(t)}{\sqrt{\Delta
t}}\frac{\Delta\hat{B}_{\alpha}(t)}{\sqrt{\Delta t}},$ (127)

at each time step, with possible measurement outcomes $N_{\alpha}(t)=0,1$. To
describe the statistics of the measurement outcomes and the change of the
state of the quantum dot for a given outcome, it is convenient to rewrite Eq.
(123) in the form

$\lvert\Psi(t+\Delta
t)\rangle=\left[\hat{M}_{0}+\hat{M}_{-}\frac{\Delta\hat{B}_{-}^{\dagger}(t)}{\sqrt{\Delta
t}}+\hat{M}_{+}\frac{\Delta\hat{B}_{+}(t)}{\sqrt{\Delta
t}}\right]\lvert\Psi(t)\rangle,$ (128)

where the measurement operators are defined as

$\hat{M}_{0}=1-\mathrm{i}\hat{H}_{\mathrm{eff}}\Delta
t,\quad\hat{M}_{-}=\sqrt{2\Delta
t}\,\hat{L}_{-},\quad\hat{M}_{+}=\sqrt{2\Delta t}\,\hat{L}_{+}.$ (129)

A measurement of the number operators, Eq. (127), performed on the state
$\lvert\Psi(t+\Delta t)\rangle$, Eq. (128), yields $N_{-}(t)=0$ and
$N_{+}(t)=1$ with probability

$p_{0}(t)=\left\langle\psi(t)\middle|\hat{M}_{0}^{\dagger}\hat{M}_{0}\middle|\psi(t)\right\rangle.$
(130)

After a measurement with this outcome, the state of the quantum dot is given
by

$\lvert\psi(t+\Delta
t)\rangle=\frac{\hat{M}_{0}\lvert\psi(t)\rangle}{\sqrt{p_{0}(t)}}=\left[1-\mathrm{i}\hat{H}_{\mathrm{eff}}\Delta
t+\sum_{\alpha=\pm}\tilde{p}_{\alpha}(t)\Delta t\right]\lvert\psi(t)\rangle,$
(131)

where we have expanded the denominator up to order $\Delta t$, and where, as
in Eq. (5), we have defined

$\tilde{p}_{\alpha}(t)=\left\langle\psi(t)\middle|\hat{L}_{\alpha}^{\dagger}\hat{L}_{\alpha}\middle|\psi(t)\right\rangle.$
(132)

The measurement outcome above and the change of the state in Eq. (131)
correspond to the case that no quantum jump has occurred during the time step,
and the states of drain and source remain unchanged. In contrast, for
$N_{-}(t)=1$ and $N_{+}(t)=1$, one fermion has been transferred from the
quantum dot to the drain. This happens with probability
$p_{-}(t)=2\tilde{p}_{-}(t)\Delta t$, and the state after the measurement is

$\lvert\psi(t+\Delta
t)\rangle=\frac{\hat{M}_{-}\lvert\psi(t)\rangle}{\sqrt{p_{-}(t)}}=\frac{\hat{L}_{-}\lvert\psi(t)\rangle}{\sqrt{\tilde{p}_{-}(t)}}.$
(133)

For $N_{-}(t)=0$ and $N_{+}(t)=0$, a fermion has been transferred from the
source to the dot, which happens with probability
$p_{+}(t)=2\tilde{p}_{+}(t)\Delta t$. Then, the state after the measurement is

$\lvert\psi(t+\Delta
t)\rangle=\frac{\hat{M}_{+}\lvert\psi(t)\rangle}{\sqrt{p_{+}(t)}}=\frac{\hat{L}_{+}\lvert\psi(t)\rangle}{\sqrt{\tilde{p}_{+}(t)}}.$
(134)

Performing measurements of the number operators, Eq. (127), in each time step
results in stochastic dynamics of the wave function of the quantum dot,
conditional to the measurement outcomes. To formulate these dynamics as an
evolution equation, we define stochastic increments $\Delta N_{\alpha}(t)=0,1$
that count the number of fermions that are transferred from the dot to the
drain and from the source to the dot during the step from $t$ to $t+\Delta t$.
Neglecting terms of order $\Delta N_{\alpha}(t)\Delta t$ [60], we can thus
write the increment of the wave function as

$\Delta\lvert\psi(t)\rangle=\left\\{\left[-\mathrm{i}\hat{H}_{\mathrm{eff}}+\sum_{\alpha=\pm}\tilde{p}_{\alpha}(t)\right]\Delta
t\right.\\\
\left.+\sum_{\alpha=\pm}\left[\frac{\hat{L}_{\alpha}}{\sqrt{\tilde{p}_{\alpha}(t)}}-1\right]\Delta
N_{\alpha}(t)\right\\}\lvert\psi(t)\rangle.$ (135)

In the limit of infinitesimal time steps, $\Delta t\to\mathrm{d}t$, the
stochastic increments $\Delta N_{\alpha}(t)\to\mathrm{d}N_{\alpha}(t)$
describe a Poisson point process with the properties specified in Sec. III.1
[100], and the evolution of the quantum dot is then described by the
stochastic Schrödinger equation (3) with $L=1$ and where we omit the
contribution from $\hat{H}_{0}$, Eq. (120), which leads to a global phase
factor for each trajectory.

###  A.2 Generalized occupation measurements

Measurements of the occupation of a quantum dot can be implemented by
modifying the above setup as outlined in Sec. III.1: Instead of tunneling from
the reservoirs to the dot as described by Eq. (122), we consider tunneling
directly from the source to the drain which, however, is possible only when
the dot is occupied [60],

$\hat{H}_{\mathrm{coup}}=\mathrm{i}\sum_{k,q}\hat{\psi}^{\dagger}\hat{\psi}\left(\chi\hat{b}_{-,k}^{\dagger}\hat{b}_{+,q}+\chi^{\ast}\hat{b}_{+,q}^{\dagger}\hat{b}_{-,k}\right).$
(136)

This coupling Hamiltonian, with a mode-independent tunneling rate $\chi$, is a
simplified version of a setup that has been realized in mesoscopic devices,
for example, in Ref. [64].

