Continuous Gaussian Measurements of the Free Boson CFT: A model for Exactly
Solvable and Detectable Measurement-Induced Dynamics

Y. Minoguchi1$\star$, P. Rabl1 and M. Buchhold2

1 Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien,
1040 Vienna, Austria

2 Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne,
Germany

⋆<EMAIL_ADDRESS>
## Abstract

Hybrid evolution protocols, composed of unitary dynamics and repeated, weak or
projective measurements, give rise to new, intriguing quantum phenomena,
including entanglement phase transitions and unconventional conformal
invariance. Defying the complications imposed by the non-linear and stochastic
nature of the measurement process, we introduce a scenario of measurement-
induced many body evolution, which possesses an exact analytical solution:
bosonic Gaussian measurements. The evolution features a competition between
the continuous observation of linear boson operators and a free Hamiltonian,
and it is characterized by a unique and exactly solvable covariance matrix.
Within this framework, we then consider an elementary model for quantum
criticality, the free boson conformal field theory, and investigate in which
way criticality is modified under measurements. Depending on the measurement
protocol, we distinguish three fundamental scenarios (a) enriched quantum
criticality, characterized by a logarithmic entanglement growth with a
floating prefactor, or the loss of criticality, indicated by an entanglement
growth with either (b) an area-law or (c) a volume-law. For each scenario, we
discuss the impact of imperfect measurements, which reduce the purity of the
wavefunction and are equivalent to Markovian decoherence, and present a set of
observables, e.g., real-space correlations, the relaxation time, and the
entanglement structure, to classify the measurement-induced dynamics for both
pure and mixed states. Finally, we present an experimental tomography scheme,
which grants access to the density operator of the system by using the
continuous measurement record only.

###### Contents

  1. 1 Introduction
    1. 1.1 Perspectives on Measurement-Induced Dynamics: A Recap
    2. 1.2 Synopsis
  2. 2 Setup
    1. 2.1 Free Boson Conformal Field Theory
    2. 2.2 Stochastic Schrödinger Equation for Continuous Measurements
    3. 2.3 Imperfect Measurements and Unconditioned Evolution
  3. 3 Gaussian measurements
    1. 3.1 Gaussian Evolution and Linear Measurements
    2. 3.2 Entanglement and Logarithmic Negativity
    3. 3.3 Relaxation, Riccati Spectrum and Purification
  4. 4 Measurement-Induced Loss of Criticality
    1. 4.1 $\Phi$-Measurements
    2. 4.2 Measurement-induced mass
      1. 4.2.1 Purity
      2. 4.2.2 Riccati Spectrum and Relaxation
      3. 4.2.3 Correlation Functions and Entanglement
  5. 5 Measurement-Enriched Criticality
    1. 5.1 $\Pi$-Measurements
    2. 5.2 Measurement Enriched Criticality
      1. 5.2.1 Correlation Functions
      2. 5.2.2 Riccati Spectrum and Relaxation
      3. 5.2.3 Entanglement and Measurement Enriched Criticality
      4. 5.2.4 Connection to $\Phi$-Measurements
  6. 6 Steady States with Extensive Entanglement Scaling
    1. 6.1 Measurement Operators
    2. 6.2 Correlation Functions
    3. 6.3 Riccati Spectrum and Relaxation
    4. 6.4 Volume Law Entanglement
  7. 7 Full Tomography of the Conditioned State
    1. 7.1 Observables and Tomography of the Unconditioned State
    2. 7.2 Tomography of the Full Conditioned Density Matrix
  8. 8 Conclusion
  9. A Ground State Covariance Matrix
  10. B Gaussian Measurements and Stochastic Schrödinger Equation
    1. B.1 Measurement Results Follow a Wiener Process
    2. B.2 Derivation of the Stochastic Schrödinger Equation
  11. C Details on the Kalman Filter
    1. C.1 Derivation of the Evolution Equation for the Conditioned First Moments and the Covariance Matrix
    2. C.2 Alternative Argument for the Deterministic Dynamics of Covariances
    3. C.3 Review of the Classical Kalman Filter
    4. C.4 Connection between Classical and Quantum Kalman Filter
  12. D Solution of the Riccati Equation
  13. E Unconditioned Observables from Averaging

##  1 Introduction

Quantum critical behavior, emerging from the competition between a set of non-
commuting operators, is a paradigmatic and fascinating phenomenon in many-body
quantum systems [1, 2, 3, 4]. The competition between non-commuting, local
operators imprints characteristic fluctuations, which leave their traces in
observables up to the largest length scales. Yet, the long-wavelength behavior
of critical systems is often well-captured by effective and non-interacting
degrees of freedom [5, 6]; a consequence of emergent symmetries at the
critical point, including scale (or conformal) invariance [7, 8, 9, 10, 11,
12].

A new type of critical quantum states has been recently discovered in hybrid
evolution protocols, which are composed of unitary time evolution and
repeated, local measurements [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Here, the
competition arises between the unitary dynamics, which lead to scrambling of
information and generate entanglement, and local measurements that extract
information from the system, and therefore reduce its entanglement over time.
Under the hybrid evolution, an initial pure state wave function remains pure,
and undergoes a phase transition from an (sub-) extensively entangled state to
a disentangled, local product state when the measurement rate exceeds a
certain threshold [13, 14]. When the unitary time evolution is implemented by
generic random circuits, and the entanglement entropy undergoes a transition
from volume- to area law scaling, the underlying phase transition has been
identified with classical universality classes [61, 13, 52, 51, 49, 54].
However, when the time-evolution is constrained by certain symmetries, e.g.,
for Gaussian circuits or a free Hamiltonian, the measurement-induced
transition has been associated with a quantum phase transition [62, 43, 44,
40, 41, 34, 39]. This enables the notion of quantum critical behavior, and
emergent conformal invariance, in a nonequilibrium wave function, generated by
the hybrid evolution protocol [61, 52, 51, 53].

In this work, we focus on quantum critical behavior, and its potential
collapse, under continuous measurements. We therefore start from an
paradigmatic model for quantum criticality, the free boson conformal field
theory (CFT) [12], and examine how continuous measurements may either enrich
or destroy its quantum critical behavior. To this end, we introduce an
elementary, and exactly solvable setup for the measurement-induced many-body
evolution of bosons, which we term Gaussian measurements. It consists of a
quadratic unitary evolution, imprinted by the Hamiltonian of the free boson
CFT, and an extensive set of continuously measured operators $\\{O_{l}\\}$,
which are linear in the boson operators. This setup preserves the Gaussianity
of an initial wave function (or density matrix), and enables an exact,
measurement-noise free, and analytically solvable expression for the
stationary state. The exact solubility is a central finding of our work and
establishes Gaussian measurements as an elementary model for the measurement-
induced evolution of bosons.

Including both perfect and imperfect measurements[48], we characterize the
measurement-induced dynamics in terms of three essential signatures of the
steady state: (i) the structure of spatial correlations, (ii) the asymptotic
relaxation time scale, i.e. the purification time in case of perfect
measurements, and (iii) the entanglement structure. Imperfect measurements
correspond to the case where only a fraction of the measurement results is
collected and thus the system evolves into a mixed state.

In order to classify the entanglement structure of mixed states, we make use
of the logarithmic negativity [63, 64, 65, 66]. We demonstrate that the
signatures (i)-(iii) are then readily generalized to the case of imperfect
measurements. We show that the characteristic properties of measurement-
induced pure states are continuously connected to those of mixed states for
small and moderate imperfections. This demonstrates the robustness of both the
measurement-induced dynamics and their signatures in the presence of non-
perfect measurements.

We then examine three elementary measurement scenarios, illustrating the fate
of quantum criticality in the presence of measurements: (a) measurements,
which preserve the conformal symmetry of the free boson CFT, and measurements
which violate the conformal symmetry and are either (b) local or (c) non-
local. Consistent with the symmetries, the quantum critical properties survive
in scenario (a). It yields a nonequilibrium stationary state with scale
invariant correlations and relaxation behavior, and a logarithmic entanglement
growth. However, the stationary state appears no longer to be universal but
rather is characterized by a floating prefactor of the logarithmic
entanglement scaling, which has been observed numerically also for monitored
fermion systems[67, 68, 37, 43, 44, 69]. But in contrast to the fermionic
setting the wavefunction exhibiting these features, is deterministic and can
be obtained analytically in the case of bosons and Gaussian measurements. In
turn, for scenarios (b) and (c), the quantum critical signatures are destroyed
by the measurements, and the violation of scale invariance leads to the
emergence of a measurement-induced mass in the spatial correlations and the
relaxation time. Interestingly, the entanglement structure of both scenarios
differs strongly from each other: in case (b) the local measurements push the
system into a stationary state, which is similar to a ground state of a gapped
boson Hamiltonian, and whose entanglement structure obeys an area law. In case
(c), however, the stationary state entanglement obeys a volume law, which is
reminiscent of excited states [70]. Depending on the structure of the
measurement operators, one therefore recovers the basic scenarios that have
been observed in hybrid protocols of random circuits with projective
measurements.

Finally, we discuss the experimental detectability of the hybrid dynamics for
Gaussian measurements. Based on the exact evolution equation, we show that a
continuous recording of the weak measurement results enables a complete
reconstruction of the trajectory density matrix. In a corresponding experiment
it is not necessary to perform additional measurements or decoding operations
on the system [71, 72], but rather one processes the measurement outcomes that
are available from the measurements itself [73, 74, 75, 76].

Figure 1:  (a) Lattice regularization of the free boson CFT. (b) Schematic
picture of measurement of operator $O_{i}$ located on size $i$ and the
associated weak measurement operator $E_{I_{i,t}}$. The measurement outcomes
are distributed following a Wiener process around the conditioned expectation
value $\langle O_{i}\rangle_{c,t}$. (c) Sketch of the algorithm how to
reconstruct the conditioned density matrix $\rho_{c,t}$. We assume that in
experiment 1 (2) the quantities $\\{\boldsymbol{\Sigma_{t}}\\}$
($\\{\boldsymbol{I}_{t}^{(i)}\\}$) were measured. Given the
$\langle\boldsymbol{s}\rangle_{c,t}^{(i)}$ for the present (initial) time the
conditioned covariance matrix $\boldsymbol{\sigma}_{c,t}$ is reconstructed
according to Eq. (22) [see dashed box (A)]. From the present
$\langle\boldsymbol{s}\rangle_{c,t}^{(i)}$ and $\boldsymbol{\sigma}_{c,t}$
together with $\boldsymbol{I}_{t}^{(i)}$, we construct
$\langle\boldsymbol{s}\rangle_{c,t+\mathrm{d}t}^{(i)}$ by numerically applying
the recursion Eq. (71) for every experimental run $i$ [see dashed box (B)].

###  1.1 Perspectives on Measurement-Induced Dynamics: A Recap

Measurement-induced dynamics generally feature a competition between a
scrambling evolution, which yields to the build up of entanglement in the
state, and measurements. The latter extract information on the state and
therefore successively reduce its entanglement. One way to implement the
scrambling evolution is by acting random unitary gates on the wave function.
Then the hybrid setup can be viewed as a quantum channel [77, 78], and the
phase transition is understood as a type of error correction threshold in the
quantum channel capacity [79]. In this picture, the unitary gates encode
information non-locally in the wave function, thereby protecting it against
the local readout provided by the measurements [79, 77, 80, 81]. When
increasing the read-out/error rate, the encoding fails and the channel
capacity goes to zero. The transition in the quantum channel capacity can also
be observed in the purification time of an initially mixed state [77, 82].

In turn, if the unitary evolution is implemented by a time-independent
Hamiltonian, the origin of the phase transition in terms of non-commuting
operators becomes directly apparent [35, 36, 37, 38, 39, 40, 41]. Repeatedly
measuring a set of local operators $\\{O_{l}\\}$ then projects the system
towards a shared eigenstate of all the operators $O_{l}$. The Hamiltonian,
however, for $[H,O_{l}]\neq 0$ is not compatible with the measurements, and
constantly pushes the system out of the eigenstate manifold. The wave function
then displays a volume law (logarithmically growing) entanglement entropy if
the evolution is dominated by a generic (integrable) Hamiltonian, and an area
law if instead the measurement-induced collapse into eigenstates dominates
[35, 36, 37, 38, 39, 40, 41]. While this competition between non-commuting
operators is reminiscent of a quantum phase transition in the ground state of
a Hamiltonian, it also reveals a crucial difference; when measuring $L$
different operators $O_{l}$, and each operator has a number of $n_{O}$
different eigenstates, then, due to the probabilistic nature of the
measurement process, the steady state will experience a macroscopic degeneracy
of $O(n_{O}^{L})$ compatible wave functions. This gives rise to an extensive
configurational entropy in the ensemble of time-evolved wave function
trajectories.

Due to the large configurational entropy, observables that are linear in the
state $\rho=|\psi\rangle\langle\psi|$, when averaged over many trajectories
with different measurement outcomes, do not reveal any information on the
phase transition or the critical state. Rather they are completely
featureless. In order to detect the features of a hybrid evolution protocol
theoretically, one needs access to observables that are non-linear in the
state, such as the entanglement entropy or certain types of connected
correlation functions [37, 40, 44]. Averages of such observables still contain
non-trivial knowledge of the state, despite the presence of this large
configurational entropy. However, nonlinear observables are significantly more
difficult, and mostly impossible, to detect experimentally. Quite generally,
this sets a big challenge for the experimental detection and classification of
measurement-induced dynamics. For the specific choice of a Clifford circuit-
based evolution, i.e. for stabilizer codes, recent proposals [82, 51] for, and
one realization [71] of the experimental detection make use of feedback
protocols, which reduce the configurational entropy to a minimum. However, the
proposed protocols so far only work for the particular case of Clifford
circuits, while general detection schemes have yet to be established. In terms
of experimental detection, our work follows a different approach and we
propose the characterization of the correlations and the entanglement
structure of the measurement-induced quantum states based on the recorded
measurement outcomes, which is feasible for Gaussian states.

###  1.2 Synopsis

Before delving into the detailed analysis, we give a brief synopsis of our
work. We start with Sec. 2.1, where we provide a review of the unitary part of
the evolution protocol, the free boson CFT. We will introduce it in the
language of continuous variables [83, 84, 85], with a special focus on the
covariance matrix, which hosts all the information on correlations in Gaussian
systems, and which will be at the heart of our analysis. We then move on to
discuss in Sec. 2.2 a special construction of the stochastic Schödinger
equation, which describes the evolution of a state under both continuous
unitary time evolution and continuous weak measurements. This is a non-linear
stochastic equation of motion for the quantum trajectories, describing the
evolution conditioned on the measurement outcomes. This framework is then
straightforwardly generalized to imperfect measurements in Sec. 2.3. Equipped
with this preliminaries we then introduce the notion of Gaussian measurements,
which are bilinears in the bosonic field operators, so the hybrid evolution
remains in the manifold of Gaussian states. The resulting equations of motion
are found in Sec. 3.1, and they are analogous to equations well known in
classical control and estimation theory. Therefore, we will refer to them as
the free boson Kalman filter. As a key consequence, in this framework the
equation of motion for the covariance matrix, a priori non-linear and
stochastic, becomes non-linear but deterministic, and is described by a matrix
Riccati differential equation. This implies that properties which inherit from
the covariance matrix like correlations and the entanglement (see Sec. 3.2)
are independent of the measurement-noise, and their values for an individual
trajectory are identical to the trajectory average. Furthermore the steady
state is obtained from solving an algebraic equation. Given the steady state,
we show in Sec. 3.3, that the asymptotic (purification) dynamics are found
from linearizing the Riccati equation, giving rise to an effective non-
hermitian Hamiltonian governing these.

We go on by constructing a first exactly solvable example of the measurement-
induced dynamics in Sec. 4.1, where the continuously measured operators are an
extensive set of linear combinations of the bosonic field operators. In Sec.
4.2 we focus on the simple limit where the fields at every site are measured
and show that this scenario breaks the underlying scale invariance of the free
boson CFT. This gives rise to an effective measurement-induced mass and
consequently to fast (system size independent) relaxation and purification,
short range correlations and area law entanglement (see Sec. 4.2.1-4.2.3). We
then turn in Sec. 5.1, to the limit where linear combinations of the canonical
conjugate of the bosonic field operators are measured in extensively many
sites. In the limit where measurements are given by the momentum operators at
every site, the steady state remains conformally invariant but strongly
differs from the ground state of the free boson CFT. We coin this phenomenon
measurement-enriched criticality and introduce it in Sec. 5.2. The key
signatures are algebraic real space correlations, slow (system size dependent)
relaxation and logarithmic entanglement growth; furthermore prefactor of the
logarithmic entanglement scaling is increased, in contrast to the free boson
CFT upon introducing the measurements (see Sec. 5.2.1-5.2.3). In Sec. 6.1 we
numerically explore a setting where the linear combinations of the field
measurements are highly non-local and drawn from a Gaussian random matrix
ensemble. We present evidence that the resulting correlations functions decay
exponentially and relax quickly (see Sec. 6.2-6.3). In contrast to Sec. 4 we
find a volume law entanglement in the steady state.

Finally, in Sec. 7 we review the experimental difficulties of observing
measurement-induced phases and criticality in general. We discuss that the
conventional approach of averaging over many experimental runs erases many
subtleties of the conditioned dynamics. For instance several quantities, such
as the entanglement entropy for a single trajectory, which are often discussed
in this context, are not accessible from such measurements. In Sec. 7.2 we
then show how, within the framework of Gaussian measurements, the full
covariance matrix, conditioned on the measurement outcomes, may be
reconstructed from the measurement records only. This enables the
determination of a large set of measurement-induced characteristica, including
entanglement, from the measurment record. The detailed derivations and
calculations are provided in the Appendices A-E.

##  2 Setup

###  2.1 Free Boson Conformal Field Theory

We consider an elementary model for quantum critical behavior in one
dimension: the free boson CFT [12]. It describes the dynamics of the massless,
free Klein-Gordon field $\Phi(x)$ in $(1+1)$ dimensions via the Hamiltonian

$H=\frac{v}{2}\int_{-\frac{L}{2}}^{\frac{L}{2}}{\rm
d}x\,\left[\Pi^{2}(x)+(\partial_{x}\Phi(x))^{2}\right].$ (1)

Here, we assume periodic boundary conditions, $\Phi(x+L)=\Phi(x)$, and a group
velocity $v$. The field $\Pi(x)$ is conjugate to $\Phi(x)$ and they obey the
commutation relation $[\Phi(x),\Pi(y)]=i\delta(x-y)$, which implies
$\Pi(x)=\frac{1}{v}\partial_{t}\Phi$. In the following, we work in discrete
space and consider the ultraviolet (UV) completion of the theory in Eq. (1) by
putting the field on a lattice with lattice spacing $a$. This maps the
continuous fields onto a discrete chain of $N=L/a$ bosonic modes with
operators $\Phi(x)\rightarrow\Phi_{i}$ and $\Pi(x)\rightarrow\Pi_{i}/a$, which
satisfy $[\Phi_{i},\Pi_{j}]=i\delta_{ij.}$. The corresponding Hamiltonian
describes a chain of coupled harmonic oscillators

$H=\frac{\omega}{2}\sum_{j=1}^{N}\Pi_{j}^{2}+(\Phi_{j+1}-\Phi_{j})^{2}+r_{N}^{2}\Phi_{i}^{2}=\frac{\omega}{2}\boldsymbol{s}^{\top}\,h\boldsymbol{s},$
(2)

with the ‘spring constant’ $\omega=v/a$. Here, we added a small effective
mass, $r_{N}=\frac{\Omega}{\omega N}$, with $\Omega\ll\omega$, which
regularizes the zero-mode of the spectrum at momentum $k=0$, enabling well-
defined inverse $H^{-1}$. The regularization scales with the system size like
the finite size gap
$\Delta\epsilon=\epsilon_{k_{1}}-\epsilon_{k=0}=O(N^{-1})$.

The Hamiltonian is a quadratic form and can be expressed in terms of the
$2N$-dimensional vector
$\boldsymbol{s}=(\Phi_{1},\ldots,\Phi_{N},\Pi_{1},\ldots,\Pi_{N})^{\top}$ and
the $2N\times 2N$ Hamiltonian matrix $h$

$h=v_{N}\oplus\mathbb{1}_{N},\ \
(v_{N})_{ij}=(2+r_{N}^{2})\delta_{i,j}-\delta_{i-1,j}-\delta_{i+1,j},\ \
(v_{N})_{1,N}=(v_{N})_{N,1}=-1.$ (3)

The matrix $h$ is a direct sum, including the lattice Laplacian $(v_{N})_{ij}$
with periodic boundary conditions. In vector notation, the canonical
commutation relations can be expressed in a compact way by using a vector
commutator (anti-commutator) defined by
$[\boldsymbol{A},\boldsymbol{B}^{\top}]=\boldsymbol{A}\boldsymbol{B}^{\top}-\boldsymbol{B}\boldsymbol{A}^{\top}$
($\\{\boldsymbol{A},\boldsymbol{B}^{\top}\\}=\boldsymbol{A}\boldsymbol{B}^{\top}+\boldsymbol{B}\boldsymbol{A}^{\top}$
). This yields

$[\boldsymbol{s},\boldsymbol{s}^{\top}]=iJ,\qquad\text{with the symplectic
form}\qquad J=\begin{pmatrix}0_{N}&\mathbb{1}_{N}\\\
-\mathbb{1}_{N}&0_{N}\end{pmatrix}.$ (4)

The free boson CFT is quadratic and therefore any Gaussian state remains
Gaussian under time evolution. For Gaussian states, all information about the
system is encoded in the linear and quadratic moments of the vector
$\boldsymbol{s}$, and higher moments can be derived from Wick’s theorem. The
linear moments are given by the average $\langle\boldsymbol{s}\rangle$, and
the quadratic moments are collected in the symmetric covariance matrix

$\boldsymbol{\sigma}=\frac{1}{2}\langle\\{\boldsymbol{s},\boldsymbol{s}^{\top}\\}\rangle-\langle\boldsymbol{s}\rangle\langle\boldsymbol{s}\rangle^{\top}=\frac{1}{2}\langle\\{(\boldsymbol{s}-\langle\boldsymbol{s}\rangle),(\boldsymbol{s}-\langle\boldsymbol{s}\rangle)^{\top}\\}\rangle=\begin{pmatrix}\sigma_{\Phi\Phi}&\sigma_{\Phi\Pi}\\\
\sigma_{\Pi\Phi}&\sigma_{\Pi\Pi}\end{pmatrix},$ (5)

where $\\{\boldsymbol{s},\boldsymbol{s}^{\top}\\}$ is the vector anti-
commutator defined above. The covariance matrix $\boldsymbol{\sigma}$ contains
all possible two-point correlation functions.

For a general Gaussian initial state, the time-evolution of the first and
second moments is given by the equation of motion

$\displaystyle\langle\dot{\boldsymbol{s}}\rangle_{t}$ $\displaystyle=$
$\displaystyle Jh\langle\boldsymbol{s}\rangle_{t},$ (6)
$\displaystyle\dot{\boldsymbol{\sigma}}_{t}$ $\displaystyle=$ $\displaystyle
Jh\boldsymbol{\sigma}_{t}+\boldsymbol{\sigma}_{t}(Jh)^{\top}.$ (7)

For the ground state of the free boson CFT, all the linear moments vanish,
$\langle{\bf s}\rangle=0$. The covariance matrix for the ground state is
independent of the frequency $\omega$ and has a block structure (see [86] and
App. A),

$\boldsymbol{\sigma}_{\rm GS}=\frac{1}{2}(v_{N}^{-1/2}\oplus v_{N}^{1/2}).$
(8)

###  2.2 Stochastic Schrödinger Equation for Continuous Measurements

In order to study the effect of measurements on the free boson CFT, we subject
the dynamics to continuous, weak measurements of a set of mutually commuting
observables $\\{O_{l}\\}$. These are represented by Hermitian operators
$O_{l}=O_{l}^{\dagger}$, and suitable choices shall be discussed later. We
model the measurement-induced evolution of the wave function in the quantum
state diffusion framework [87, 88, 89, 90]. In this framework, the operators
$O_{l}$ are not measured directly but coupled to an auxilliary system, a
meter, which is then read out stroboscopically at time intervals ${\rm d}t$.

For each time interval ${\rm d}t$, the update of the wave function ${\rm
d}|\psi_{t}\rangle_{c}\equiv|\psi_{t+{\rm
d}t}\rangle_{c}-|\psi_{t}\rangle_{c}$ is described by the stochastic
Schrödinger equation (SSE)

${\rm
d}|\psi_{t}\rangle_{c}=\left(-iH\mathrm{d}t-\sum_{l}\frac{\gamma_{l}}{2}\left(O_{l}-\langle
O_{l}\rangle_{c,t}\right)^{2}\mathrm{d}t+\sum_{l}\sqrt{\gamma_{l}}(O_{l}-\langle
O_{l}\rangle_{c,t}){\rm d}W_{l,t}\right)|\psi_{t}\rangle_{c}.$ (9)

Here, $\gamma_{l}$ are the measurement strengths, i.e., the rates at which the
continuous readout of observable $O_{l}$ changes the state over time. The
nonlinear and probabilistic nature of the measurement evolution appear in the
quantum state diffusion via the dependence on the instantaneous quantum
mechanical expectation value $\langle
O_{l}\rangle_{c,t}=\langle\psi_{t}|O_{l}|\psi_{t}\rangle_{c}$ and the Wiener
process ${\rm d}W_{l,t}$. The latter is a $\delta$-correlated white noise,
which has zero mean, $\mathbb{E}\,({\rm d}W_{l,t})=0$, and variance
$\mathbb{E}\,({\rm d}W_{l,t}{\rm
d}W_{m,t^{\prime}})=\delta(t-t^{\prime})\delta_{l,m}{\rm d}t$.

The SSE combines the unitary Hamiltonian time evolution with the infinitesimal
update due to continuous measurements. Each term $\sim O_{l}-\langle
O_{l}\rangle_{c,t}$ pushes the state closer to an eigenstate of the measured
operator $O_{l}$, and is annealed in an eigenstate with
$O_{l}|\psi_{t}\rangle=\langle O_{l}\rangle_{c,t}|\psi_{t}\rangle_{c}$. If the
operators $O_{l}$ are pairwise commuting, $[O_{l},O_{m}]=0$, then there exists
a set of joint eigenstates. In the absence of a Hamiltonian, $H=0$, the system
continuously involves (collapses) into such an eigenstate, which is compatible
with the initial conditions. If, however, $H$ is nonzero and does not commute
with the measurement operators $[H,O_{l}]\neq 0$, the unitary evolution
competes with the measurements and prevents an asymptotic collapse of the wave
function.

We will now briefly motivate the form of the SSE from a rigorous formulation
of weak measurements [73, 76, 91, 74]. For simplicity, we focus on single
observable $O$. Without specifying the microscopic realization of the
auxiliary system, i.e., the meter, we expect that after each measurement of
the meter we receive an outcome in terms of a number $I_{t}$, which we call
‘current’. The measurement setup shall be designed in such a way that the
overlap between the state $|\psi_{t}\rangle$ and an eigenstate
$O|\lambda_{O}\rangle=\lambda_{O}|\lambda_{O}\rangle$ of the observable is
given by a Gaussian
$|\langle\psi_{t}|\lambda_{O}\rangle|^{2}~{}\sim\exp(-\alpha(I_{t}-\frac{\gamma}{2}\lambda_{O})^{2})$,
whose center is determined by the current $I_{t}$ (for some $\alpha>0$).
Formally, this is expressed by a set of positive operators $\\{E_{I_{t}}\\}$
with

$E_{I_{t}}=\left(\frac{8{\rm
d}t}{\pi\gamma}\right)^{\frac{1}{4}}\mathrm{exp}\left[-\frac{4{\rm
d}t}{\gamma}\left(I_{t}-\frac{\gamma}{2}O\right)^{2}\right],$ (10)

with the completeness relation $\int{\rm
d}I_{t}\,E^{\dagger}_{I_{t}}E_{I_{t}}=\mathbb{1}$. After measuring the outcome
$I_{t}$, the state is [73]

$|\psi_{t+{\rm d}t}\rangle_{c}=\frac{e^{-iH{\rm
d}t}E_{I_{t}}|\psi_{t}\rangle_{c}}{\rVert
E_{I_{t}}|\psi_{t}\rangle_{c}\rVert}.$ (11)

The probability for measuring a specific value $I_{t}$ is ${\rm
Pr}(I_{t})={}_{c}\langle\psi_{t}|E^{\dagger}_{I_{t}}E_{I_{t}}|\psi_{t}\rangle_{c}$,
which yields the average $\mathbb{E}\,(I_{t})=\int{\rm d}I_{t}I_{t}{\rm
Pr}(I_{t})=\frac{\gamma}{2}\langle O\rangle_{c,t}$ and the variance
$\mathbb{E}(I_{t}-\mathbb{E}(I_{t}))^{2}=\frac{\gamma}{16{\rm d}t}$. Since
$E_{I_{t}}$ is Gaussian, $I_{t}$ is equivalent to an Itō random process

$I_{t}{\rm d}t=\frac{\gamma}{2}\langle
O\rangle_{c,t}\mathrm{d}t+\frac{1}{4}\sqrt{\gamma}\mathrm{d}W_{t},$ (12)

with the Wiener increment ${\rm d}W_{t}^{2}={\rm d}t$, see App. B.1. The SSE
for the quantum state diffusion process is obtained from Eq. (11) by sending
${\rm d}t\rightarrow 0$ and expanding $E_{I_{t}}$ up to order ${\rm d}t$ [for
details see App. B.2].

Equation (9) is the conventional form of the SSE, for the dependence on the
measurement outcomes $I_{t}$ appears only implicitly. In order to make it
explicit, one may solve Eq. (12) for the Wiener increment ${\rm d}W_{t}$ and
then insert the solution back into Eq. (9). Reinstating the index $l$ this
yields

${\rm d}|\psi_{t}\rangle_{c}={\rm
d}t\left[-iH-\sum_{l}\frac{\gamma_{l}}{2}\left(O_{l}-\langle
O_{l}\rangle_{c,t}\right)^{2}-4\sum_{l}(O_{l}-\langle
O_{l}\rangle_{c,t})\left(I_{l,t}-\frac{\gamma_{l}}{2}\langle
O_{l}\rangle_{c,t}\right)\right]|\psi_{t}\rangle_{c}.$ (13)

This evolution equation has the following, appealing interpretation: Given
that the Hamiltonian $H$, the measurement operators $O_{l}$, the rates
$\gamma_{l}$, the current state $|\psi_{t}\rangle_{c}$ and  the measurement
results $I_{l,t}$ are known, then the evolution step $t\rightarrow t+{\rm d}t$
in Eq. (13) is deterministic. By recording the stream of currents $I_{l,t}$
and by feeding it into the SSE, one can therefore track the evolution of the
state [92, 93]. Precisely this identity is the idea behind the experimental
detection scheme, that we discuss in Sec. 7.1.

###  2.3 Imperfect Measurements and Unconditioned Evolution

The SSE in (9) predicts that weak continuous measurements preserve the purity
of the wave function. Indeed, a complete readout of the auxiliary system at
each infinitesimal time step ${\rm d}t$ suppresses any build-up of
entanglement between the state $|\psi_{t}\rangle_{c}$ and the meter, and
ensures that the state remains pure. In reality, however, the measurement
process may be imperfect, for instance due to unwanted interactions with an
external bath or an incomplete readout. This will foster the wave function to
start entangling with the environment, and the state to become impure. Here,
we revisit the description for a continuous measurement evolution with
imperfect measurements. This yields a stochastic master equation (SME), which
we will use in the following sections in order to discuss the impact of
imperfect measurements on both the measurement-induced dynamics, including the
critical properties of the system, and on potential observables.

An incomplete measurement of the operator $O$ is commonly described by
coupling $O$ simultaneously to two different meters, one which is read out
(i.e., subject to a perfect measurement), and another one which is not read
out [75, 73, 76, 74]. No information is collected from the latter and the
state must be described by a statistical ensemble including all measurement
outcomes, i.e. the statistical average over all measurement outcomes of the
second meter. To perform the average, we consider the conditioned density
matrix $\rho_{c,t}=|\psi_{t}\rangle_{c}\langle\psi_{t}|$. Its evolution
equation is obtained from the Itō convention ${\rm d}\rho_{c,t}={\rm
d}|\psi_{t}\rangle_{c}\langle\psi_{t}|+|\psi_{t}\rangle_{c}{\rm
d}\langle\psi_{t}|+{\rm d}|\psi_{t}\rangle_{c}{\rm d}\langle\psi_{t}|$ in
combination with the SSE in Eq. (13). Before averaging over the second meter,
the master equation is [74]

$\displaystyle{\rm d}\rho_{c,t}$ $\displaystyle=$
$\displaystyle-i[H,\rho_{c,t}]{\rm
d}t-\frac{\gamma_{1}+\gamma_{2}}{2}[O,[O,\rho_{c,t}]]{\rm
d}t+\sum_{j=1,2}\sqrt{\gamma_{j}}{\rm d}W_{j,t}\\{O-\langle
O\rangle,\rho_{c,t}\\}.$ (14)

Here, the $\gamma_{j}$ and the ${\rm d}W_{j,t}$ are the measurement strengths
and the Wiener increments for the perfect (unread) measurement $j=1$ ($j=2$).
The Wiener increments for the two different meters are uncorrelated.