As above, we integrate the Schrödinger equation for the dot and reservoirs in
small time steps, leading to

$\Delta\lvert\Psi(t)\rangle=\left[-\mathrm{i}\hat{H}_{\mathrm{eff}}\Delta
t+\sqrt{2}\hat{L}\Delta\hat{B}(t)\right]\lvert\Psi(t)\rangle,$ (137)

where $\hat{n}=\hat{\psi}^{\dagger}\hat{\psi}$ and
$\hat{L}=\sqrt{\gamma}\,\hat{n}$. The measurement rate is
$\gamma=2\pi\rho_{-}(\omega_{0})\rho_{+}(\omega_{0})\left\lvert\chi\right\rvert^{2}\left(\mu_{+}-\mu_{-}\right)$,
where $\mu_{\alpha}$ are the chemical potentials of the reservoirs, and the
effective Hamiltonian is given by

$\hat{H}_{\mathrm{eff}}=\hat{H}_{0}-\mathrm{i}\hat{L}^{\dagger}\hat{L}.$ (138)

We have again introduced quantum Itô increments,

$\Delta\hat{B}(t)=\sqrt{\frac{\gamma}{2}}\chi^{*}\int_{t}^{t+\Delta
t}\mathrm{d}t^{\prime}\sum_{k,q}\hat{b}_{-,k}^{\dagger}\hat{b}_{+,q}\mathrm{e}^{-\mathrm{i}\left(\omega_{+,q}-\omega_{-,k}\right)t^{\prime}}.$
(139)

which obey canonical anticommutation relations,

$\left\\{\Delta\hat{B}(t),\Delta\hat{B}^{\dagger}(t^{\prime})\right\\}=\Delta
t\,\delta_{t,t^{\prime}}.$ (140)

From here on, the derivation of the stochastic Schrödinger equation describing
the dynamics of the quantum dot is analogous to the one for fermion counting
outlined above.

##  Appendix B Regularization, causality structure, and normalization in
Keldysh field theory

Here, we address points (i) and (ii) listed below Eq. (46). Further, we
discuss the causality structure and normalization of the Keldysh partition
function in the replica limit $R\to 1$.

###  B.1 Regularization of the Keldysh action

(i) In the construction of the Keldysh functional integral, in Eq. (36), time
evolution from $t_{0}$ to $t$ is split into $N$ time steps of duration $\Delta
t=(t-t_{0})/N$. At each discrete time $t_{n}=t_{0}+\Delta tn$, a resolution of
the identity in terms of coherent states,
$\lvert\pm\psi_{\pm,n}\rangle=\mathrm{e}^{\pm\sum_{r=1}^{R}\sum_{l=1}^{L}\psi_{\pm,r,l,n}\hat{\psi}_{r,l}^{\dagger}}\lvert\Omega\rangle$,
where $\lvert\Omega\rangle$ is the vacuum state, is inserted. If a measurement
is performed between times $t_{n}$ and $t_{n+1}$, we thus obtain a matrix
element of the measurement operator between the coherent states
$\lvert\psi_{\pm,n}\rangle$ and $\lvert\psi_{\pm,n+1}\rangle$. For generalized
occupation measurements with measurement operators given in Eq. (16), the
matrix elements on the forward and backward branches contain the factors

$\begin{split}\left\langle\psi_{+,n+1}|\hat{n}_{r,l}|\psi_{+,n}\right\rangle&=\bar{\psi}_{+,r,l,n+1}\psi_{+,r,l,n}\left\langle\psi_{+,n+1}|\psi_{+,n}\right\rangle,\\\
\left\langle-\psi_{-,n}|\hat{n}_{r,l}|-\psi_{-,n+1}\right\rangle&=\bar{\psi}_{-,r,l,n}\psi_{-,r,n+1,l}\left\langle-\psi_{-,n}|-\psi_{-,n+1}\right\rangle,\end{split}$
(141)

where the fields that represent the operators $\hat{n}_{r,l}$ are one discrete
time step apart. However, as shown in Ref. [30], in the functional integral,
we can replace

$\begin{split}\bar{\psi}_{+,r,l,n+1}\psi_{+,r,l,n}&\to\bar{\psi}_{+,r,l,n}\psi_{+,r,l,n}+1,\\\
\bar{\psi}_{-,r,l,n}\psi_{-,r,l,n+1}&\to\bar{\psi}_{-,r,l,n}\psi_{-,r,l,n}+1,\end{split}$
(142)

where now the fields are evaluated at equal times. This is a prerequisite for
introducing fermionic bilinears at equal times as new variables. The vertex in
Eq. (44) is thus modified as

$V[\bar{\psi}_{r},\psi_{r}]=\frac{1}{n}\left(\bar{\psi}_{+,r}\psi_{+,r}+1\right)\left(\bar{\psi}_{-,r}\psi_{-,r}+1\right).$
(143)

For fermion counting, with the linear measurement operators in Eq. (15), this
issue does not arise.

(ii) In the discrete-time formulation of the theory, the Green’s function is
defined as
$\mathrm{i}G_{n,n^{\prime}}=\left\langle\psi_{n}\bar{\psi}_{n^{\prime}}\right\rangle$
[78]. Working in the basis introduced through the Larkin-Ovchinnikov rotation,
Eq. (45), the continuous-time limit is given by

$G(t,t^{\prime})=\lim_{n,n^{\prime},N\to\infty}\left(G_{n,n^{\prime}}+\frac{\mathrm{i}}{2}\delta_{n,n^{\prime}}\sigma_{x}\right),$
(144)

where the limit is taken such that $t_{n}\to t$ and $t_{n^{\prime}}\to
t^{\prime}$. In the last term, we leave the identity matrices in replica and
lattice space implicit. The shift
$\mathrm{i}\delta_{n,n^{\prime}}\sigma_{x}/2$ on the right-hand side leads to
the continuous-time Green’s function at equal times being given by the
symmetrized limit as in Eq. (46),

$G(t,t)=\frac{1}{2}\left[G(t,t+0^{+})+G(t,t-0^{+})\right].$ (145)

In turn, this implies that the Keldysh Green’s function is continuous at
$t=t^{\prime}$, and that the anti-Keldysh component of the Green’s function
vanishes at all times—which is not the case for the discrete-time Green’s
function $G_{n,n^{\prime}}$ that has a nonvanishing anti-Keldysh component at
$n=n^{\prime}$ [78].