The imperfect measurement, i.e., the average over the outcomes of the second
meter, is equivalent to the average over ${\rm d}W_{2,t}$, eliminating it from
Eq. (14). Introducing the total measurement strength
$\gamma=\gamma_{1}+\gamma_{2}$ and the measurement efficiency
$\eta=\gamma_{1}/\gamma$, this yields the partially averaged SME (generalized
to a set of measurement operators $\\{O_{l}\\}$)

$\displaystyle{\rm d}\rho_{c,t}$ $\displaystyle=$
$\displaystyle-i[H,\rho_{c,t}]{\rm
d}t-\sum_{l}\frac{\gamma_{l}}{2}[O_{l},[O_{l},\rho_{c,t}]]{\rm
d}t+\sum_{l}\sqrt{\eta_{l}\gamma_{l}}\\{O_{l}-\langle
O_{l}\rangle,\rho_{c,t}\\}{\rm d}W_{l,t}.$ (15)

In the limit of a perfect measurement ($\eta=1$) this evolution leaves
$\rho_{c,t}$ in a pure state, while imperfect measurements ($\eta<1$)
immediately destroy the purity of the state, thereby acting in a similar way
as a regular bath. For $\eta=0$ one finds the purely deterministic quantum
master equation of the unconditioned evolution [94]

$\dot{\rho}_{t}=\mathcal{L}\rho_{t}\qquad{\rm
with}\qquad\mathcal{L}\rho_{t}=-i[H,\rho_{t}]-\sum_{l}\frac{\gamma_{l}}{2}[O_{l},[O_{l},\rho_{t}]],$
(16)

where all the jump operators $O_{l}$ are hermitian and
$\rho_{t}=\mathbb{E}\rho_{c,t}$. This limit is then equivalent to the coupling
of the system to a Markovian dephasing bath.

##  3 Gaussian measurements

The SSE (9) and the SME (15) provide the theoretical description for a general
continuous measurement scenario, including an interacting Hamiltonian and
arbitrary measurement operators $O_{l}$. Solving for the nonlinear and
stochastic measurement-induced evolution is generally demanding and in most
cases analytically intractable. For this reason, exactly solvable reference
systems, for which the steady state solution is known are rare [13, 52, 51].

Here we introduce an analytically solvable setup, the Gaussian measurement
[73, 74]. It describes the evolution of a state under the quadratic
Hamiltonian Eq. (2) and linear measurement operators. The central object of
the theory will be the connected two-point correlation function, or covariance
matrix, $\boldsymbol{\sigma}_{c,t}$. It plays a central role for this setup
because (i) for Gaussian measurements it follows a closed, deterministic
evolution equation, which is analytically solvable, and (ii) due to the
Gaussian nature of the evolution (enabling Wick’s theorem), all higher order
connected correlation functions can be expressed uniquely by the covariance
matrix. The Gaussian measurement represents an extension of Gaussian theories,
existing for both pure and mixed states, to a continuous measurement scenario.
This provides a paradigmatic model for quantum criticality in the presence of
measurements, similar to Gaussian Hamiltonians (Lindblad master equations) for
closed (open) systems.

###  3.1 Gaussian Evolution and Linear Measurements

First, we provide the formal solution for a general Gaussian measurement
scenario, which consists of a quadratic Hamiltonian $H$ and a general set of
$N_{m}$ linear measurement operators $O_{l}$. They have unique expression in
terms of the bosons via hermitian (real) matrices $h$ ($M$),

$H=\sum_{i,j=1}^{2N}h_{ij}s_{i}s_{j}\qquad\text{ and }\qquad
O_{l}=\sum_{j=1}^{2N}M_{lj}s_{j}.$ (17)

The SSE in Eq. (9) then is at most quadratic in the operators $s_{i}$, and
therefore preserves the Gaussian nature of a state under time evolution. The
conditioned density matrix $\rho_{c,t}=|\psi_{t}\rangle_{c}\langle\psi_{t}|$
of a Gaussian state is also of the Gaussian form

$\rho_{c,t}=\rho_{c,t}[\langle\boldsymbol{s}\rangle_{c,t},\boldsymbol{\sigma}_{c,t}]=\frac{1}{Z_{c,t}}\exp\left[-({\bf
s}-\langle{\bf s}\rangle_{c,t})^{\top}\mathcal{H}_{c,t}({\bf s}-\langle{\bf
s}\rangle_{c,t})\right].$ (18)

Here, $\mathcal{H}_{c,t}$ denotes the quadratic, modular Hamiltonian, which is
connected to the conditioned covariance matrix, via the identity
$\boldsymbol{\sigma}_{c,t}=\frac{1}{2}\coth(iJ\mathcal{H}_{c,t}/2)iJ$ [95].
According to Eq. (5), the covariance matrix is
$\boldsymbol{\sigma}_{c,t}=\frac{1}{2}{\rm
Tr}(\\{\boldsymbol{s}-\langle\boldsymbol{s}\rangle_{c,t},(\boldsymbol{s}-\langle\boldsymbol{s}\rangle_{c,t})^{\top}\\}\rho_{c,t})$,
with the first moments $\langle{\bf s}\rangle_{c,t}={\rm
Tr}(\boldsymbol{s}\rho_{c,t})$. The partition function $Z_{c,t}=\sqrt{{\rm
Det}(\boldsymbol{\sigma}_{c,t}+iJ/2)}$ guarantees that
$\text{Tr}(\rho_{c,t})=1$. While the density matrix in Eq. (18) looks
appealing, one should recall that it is conditioned on a single trajectory
$|\psi_{t}\rangle_{c}$ and that $Z_{c,t}$, $\langle{\bf s}\rangle_{c,t}$ and
$\mathcal{H}_{c,t}$ are time-dependent, stochastic quantities.

Instead of computing $\rho_{c,t}$, a more suitable approach for measurement-
induced dynamics is to focus directly on the connected correlation functions
$\boldsymbol{\sigma}_{c,t}$ from Eq. (5) [62, 96, 40]. As a key feature of
Gaussian measurements, the evolution of $\boldsymbol{\sigma}_{c,t}$ becomes
independent of the measurement outcomes $I_{t}$, i.e., it is noise-free, and
therefore does not depend on individual trajectories. The time-evolution of
$\langle\boldsymbol{s}\rangle_{c,t}$ and $\boldsymbol{\sigma}_{c,t}$ is
obtained by applying the SME in Eq. (15) (see App. C.1), yielding the
equations, which we will refer to as the free boson Kalman filter,

$\displaystyle{\rm d}\langle\boldsymbol{s}\rangle_{c,t}$ $\displaystyle=$
$\displaystyle Jh\langle\boldsymbol{s}\rangle_{c,t}{\rm
d}t+2\boldsymbol{\sigma}_{c,t}M^{\top}\sqrt{\eta\Gamma}{\rm
d}\boldsymbol{W}_{t},$ (19) $\displaystyle\dot{\boldsymbol{\sigma}}_{c,t}$
$\displaystyle=$ $\displaystyle
Jh\boldsymbol{\sigma}_{c,t}+\boldsymbol{\sigma}_{c,t}(Jh)^{\top}+JM^{\top}\Gamma
MJ^{\top}-4\boldsymbol{\sigma}_{c,t}M^{\top}\eta\Gamma
M\boldsymbol{\sigma}_{c,t}.$ (20)

Here ${\rm d}\boldsymbol{W}_{t}=({\rm d}W_{1,t},\ldots,{\rm
d}W_{N_{m},t})^{\top}$ is the vector of Wiener increments obeying the Itō rule
${\rm d}\boldsymbol{W}_{t}{\rm
d}\boldsymbol{W}^{\top}_{t}=\mathbb{1}_{N_{m}}{\rm d}t$. The measurement
strength and efficiency is encoded in the diagonal matrices $\Gamma={\rm
Diag}(\gamma_{1},\ldots,\gamma_{N_{m}})>0$ and $\eta={\rm
Diag}(\eta_{1},\ldots,\eta_{N_{m}})>0$, which we allow to be different for
each measured operator $O_{l}$, $l=1,...,N_{m}$. Within this general
framework, the measurement outcomes, as in Eq. (12), follow a Wiener random
process

$\boldsymbol{I}_{t}=\frac{1}{2}\Gamma
M\langle\boldsymbol{s}\rangle_{c,t}+\frac{1}{4}\sqrt{\frac{\Gamma}{\eta}}\boldsymbol{\xi}_{t},$
(21)

with the vector of outcomes
$\boldsymbol{I}_{t}=(I_{1,t}\ldots,I_{N_{m},t})^{\top}$,
$M\langle\boldsymbol{s}\rangle_{c,t}=\langle\boldsymbol{O}\rangle_{c,t}$ and
$\boldsymbol{\xi}_{t}=\frac{{\rm d}\boldsymbol{W}_{t}}{{\rm d}t}$.

The evolution of the first moments in Eq. (19) is stochastic, but the second
moments in Eq. (20) do not depend on ${\rm d}W_{t}$ and thus are
deterministic. We emphasize that this result is obtained without performing a
stochastic average. Rather the stochastic terms in the evolution equation of
$\boldsymbol{\sigma}_{c,t}$, all appear as cubic powers of boson operators and
their quantum mechanical averages add up to zero in a Gaussian state (due to
Wick’s theorem, see App. C.1). This yields a nonlinear but closed evolution
equation, which is of the Riccati form [97]. It has in general a unique and
well-defined steady state [98]. (In App. C.2 an alternative argument for the
deterministic evolution is discussed.)

The deterministic evolution of $\boldsymbol{\sigma}_{c,t}$ has very important
consequences: as long as one remains in the Gaussian manifold, Wick’s theorem
can be applied to express any correlation function, i.e., containing arbitrary
powers in the boson fields, in terms of $\langle{\bf s}\rangle_{c,t}$ and
$\boldsymbol{\sigma}_{c,t}$. However, since Eq. (20) predicts that
$\boldsymbol{\sigma}_{c,t}$ is deterministic, the only stochastic elements
entering higher order correlations are terms $\sim\langle{\bf
s}\rangle_{c,t}$. For connected correlators, these terms, however, appear at
most to linear order. Therefore, any steady state correlation function can be
expressed in terms of $\boldsymbol{\sigma}_{c,t}$ and, at most, a linear
trajectory average of $\sim\langle{\bf s}\rangle_{c,t}$. One consequence,
which will be crucial later on, is that for any $n\in\mathbb{N}$,

$\mathbb{E}\,(\boldsymbol{\sigma}_{c,t}^{n})=\boldsymbol{\sigma}_{c,t}^{n}.$
(22)

In case of imperfect measurements the non-linear part of Eq. (20) is
suppressed with decreasing measurement efficiency $\eta<1$. In the extreme
limit $\eta=0$, the non-linearity disappears, which yields a linear equation
of motion for the covariance matrix. This is known as the Lyupanov equation.
It is equivalent to the covariance matrix for the unconditioned state,
$\rho_{t}=\mathbb{E}\rho_{c,t}$, which obeys Eq. (16).

The equation of motion Eq. (20) has a unique steady state
$\boldsymbol{\sigma}\equiv\boldsymbol{\sigma}_{c,t\rightarrow\infty}$, which
is independent of the initial conditions $\boldsymbol{\sigma}_{c,t=0}$ and,
importantly, is the same for each trajectory. We therefore drop the index for
the conditioned evolution in the following. The steady state obeys time-
translational invariance, $\dot{\boldsymbol{\sigma}}=0$, such that Eq. (20)
yields

$0=Jh\boldsymbol{\sigma}+\boldsymbol{\sigma}(Jh)^{\top}+JM^{\top}\Gamma
MJ^{\top}-4\boldsymbol{\sigma}M^{\top}\eta\Gamma M\boldsymbol{\sigma}.$ (23)

Before we discuss the steady state solution of Eq. (20) for particular cases,
we add that a similar set of equations emerges in the Kalman filter from
classical control and estimation theory [99, 98, 100, 101]. This connection is
discussed in App. C.3-C.4. The Kalman filter has also been applied to sensing
[102, 103, 104] and measurement based feedback control in quantum optics for
single particle systems [92, 105, 93, 106, 74, 76]. Here we generalized the
Riccati equation to Gaussian measurements in a many-body framework beyond
[107, 108], which enables an analytical solution for the measured free boson
CFT for different kinds of measurements.

###  3.2 Entanglement and Logarithmic Negativity

A convenient tool for the characterization of measurement-induced dynamics and
a quantifier for how close the state $\rho_{c,t}$ is to a product state, is
the entanglement between a (connected) subsystem
$A=\\{1,\ldots,n_{A}<N\\}\subset L$ and its complement $\bar{A}={L\setminus
A}$, where $L$ denotes the list of all sites. In case of perfect measurements
($\eta=1$) the steady state is pure and a convenient measure for the
entanglement of the state are the $n$-th order Rényi entropies and the von
Neumann entanglement entropy [14, 61, 15]. We want to perform an entanglement
classification for both dynamics in the presence of perfect and imperfect
measurements ($\eta<1$). The latter yields a mixed steady state for which the
notion of entanglement is more subtle [109, 63, 110]. We will focus on the
logarithmic negativity $\mathcal{N}_{A}[\rho]$ [64, 65], which is generally
(besides known exceptions [111, 112, 113]) a good measure for entanglement in
mixed states. In the limit of a pure state $\rho^{2}=\rho$, the logarithmic
negativity becomes the $\frac{1}{2}$-Rényi entanglement entropy
$\mathcal{N}_{A}[\rho]=S_{A}^{(1/2)}[\rho]=2\ln{\rm Tr}(\rho_{A}^{1/2})$. The
logarithmic negativity with respect to the subregion $A$ is

$\mathcal{N}_{A}[\rho]=\ln{\rm Tr}(|\rho^{\top_{\bar{A}}}|),\quad\text{ where
}\quad\langle
i_{A},j_{\bar{A}}|\rho^{\top_{\bar{A}}}|k_{A},l_{\bar{A}}\rangle=\langle
i_{A},l_{\bar{A}}|\rho|k_{A},j_{\bar{A}}\rangle$ (24)

denotes the partial transposition with respect to $A$, where we used
$|i_{A},j_{\bar{A}}\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$. For
a Gaussian density matrix
$\rho[\langle\boldsymbol{s}\rangle,\boldsymbol{\sigma}]$, the logarithmic
negativity only depends on the covariance matrix $\boldsymbol{\sigma}$ [83,
84, 85].

In order to compute $\mathcal{N}_{A}[\rho]$ for a Gaussian state [83, 84],
first the partial transposition for the covariance matrix is performed,

$\tilde{\boldsymbol{\sigma}}=\mathcal{T}_{\bar{A}}\boldsymbol{\sigma}\mathcal{T}_{\bar{A}}\quad\text{
with
}\quad\mathcal{T}_{\bar{A}}=\mathbb{1}_{N}\oplus(\mathbb{1}_{A}\oplus(-\mathbb{1}_{\bar{A}})).$
(25)

Then, the partially transposed covariance matrix $\tilde{\boldsymbol{\sigma}}$
is diagonalized according to the Williamson decomposition (i.e. a symplectic
transformation $S$ with $S^{\top}JS=J$)

$\tilde{\boldsymbol{\sigma}}=SDS^{\top}\text{ with }D={\rm
Diag}(\tilde{\nu}_{1},\tilde{\nu}_{1},\ldots,\tilde{\nu}_{N},\tilde{\nu}_{N}).$
(26)

The logarithmic negativity then is

$\mathcal{N}_{A}[\boldsymbol{\sigma}]=\sum_{n=1}^{N}\ln\max\left\\{1,\frac{1}{2\tilde{\nu}_{n}}\right\\}.$
(27)

Here, we focus on the half-chain logarithmic negativity, i.e., the subset
$A={1,\ldots,N/2}$ with $N$ being the length of the full system. Commonly, one
distinguishes three characteristic scenarios (neglecting a constant offset)
[86, 114, 115]

$\mathcal{N}_{A}[\boldsymbol{\sigma}]=\begin{cases}a\cdot
N^{0}&{\rm``area\;law"}\\\ \frac{c}{2}\ln(N)&{\rm``critical"}\\\ b\cdot
N&{\rm``volume\;law"}\end{cases}.$ (28)

The ‘critical’ scenario typically appears for the ground state wave function
of a critical (conformally invariant) theory. In this case the prefactor $c$
is the central charge, which is a universal property of the underlying CFT,
and $c=1$ in the case of free bosons. In the measurement-induced dynamics, we
will encounter the situation, where the entanglement entropy obeys critical
scaling even though the system is not in the ground state.

Determining the logarithmic negativity from the covariance matrix is
particularly appealing for continuous Gaussian measurements since
$\boldsymbol{\sigma}_{c,t}$ is a deterministic quantity. The same is true for
the partially transposed covariance matrix $\tilde{\boldsymbol{\sigma}}_{c,t}$
and, consequently, for the logarithmic negativity:

$\mathbb{E}\,\mathcal{N}_{A}[\boldsymbol{\sigma}]=\mathcal{N}_{A}[\mathbb{E}\,\boldsymbol{\sigma}]=\mathcal{N}_{A}[\boldsymbol{\sigma}].$
(29)

This result may a priori be surprising: For a stochastic quantity, such as
$\rho_{c,t}$ (Eq. (18)), and an arbitrary function $f$, one generally finds
the inequality $\mathbb{E}f(\rho_{c,t})\leq f(\mathbb{E}\rho_{c,t})$. The
logarithmic negativity of a Gaussian state is an important example, for which
the inequality is tight. The reason is that the fluctuating first moments of
the density matrix in Eq. (18) do not enter its computation. A similar result
in Eq. (29) has been derived recently also in a Gaussian replica field theory
for measurement-induced dynamics for Rényi entanglement entropies in Ref.
[40]. It is a special property of Gaussian measurements and does not hold for
general measurement setups [14, 116, 51, 52, 61].

###  3.3 Relaxation, Riccati Spectrum and Purification

Besides the entanglement structure, measurement-induced phases can be
characterized by their relaxation towards the steady state [77, 40]. In our
setup, the relaxation can be inferred from a set of conditioned observables
$\langle O\rangle_{c,t}$, e.g., from the outcomes of the continuous
measurements. By taking the Fourier transform, $\tilde{O}(\nu)=\int{\rm
d}te^{i\nu t}\langle O\rangle_{c,t}$ we obtain the corresponding power
spectrum $S(\nu,O)\equiv|\tilde{O}(\nu)|^{2}$, which depends on the observable
$O$ and the system size $N$. The power spectrum is characterized by its poles
$\lambda_{n}\in\mathcal{R}_{N,O}$, with $\lambda_{n}=-\kappa_{n}\pm
i\epsilon_{n}$, and $\kappa_{n}>0$ due to stability. The real part
$\kappa_{n}\geq 0$ describes the relaxation back towards the steady state,
while the imaginary part leads to coherent oscillations with frequency
$\epsilon_{n}$. Focusing on finite size systems, the poles are separated by a
finite size gap and there will be no branch cuts. We consider the spectrum of
the problem to be the union of the poles $\mathcal{R}_{{\rm
spec},N}=\\{\mathcal{R}_{N,O}\\}_{O}$ with respect to all possible
observables. For a deterministic, Hamiltonian (Lindbladian) time evolution,
$\mathcal{R}_{{\rm spec},N}$ then coincides with the spectrum of the
Hamiltonian (Lindbladian). Due to the additional nonlinear dependence of the
measurements on the state $\rho_{c,t}$, the measurement-induced spectrum may
in general also be state-dependent.

For the free boson Kalman filter, the relaxation is described by the Riccati
equation (20), which is generally nonlinear in the covariance matrix. However,
since the steady state of the Riccati equation is unique, the asymptotic
relaxation $t\rightarrow\infty$ becomes largely independent of the initial
state. In particular, if one considers small perturbations
$\delta\boldsymbol{\sigma}_{c,t}=\boldsymbol{\sigma}_{c,t}-\boldsymbol{\sigma}$
around the steady state, their relaxation is well-described by the linearized
equation. For small
$\|\delta\boldsymbol{\sigma}_{c,t}\|\ll\|\boldsymbol{\sigma}\|$, we then
expand Eq. (20) up to first order in $\delta\boldsymbol{\sigma}_{c,t}$. This
yields

$\delta\dot{\boldsymbol{\sigma}}_{c,t}\simeq Jh_{\rm
eff}\delta\boldsymbol{\sigma}_{c,t}+\delta\boldsymbol{\sigma}_{c,t}(Jh_{\rm
eff})^{\top},\text{ with }h_{\rm
eff}=h+4J\boldsymbol{\sigma}M^{\top}\eta\Gamma M.$ (30)

The structure is reminiscent of the Hamiltonian evolution of Eq. (7) but now
with an effective, i.e. non-hermitian, Hamiltonian $h_{\rm eff}^{\top}\neq
h_{\rm eff}$ which depends on state steady state. The eigenvalues
$\lambda_{n}\in\mathcal{R}_{{\rm Ricc},N}={\rm spec}(Jh_{\rm eff})$ are
complex $\lambda_{n}=-\kappa_{n}\pm i\epsilon_{n}$ and we will refer to it as
the Riccati spectrum. At sufficiently late times, when the covariance matrix
is close to the steady state, the Riccati spectrum equals the previously
defined operator spectrum $\mathcal{R}_{{\rm Ricc},N}=\mathcal{R}_{{\rm
spec},N}$ (each operator can be expressed in terms of the covariance matrix).
The asymptotic relaxation timescale $\tau_{\rm relax}^{-1}={\rm
min}\\{\kappa_{n}:n\\}$ is then determined by the inverse of the smallest
relaxation rate. For perfect measurements $\eta=1$ the relaxation time will
reduce to the purification time $\tau_{\rm relax}\rightarrow\tau_{\rm pure}$
[77, 82].

##  4 Measurement-Induced Loss of Criticality

The free boson CFT described by the Hamiltonian in Eq. (2) describes a
critical, scale invariant system. One potential consequence of performing weak
continuous measurements is the violation of scale invariance, and the
associated loss of critiality, in the dynamics. The violation of scale
invariance is then associated with the generation of a (mass) scale from
measurements. We will now discuss one possible scenario where this occurs: The
continuous observation of the fields $\Phi_{i}$. Before we consider an
explicit scenario, we start with a general measurement, which involves the
fields $\\{\Phi_{i}\\}$ but not $\\{\Pi_{i}\\}$. It is expressed by $N$
measurement operators of the form $O_{i}=\sum_{j}(m_{\phi})_{ij}\Phi_{j}$,
where $m_{\phi}$ is still an arbitrary $N\times N$ matrix. We will then focus
below on a choice of $m_{\phi}$ that breaks the discrete scale invariance of
the stochastic Schrödinger equation, and thereby generates an effective,
measurement-induced mass.

###  4.1 $\Phi$-Measurements

The general solution for measurements of the operators $O_{i}$ defined above
can be obtained readily from the free boson Kalman filter, Eq. (20). The
measurement operators above then correspond to the matrix

$M=\begin{pmatrix}m_{\phi}&0\end{pmatrix}.$ (31)

The measurement efficiencies and rates are $\eta={\rm
Diag}(\eta_{1},\ldots,\eta_{N})$ and $\Gamma={\rm
Diag}(\gamma_{1},\ldots,\gamma_{N})$. In terms of the submatrices
$\sigma_{\alpha\beta}=(\boldsymbol{\sigma})_{\alpha\beta}$ (with
$\alpha,\beta=\Phi,\Pi$) the Riccati equation yields the steady state solution

$\displaystyle(\Phi,\Phi):\qquad 0$ $\displaystyle=$
$\displaystyle\omega(\sigma_{\Pi\Phi}+\sigma_{\Phi\Pi})-4\sigma_{\Phi\Phi}\mu_{m}\sigma_{\Phi\Phi},$
(32) $\displaystyle(\Phi,\Pi):\qquad 0$ $\displaystyle=$
$\displaystyle\omega(\sigma_{\Pi\Pi}-\sigma_{\Phi\Phi}v_{N})-4\sigma_{\Phi\Phi}\mu_{m}\sigma_{\Phi\Pi},$
(33) $\displaystyle(\Pi,\Pi):\qquad 0$ $\displaystyle=$
$\displaystyle\gamma_{m}-\omega(v_{N}\sigma_{\Phi\Pi}+\sigma_{\Pi\Phi}v_{N})-4\sigma_{\Pi\Phi}\mu_{m}\sigma_{\Phi\Pi}.$
(34)

Here, we defined $\gamma_{m}=m_{\phi}^{\top}\Gamma m_{\phi}$ and
$\mu_{m}=m_{\phi}^{\top}\eta\Gamma m_{\phi}$. Equation (34) contains only
$\sigma_{\Phi\Pi}$ (and $\sigma_{\Pi\Phi}=\sigma_{\Phi\Pi}^{\top}$) and can be
solved (for details see App. D). The solution of $\sigma_{\Phi\Phi}$ and
$\sigma_{\Pi\Pi}$ is then readily obtained from the remaining equations, which
yield

$\displaystyle\sigma_{\Phi\Pi}$ $\displaystyle=$
$\displaystyle-\frac{\omega}{4}\mu_{m}^{-1}v_{N}+\frac{1}{2}\mu_{m}^{-\frac{1}{2}}\left(\gamma_{m}+\frac{\omega^{2}}{4}v_{N}\mu_{m}^{-1}v_{N}\right)^{\frac{1}{2}},$
(35) $\displaystyle\sigma_{\Phi\Phi}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\omega}}{2}\mu_{m}^{-\frac{1}{2}}\left(\sigma_{\Phi\Pi}+\sigma_{\Pi\Phi}\right)^{\frac{1}{2}},$
(36) $\displaystyle\sigma_{\Pi\Pi}$ $\displaystyle=$
$\displaystyle\left(v_{N}+\frac{4}{\omega}\sigma_{\Pi\Phi}\mu_{m}\right)\sigma_{\Phi\Phi},$
(37)

where the matrices $\mu_{m}^{-1}$ and $\mu_{m}^{-\frac{1}{2}}$ are the
(pseudo-) inverse of $\mu_{m}$ and $\mu_{m}^{\frac{1}{2}}$.

###  4.2 Measurement-induced mass

Now we consider the specific choice $O_{i}=\Phi_{i}$ for the measurement
operators. It breaks the discrete scale invariance of the stochastic
Schrödinger equation (9) and for identical $\gamma_{i}=\gamma$ and
$\eta_{i}=\eta_{0}$ it yields the matrices

$m_{\phi}=\mathbb{1}_{N},\qquad\eta=\eta_{0}\mathbb{1}_{N}\quad\text{and}\quad\Gamma=\gamma\mathbb{1}_{N}.$
(38)

For this uniform distribution of measurement rates, the system still possesses
discrete translational invariance, and it is convenient to express the
covariance matrix in Eqs. (35-37) in momentum space,
$\sigma_{\alpha\beta}(q,k)=\delta(q+k)\sigma_{\alpha\beta}(q)$, with $q\in
Q=[-\pi,\pi]$. This yields:

$\displaystyle 2\eta_{0}\gamma\sigma_{\Phi\Pi}(q)$ $\displaystyle=$
$\displaystyle\left(\eta_{0}\gamma^{2}+\frac{\omega^{2}}{4}\sin^{4}(q/2)\right)^{\frac{1}{2}}-\frac{\omega}{2}\sin^{2}(q/2),$
(39) $\displaystyle\sigma_{\Phi\Phi}(q)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\omega\sigma_{\Phi\Pi}(q)}{2\gamma\eta_{0}}},$ (40)
$\displaystyle\sigma_{\Pi\Pi}(q)$ $\displaystyle=$
$\displaystyle\left(\sin^{2}(q/2)+\frac{4\eta_{0}\gamma}{\omega}\sigma_{\Phi\Pi}(q)\right)\sigma_{\Phi\Phi}(q)$
(41)

This result is consistent with [107, 108] where coarse grained continuous
measurements of a Bose-Einstein condensate were proposed for imaging and the
backaction on the Bogoliubov modes was analyzed in detail.

####  4.2.1 Purity

In order to understand the impact of monitoring on the steady state
$\rho_{c,t\rightarrow\infty}$, we start with an analysis of its purity. For a
Gaussian state, the purity is computed from the covariance matrix via [83, 84,
85]

$-\ln\text{Tr}(\rho_{c,t\rightarrow\infty}^{2})=\ln{\rm
Det}(2\boldsymbol{\sigma})=S_{2}[{\boldsymbol{\sigma}}],$ (42)

where $S_{2}[{\boldsymbol{\sigma}}]$ is the second Rényi entropy. In the
steady state described by Eqs. (40-41), each momentum mode contributes one
eigenvalue $\frac{1}{\eta_{0}}$, and therefore

$S_{2}[\boldsymbol{\sigma}]=-\frac{N}{2}\ln\eta_{0}.$ (43)

For a perfect measurement setup $\eta_{0}=1$ and the state remains pure under
the stochastic evolution. For $\eta_{0}<1$, however, the system evolves into a
mixed state and the impurity scales extensively in the size $\sim N$. For
mixed steady states, the $n\geq 1$ Rényi entropies serve no longer as good
quantifiers for the measurement-induced dynamics [17] and we therefore make
use of the logarithmic negativity.

Figure 2: (a) Riccati spectrum $\mathcal{R}_{{\rm Ricc},N}$ for $\Phi$-field
measurements and for different measurement strengths $\gamma$. Markers
correspond to results obtained from a numerical diagonalization of Eq. (30)
with finite $N=30$ and $\eta_{0}=1$; the dashed lines correspond to the
analytical result from Eq. (45). The predominant feature is an emerging gap in
the imaginary part of the eigenvalues corresponding to the measurement-induced
mass. (b) Numerical simulation of the conditioned dynamics of the field
fluctuations at $i=N/2$ after a mass quench $r=\Omega/\omega=10$ and
$\eta_{0}=0.9$ for different system sizes $N$. The inset shows the power
spectrum for different system sizes $N$, where a trivial $\sim\delta(\nu)$ has
been omitted. (c) Steady state real space field correlations
$\sigma_{\Phi_{l}\Phi_{j}}$ for $\eta_{0}=1$ and $N=1000$ with periodic
boundary conditions for different measurement strengths $\gamma$. The solid
lines correspond to exact numerical results obtained from the solution of Eq.
(23). The dashed lines correspond to the approximate analytical result of Eq.
(48). (d) Real space correlations $\sigma_{\Phi_{l}\Phi_{j}}$ under
measurement imperfections for $\gamma/\omega=0.1$ (blue line in (c)) and
different $\eta_{0}$. Solid (dashed) lines corresponds to exact (approximate)
results.

####  4.2.2 Riccati Spectrum and Relaxation

The measurement-induced evolution approaches a unique steady state
$\boldsymbol{\sigma}$, which is independent of the initial state. At large
times, the asymptotic relaxation towards $\boldsymbol{\sigma}$ is described by
the effective Hamiltonian $h_{\rm eff}$ in Eq. (30). The first consequence of
the measurement-induced violation of scale invariance is the generation of a
relaxation scale $\kappa>0$. It appears as a mass (or gap) in the spectrum of
$h_{\rm eff}$ and imposes a rapid, exponential relaxation towards the steady
state.

For the steady state $\boldsymbol{\sigma}$ from Eqs. (39-41) the effective
Hamiltonian does not couple different momentum sectors and can be expressed
for each momentum mode $q$ individually,

$Jh_{\rm eff}=Jh-4\boldsymbol{\sigma}_{0}M^{\top}\eta\Gamma
M=\begin{pmatrix}-4\sigma_{\Phi\Phi}(q)\eta_{0}\gamma&\omega\\\
-\omega\sin^{2}(q/2)-4\sigma_{\Pi\Phi}(q)\eta_{0}\gamma&0\end{pmatrix}.$ (44)

It has complex eigenvalues

$\lambda(q)=-\kappa(q)\pm i\epsilon(q)\quad{\rm
with}\quad\epsilon(q)=\sqrt{\kappa^{2}(q)+h^{2}(q)},$ (45)

which are composed of the relaxation rate for momentum $q$,
$\kappa^{2}(q)=2\eta_{0}\gamma\omega\sigma_{\Pi\Phi}(q)$, and the free boson
dispersion $h(q)=2\omega|\sin(q/2)|$ shown in Fig. 2(a). Here, the largest
(smallest) decay rates correspond to the long (short) wavelength modes
$q\rightarrow 0$ ($q\rightarrow\pm\pi)$,

$\displaystyle\kappa_{0}$ $\displaystyle=\lim_{q\rightarrow
0}\kappa(q)=\omega\sqrt{\sqrt{\eta_{0}}\gamma/\omega},$ (46)
$\displaystyle\kappa_{\pi}$
$\displaystyle=\lim_{q\rightarrow\pm\pi}\kappa(q)=\omega\sqrt{\sqrt{\left(\gamma/\omega\right)^{2}\eta_{0}+4}-2}<\kappa_{0}.$
(47)

Both rates $\kappa_{\pi}$ and $\kappa_{0}$ are independent of the system size
$N$, which leads to an exponential decay of any perturbation in the
thermodynamics limit $N\rightarrow\infty$. The same measurement-induced gap
scale also enters the coherent oscillation frequency $\epsilon(q)$ thereby
generating an effective measurement-induced mass $m_{0}=\kappa_{0}/\omega$,
which will have consequences for the static properties like the correlations.
The slowest relaxation time scale is size independent $\tau_{\rm
relax}=\kappa_{\pi}^{-1}\sim N^{0}$. This matches the measurement-induced
purification dynamics for a system in an area law phase described in Refs.
[77, 82], where for perfect measurements a pure state is reached on a
timescale which is independent of the system size.

The gap $\kappa(q)$ remains robust also for imperfect measurements
$0<\eta_{0}<1$. Upon decreasing the efficiency $\eta_{0}$ both rates
$\kappa_{0,\pi}$ are reduced and the steady state is approached slower. The
relaxation rate vanishes in the limit $\eta_{0}\rightarrow 0$ in which a fully
mixed state is approached. Imperfect measurements thus suppress the relaxation
towards the steady state.