For most applications of Keldysh field theory, the shift on the right-hand
side of Eq. (144) is irrelevant and can be omitted. However, here we have to
take it into account to ensure normalization of the Keldysh partition function
as discussed below. The shift can be incorporated in the action by switching
to a regularization in which the free discrete-time Green’s function is
$G^{\mathrm{(reg)}}_{n,n^{\prime}}=G_{n,n^{\prime}}+\mathrm{i}\delta_{n,n^{\prime}}\sigma_{x}/2$
[58]. Changing the regularization is achieved through the following relation:

$\begin{split}Z_{R}(t)&=\int\frac{\mathrm{D}[\bar{\psi},\psi]}{\mathop{\mathrm{Det}}\\!\left(-\mathrm{i}G_{0}^{-1}\right)}\,\mathrm{e}^{\mathrm{i}\left(\bar{\psi}G_{0}^{-1}\psi+\gamma\int\mathrm{d}^{2}\mathbf{x}\,\mathcal{L}_{M}[\bar{\psi}_{l},\psi_{l}]\right)}\\\
&=\int\frac{\mathrm{D}[\bar{\psi},\psi]}{\mathop{\mathrm{Det}}\\!\left(-\mathrm{i}G_{0}^{\mathrm{(reg)}-1}\right)}\,\mathrm{e}^{\mathrm{i}\left(\bar{\psi}G_{0}^{\mathrm{(reg)}-1}\psi+\gamma\int\mathrm{d}^{2}\mathbf{x}\,\mathcal{L}_{M}^{\mathrm{(reg)}}[\bar{\psi}_{l},\psi_{l}]\right)},\end{split}$
(146)

where we make the determinants that are usually absorbed in the integration
measure explicit [78]. The regularized measurement Lagrangian can be
constructed order by order in an expansion in $\gamma$. To first order,
$\mathcal{L}_{M}^{\mathrm{(reg)}}$ is obtained by adding the sum over all
partial contractions, where each contraction is
$\left\langle\psi_{n}\bar{\psi}_{n^{\prime}}\right\rangle=\delta_{n,n^{\prime}}\sigma_{x}/2$,
to the original measurement Lagrangian $\mathcal{L}_{M}$. Since these
contractions are diagonal in replica space, they can be added to each factor
of the product over replicas in Eq. (42), that is, to the vertices Eqs. (43)
and (143). As stated in the main text, the vertices for fermion counting do
not change under the regularization. The regularized vertex for occupation
measurements is given in Eq. (48).

###  B.2 Causality structure and normalization of the Keldysh partition
function

For closed quantum systems undergoing unitary dynamics without measurements,
the Keldysh partition function is normalized to unity,
$Z(t)=\mathop{\mathrm{tr}}(\hat{\rho}(t))=1$, as follows from the
normalization of the density matrix $\hat{\rho}(t)$. However, the replica
Keldysh partition function $Z_{R}(t)$ is defined in Eq. (36) in terms of $R$
copies of the nonnormalized density matrix $\hat{D}(t)$. In the replica limit
$R\to 1$, the completeness relation of measurement operators implies again
that $Z_{R}(t)\to 1$. For $R>1$, on the other hand, the replica Keldysh
partition function is not normalized. This is because the products of
measurement operators over $r$ replicas, which occur in
$\otimes_{r=1}^{R}\hat{D}_{r}(t)$, do not obey a completeness relation.

In Keldysh field theory for closed systems, the normalization of the partition
function is reflected in the causality structure of the action, in particular,
in the property that the action vanishes for $\psi_{+}=\psi_{-}$ and
$\bar{\psi}_{+}=\bar{\psi}_{-}$ [77, 78, 79, 80]. To see this, we consider a
system with Hamiltonian $\hat{H}$. The Keldysh partition function is
$Z(t)=\int\mathrm{D}[\bar{\psi},\psi]\,\mathrm{e}^{\mathrm{i}S}$ where
$S=\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\mathcal{L}[\bar{\psi},\psi]$, and with
the Lagrangian

$\mathcal{L}[\bar{\psi},\psi]=\sum_{\sigma=\pm}\sigma\left(\bar{\psi}_{\sigma}\partial_{t}\psi_{\sigma}-H[\bar{\psi}_{\sigma},\psi_{\sigma}]\right).$
(147)

From $Z(t)=\mathop{\mathrm{tr}}(\hat{\rho}(t))=1$ it follows that

$\partial_{t}Z=\mathrm{i}\left\langle\mathcal{L}[\bar{\psi},\psi]\right\rangle=\mathrm{i}\sum_{\sigma=\pm}\sigma\left\langle\bar{\psi}_{\sigma}\partial_{t}\psi_{\sigma}-H[\bar{\psi}_{\sigma},\psi_{\sigma}]\right\rangle=0.$
(148)

Upon taking the sum over $\sigma=\pm$, both terms in the average vanish
individually. Indeed, the cyclic property of the trace implies that the
expectation value of any operator $\hat{O}$ at time $t$ can be evaluated
equivalently on the forward and on the backward branch of the Keldysh contour,
and, therefore,

$\left\langle\mathcal{L}[\bar{\psi},\psi]\right\rangle=\left\langle\left.\mathcal{L}[\bar{\psi},\psi]\right.\rvert_{\psi_{+}=\psi_{-},\bar{\psi}_{+}=\bar{\psi}_{-}}\right\rangle.$
(149)

Then, $\left\langle\mathcal{L}[\bar{\psi},\psi]\right\rangle=0$ follows for
the Lagrangian of a closed system in Eq. (147) from the causality structure,

$\left.\mathcal{L}[\bar{\psi},\psi]\right.\rvert_{\psi_{+}=\psi_{-},\bar{\psi}_{+}=\bar{\psi}_{-}}=0.$
(150)

This shows that the causality structure of the action guarantees the
normalization of the Keldysh partition function.

As mentioned in Sec. V.2, the term $\bar{\psi}_{2}\psi_{1}$ in the regularized
vertex for occupation measurements, Eq. (48), violates the usual causality
structure. Inverting the Larkin-Ovchinnikov rotation, Eq. (45), shows that
this term couples the forward and backward branches. The above argument based
on the cyclic property of the trace does, in general, not apply, and Eqs.
(149) and (150) do not hold for such terms. To see that the partition function
is normalized even though the action does not obey the usual causality
structure, one should note that according to Eq. (148), normalization is
ensured when the expectation value of the Lagrangian vanishes. We will now
confirm that this condition is met.

The free evolution described by the first term in the Keldysh action in Eq.
(41) obeys the causality structure for any value of $R$, and, therefore, we
only need to consider the measurement Lagrangian Eq. (42). For $R=1$, we thus
require

$\left\langle\mathrm{i}\mathcal{L}_{M}[\bar{\psi},\psi]\right\rangle=\sum_{n=\pm}\left\langle
V_{n}[\bar{\psi},\psi]\right\rangle-1=0.$ (151)

We first check this condition for fermion counting. Using the form of the
vertices given in Eq. (47), we obtain

$\left\langle\mathrm{i}\mathcal{L}_{M}[\bar{\psi}_{l},\psi_{l}]\right\rangle=\mathrm{i}\left[G_{l,l}^{R}(t,t)-G_{l,l}^{A}(t,t)\right]-1.$
(152)

To evaluate the Green’s functions at equal times, we express them through
fermionic field operators [77]:

$\begin{split}\mathrm{i}G_{l,l^{\prime}}^{R}(t,t^{\prime})&=\left\langle\psi_{1,l}(t)\bar{\psi}_{1,l^{\prime}}^{*}(t^{\prime})\right\rangle=\theta(t-t^{\prime})\left\langle\left\\{\hat{\psi}_{l}(t),\hat{\psi}_{l^{\prime}}^{\dagger}(t^{\prime})\right\\}\right\rangle,\\\
\mathrm{i}G^{A}_{l,l^{\prime}}(t,t^{\prime})&=\left\langle\psi_{2,l}(t)\bar{\psi}_{2,l^{\prime}}^{*}(t^{\prime})\right\rangle=-\theta(t^{\prime}-t)\left\langle\left\\{\hat{\psi}_{l}(t),\hat{\psi}_{l^{\prime}}^{\dagger}(t^{\prime})\right\\}\right\rangle,\\\
\mathrm{i}G_{l,l^{\prime}}^{K}(t,t^{\prime})&=\left\langle\psi_{1,l}(t)\bar{\psi}_{2,l^{\prime}}^{*}(t^{\prime})\right\rangle=\left\langle\left[\hat{\psi}_{l}(t),\hat{\psi}_{l^{\prime}}^{\dagger}(t^{\prime})\right]\right\rangle,\end{split}$
(153)

where two-time averages are defined as in Eq. (20). The values of the Green’s
functions in the symmetrized limit of equal times, Eq. (145), follow directly
from fermionic anticommutation relations and the expectation value
$\left\langle\hat{n}_{l}\right\rangle_{\mathrm{ss}}$ in the unconditional
steady state. According to Sec. III.1,
$\left\langle\hat{n}_{l}\right\rangle_{\mathrm{ss}}=n$ for occupation
measurements, with $n=1/2$ for fermion counting with equal rates of loss and
gain. We thus obtain

$\mathrm{i}G_{l,l^{\prime}}^{R/A}(t,t)=\pm\frac{1}{2}\delta_{l,l^{\prime}},\quad\mathrm{i}G_{l,l^{\prime}}^{K}(t,t)=\left(1-2n\right)\delta_{l,l^{\prime}}.$
(154)

Inserting these values in Eq. (152) yields
$\left\langle\mathrm{i}\mathcal{L}_{M}[\bar{\psi}_{l},\psi_{l}]\right\rangle=0$,
confirming that the Keldysh partition function is normalized.

Finally, we check Eq. (151) for occupation measurements. Since the regularized
vertex Eq. (48) is quartic, the expectation value in Eq. (151) cannot be
evaluated exactly. At lowest order in $\gamma$, we can use Wick’s theorem to
obtain

$\left\langle\mathrm{i}\mathcal{L}_{M}[\bar{\psi}_{l},\psi_{l}]\right\rangle=\frac{1}{n}\left[\frac{1}{4}-\frac{\mathrm{i}}{2}G^{K}_{l,l}(t,t)+G^{R}_{l,l}(t,t)G^{A}_{l,l}(t,t)\right]-1,$
(155)

where now the Green’s functions are those of the system without measurements.
However, Eq. (154) still applies, and we again find
$\left\langle\mathrm{i}\mathcal{L}_{M}[\bar{\psi}_{l},\psi_{l}]\right\rangle=0$.

##  Appendix C Symmetry of the measurement Lagrangian under
$\mathcal{R}_{\Phi}$

Rotations of the form
$\mathcal{R}_{\Phi}=\mathrm{e}^{\mathrm{i}\Phi\sigma_{x}/2}$ with a traceless
Hermitian matrix $\Phi$ are a symmetry of the measurement Lagrangian for
fermion counting, Eq. (51). To prove this statement, we first perform two
auxiliary calculations. We define the adjoint action of a matrix $A$ on a
matrix $B$ as $\mathcal{A}_{A}B=\left[A,B\right]$. Then,

$\begin{split}\mathcal{R}_{\Phi}^{-1}\tau_{\pm}\mathcal{R}_{\Phi}&=\sum_{n=0}^{\infty}\frac{\left(-\mathrm{i}\right)^{n}}{n!}\mathcal{A}_{\Phi\sigma_{x}/2}^{n}\tau_{\pm}=\sum_{n=0}^{\infty}\frac{\left(\mp\mathrm{i}\Phi\right)^{n}}{n!}\tau_{\pm}=\tau_{\pm}\mathrm{e}^{\mp\mathrm{i}\Phi},\end{split}$
(156)

where we have used that $\tau_{\pm}$ is invariant—up to a change of sign—under
multiplication by $\sigma_{x}$ from both the left and the right,
$\sigma_{x}\tau_{\pm}=\pm\tau_{\pm}$ and $\tau_{\pm}\sigma_{x}=\mp\tau_{\pm}$,
which can be combined to $\mathcal{A}_{\sigma_{x}}\tau_{\pm}=\pm 2\tau_{\pm}$.
Next, for a matrix $A$ in Keldysh and replica space, we consider

$\mathop{\mathrm{tr}_{K}}\\!\left(\mathcal{R}_{\Phi}\tau_{\pm}A\mathcal{R}_{\Phi}^{-1}\right)\\\
\begin{aligned}
&=\sum_{n=0}^{\infty}\frac{\mathrm{i}^{n}}{n!}\mathop{\mathrm{tr}_{K}}\\!\left(\mathcal{A}_{\Phi\sigma_{x}/2}^{n}\tau_{\pm}A\right)=\sum_{n=0}^{\infty}\frac{\mathrm{i}^{n}}{n!}\mathcal{A}_{\Phi/2}^{n}\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{x}^{n}\tau_{\pm}A\right)\\\
&=\sum_{n=0}^{\infty}\frac{\left(\pm\mathrm{i}\right)^{n}}{n!}\mathcal{A}_{\Phi/2}^{n}\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\pm}A\right)=\mathrm{e}^{\pm\mathrm{i}\Phi/2}\mathop{\mathrm{tr}_{K}}(\tau_{\pm}A)\mathrm{e}^{\mp\mathrm{i}\Phi/2},\end{aligned}$
(157)