The measurement-induced relaxation scale $\kappa(q)$ can be measurement in an
experiment, for instance by monitoring the relaxation of an observable $O$
towards its steady state value. In general, the relaxation scale should enter
any observable but for concreteness we consider the $\Phi\Phi$-covariance,
$(\boldsymbol{\sigma}_{c,t})_{\Phi_{i}\Phi_{i}}$ with $i=N/2$. It can be
obtained from the collected measurement outcomes $\\{\boldsymbol{I}_{t}\\}$,
see Sec. 7.2. We implement a setup, in which we start from the ground state
$\boldsymbol{\sigma}_{c,t=0}=\boldsymbol{\sigma}_{{\rm GS},r}$ of a gapped
oscillator chain in Eq. (2), with a finite mass $r_{N}\rightarrow
r=\Omega/\omega$. Then the mass is suddenly set to zero and a continuous
monitoring according to Eq. (38) is implemented. We then track the relaxation
of $(\boldsymbol{\sigma}_{c,t})_{\Phi_{i}\Phi_{i}}$ over time, which is
plotted in Fig. 2(b). The power spectrum is depicted in the inset of Fig. 2(b)
and shows that the relaxation is system size independent. From the power
spectrum the dominant poles $\lambda_{n}$ for this special quench can be
extracted and their finite size scaling is shown in the inset of Fig. 3(d).
These poles match the predictions of the Riccati spectrum $\mathcal{R}_{{\rm
Ricc},N}$ and demonstrate that the relaxation rate to the steady state is
system size independent, as predicted by the relaxation scales $\kappa(q)$.

####  4.2.3 Correlation Functions and Entanglement

The measurement-induced mass not only sets a relaxation time scale but also a
correlation length, which appears, e.g., in the covariance matrix. We
illustrate this for the correlation function $\sigma_{\Phi_{l}\Phi_{j}}$
between the modes $\Phi_{l}$ and $\Phi_{j}$, for which the steady state is
provided by Eq. (40). In order to approximate its Fourier transform into real
space, we provide an analogy to the ground state of the hermitian Hamiltonian
$H$ with dispersion $h(q)$: then the real space correlation function in the
ground state is $(\sigma_{\rm
GS})_{\Phi_{l}\Phi_{j}}=\frac{\omega}{4\pi}\int_{Q}{\rm d}qe^{-iq|l-j|}/h(q)$.
In the presence of measurements, the covariance matrix, instead of being a
ground state, then corresponds to the dark state of the effective Hamiltonian
$h_{\text{eff}}$. It has a modified dispersion $h(q)\rightarrow\epsilon(q)$,
displayed in Eq. (45). To probe how physical this spectral information is we
attempt the ansatz

$\sigma_{\Phi_{l}\Phi_{j}}=\frac{\omega}{4\pi}\int_{Q}{\rm
d}q\frac{e^{iq|l-j|}}{\epsilon(q)}\simeq\frac{1}{2\pi}K_{0}(\kappa_{0}|l-j|/\omega)\sim\frac{e^{-\kappa_{0}|l-j|/\omega}}{\sqrt{\kappa_{0}|l-j|/\omega}}.$
(48)

In the second step, we have replaced
$\epsilon(q)\rightarrow\sqrt{\kappa_{0}^{2}+\omega^{2}q^{2}}$, as relevant for
$|l-j|\gg 1$. This yields the $0$-th modified Bessel function, $K_{0}(x)$,
which decays exponentially
$K_{0}(x)\simeq\sqrt{\frac{\pi}{2}}\frac{e^{-x}}{\sqrt{x}}$ for large
$\kappa_{0}|l-j|/\omega\gg 1$. Remarkably the solution of our ansatz in Eq.
(48) is in good agreement with the true correlation function obtained from the
Fourier transformation of $\sigma_{\Phi\Phi}(q)$, shown in Fig. 2(c). It
confirms that the inverse correlation length $\xi^{-1}\sim\kappa_{0}$ is set
by the relaxation rate $\kappa_{0}$.

For imperfect measurements, $0<\eta_{0}<1$, we found the dependence
$\kappa_{0}\sim\eta_{0}^{1/4}$. Therefore the correlation length
$\xi=\omega/\kappa_{0}$ increases with decreasing $\eta_{0}$, as shown in Fig.
2(d). This leaves us with the picture that continuous measurements of
$\\{\Phi_{l}\\}$ lead to a localization of the wave function in the real
space, where the degree of localization is proportional to the measurement
strength, i.e. $\xi\sim\gamma_{0}^{-1/2}$. Imperfect measurements, however,
water down the localization and lead to a growth of the localization length
with growing measurement inefficiency.

Exponentially decaying correlation functions also leave their fingerprints in
the entanglement structure of the system. The logarithmic negativity
$\mathcal{N}_{A}[{\boldsymbol{\sigma}}]$ grows for small subsystem sizes $A$
but saturates at length scales of the order of the correlation length. At
sizes $L>\xi\sim\kappa_{0}^{-1}$ it then obeys an area law [114, 115]

$\mathcal{N}_{A}[\boldsymbol{\sigma}]\simeq\frac{1}{2}\ln\left(\omega/\kappa_{0}\right),$
(49)

which is shown by the blue dashed curve in Fig. 4(a) below.

In general we find that an arbitrary weak, measurement-induced breaking of
scale invariance in the SSE (9) represents a relevant perturbation and
generates a mass $\kappa_{0}$, which then dominates correlation functions and
the relaxation behavior at large distances and large times. Overall, this may
not be surprising for a Gaussian theory, for which introducing an additional
scale usually corresponds to adding a relevant operator. Then the saturation
of the entanglement entropy to an area law and an exponentially fast
relaxation towards the steady state (including exponentially fast purification
in the perfect measurement limit $\eta_{0}=1$ [77, 82]) is a direct
consequence of the measurement-induced scale $\kappa_{0}$.

##  5 Measurement-Enriched Criticality

Another possible measurement scenario arises when the measurements are
compatible with the scale invariance of the Hamiltonian in the SSE (9), and
therefore with the conformal symmetry of the free boson CFT. In this case, we
would expect, as for a ground state, that scale invariance has an impact on
the correlation functions and the entanglement structure of the steady state.
However, if the measurement operators $\\{O_{l}\\}$ do not commute with the
Hamiltonian, the steady state will neither be the ground state nor any excited
eigenstate of the Hamiltonian. Rather it corresponds to a new type of state,
which gives rise to several modifications of the critical properties of the
theory. We term this scenario measurement-enriched criticality. For perfect
measurements, it corresponds to quantum criticality in a pure but non-ground
state. Although there exist several possible choices for measurement
operators, which preserve scale invariance, to be concrete, we consider
measurements of the operators $O_{l}=\Pi_{l}$.

###  5.1 $\Pi$-Measurements

We start again by considering a general measurement involving the operators
$\Pi_{l}$. This can be expressed in terms of a $N\times N$ matrix $m_{\pi}$
with

$O_{l}=\sum_{j=1}^{N}(m_{\pi})_{lj}\Pi_{j}.$ (50)

The measurement strength and efficiency are collected in the diagonal matrices
$\Gamma,\eta$, and again we define two matrices
$\gamma_{m}=m_{\pi}^{\top}\Gamma m_{\pi}$ and
$\mu_{m}=m_{\pi}^{\top}\eta\Gamma m_{\pi}$ for a compact notation. The Riccati
equation (23) then has the steady state solution

$\displaystyle\sigma_{\Phi\Pi}$ $\displaystyle=$
$\displaystyle\frac{\omega}{4}\mu_{m}-\frac{1}{2}\mu_{m}^{\frac{1}{2}}\left(\gamma_{m}+\frac{\omega^{2}}{4}\mu_{m}\right)^{\frac{1}{2}},$
(51) $\displaystyle\sigma_{\Pi\Pi}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\omega}}{2}\mu_{m}^{\frac{1}{2}}(-v_{N}\sigma_{\Phi\Pi}-\sigma_{\Pi\Phi}v_{N})^{\frac{1}{2}},$
(52) $\displaystyle\sigma_{\Phi\Phi}$ $\displaystyle=$
$\displaystyle\left(\mathbb{1}_{N}-\frac{4}{\omega}\sigma_{\Phi\Pi}\mu_{m}\right)\sigma_{\Pi\Pi}v_{N}^{-1}.$
(53)

###  5.2 Measurement Enriched Criticality

Now we consider the aforementioned scenario of $O_{i}=\Pi_{i}$ and a uniform
measurement strength and efficiency. This corresponds to

$m_{\pi}=\mathbb{1}_{N},\qquad\eta=\eta_{0}\mathbb{1}_{N}\quad{\rm
and}\qquad\Gamma=\gamma\mathbb{1}_{N}.$ (54)

In this case, the solutions from Sec. 5.1 have a very peculiar form. They can
be parametrized in terms of three real numbers $A,B,C\in\mathbb{R}$,

$\boldsymbol{\sigma}=\frac{1}{2}\begin{pmatrix}Av_{N}^{-\frac{1}{2}}&B\mathbb{1}_{N}\\\
B\mathbb{1}_{N}&Cv_{N}^{\frac{1}{2}}\end{pmatrix}\qquad{\rm with}\qquad
AC-B^{2}=\eta_{0}^{-1}.$ (55)

The exact dependence of the coefficients on the measurement parameter is

$\displaystyle
B=\frac{1}{2\eta_{0}\gamma}\left(\left(4\gamma^{2}\eta_{0}+\omega^{2}\right)^{\frac{1}{2}}-\omega\right),\
\ C=\sqrt{\frac{\omega B}{\eta_{0}\gamma}}\ \text{ and}\
A=\left(1+\frac{2\eta_{0}\gamma}{\omega}B\right)C.$ (56)

The parameter $B\sim\gamma$ describes a continuous deformation of the ground
state covariance matrix of the free boson CFT [see Eq. (8)]. The latter is
recovered for $B=0$ (which implies $\eta_{0}=1$). In this case, $A=1/C\neq 1$,
which would correspond to another variant of the free boson CFT, namely the
Luttinger liquid with a Luttinger parameter $K=A$ [40]. For imperfect
measurements, $\eta_{0}<1$, and in the limit $\gamma\rightarrow 0$ the steady
state is proportional to the ground state of the free boson CFT and we obtain
$\boldsymbol{\sigma}=\eta_{0}^{-1/2}\boldsymbol{\sigma}_{\rm GS}$. The
determinant of ${\boldsymbol{\sigma}}$ is then $\sim\eta_{0}^{-N}$, and we
again obtain the steady state purity
$S_{2}[\boldsymbol{\sigma}]=-\frac{N}{2}\ln\eta_{0}$. Furthermore, due to the
second equality in Eq. (55), this scaling of the determinant of
$\boldsymbol{\sigma}$ and the value of the purity holds for any $\gamma>0$.

####  5.2.1 Correlation Functions

The diagonal blocks of the covariance matrix in Eq. (55) are almost identical
to the correlation functions $\sigma_{\Phi\Phi}$ and $\sigma_{\Pi\Pi}$ in the
ground state. They only deviate from that by a multiplication with the
constants $A$ and $C$. The correlation function $\sigma_{\Phi\Phi}$ has a
divergent short-distance behavior for $N\rightarrow\infty$ and we therefore
consider the relative covariance matrix,

$\begin{split}\sigma_{\Phi_{l}\Phi_{j}}-\sigma_{\Phi_{l}\Phi_{l}}&\simeq\frac{A}{2\pi}\left[{\rm
Ci}(|l-j|)-\gamma_{\rm
EM}-\log(|l-j|)\right]\sim-\frac{A}{2\pi}\log(|l-j|).\end{split}$ (57)

Here, ${\rm Ci}(x)=-\int_{x}^{\infty}{\rm d}t\cos(t)/t\approx x^{-2}$, which
is subleading in the limit $x\gg 1$ and $\gamma_{\rm EM}$ is the Euler-
Mascheroni constant. The $\Pi_{l}-\Pi_{j}$-covariance matrix is

$\sigma_{\Pi_{l}\Pi_{j}}\simeq-\frac{C}{2\pi|l-j|^{2}}.$ (58)

A qualitative modification is found, however, for the $\Phi-\Pi$ correlations
$\sigma_{\Phi_{l}\Pi_{j}}=\delta_{lj}B$. The functions are local and vanish
for $|l-j|>0$, but acquire a short-distance or ultraviolet correction. As we
show below, this short-range correction will have a profound consequence on
the entanglement structure at large distances.

Figure 3:  (a) Riccati spectrum for $\Pi$-measurements with different
measurement strengths $\gamma$ at fixed $\eta_{0}=0.95$. The markers
correspond to the numerical diagonalization of Eq. (59) for a system of size
$N=30$. The dashed lines corresponds to the analytical result in Eq. (60). (b)
Inset: Quench dynamics under continuous $\Pi$-measurements for an initial mass
quench with $r=\Omega/\omega=10$ and $\eta_{0}=0.9$ for different system sizes
$N$. Power spectrum for the quench dynamics of the inset. In (c) we zoom in on
the region in the dashed box. The finite size scaling of the width (position),
characteristic of the real (imaginary) part of the dominant pole is depicted.
(d) Finite size scaling of the poles from the relaxation dynamics
reconstructed from fitting the power spectrum of (c). The two lower blue lines
depict the corresponding poles for the $\Phi$ measurements discussed in Sec.
4.2.2.

####  5.2.2 Riccati Spectrum and Relaxation

Next, we address the relaxation dynamics close to the steady state, i.e., the
spectrum of the effective Hamiltonian [Eq. (30) for the steady state in Eq.
(55)]

$Jh_{\rm eff}=\begin{pmatrix}0&(\omega-2B\eta_{0}\gamma)\mathbb{1}_{N}\\\
-\omega v_{N}&-2C\eta_{0}\gamma v_{N}^{\frac{1}{2}}\end{pmatrix}.$ (59)

It is diagonal in momentum space and has eigenvalues [see Fig. 3(a)]

$\lambda(q)=-2\omega|\sin(q/2)|\left(h_{-}\pm ih_{+}\right),\qquad{\rm
with}\qquad
h_{\pm}=\left(\frac{1}{2\omega}\sqrt{4\eta_{0}\gamma^{2}+\omega^{2}}\pm\frac{1}{2}\right)^{\frac{1}{2}}.$
(60)

For $\gamma\rightarrow 0$, $\lambda(q)$ approaches the Hamiltonian spectrum
$\lim_{\gamma\rightarrow 0}h_{-}=0$ and $\lim_{\gamma\rightarrow 0}h_{+}=1$.
For $\gamma>0$, the spectrum remains qualitatively similar; it is massless and
linear in $q$ for small momenta. Measuring the fields $\Pi_{l}$, then
introduces an overall constant $h_{+}\geq 1$, i.e., an effective index of
refraction, which increases the group velocity $\tilde{v}=h_{+}v$. In the
limit $\gamma\gg\omega$ it approaches
$h_{+}\sim\sqrt{\sqrt{\eta_{0}}\gamma/\omega}$, which coincides with the
measurement-induced mass scale from the $\Phi$-measurement. The real part of
the spectrum, describing the relaxation rate towards the steady state
$\boldsymbol{\sigma}$, now scales with the momentum
$\kappa(q)=2h_{-}\omega|\sin(q/2)|\simeq h_{-}\omega|q|$ for long wavelengths
$|q|\ll 1$. This transfers the common dynamical scaling behavior of quantum
critical phenomena, i.e., $\epsilon(q)\sim|q|$ also to the relaxation dynamics
$\kappa(q)\sim|q|^{z}$, with a dynamical critical exponent $z=1$.

Each momentum mode then relaxes with its own, characteristic time scale,
$\tau(q)\sim(\omega h_{-}|q|)^{-1}$ for small momenta, i.e. large distances.
In the thermodynamic limit $N\rightarrow\infty$, when the momenta $q$ are
continuous, there is no upper bound for the relaxation time and the steady
state is approached algebraically in time $\sim t^{-1}$. For any finite
system, $N<\infty$, the largest relaxation time is set by the smallest
momentum scale $q_{\text{min}}=\pm\frac{2\pi}{N}$,
$\tau(q)\leq\tau(q_{\text{min}})=\frac{N}{2\pi\omega h_{-}}$. Therefore, the
overall relaxation time grows linearly with the system size. For perfect
measurements, $\eta_{0}=1$, the steady state is again pure and the relaxation
time is identical to the purification time, which implies a purification time
scale that diverges linearly with the system size. This linear in system size
scaling has also been observed at the critical point for a measurement-induced
phase transition [77, 117], and we emphasize that the system for this type of
measurements still purifies in the thermodynamic limit. Only this purification
is not associated with a characteristic scale.