These two results and
$\mathop{\mathrm{det}_{R}}\\!\left(\mathrm{e}^{\mp\mathrm{i}\Phi}\right)=\mathrm{e}^{\mp\mathrm{i}\mathop{\mathrm{tr}_{R}}(\Phi)}=1$
lead to

$\mathop{\mathrm{det}_{R}}\\!\left[\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\pm}\mathcal{R}_{\Phi}\mathcal{G}\mathcal{R}_{\Phi}^{-1}\right)\right]=\mathop{\mathrm{det}_{R}}\\!\left[\mathop{\mathrm{tr}_{K}}\\!\left(\mathcal{R}_{\Phi}\tau_{\pm}\mathrm{e}^{\mp\mathrm{i}\Phi}\mathcal{G}\mathcal{R}_{\Phi}^{-1}\right)\right]\\\
\begin{aligned}
&=\mathop{\mathrm{det}_{R}}\\!\left[\mathrm{e}^{\pm\mathrm{i}\Phi/2}\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\pm}\mathrm{e}^{\mp\mathrm{i}\Phi}\mathcal{G}\right)\mathrm{e}^{\mp\mathrm{i}\Phi/2}\right]\\\
&=\mathop{\mathrm{det}_{R}}\\!\left(\mathrm{e}^{\mp\mathrm{i}\Phi}\right)\mathop{\mathrm{det}_{R}}\\!\left[\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\pm}\mathcal{G}\right)\right]=\mathop{\mathrm{det}_{R}}\\!\left[\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\pm}\mathcal{G}\right)\right],\end{aligned}$
(158)

showing that the transformation
$\mathcal{G}\mapsto\mathcal{R}_{\Phi}\mathcal{G}\mathcal{R}_{\Phi}^{-1}$ is
indeed a symmetry of Eq. (51) for $R>1$.

The above derivation relies on $\tau_{\pm}$ being invariant under
multiplication by $\sigma_{x}$. This invariance does not apply to
multiplication by $\sigma_{y}$ or $\sigma_{z}$. Therefore, rotations with
$\sigma_{x}$ replaced by $\sigma_{y}$ or $\sigma_{z}$ are not symmetries of
the measurement Lagrangian.

##  Appendix D Fluctuation expansion of the Keldysh action

To obtain the variation of the Keldysh action with respect to $\mathcal{G}$
and thereby find the saddle-point manifolds described by Eqs. (64) and (65),
and to derive the action that governs the Gaussian theory discussed in Sec.
VI, we perform an expansion of the Keldysh action around its saddle points.

###  D.1 Fluctuation expansion of the measurement Lagrangian for fermion
counting

The existence of a manifold of saddle points rather than just a single
isolated saddle point is a direct consequence of the symmetries of the action.
Therefore, to establish the full manifold, it is sufficient to show that a
single point on the manifold is indeed a saddle point. For the manifold
described by Eqs. (62) and (64), a convenient choice is given by
$Q=\sigma_{z}$, where we omit the identity $1_{R}$ in replica space for
brevity. We parametrize fluctuations around this configuration as
$\mathcal{G}=-\mathrm{i}\left(\sigma_{z}+\delta
Q_{\mathcal{G}}\right)\\!\big{/}2$. Inserting this expression in Eq. (51), we
obtain

$\mathrm{i}\mathcal{L}_{M}=\frac{1}{2^{R}}\sum_{\alpha=\pm}\mathop{\mathrm{det}_{R}}\\!\left[1-\mathrm{i}\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\alpha}\delta
Q_{\mathcal{G}}\right)\right]-1.$ (159)

By expanding the determinant using

$\mathop{\mathrm{det}}(1+\epsilon
X)=1+\epsilon\mathop{\mathrm{tr}}(X)-\frac{\epsilon^{2}}{2}\left[\mathop{\mathrm{tr}}\\!\left(X^{2}\right)-\mathop{\mathrm{tr}}(X)^{2}\right]+O\\!\left(\epsilon^{3}\right),$
(160)

we find, to zeroth and first order in $\delta Q_{\mathcal{G}}$,

$\mathrm{i}\mathcal{L}_{M}^{(0)}=\frac{1}{2^{R-1}}-1,\qquad\mathrm{i}\mathcal{L}_{M}^{(1)}=\frac{1}{2^{R}}\mathop{\mathrm{tr}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G}}{}\right).$ (161)

There are two contributions of second order. The first one is

$\begin{split}\mathrm{i}\mathcal{L}_{M}^{(2,1)}&=\frac{1}{2^{R+1}}\sum_{\alpha=\pm}\mathop{\mathrm{tr}_{R}}\\!\left[\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\alpha}\delta
Q_{\mathcal{G}}\right)^{2}\right]\\\
&=-\frac{1}{2^{R+2}}\sum_{r,r^{\prime}=1}^{R}\left[\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G},r,r^{\prime}}\right)\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G},r^{\prime},r}\right)\vphantom{\sigma_{y}}\right.\\\
&\mathrel{\hphantom{=}}\left.-\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{y}\delta
Q_{\mathcal{G},r,r^{\prime}}\right)\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{y}\delta
Q_{\mathcal{G},r^{\prime},r}\right)\right].\end{split}$ (162)

For fixed replica indices $r$ and $r^{\prime}$, $\delta
Q_{\mathcal{G},r,r^{\prime}}$ is a $2\times 2$ matrix in Keldysh space. To
simplify Eq. (162) we can, therefore, use the following relation between
determinants and traces of $2\times 2$ matrices $A$ and $B$:

$\det(A+B)=\det(A)+\det(B)+\mathop{\mathrm{tr}}(A)\mathop{\mathrm{tr}}(B)-\mathop{\mathrm{tr}}(AB).$
(163)

This allows us to express the products of traces in Eq. (162) as

$\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G},r,r^{\prime}}\right)\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G},r^{\prime},r}\right)=\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G},r,r^{\prime}}\sigma_{z}\delta
Q_{\mathcal{G},r^{\prime},r}\right)\\\
-\mathop{\mathrm{det}_{K}}\\!\left(\delta Q_{\mathcal{G},r,r^{\prime}}+\delta
Q_{\mathcal{G},r^{\prime},r}\right)+\mathop{\mathrm{det}_{K}}\\!\left(\delta
Q_{\mathcal{G},r,r^{\prime}}\right)+\mathop{\mathrm{det}_{K}}\\!\left(\delta
Q_{\mathcal{G},r^{\prime},r}\right),$ (164)

and analogously for the terms involving $\sigma_{y}$. The terms that contain
$\mathop{\mathrm{det}_{K}}$ cancel in the difference between terms involving
$\sigma_{z}$ and $\sigma_{y}$ in Eq. (162). We thus obtain

$\mathrm{i}\mathcal{L}_{M}^{(2,1)}=-\frac{1}{2^{R+2}}\mathop{\mathrm{tr}}\\!\left[\left(\sigma_{z}\delta
Q_{\mathcal{G}}\right)^{2}-\left(\sigma_{y}\delta
Q_{\mathcal{G}}\right)^{2}\right].$ (165)

The second contribution of second order reads

$\begin{split}\mathrm{i}\mathcal{L}_{M}^{(2,2)}&=-\frac{1}{2^{R+1}}\sum_{\alpha=\pm}\mathop{\mathrm{tr}_{R}}\\!\left[\mathop{\mathrm{tr}_{K}}\\!\left(\tau_{\alpha}\delta
Q_{\mathcal{G}}\right)\right]^{2}\\\
&=\frac{1}{2^{R+2}}\left[\mathop{\mathrm{tr}}\\!\left(\sigma_{z}\delta
Q_{\mathcal{G}}\right)^{2}-\mathop{\mathrm{tr}}\\!\left(\sigma_{y}\delta
Q_{\mathcal{G}}\right)^{2}\right].\end{split}$ (166)

Using the first-order contribution in Eq. (161), we can derive the saddle-
point equation $\delta S/\delta\mathcal{G}=\mathrm{i}2\left(\delta S/\delta
Q_{\mathcal{G}}{}\right)=0$, leading to $\Sigma=\gamma Q/2^{R-1}$ with
$Q=\sigma_{z}$ as expected.

###  D.2 Fluctuation expansion of the measurement Lagrangian for occupation
measurements

To obtain the saddle-point manifold and derive the Gaussian theory of Sec. VI,
it is sufficient to expand the measurement Lagrangian in fluctuations around a
particular saddle point, which can be chosen as $Q=\Lambda$ for occupation
measurements. However, the derivation of the NLSM for occupation measurements
requires an expansion around $Q_{0}$ given in Eq. (95), which reduces to
$Q_{0}=\Lambda$ for $\mathcal{R}_{\phi}=\mathcal{R}_{\theta}=1$. From
$\Lambda$ given in Eq. (63), $Q_{0}$ inherits the properties $Q_{0}^{2}=1$,
$\mathop{\mathrm{tr}_{K}}(Q_{0})=0$, and
$\mathop{\mathrm{det}_{K}}(Q_{0})=-1$. Inserting
$\mathcal{G}=-\mathrm{i}\left(Q_{0}+\delta Q_{\mathcal{G}}\right)\\!\big{/}2$
in Eq. (52), we obtain

$\mathrm{i}\mathcal{L}_{M}=\frac{1}{\left(4n\right)^{R}}\left\\{\mathop{\mathrm{det}}\\!\left[1-\sigma_{x}\left(Q_{0}+\delta
Q_{\mathcal{G}}\right)\right]\right.\\\
\left.+\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{x}\delta
Q_{\mathcal{G}}\right)+2R\left(1-2\rho_{0}\right)\right\\}-1,$ (167)

where $\rho_{0}$ is defined in Eq. (100). A more convenient form for the
expansion of the determinant in $\delta Q_{\mathcal{G}}$ is given by

$\mathop{\mathrm{det}}\\!\left[1-\sigma_{x}\left(Q_{0}+\delta
Q_{\mathcal{G}}\right)\right]\\\
=\left[-\mathop{\mathrm{det}_{K}}(\sigma_{x}-Q_{0})\right]^{R}\mathop{\mathrm{det}}\\!\left[1-\left(\sigma_{x}-Q_{0}\right)^{-1}\delta
Q_{\mathcal{G}}\right].$ (168)

To calculate the inverse $\left(\sigma_{x}-Q_{0}\right)^{-1}$, we use the
following relation, which is valid for any invertible $2\times 2$ matrix $A$:

$A^{-1}=\frac{1}{\det(A)}\left[\mathop{\mathrm{tr}}(A)-A\right].$ (169)

Further, using Eq. (163), we obtain
$\mathop{\mathrm{det}_{K}}(\sigma_{x}-Q_{0})=-4\rho_{0}$. We thus find

$\frac{1}{\left(4n\right)^{R}}\mathop{\mathrm{det}}\\!\left[1-\sigma_{x}\left(Q_{0}+\delta
Q_{\mathcal{G}}\right)\right]\\\
=\frac{\rho_{0}^{R}}{n^{R}}\mathop{\mathrm{det}}\\!\left(1+\frac{Q_{0}-\sigma_{x}}{4\rho_{0}}\delta
Q_{\mathcal{G}}\right).$ (170)

Then, employing Eq. (160) to expand $\mathrm{i}\mathcal{L}_{M}$ to second
order in $\delta Q_{\mathcal{G}}$, we obtain

$\begin{split}\mathrm{i}\mathcal{L}_{M}^{(0)}&=\frac{\rho_{0}^{R}}{n^{R}}+\frac{2R}{\left(4n\right)^{R}}\left(1-2\rho_{0}\right)-1,\\\
\mathrm{i}\mathcal{L}_{M}^{(1)}&=\frac{\rho_{0}^{R-1}}{4n^{R}}\mathop{\mathrm{tr}}\\!\left[\left(Q_{0}-\sigma_{x}\right)\delta
Q_{\mathcal{G}}\right]+\frac{1}{\left(4n\right)^{R}}\mathop{\mathrm{tr}}\\!\left(\sigma_{x}\delta
Q_{\mathcal{G}}\right),\\\
\mathrm{i}\mathcal{L}_{M}^{(2,1)}&=-\frac{\rho_{0}^{R-2}}{32n^{R}}\mathop{\mathrm{tr}}\\!\left\\{\left[\left(Q_{0}-\sigma_{x}\right)\delta
Q_{\mathcal{G}}\right]^{2}\right\\},\\\
\mathrm{i}\mathcal{L}_{M}^{(2,2)}&=\frac{\rho_{0}^{R-2}}{32n^{R}}\mathop{\mathrm{tr}}\\!\left[\left(Q_{0}-\sigma_{x}\right)\delta
Q_{\mathcal{G}}\right]^{2}.\end{split}$ (171)

To show that $Q=\Lambda$ is a saddle point, we can replace $Q_{0}$ by
$\Lambda$ in the expansion above, leading to $\rho_{0}=n$. Then, the solution
to $\delta S/\delta\mathcal{G}=0$ is given by

$\Sigma=\frac{\gamma}{2n}\left\\{\Lambda-\left[1-\frac{1}{\left(4n\right)^{R-1}}\right]\sigma_{x}\right\\}.$
(172)

The Pauli matrix $\sigma_{x}$ in Keldysh space corresponds to a nonvanishing
anti-Keldysh component of the self-energy and thus violates the usual
causality structure. However, in the replica limit $R\to 1$, the coefficient
of $\sigma_{x}$ vanishes, and we recover the expected causal result
$\Sigma=\gamma Q/(2n)$ with $Q=\Lambda$.