In order to discuss a potential experimental probe of the relaxation dynamics,
we consider the same scenario as in Sec. 4.2.2. The system is prepared in the
ground state of a gapped boson chain and then the mass gap is switched off and
the measurements are switched on. Here, we instead monitor the half-chain
momentum fluctuations $(\boldsymbol{\sigma}_{c,t})_{\Pi_{i}\Pi_{i}}$ at site
$i=N/2$, since this is what can be reconstructed from the measurement results
[see Sec. 7.2]. The conditioned dynamics for different system sizes $N$ are
depicted in the inset of Fig. 3(b). As we expect from our previous analysis,
the early time dynamics are system size independent, since they correspond to
small momentum modes. Close to the steady state, however, we observe
characteristic oscillations and a decay rate, which both depend on system
size. In Fig. 3(b) the power spectrum is plotted and a focus on its low-
frequency part is shown in Fig. 3(c). This confirms the system size dependence
of the low frequency part. However, one also observes that for larger system
sizes, (i) the low-frequency peaks of the spectrum move towards $\nu=0$ and
(ii) the weight of the peaks decreases continuously. Therefore the system size
dependence is continuously decreasing. We can extract the poles leading to the
peaks in the spectrum by fitting a complex Lorentzian to the low frequency
part of the spectrum. In Fig. 3(d) the scaling of the imaginary and real part
of the dominant low frequency poles with system size is shown for $\Pi$
($\Phi$)-measurement. It shows the characteristic scaling $\sim N$ ($\sim
N^{0}$), matching our discussion above.

Figure 4:  (a) Finite size scaling of the half system logaritmic negativity in
the steady state for $\Pi$ ($\Phi$) measurements. The steady states have been
obtained analytically for different system sizes $N$. For the $\Pi$
($\Phi$)-measurements we obtain a critical (area law) scaling. The markers
with dark edges correspond to the steady state obtained from the numerical
solution of Eq. (23). The black dashed line (on top of the blue) corresponds
to the analytical result in Eq. (49). (b) Measurement enriched criticality
from sliding logarithmic prefactors $c_{\gamma}^{(\alpha)}$ with respect to
the measurement strength $\gamma$. The purple (green) line corresponds to the
prefactor $c^{(1)}$ ($c^{(1/2)}$) in the finite size scaling of the $1$-Rényi
entropy ($\frac{1}{2}$-Rényi entropy/logarithmic negativity). The dashed green
(purple) line corresponds to the maximum attainable scaling constant
$c^{(1/2)}_{\rm max}\approx 1.25$ ($c^{(1)}_{\rm max}\approx 1.29$). (c) Data
collapse (non-collapse) of the mutual negativity for $\Pi$
($\Phi$)-measurements for different $\gamma$. (d) Collapse of mutual
negativity for imperfect $\Phi$-measurements at $\gamma/\omega=100$ and
different measurement efficiencies $\eta_{0}$. (e) Dependence of the scaling
prefactor $c^{(1/2)}$ in the logarithmic negativity on the measurement
efficiency $\eta_{0}$ for a fixed measurement strength $\gamma$. The dashed
line corresponds to the linear decay of $c^{(1/2)}$ with $\eta_{0}$.

####  5.2.3 Entanglement and Measurement Enriched Criticality

Let us now turn to the entanglement structure of the steady state, which we
characterize again by the half-chain logarithmic negativity $\mathcal{N}_{A}$.
For this measurement scenario, we observe that $\mathcal{N}_{A}$ scales
logarithmically with the system size [see Fig. 4(a)]

$\mathcal{N}_{A}[\boldsymbol{\sigma}]=\frac{1}{2}c^{(1/2)}_{\gamma}\ln(N)+{\rm
const}.$ (61)

Here, the prefactor of the log-scaling $c^{(1/2)}_{\gamma}$ of the logarithmic
negativity ($\frac{1}{2}$-Rényi entropy), turns out to increase monotonously
with $\gamma$.

We now focus on perfect measurements, $\eta_{0}=1$. Then $c_{\gamma}^{(1/2)}$
approaches the ground state value $c^{(1/2)}_{\gamma\rightarrow 0^{+}}=1$ in
the limit of vanishing measurement strength and converges to an upper
threshold of $c^{(1/2)}_{\rm max}\approx 1.25$ in the limit of infinite
measurement strength. Its detailed $\gamma$-dependence is displayed by the
green line in Fig. 4(b). Although the steady state for $\gamma>0$ is not a
ground state of any familiar free boson CFT, its entanglement structure is
consistent with a quantum critical, i.e., conformally invariant, ground state
in one dimension.

In order to test the conformal symmetry of the steady state, we rely on the
characteristic form of four-point correlation functions in a conformal field
theory. One possibility for such a four-point function is the mutual
(logarithmic) negativity [13, 53, 67, 17, 22]

$\mathcal{M}_{A,B}[\chi]=\mathcal{N}_{A}[\boldsymbol{\sigma}]+\mathcal{N}_{B}[\boldsymbol{\sigma}]-\mathcal{N}_{A\cup
B}[\boldsymbol{\sigma}],$ (62)

for the two intervals $A=[i_{1},i_{2}]$ and $B=[i_{3},i_{4}]$. From intervals
$A,B$ one can compute the cross ratio $\chi=i_{12}i_{34}/(i_{13}i_{24})$,
where $i_{ab}=\sin(\pi|i_{a}-i_{b}|/N)$. For a conformally invariant state,
the mutual negativity $\mathcal{M}_{A,B}\equiv\mathcal{M}(\chi)$ depends only
on the value of the cross ratio $\chi$ for any randomly chosen pair $A$ and
$B$. Indeed, as shown in Fig. 4(c), for $\Pi$ measurements, the mutual
negativity shows a scaling collapse as a function of $\chi$, indicating
conformal invariance [13, 53, 96]. The collapse is absent for measurements,
which induced a mass [see blue markers in Fig. 4(c)].

For ground states in $(1+1)$-dimensional systems, the prefactor of the
entanglement entropy (the central charge) $c$ is intimately linked to the
symmetries of the Hamiltonian. A sliding prefactor $c^{(1/2)}_{\gamma}$ (see
also [96, 62]) as depicted in Fig. 4(b) is peculiar. When interpreted as a
central charge, it would indicate that continuous monitoring would prepare the
system in the ground state of a CFT but with a larger central charge
$c_{\gamma}^{(1/2)}>1$. However, we believe that it is more likely that the
modification of $c^{(1/2)}_{\gamma}$ is rooted in the fundamentally non-
equilibrium character of the steady state, which may not correspond to a
ground state of any established CFT. This interpretation is strengthened when
considering another entanglement measure, e.g., we show the $1$-Rényi entropy
(or von Neumann entanglement entropy) as the purple line in Fig. 4(b). It
shows that the logarithmic scaling factor
$S^{(1)}_{A}[\boldsymbol{\sigma}]=\frac{1}{3}c_{\gamma}^{(1)}\ln(N)$ differs
from the one of the logarithmic negativity (see [118]). If indeed the
prefactor in the log-scaling of the entanglement measures was universal, i.e.,
as the central charge for a CFT in the ground state, we would expect
$c_{\gamma}^{(1/2)}=c_{\gamma}^{(1)}$.

We will now turn to the entanglement structure for imperfect measurements,
$0<\eta_{0}<1$. In Fig. 4(d), we depict the mutual negativity under imperfect
measurements, which indicates that the steady state is still conformally
invariant. Upon decreasing the efficiency there is still a data collapse,
however less crisp as for $\eta_{0}=1$. The logarithmic negativity
$\mathcal{N}_{A}[\boldsymbol{\sigma}]$ for fixed measurement strength $\gamma$
but decreasing efficiency $\eta_{0}$ remains a logarithmic function of the
system size and we can assign an effective logarithmic prefactor
$c_{\gamma}^{(1/2)}$ even for $\eta_{0}<1$. In Fig. 4(e) the deterioration of
this prefactor with $\eta_{0}$ is shown. We find that decreasing the
measurement efficiency $\eta_{0}$ leads to a monotonous decrease of
$c^{(1/2)}_{\gamma}$, counteracting the $\gamma$-induced increase.

####  5.2.4 Connection to $\Phi$-Measurements

A critical measurement-induced evolution can also be obtained by general
position measurements, such as in Eq. (31). This is realized, for instance, by
measuring the operators $O_{i}=\Phi_{i+1}-\Phi_{i}$ with a uniform measurement
strength $\gamma$, which corresponds to

$(m_{\phi})_{ij}=\delta_{i,j}-\delta_{i-1,j}-\delta_{iN}\delta_{j1},\qquad\eta_{\phi}=\eta_{0}\mathbb{1}_{N}\quad{\rm
and}\quad\Gamma_{\phi}=\gamma\mathbb{1}_{N}.$ (63)

Inserting this into Eqs. (39-41) we obtain a covariance matrix as in Eq. (55),
only with the constants $A$ and $C$ exchanged. Therefore, measuring
$\\{O_{i}=\Pi_{i}\\}$ or $\\{O_{i}=\Phi_{i+1}-\Phi_{i}\\}$ yields similar
steady states. This may not be too surprising since taking the continuum limit
of the latter yields $O_{i}\rightarrow O(x)=\partial_{x}\Phi(x)$. Rewriting
the Hamiltonian in Eq. (1) in terms of its dual field yields

$H=\frac{v}{2}\int_{-\frac{L}{2}}^{\frac{L}{2}}{\rm
d}x\,\left[(\partial_{x}\theta(x))^{2}+\Pi^{2}_{\theta}(x)\right],$ (64)

with the dual field $\theta(x)=\int_{-L/2}^{x}{\rm d}y\,\Pi(y)$ or
equivalently $\Pi=\partial_{x}\theta$ [119, 3]. The momentum conjugate to the
dual field $\theta$ is $\Pi_{\theta}=\partial_{x}\Phi$ and they satisfy the
canonical commutation relation $[\theta(x),\Pi_{\theta}(y)]=i\delta(x-y)$.
Since the $\theta(x)\leftrightarrow\Phi(x)$ is a symmetry of Eq. (1),
measuring $O(x)=\partial_{x}\Phi(x)=\Pi_{\theta}(x)$ is equivalent to the
momentum measurement discussed above.

##  6 Steady States with Extensive Entanglement Scaling

In the previous sections, we have discussed measurement protocols, which lead
to steady states with either logarithmic growth of the entanglement entropy or
an area law entanglement. In both cases, the entanglement structure is similar
to that of a ground state wave function, either for a gapless or a gapped
Hamiltonian. For generic measurement-induced dynamics, however, an extensive
scaling of the entanglement entropy has been reported [14, 15, 13]. An
extensive entanglement entropy, obeying a volume law growth, is typically
associated with an excited state, located in the center of the spectrum [120,
121, 122]. For Gaussian measurements it is a priori not clear, whether or not
states with an extensive entanglement entropy can be sustained from the
interplay of local measurements and local Hamiltonians due to the structure of
the reduced Hilbert space. For example, volume law entangled states for free
fermions [117] as well as for free bosons [123] have been ruled out.

In order to mimic the Hilbert space structure of generic, interacting systems
within the framework of Gaussian measurements, we will now turn to a
measurement protocol where the measured operators $\\{O_{l}\\}$ are chosen
from a random matrix ensemble. This setting is strongly idealized since the
non-local measurements are experimentally challenging to implement. In this
setting the measurement operators have a highly non-local structure in the
eigenbasis of the Hamiltonian. The measurement-induced stationary states then
correspond to highly excited states in the spectrum of the Hamiltonian and
share many properties, which are reminiscent of the eigenstates of random
matrices [124, 125, 126]. Those properties include short range correlations,
finite relaxation times as well as extensive entanglement scaling [70].

###  6.1 Measurement Operators

For concreteness we will consider measurement operators
$O_{l}=\sum_{j=1}^{N}m_{lj}\Phi_{j}$, where $m$ is a stationary but random
matrix, which is drawn from the Gaussian orthogonal ensemble (GOE). It is
sampled from to the distribution function
$\mathrm{Pr}(m)\sim\exp[-\mathrm{Tr}(m^{\top}m)]$. For this choice of random
measurement operators, the Riccati equation (23) has no particular symmetries,
which would allow us to determine the steady state solution analytically. We
thus solve it for the steady state numerically.

Figure 5: (a) Averaged real space correlations
$\overline{\sigma_{\Phi_{i}\Phi_{l}}}$ for a system size of $N=200$,
$\eta_{0}=1$ and averaged over $n_{\rm sample}=500$ random measurement
settings. (b) Scaling of the correlation length $\xi$ with increasing $\gamma$
(green markers) and $\eta_{0}$ (purple markers) for $N=200$ and $n_{\rm
sample}=500$. The shaded region corresponds to the variance between
realizations. The systematic errors for the green line can be attributed to
heuristics in finding the exponential tails (darker dashed lines) in (a). The
green dashed line corresponds to the fit $\xi\sim\gamma^{-\alpha}$ with
$\alpha=0.478$ , whereas the purple dashed line corresponds to
$\xi\sim\eta_{0}^{-\beta}$ with $\beta=0.226$. (c) Riccati spectrum for one
realization of measurement operators with real (imaginary) gaps $\kappa_{0}$
($\omega_{0}$). (d) Scaling of averaged $\overline{\omega}_{0}$ (orange
markers) and $\overline{\kappa}_{0}$ (blue markers) with $\gamma$ for a system
size of $N=200$ and $n_{\rm sample}=500$. The orange (blue) dashed line
corresponds to $\overline{\omega}_{0}\sim\gamma^{\delta}$ with $\delta=0.478$
($\overline{\kappa}_{0}\sim\gamma^{\epsilon}$ with $\epsilon=0.968$). (e)
Finite size scaling of half chain logarithmic negativity for half system
averaged over $n_{\rm sample}=500$ realizations. (f) Dependence of the linear
scaling constant $b$ on $\gamma$ (brown) and $\eta_{0}$ (turquoise) obtained
from the finite size scaling over system sizes from (e). The turquoise dashed
line indicates a linear decay. For small measurement efficiencies the
entanglement vanishes completely, in accordance to entanglement "sudden-
death".

###  6.2 Correlation Functions

In order to discuss the real space correlations in the steady state, we
compute the stationary state covariance matrix and average it over several
realizations of random measurement matrices $m$. The averaged correlation
function $\overline{\sigma_{\Phi_{i}\Phi_{l}}}$ for $|i-l|\gg 1$ decays
exponentially in the distance, featuring a correlation length $\xi$, which is
shown in Fig. 5(a). This is also characteristic for excited states of many
body systems which are well captured by eigenstates of random matrices [124,
125, 126].

The correlation length $\xi$ displays a power-law-dependence on both the
measurement strength $\gamma$ and the measurement imperfection $\eta_{0}$,
which is shown in Fig. 5(b). For perfect measurements, $\eta_{0}=1$, one finds
that the correlation length decays with the measurement strength
$\xi\sim\gamma^{-\alpha}$ with an exponent $\alpha=0.478$. This is similar to
what one finds for local $\Phi$-measurements in Sec. 4, where
$\xi\sim\gamma^{-1/2}$ [see Eq. (46)]. In turn, for fixed
$\gamma/\omega=0.01$, the correlation length decays as
$\xi\sim\eta_{0}^{-\beta}$ with $\beta=0.226$, which is again comparable with
local $\Phi$-measurements with $\xi\sim\eta_{0}^{-1/4}$. From the viewpoint of
correlations functions, random measurements thus have a comparable effect as
local $\Phi$-measurements. Both break the scale invariance of the Hamiltonian
and induce a non-zero correlation length, which decreases proportional to the
measurement strength.

###  6.3 Riccati Spectrum and Relaxation

Analogous to the non-zero correlation function $\xi$, random measurements also
induce a non-zero relaxation rate in the dynamics towards the steady state.
The Riccati spectrum for one realization of $m$ is displayed in Fig. 5(c),
which reveals a gap in the real ($\kappa_{0}$) and imaginary parts
($\omega_{0}$) of the eigenvalues $\lambda_{n}$ [cf. Eq. (45)]. The behavior
of both $\omega_{0}$ and $\kappa_{0}$ as a function of $\gamma$, and averaged
over several realizations of $m$, is shown in Fig. 5(d). For small coupling
constants $\gamma\ll\omega$, $\overline{\omega}_{0}\sim\gamma^{\delta}$ with
$\delta=0.478$, which is consistent with the behavior of the correlation
length and the identification $\omega_{0}\sim\xi$. The average relaxation
rate, however, scales as $\overline{\kappa}_{0}\sim\gamma^{\epsilon}$ with
$\epsilon=0.968$. This is twice the value of the correlation length and
therefore does not allow the identification
$\overline{\kappa}_{0}\sim\overline{\omega}_{0}\sim\xi$, which we encountered
in Eq. (46) of Sec. 4. Rather it yields $\overline{\kappa}_{0}\sim\xi^{2}$,
which corresponds to a dynamical critical exponent of $z=2$, and to a
classical (finite temperature) relaxation dynamics. It is thus consistent with
the picture of relaxation into a highly excited state in the middle of the
spectrum.

When increasing the measurement strength to values $\gamma\sim\omega$, the
corresponding correlation length becomes smaller than the distance between two
neighboring lattice sites, which also yields a breakdown of the scaling
relations for $\overline{\omega}_{0}$ and $\overline{\kappa}_{0}$.