###  D.3 Expansion of the full action

Next, we expand the action Eq. (57) in fluctuations around the particular
saddle points given by Eq. (64) and (65) with
$\mathcal{R}_{\Phi}=\mathcal{R}_{\phi}=1$. For simplicity, we set $R=1$ in all
numerical factors. Then, fluctuations around the saddle points can be
parametrized as $\Sigma=\gamma\left(\Lambda+\delta
Q_{\Sigma}{}\right)\\!\big{/}(2n)$, where $n=1/2$ for fermion counting or
occupation measurements at half filling.

The saddle-point self-energy $\Sigma=\gamma\Lambda/(2n)$ obeys the familiar
causality structure of propagators in Keldysh field theory. Therefore, at the
saddle point,
$\mathop{\mathrm{Tr}}\\!\left[\ln\\!\left(G^{-1}\right)\right]=0$ and
$\mathop{\mathrm{Tr}}\\!\left(\mathcal{G}\Sigma\right)=0$ [78]. For fermion
counting with $A=0$ in Eq. (57), this implies that $\mathrm{i}S_{0}^{(0)}=0$.
Further, according to Eq. (161), also $\mathrm{i}\mathcal{L}_{M}^{(0)}=0$, so
that the Keldysh action vanishes at the saddle point, in agreement with the
normalization $Z_{R}(t)=1$ of the Keldysh partition function in the limit
$R\to 1$. For occupation measurements, the matrix $A$ given in Eq. (56)
involves an anti-Keldysh component, leading to a nonvanishing contribution

$\mathrm{i}S_{0}^{(0)}=\mathrm{i}\mathop{\mathrm{Tr}}\\!\left(GA\right)=-\frac{\gamma
LT}{2n}\left(1-2n\right),$ (173)

where we have set $G=\Lambda$, $L$ is the system size and $T=t-t_{0}$ is the
total evolution time. However, this contribution is cancelled by the
measurement action: from Eq. (171), with $\rho_{0}=n$ at the saddle point
$Q_{0}=\Lambda$, we obtain

$\mathrm{i}\gamma
S_{M}^{(0)}=\mathrm{i}\gamma\int\mathrm{d}^{2}\mathbf{x}\,\mathcal{L}_{M}^{(0)}=\frac{\gamma
LT}{2n}\left(1-2n\right),$ (174)

such that $\mathrm{i}\left(S_{0}^{(0)}+\mathrm{i}\gamma S_{M}^{(0)}\right)=0$.

In an expansion around a saddle point, there is no contribution of first
order. To second order in $\delta Q_{\Sigma}$, we find

$\mathrm{i}S_{0}^{(2)}=\frac{1}{4\tau_{0}}\mathop{\mathrm{Tr}}\\!\left(\frac{1}{2\tau_{0}}G\delta
Q_{\Sigma}{}G\delta Q_{\Sigma}{}-\delta Q_{\Sigma}{}\delta
Q_{\mathcal{G}}{}\right),$ (175)

where $\tau_{0}=n/\gamma$. The dressed Green’s function in momentum space is
defined as

$\begin{split}G_{q}(\omega)&=\left[\omega-\xi_{q}+\mathrm{i}\Lambda/(2\tau_{0})\right]^{-1}\\\
&=\frac{1}{2}G^{R}_{q}(\omega)\left(1+\Lambda\right)+\frac{1}{2}G^{A}_{q}(\omega)\left(1-\Lambda\right),\end{split}$
(176)

where $\xi_{q}=-2J\cos(q)$ and retarded $G^{R}$ and advanced $G^{A}$ Green’s
functions are given by

$G^{R/A}_{q}(\omega)=\frac{1}{\omega-\xi_{q}\pm\mathrm{i}/(2\tau_{0})}.$ (177)

Indeed, as stated in the main text in Eq. (68), only the diffusion block
$\mathcal{B}_{l}(t)=G^{R}_{l}(t)G^{A}_{-l}(-t)$ contributes to Eq. (175).

##  Appendix E Derivation of the NLSM

As explained in Sec. VII, the parametrization in Eqs. (93) and (95) is chosen
such that the symmetries of the action that are broken spontaneously by the
saddle point $Q=\Lambda$ and thus give rise to Goldstone modes appear
explicitly. For fermion counting, these are rotations of the form
$\mathcal{R}_{\Phi}$, Eq. (94); for generalized occupation measurements, also
rotations of the form $\mathcal{R}_{\phi}$, Eq. (96), are a symmetry of the
action. The NLSM describes fluctuations of the corresponding massless
Goldstone modes, and is obtained by integrating out the massive fluctuations
described by $\Theta$, $\theta$, and, for fermion counting, $\phi$.

To derive the NLSM, we begin by taking the spatial continuum limit, in which
the matrix
$H_{l,l^{\prime}}=-J\left(\delta_{l+1,l^{\prime}}+\delta_{l,l^{\prime}+1}\right)$
that describes hopping on the lattice is replaced by a differential operator
containing derivatives with respect to a continuous spatial coordinate $x$.
Inserting Eq. (92) in the action in Eq. (57) and performing an expansion in
spatial and temporal derivatives yields a Lagrangian of the form [78]

$\mathrm{i}\mathcal{L}_{0}[Q]=\mathop{\mathrm{Tr}}\\!\left[\frac{1}{2}\Lambda\mathcal{R}^{-1}\partial_{t}\mathcal{R}-\frac{\nu}{8}\left(\partial_{x}Q\right)^{2}\right]-\frac{\gamma}{2n}\left(1-2n\right),$
(178)

where $\nu=2nJ^{2}/\gamma$. In the term $\mathrm{i}\mathop{\mathrm{Tr}}(GA)$
in Eq. (57), we have neglected derivatives of $\mathcal{R}$, which would be
multiplied by an additional small parameter $A\sim\gamma$. Rotations of the
form $\mathcal{R}_{\Phi}$, Eq. (94), being a symmetry of the action and, in
particular, commuting with $A$, drop out of
$\mathrm{i}\mathop{\mathrm{Tr}}(GA)$. Then, expanding
$\mathrm{i}\mathop{\mathrm{Tr}}(GA)$ in the massive mode $\Theta$ leads to
terms of second order $\gamma$, which should be dropped on our order of
approximation. The same reasoning can be repeated for the rotations given in
Eq. (96). Consequently, in Eq. (178) we have replaced the term
$\mathrm{i}\mathop{\mathrm{Tr}}(GA)$ by its value on the saddle point given in
(173).