###  6.4 Volume Law Entanglement

We will now turn to the entanglement structure of the steady state with random
measurements, which we characterize again in terms of the logarithmic
negativity. In Fig. 5(e) we show the scaling of the half-system logarithmic
negativity with system size $N$ for $\eta_{0}=1$. It reveals an extensive,
volume law scaling of the entanglement,

$\overline{\mathcal{N}_{A}[\boldsymbol{\sigma}]}\sim bN,$ (65)

with linear scaling parameter $b$ and neglecting a constant offset. Volume law
entanglement, combined with exponentially decaying correlations is a
characteristic that is usually found in excited states. Here, it is fully
attributed to the random measurements $\\{O_{l}\\}$.

Varying the measurement strength $\gamma$ preserves the volume law structure
[see Fig. 5(e)] but modifies the linear prefactor $b$, which is displayed in
Fig. 5(f). The prefactor grows proportional to the measurement strength
$\gamma$ for weak measurements $\gamma<\omega$ and then saturates around
$\gamma\sim\omega$. This is around the value, at which the correlation length
approaches the value of the lattice spacing, and when increasing $\gamma$
therefore no longer affects the steady state wave function. The monotonous
growth of the entanglement entropy with $\gamma$ has also been observed for
the critical setup in Sec. 5. In both cases the entanglement structure
indicates that the measurements increase the non-locality of the wave
function, which then leads to a growth of the entanglement entropy.

The volume law in the logarithmic negativity is robust against moving away
from perfect measurements and reducing the measurement efficiency
$\eta_{0}<1$. For sufficiently large values of $\eta_{0}\approx 1$, the
entanglement decreases linearly in $\eta_{0}$, i.e., $b\sim\eta_{0}$, see Fig.
5(f). When reaching smaller values of $\eta_{0}\approx 0.5$, it starts to
decrease faster than linear and finally vanishes $b=0$. This corresponds to
the phenomenon of entanglement "sudden-death" [127] and is attributed to the
high impurity of the steady state for small measurement efficiencies. The
point at which the logarithmic negativity vanishes completely is not universal
and depends on the specific measure of mixed state entanglement.

In summary, fully randomized measurements $O_{l}=\sum_{j}m_{lj}\Phi_{j}$ lead
to a steady state wave function (or density matrix), which mimics the behavior
of a generic wave function in ergodic quantum mechanical systems: short-ranged
correlations, finite temperature relaxation dynamics with a dynamical critical
exponent $z=2$, and a volume law entanglement structure. From this viewpoint,
random measurement matrices in the Gaussian measurement framework act
analogously to free random matrix Hamiltonians in isolated systems, and may be
used as proxies for measurement-induced dynamics in generic systems.

##  7 Full Tomography of the Conditioned State

Among the key signatures that have been established for a theoretical
characterization of measurement-induced phases and criticality in continuously
or stroboscopically observed quantum systems, are the entanglement scaling or
the timescale of purification of the conditioned quantum state. However, such
quantities require in essence the knowledge of the full conditioned quantum
many-body state, which is notoriously difficult, if yet possible to verify
experimentally. The difficulty arises from both the exponential scaling of the
number of measurements required for a full state tomography and the fact that
we are interested in properties of the conditioned state $\rho_{c,t}$, which
is the result of a specific sequence of random measurement outcomes. For
discrete measurements, it will take exponentially many trials to re-obtain the
very same measurement sequence and therefore to obtain multiple copies of
$\rho_{c,t}$ to perform measurements on. For continuously monitored systems,
the same trajectory of the measurement current will never appear twice. Thus,
the question how to overcome this “post-selection barrier" in general and to
detect measurement-induced criticality and phases in real experiments remains
an open problem of utmost importance in this field [82, 71, 29].

We demonstrate in the following that also in this respect the free boson
Kalman filter assumes a special role. For Gaussian measurements, the
entanglement structure, the relaxation (and purification) dynamics, and the
spatial correlations are encoded in the covariance matrix
$\boldsymbol{\sigma}_{c,t}$. Not only can it be solved analytically, in this
setting it is possible to reconstruct the conditioned density matrix
$\rho_{c,t}$ and $\boldsymbol{\sigma}_{c,t}$ from the knowledge of the
measurement outcomes and the measured observables. Thereby one can verify the
very characteristics of measurement-induced phases discussed above. This is an
alternative to the approaches presented in [71] and [72] where the
postselection barrier has been treated with a decoding procedure, which
requires additional operations on the system. In order to demonstrate this, we
will now concentrate on the actual aspect of a measurement, namely to learn
something about the system at hand. In particular, we will describe a
measurement protocol to determine conditioned expectation values and the full
conditioned state $\rho_{c,t}$, which for Gaussian states is equivalent to the
knowledge of $\boldsymbol{\sigma}_{c,t}$ and
$\langle\boldsymbol{s}\rangle_{c,t}$ along a single measurement trajectory.

###  7.1 Observables and Tomography of the Unconditioned State

During each realization of a continuously measured trajectory, a stream of
currents $\boldsymbol{I}_{t}$ is recorded. In general, quantum mechanical
observables are measured via the averaging of the measurement results over
many experimental runs, and therefore we have to understand
$\mathbb{E}(O)\simeq n^{-1}\sum_{i=1}^{n}O^{(i)}$ as a statistical average
over $n\gg 1$ experimental runs, where $O^{(i)}$ denotes the outcome of the
$i$-th experiment. In the same spirit, for a set of continuously measured
operators $\\{O_{l}\\}$, one records the currents $\\{I^{(i)}_{l,t}\\}$,
discussed in Eq. (12), in each experimental realization. Averaging the
measured currents over many experimental runs then allows us to determine the
unconditioned observables

$\mathbb{E}(I_{l,t})=\langle O_{l}\rangle_{t}={\rm Tr}(O_{l}\rho_{t}).$ (66)

Here, the state $\rho_{t}=\mathbb{E}\rho_{c,t}=e^{\mathcal{L}t}\rho_{t=0}$
evolves according to the master equation in Eq. (16) and corresponds to the
unconditioned, trajectory averaged density matrix. Similarly, when averaging
the fluctuations of the measured currents over many runs, we obtain

$\mathbb{E}(\boldsymbol{I}_{t+0^{+}}\boldsymbol{I}_{t}^{\top})=\frac{1}{4}\eta\Gamma
M\boldsymbol{\Sigma}_{t}M^{\top}\Gamma^{\top}\eta^{\top}.$ (67)

The left of Eq. (67) is what is obtained in the experiment, which can then be
used to compute the unconnected correlator

$\boldsymbol{\Sigma}_{t}=\frac{1}{2}\langle\\{\boldsymbol{s},\boldsymbol{s}^{\top}\\}\rangle_{t},$
(68)

if the measurement operators $M$ and $\Gamma,\eta$ are know. The unconnected
correlator corresponds to the expectation value with respect to the
unconditioned density matrix $\langle\ldots\rangle_{t}={\rm
Tr}(\ldots\rho_{t})$. The derivation of Eq. (67) is discussed in App. E.

Note that $\boldsymbol{\Sigma}_{t}$ does not depend on a specific measurement
record and therefore is the same for any large set of trajectories. This
provides some freedom for implementing an experimental detection scheme. For
instance, in order to determine $\boldsymbol{\Sigma}_{t}$ for a given
measurement setting, say $\Pi$-measurements, one can run the measurement-
induced evolution exclusively with $\Pi$-measurements up to time $t-dt$. Then
a complete set of measurements, i.e., of the $2N$ observables
$\\{O_{l}=\Phi_{l},\Pi_{l}\\}_{l=1}^{N}$, is implemented at time $t$ and this
experiment is repeated $n\gg 1$ times. In this way we can determine the full
$\boldsymbol{\Sigma}_{t}$ at an arbitrary point in time. In a Gaussian
setting, this corresponds to a full tomography of the unconditioned state,
which can be done with a number of measurements that scales only polynomially
in the system size and the evolution time. This information does not yet
reveal any characteristic behavior of the measurement-induced dynamics, since
unconditioned observables correspond to the limit of completely imperfect
measurements, $\eta_{0}=0$. Nevertheless, the knowledge of
$\boldsymbol{\Sigma}_{t}$ will allow us to reconstruct the covariance matrix
$\boldsymbol{\sigma}_{c,t}$, which we demonstrate below.

###  7.2 Tomography of the Full Conditioned Density Matrix

The previous examples highlight that the standard method of obtaining
expectation values from measurement results, via averaging over experimental
measurement outcomes, only provide access to unconditioned expectation values
and observables. However, the actual object of interest is the conditioned
state of the system, $\rho_{c,t}$ (or equivalently $\boldsymbol{\sigma}_{c,t}$
and $\langle\boldsymbol{s}\rangle_{c,t}$), during a single experimental run.
In order to extract $\boldsymbol{\sigma}_{c,t}$ from the measured current
$\boldsymbol{I}_{t}$, we follow the protocol sketched in Fig. 1(c) and make
use of Eq. (22), i.e., the deterministic (and therefore integrable) structure
of the free boson Kalman filter. From this equation we obtain the identity

$\boldsymbol{\sigma}_{c,t}=\mathbb{E}(\boldsymbol{\sigma}_{c,t})=\boldsymbol{\Sigma}_{t}-\mathbb{E}(\langle\boldsymbol{s}\rangle_{c,t}\langle\boldsymbol{s}\rangle_{c,t}^{\top}),$
(69)

which shows the subtle interplay between conditioned and unconditioned means
in the covariance matrix. This expression is remarkable in the regard that the
only conditioned quantities the covariance matrix depends on are the first
moments, which we will harness in the following. The first term,
$\boldsymbol{\Sigma}_{t}$, can be obtained in the ‘standard’ way from
measurements of the averaged current fluctuations described above [see Eq.
(67)]. The second term,
$\mathbb{E}(\langle\boldsymbol{s}\rangle_{c,t}\langle\boldsymbol{s}\rangle_{c,t}^{\top})$,
denotes the average over products of conditioned expectation values, which
evolve according to Eq. (19). Substituting the noise increments through the
measured currents, i.e.,

${\rm
d}\boldsymbol{W}_{t}=4\Gamma^{-\frac{1}{2}}\eta^{\frac{1}{2}}\left(\boldsymbol{I}_{t}-\frac{\Gamma}{2}M\langle\boldsymbol{s}\rangle_{c,t}\right){\rm
d}t,$ (70)

we obtain

$\langle\boldsymbol{s}\rangle_{c,t+{\rm
d}t}^{(i)}=\langle\boldsymbol{s}\rangle_{c,t}^{(i)}+\left\\{Jh\langle\boldsymbol{s}\rangle_{c,t}^{(i)}+2\boldsymbol{\sigma}_{c,t}M^{\top}\eta\left(\boldsymbol{I}_{t}^{(i)}-\frac{\Gamma}{2}M\langle\boldsymbol{s}\rangle_{c,t}^{(i)}\right)\right\\}{\rm
d}t.$ (71)

For a given measurement record $\boldsymbol{I}_{t}^{(i)}$, this equation
describes a deterministic update of $\langle\boldsymbol{s}\rangle_{c,t}$. It,
however, depends explicitly on the unknown $\boldsymbol{\sigma}_{c,t}$, and
therefore, Eq. (71) and Eq. (69) must be solved iteratively. To do so one
performs a large number of $n\gg 1$ experiments, each prepared in the same
initial state with known values for $\langle{\boldsymbol{s}}\rangle_{c,t=0}$
and ${\boldsymbol{\sigma}}_{c,t=0}$. For each experiment the measurement
outcomes $\\{I^{(i)}_{l,t}\\}$ are recorded and used to calculate the values
of $\langle\boldsymbol{s}\rangle_{c,t+{\rm d}t}^{(i)}$ from the known values
of $\langle\boldsymbol{s}\rangle_{c,t}^{(i)}$ and
${\boldsymbol{\sigma}}_{c,t}$. For the updated value of
${\boldsymbol{\sigma}}_{c,t+{\rm d}t}$ we incur the identity Eq. (69) with the
predetermined values of the unconditioned correlations
$\\{\boldsymbol{\Sigma}_{\tau}\\}_{\tau\leq t}$ and the ensemble average
performed over the set of $n$ conditioned first order moments
$\langle\boldsymbol{s}\rangle_{c,t}^{(i)}$. With this, all the values for
$\langle\boldsymbol{s}\rangle_{c,t}^{(i)}$ and $\boldsymbol{\sigma}_{c,t}$ can
be constructed iteratively for a given measurement trajectory. This is
equivalent to a reconstruction of the full conditioned state $\rho_{c,t}$ [see
Eq. (18)], from which all other measurement-induced characteristics, such as
the scaling of the entanglement, can be obtained.

Although the described procedure still requires many measurements and accurate
knowledge about the experimental setup, it can be done with a polynomial
number of measurements and thus overcomes the usual double exponential barrier
encountered in the reconstruction of conditioned quantum many-body states. The
reason for this surprising result is rooted in the fact that (i) Gaussians
states can be described by a polynomial number of parameters that can be
numerically integrated for rather large system sizes, and that (ii) due to the
deterministic dynamics of the covariance matrix we do not have the problem of
postselection. This means that we do not have to wait for an equivalent
measurement pattern before the state tomography can be resumed.

Finally, let us point out that in many situations of interest, it might be
enough to verify that the theoretically simulated estimate
$\hat{\boldsymbol{\sigma}}_{c,t}$, which can be calculated directly from Eq.
(20), is compatible with the obtained measurement results. In this case, the
calculated value $\hat{\boldsymbol{\sigma}}_{c,t}$, and not the measured value
of the covariance matrix, is used in Eq. (71) to evaluate the update for the
first moments. Only at the final time, the covariance matrix
$\boldsymbol{\sigma}^{*}_{c,t}$ is explicitly constructed from the ensemble
average in Eq. (69). To do so, $\boldsymbol{\Sigma}_{t}$ needs to be known
only at a single point in time $t$, which substantially reduces the
experimental cost. If
$\|\boldsymbol{\sigma}_{c,t}^{*}-\hat{\boldsymbol{\sigma}}_{c,t}\|\ll 1$ we
may conclude that
$\boldsymbol{\sigma}^{*}_{c,t},\hat{\boldsymbol{\sigma}}_{c,t}\rightarrow\boldsymbol{\sigma}_{c,t}$
almost surely.

##  8 Conclusion

We have demonstrated that bosonic Gaussian measurements represent a rather
unique class of systems, whose conditioned dynamics under continuous
measurements is exactly solvable. Starting from the unitary free boson CFT, we
examined three different types of continuous measurements, and found that the
corresponding stationary states cover the three paradigmatic cases of area
law, volume law, or a logarithmic scaling of the entanglement. Despite its
simplicity, the Gaussian measurement framework thus represents an elementary
model, which covers the conventional measurement-induced phases (volume and
area law) and quantum criticality under measurements. Common for a Gaussian
theory, it lacks the non-linear competition that leads to a measurement-
induced phase transition between the different regimes. However, similar to
Gaussian Hamiltonians (or Lindbladians), the Gaussian measurements capture the
characteristics in each regime, and provide insights on and foster our
understanding of the emergent behavior of more complicated, generic bosonic
measurement dynamics.

In addition to the established classifier of measurement-induced dynamics, the
entanglement structure of the steady state, we provide a set of additional
observables. Both the spatial correlation functions and the relaxation
behavior unambigously distinguish measurement-induced (or -enriched)
criticality, and therefore conformally invariant behavior, from the gapped
counterparts. This provides additional robust classifiers, which also work
reliably for mixed state phases in the presence of imperfect measurements, or
other environmental dephasing processes. The fact that all three observables
are determined from the deterministic covariance matrix underpins the direct
relation between correlations, relaxation behavior, and the entanglement
structure in hybrid setups and points out their significance [82, 67, 37, 41,
40].

For the Gaussian measurements, the role of the covariance matrix is
highlighted further by the fact that it can be partly (or completely)
reconstructed on the basis of the already existing continuous measurement
records. In a broader scope, this result shows that correlations between the
measurement outcomes in a given continuously measured evolution protocol
reveal information on the underlying wave function or density matrix.
Statistical correlations between measurement outcomes at different times and
(or) positions are nonlinear functions of the state of the system and thus are
the type of quantities that contain nontrivial information. When this
information is sufficient to distinguish different measurement-induced phases
(as it is for Gaussian measurements), then this provides a robust experimental
detection scheme.

Further directions, which may build up on the formalism and the results
discussed here include the analysis of the apparent conformal invariance in
the critical regime, and the impact of a dissipative environment on the hybrid
dynamics. Conformal invariance is associated with both quantum and classical
critical phenomena, and typically described by a universal ground or
stationary state. While we observe scale invariant correlations and relaxation
dynamics with universal exponents in the critical regime, the floating
prefactor of the logarithmic entanglement growth as well as the non-
universality of the central charges depending on the entanglement measure,
indicate that some aspects of universality in the steady state are violated
under measurements. Whether the hybrid dynamics yields indeed a quantum
critical steady state, or only an apparently critical state, therefore is
still an open question, which is relevant for the similar scenario of
continuously observed fermions [67, 37, 68, 44].