The expansion of the derivative terms in Eq. (178) in $\Theta$ is detailed in
Ref. [58] and yields

$\mathrm{i}\mathcal{L}_{0}[Q]=\mathop{\mathrm{Tr}}\\!\left[\frac{1}{2}\Lambda\mathcal{R}_{0}^{-1}\partial_{t}\mathcal{R}_{0}-\frac{\nu}{8}\left(\partial_{x}Q_{0}\right)^{2}\right]\\\
+\frac{1}{4}\mathop{\mathrm{tr}_{K}}(Q_{0}\sigma_{z})\mathop{\mathrm{tr}_{R}}\\!\left[\Theta
U^{-1/2}\left(\partial_{t}U\right)U^{-1/2}\right]\\\
-\nu\rho_{0}\left(1-\rho_{0}\right)\mathop{\mathrm{tr}_{R}}\\!\left[\left(\partial_{x}U^{-1}\right)\left(\partial_{x}U\right)\right]-\frac{\gamma}{2n}\left(1-2n\right),$
(179)

where $\mathcal{R}_{0}=\mathcal{R}_{\phi}\mathcal{R}_{\theta}$,
$U=\mathrm{e}^{\mathrm{i}\Phi}$, and $\rho_{0}$ is defined in Eq. (100).

We focus now for the moment on the replica-symmetric sector of the theory,
setting $R=1$. Since $\theta$ is a massive mode for both fermion counting and
occupation measurements, we expand $\mathcal{L}_{0}$ to second order in
$\theta$. Further, we drop terms that contain $\partial_{t}\theta$ and
$\partial_{t}\phi$ and no other fields. Such terms contribute to the action
only at the initial and final time. However, these contributions vanish. We
thus find, with $f=1-2n$,

$\mathrm{i}\mathcal{L}_{0}[Q_{0}]=-\frac{\mathrm{i}}{4}\left(2+f\theta\right)\theta\partial_{t}\phi-\frac{\nu}{4}\left\\{\left[1-f^{2}+2f\theta\right.\right.\\\
\left.\left.-\left(1-f^{2}\right)\theta^{2}\right]\left(\partial_{x}\phi\right)^{2}-\mathrm{i}2f\left(f-\theta\right)\left(\partial_{x}\phi\right)\left(\partial_{x}\theta\right)\right.\\\
\left.+\left(1+f^{2}\right)\left(\partial_{x}\theta\right)^{2}\right\\}-\frac{f\gamma}{2n}.$
(180)

Returning to $R>1$, we next consider the contribution from the measurement
Lagrangian, given in Eqs. (51) and (52) for fermion counting and occupation
measurements, respectively, where we insert Eq. (92). The parametrization of
rotations in Eq. (93) is chosen such that $\mathcal{R}_{\Phi}$, being a
symmetry of the Lagrangian for both models, drops out. For fermion counting,
$\mathcal{R}_{\Phi}$ is the only symmetry. Therefore, we can expand $Q$ to
second order in the massive modes $\Theta$, $\phi$, and $\theta$,

$\mathcal{R}_{\Theta}Q_{0}\mathcal{R}_{\Theta}^{-1}=\sigma_{z}+\delta Q,$
(181)

where

$\delta
Q=\phi\sigma_{y}-\theta\sigma_{x}-\frac{\theta^{2}}{2}\sigma_{z}-\frac{\phi^{2}}{2}\sigma_{z}-\frac{\Theta^{2}}{2}\sigma_{z}-\Theta\left(\sigma_{x}+\sigma_{z}\theta\right).$
(182)

Then, using Eqs. (161), (165), and (166), we obtain

$\mathrm{i}\mathcal{L}_{M}[Q]=-\frac{1}{2}\left[\mathop{\mathrm{tr}_{R}}\\!\left(\Theta^{2}\right)+\theta^{2}+\phi^{2}\right],$
(183)

where we have set $R=1$ in all numerical factors, such that
$\mathcal{L}_{M}^{(0)}$, given in Eq. (161), vanishes.

For occupation measurements, also $\mathcal{R}_{\phi}$ is a symmetry.
Therefore, we cannot expand $Q$ in $\phi$ as we did for fermion counting, and
Eqs. (181) and (182) have to be replaced by

$\mathcal{R}_{\Theta}Q_{0}\mathcal{R}_{\Theta}^{-1}=Q_{0}+\delta Q,$ (184)

where

$\delta
Q=\frac{\mathrm{i}}{2}\Theta\left[\sigma_{y},Q_{0}\right]-\frac{\Theta^{2}}{4}\left(Q_{0}-\sigma_{y}Q_{0}\sigma_{y}\right).$
(185)

Inserting Eq. (184) in Eq. (171) we obtain an expansion of the measurement
Lagrangian in $\Theta$,

$\mathrm{i}\mathcal{L}_{M}=\frac{f}{2n}-\frac{1}{32n\rho_{0}}\mathop{\mathrm{tr}_{K}}\\!\left(\sigma_{z}Q_{0}\right)^{2}\mathop{\mathrm{tr}_{R}}\\!\left(\Theta^{2}\right).$
(186)

To find the replica-symmetric Lagrangian for fermion counting, we can simplify
Eq. (180) further by setting $f=0$, performing an expansion to second order in
both $\phi$ and $\theta$, and omitting the last term that stems from
$\mathrm{i}\mathop{\mathrm{tr}}(GA)$. Then, combining Eq. (180) with Eq. (183)
yields Eq. (97). For occupation measurements, combining Eq. (180) with Eq.
(186) we obtain Eq. (98). In the replicon sector described by the modes $\Phi$
and $\Theta$, integration over $\Theta$ leads for both models to Eq. (99).

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