Environmental dissipation, even if weak, will inevitably be present in
experimental realizations of continuously measured evolution protocols.
However, the interplay between measurements and dissipation is still barely
understood, especially for dissipation induced by non-Hermitian Lindblad
operators, such as particle gain and loss. Including such processes into the
measurement-induced evolution is particularly relevant for the connection of
measurement-induced phase transitions with quantum error correction [78, 77,
80], for which environmental dissipation represents a severe limitation [128,
129, 130, 131]. Our work, with a simple but powerful formalism, may lay the
foundation for this analysis of even more general boson models, which combine
unitary evolution, measurements and dissipation.

## Acknowledgement

We thank S. Diehl, R. Küng, V. Eisert and B. Lapierre for fruitful
discussions.

##### Funding information

Y. M. and P. R. were supported by the Austrian Science Fund (FWF) through
Grant No. P32299 (PHONED). M. B. acknowledges funding from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) via grant DI 1745/2-1
under DFG SPP 1929 GiRyd.

##  Appendix A Ground State Covariance Matrix

The steady state covariance matrix in Eq. (55) is very similar to the ground-
state covariance matrix in Eq. (8), which therefore has a prominent role in
this work. In order to be self contained we restate this result, which can,
among others, be found in [86]. The Hamiltonian is diagonalized by the
symplectic transformation

$H=\frac{\omega}{2}\sum_{ij}\Phi_{i}(v_{N})_{ij}\Phi_{j}+\Pi_{i}\delta_{ij}\Pi_{j}=\frac{\omega}{2}\sum_{n}\epsilon_{n}^{2}\Phi_{n}^{2}+\Pi_{n}^{2}=\frac{\omega}{2}\sum_{n}\epsilon_{n}(\tilde{\Phi}_{n}^{2}+\tilde{\Pi}_{n}^{2}),$
(72)

where $\Phi_{n}=\sum_{j}O_{nj}\Phi_{j}$ and $\Pi_{n}=\sum_{j}O_{nj}\Pi_{j}$
with the same orthogonal matrix $O\in O(N)$. Here we used the fact that
position and momentum operators are uncoupled, such that the symplectic
transformation $S\in{\rm Sp}(2N,\mathbb{R})$, which diagonalized $H$,
simplifies to $S=O\oplus O$. $v_{N}$ is diagonalized by these orthogonal
transformations with eigenvalues $\\{\epsilon_{n}^{2}\\}$. Further, we
rescaled $\tilde{\Phi}_{n}=\Phi_{n}/\sqrt{\epsilon_{n}}$ and
$\tilde{\Pi}_{n}=\sqrt{\epsilon_{n}}\Pi_{n}$ with
$\langle\tilde{\Pi}_{n}^{2}\rangle_{\rm GS}=\epsilon_{n}/2$ and
$\langle\tilde{\Phi}_{n}^{2}\rangle_{\rm GS}=1/(2\epsilon_{n})$. Therefore,
the real space correlations in the ground state are

$\sigma_{\Phi_{i}\Phi_{j}}=\langle\Phi_{i}\Phi_{j}\rangle_{\rm
GS}=\frac{1}{2}\sum_{n}O_{in}\epsilon_{n}^{-1}O_{nj}=\frac{1}{2}(v_{N}^{-\frac{1}{2}})_{ij}$
(73)

and the momentum correlations are found in an analogous way. This gives the
expression in Eq. (8).

##  Appendix B Gaussian Measurements and Stochastic Schrödinger Equation

###  B.1 Measurement Results Follow a Wiener Process

In the main text we claimed in Eq. (12) that the measurement outcomes $I_{t}$
are distributed in a Gaussian way around the mean $\langle O\rangle_{c,t}$. In
the following we will briefly sketch the derivation of this result. We start
by considering a representation of the trace ${\rm Tr}(\ldots)=\int{\rm
d}o\,\langle o|\ldots|o\rangle$ in terms of the eigenvectors
$O|o\rangle=o|o\rangle$. The probability for outcome $I_{t}$ is then expressed
as

$\begin{split}{\rm Pr}(I_{t})={\rm
tr}\left(E^{\dagger}_{I_{t}}E_{I_{t}}\rho_{c,t}\right)&=\left(\frac{8{\rm
d}t}{\pi\gamma}\right)^{\frac{1}{2}}\int\mathrm{d}o\,\langle
o|\rho_{c,t}|o\rangle\,\mathrm{exp}\left[-\frac{8{\rm
d}t}{\gamma}\left(I_{t}-\frac{\gamma}{2}o\right)^{2}\right]\\\
&\simeq\left(\frac{8{\rm
d}t}{\pi\gamma}\right)^{\frac{1}{2}}\int\mathrm{d}o\,\delta(o-\langle
O\rangle_{c,t})\,\mathrm{exp}\left[-\frac{8{\rm
d}t}{\gamma}\left(I_{t}-\frac{\gamma}{2}o\right)^{2}\right]\\\
&=\left(\frac{8{\rm
d}t}{\pi\gamma}\right)^{\frac{1}{2}}\,\mathrm{exp}\left[-\frac{8{\rm
d}t}{\gamma}\left(I_{t}-\frac{\gamma}{2}\langle
O\rangle_{c,t}\right)^{2}\right].\end{split}$ (74)

From the first to the second line we used that for small time increments ${\rm
d}t$ with $(\gamma{\rm d}t)^{-1}\gg\Delta O_{t}^{2}$, where $\Delta
O^{2}_{t}=\langle O^{2}\rangle_{c,t}-\langle O\rangle_{c,t}^{2}$ denotes the
width of $\langle o|\rho_{c,t}|o\rangle$, the Gaussian measurement profile is
much wider than the wave function, such that we approximate the latter to be
delta-localized. From this it follows that the measurement results are
distributed in a Gaussian way around the mean $\langle O\rangle_{c,t}$ and
therefore the measurement results follow the Wiener process from Eq. (12) in
the main text.

###  B.2 Derivation of the Stochastic Schrödinger Equation

In Eq. (9) the SSE was introduced and subsequently its derivation was
sketched. Here we explicitly spell out the crucial steps of the derivation,
which we omitted in the main text. This presentation is inspired by [76, 91,
74]. The starting point is the state update in Eq. (11). Expanding the
measurement operator to lowest order in ${\rm d}t$, using that Eq. (12) is a
Wiener process and applying the Itō rule ${\rm d}W_{t}^{2}={\rm d}t$ we obtain

$E_{I_{t}}|\psi_{t}\rangle_{c}\simeq\left(1-iH{\rm
d}t-\frac{\gamma}{2}(O^{2}-4O\langle O\rangle_{c,t}){\rm
d}t+\sqrt{\gamma}O{\rm d}W_{t}\right)|\psi_{t}\rangle_{c},$ (75)

where we neglected terms of order $O({\rm d}t^{3/2})$. The norm of Eq. (75) to
lowest order is

$\lVert E_{I_{t}}|\psi_{t}\rangle_{c}\rVert\simeq 1+2\sqrt{\gamma}\langle
O\rangle_{c,t}{\rm d}W_{t}+4\gamma\langle O\rangle_{c,t}^{2}{\rm d}t$ (76)

and inverting this expression with a subsequent expansion yields

$\lVert E_{I_{t}}|\psi_{t}\rangle_{c}\rVert^{-1}=1-\frac{\gamma}{2}\langle
O\rangle_{c,t}^{2}{\rm d}t-\sqrt{\gamma}\langle O\rangle_{c,t}{\rm d}W_{t}$
(77)

to lowest order in ${\rm d}t$. Combining Eq. (77) and Eq. (75) we obtain the
discrete stochastic Schrödinger equation

$|\psi_{t+{\rm
d}t}\rangle_{c}\simeq\left(1-i\left(H-i\frac{\gamma}{2}\left(O-\langle
O\rangle_{c,t}\right)^{2}\right){\rm d}t+\sqrt{\gamma}(O-\langle
O\rangle_{c,t}){\rm d}W_{t}\right)|\psi_{t}\rangle_{c},$ (78)

and from ${\rm d}|\psi_{t}\rangle_{c}=|\psi_{t+{\rm
d}t}\rangle_{c}-|\psi_{t}\rangle_{c}$ we obtain the expression in Eq. (9) for
a single measurement. This derivation is straightforwardly extended to the
simultaneous measurement of multiple observables $O_{i}$ with strengths
$\gamma_{i}$ and independent outcomes $\\{I_{i,t}\\}$.

##  Appendix C Details on the Kalman Filter

###  C.1 Derivation of the Evolution Equation for the Conditioned First
Moments and the Covariance Matrix

In this section we derive the equations of motion for the first moments and
the covariance matrix from Eq. (19) and Eq. (20) of the main text. From the
SME in Eq. (15) we derive the equation of motion for a generic operator $A$,
which becomes

$\begin{split}{\rm d}\langle
A\rangle_{c,t}&=-\left(i\langle[A,H]\rangle_{c,t}+\sum_{i}\frac{\gamma_{i}}{2}\langle[O_{i},[O_{i},A]]\rangle_{c,t}\right){\rm
d}t\\\
&\quad+\sum_{i}\sqrt{\eta_{i}\gamma_{i}}\langle\left\\{O_{i},A\right\\}-\langle
O_{i}\rangle_{c,t}A\rangle_{c,t}{\rm d}W_{j,t}.\end{split}$ (79)

We first start by considering the unitary limit $\gamma_{i}=0$. In order to
examine the unitary evolution of the first moments we start by considering the
commutator

$\dot{s}_{i}=-i[s_{i},H]=-\frac{i}{2}\sum_{a,b}h_{ab}[s_{i},s_{a}s_{b}]=-\frac{i}{2}\sum_{a,b}iJ_{ia}h_{ab}s_{b}+iJ_{ib}h_{ba}s_{a})=(Jh\boldsymbol{s})_{i},$
(80)

from which Eq. (6) follows. The equations of motion of the operators centered
around the mean, $\tilde{s}_{i}=s_{i}-\langle s_{i}\rangle$, are the same.
From this the equation of motion for the unitary evolution of the covariance
matrix,
$\dot{\boldsymbol{\sigma}}_{ij}=(\langle\\{\dot{\tilde{s}}_{i},\tilde{s}_{j}\\}\rangle+\langle\\{\tilde{s}_{i},\dot{\tilde{s}}_{j}\\}\rangle)/2$,
is found to be Eq. (7) from the main text. Let us now turn to $\gamma_{i}>0$,
where we start by considering the first moments with the last term in Eq.
(79), which becomes

$\displaystyle\sum_{i}\sqrt{\eta_{i}\gamma_{i}}\langle\\{O_{i},s_{j}\\}-2\langle
O_{i}\rangle_{c,t}s_{j}\rangle_{c,t}{\rm d}W_{i,t}$ $\displaystyle=$
$\displaystyle\sum_{i}\sqrt{\eta_{i}\gamma_{i}}M_{ik}(\langle
s_{k}s_{j}\rangle_{c,t}+\langle s_{j}s_{k}\rangle_{c,t}-2\langle
s_{j}\rangle_{c,t}\langle s_{k}\rangle_{c,t}){\rm d}W_{i,t}$ (81)
$\displaystyle=$
$\displaystyle(2\boldsymbol{\sigma}_{c,t}M^{\top}\sqrt{\eta\Gamma}{\rm
d}\boldsymbol{W}_{t})_{j}.$

The double commutator $[s_{i},[s_{i},s_{j}]]=0$ from the second term in Eq.
(79) vanishes and therefore does not contribute to the final result of Eq.
(19) in the main text. We will now turn to the equation of motion for the
covariances. First of all we note that the equation of motion for the first
moment is now a stochastic differential equation of the Itō kind. From this we
obtain the evolution of the covariances from

$\displaystyle{\rm d}(\boldsymbol{\sigma}_{c,t})_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\langle\\{{\rm
d}s_{i},s_{j}\\}\rangle_{c,t}+\langle\\{s_{i},{\rm
d}s_{j}\\}\rangle_{c,t})-\langle s_{i}\rangle_{c,t}{\rm d}\langle
s_{j}\rangle_{c,t}-{\rm d}\langle s_{i}\rangle_{c,t}\langle
s_{j}\rangle_{c,t}-{\rm d}\langle s_{i}\rangle_{c,t}{\rm d}\langle
s_{j}\rangle_{c,t}$ (82) $\displaystyle=$
$\displaystyle\frac{1}{2}(\langle\\{{\rm
d}\tilde{s}_{i},\tilde{s}_{j}\\}\rangle_{c,t}+\langle\\{\tilde{s}_{i},{\rm
d}\tilde{s}_{j}\\}\rangle_{c,t})-{\rm d}\langle s_{i}\rangle_{c,t}{\rm
d}\langle s_{j}\rangle_{c,t},$

where the last term in both lines is attributed to the Itō correction to the
differential increment, ${\rm d}(fg)={\rm d}fg+f{\rm d}g+{\rm d}f{\rm d}g$.
The unitary evolution will be unaltered, but there will be a correction
attributed to the double commutator, which we obtain by considering

$\begin{split}\sum_{i}\frac{\gamma_{i}}{2}\langle[O_{i},[O_{i},\tilde{s}_{j}\tilde{s}_{k}]]\rangle_{c,t}&=-\sum_{i,m,n}\frac{\gamma_{i}}{2}M_{im}M_{in}\langle[s_{m},[s_{n},\tilde{s}_{j}\tilde{s}_{k}]]\rangle_{c,t}\\\
&=\sum_{i,m,n}\frac{\gamma_{i}}{2}M_{im}M_{in}(J_{nj}J_{mk}+J_{nk}J_{mj})=(JM^{\top}\Gamma
MJ^{\top})_{jk},\end{split}$ (83)

which gives the inhomogeneous contribution in Eq. (20). The most important
feature of the linear measurements is that the stochastic contribution in the
equation of motion for the covariances cancels. This is seen by noting that
$\langle\tilde{s}_{i}\rangle=0$ and since the state is at all times in the
manifold of Gaussian states, Wick’s theorem applies. Therefore, the prefactor
in the stochastic contribution in Eq. (79),

$\begin{split}\langle\left\\{s_{i},\tilde{s}_{j}\tilde{s}_{k}\right\\}-\langle
s_{i}\rangle_{c,t}\tilde{s}_{j}\tilde{s}_{k}\rangle_{c,t}&=\langle
s_{i}\tilde{s}_{j}\tilde{s}_{k}\rangle_{c,t}+\langle\tilde{s}_{j}\tilde{s}_{k}s_{i}\rangle_{c,t}-2\langle
s_{i}\rangle_{c,t}\langle\tilde{s}_{j}\tilde{s}_{k}\rangle_{c,t}\\\
&=\quad\langle
s_{i}\tilde{s}_{j}\rangle_{c,t}\langle\tilde{s}_{k}\rangle_{c,t}+\langle
s_{i}\tilde{s}_{k}\rangle_{c,t}\langle\tilde{s}_{j}\rangle_{c,t}+\langle\tilde{s}_{j}s_{i}\rangle_{c,t}\langle\tilde{s}_{k}\rangle_{c,t}+\langle\tilde{s}_{k}s_{i}\rangle_{c,t}\langle\tilde{s}_{j}\rangle_{c,t}\\\
&\qquad+\langle
s_{i}\rangle_{c,t}\langle\tilde{s}_{j}\tilde{s}_{k}\rangle_{c,t}+\langle\tilde{s}_{j}\tilde{s}_{k}\rangle_{c,t}\langle
s_{i}\rangle_{c,t}-2\langle
s_{i}\rangle_{c,t}\langle\tilde{s}_{j}\tilde{s}_{k}\rangle_{c,t}=0,\end{split}$
(84)

cancels. Finally, using the Itō rule we see that

${\rm d}\langle\boldsymbol{s}\rangle_{c,t}{\rm
d}\langle\boldsymbol{s}\rangle_{c,t}^{\top}=4\boldsymbol{\sigma}_{c,t}M^{\top}\eta\Gamma
M\boldsymbol{\sigma}_{c,t}+O({\rm d}t^{3/2}),$ (85)

which concludes the derivation of Eq. (20).

###  C.2 Alternative Argument for the Deterministic Dynamics of Covariances

In App. C.1 we showed that the evolution of the first moments is noisy,
whereas from Wick’s theorem it follows that the dynamics of the covariance
matrix is deterministic. In the following we will present an alternative
argument for this remarkable property of Gaussian measurements. We start with
the measurement operator $O=a\Phi+b\Pi=m^{\top}\boldsymbol{s}$ focusing on a
single site for simplicity. As above we defined
$\boldsymbol{s}=(\Phi,\;\Pi)^{\top}$ and $m=(a,\;b)^{\top}$. The measurement
operator Eq. (10) becomes

$E_{I_{t}}\sim\exp\left(-\frac{1}{2}{\rm
d}t(\boldsymbol{s}-\boldsymbol{j}_{t})^{\top}Q(\boldsymbol{s}-\boldsymbol{j}_{t})\right),$
(86)

where we have defined the quadratic form

$Q=2\gamma m\,m^{\top}\qquad\text{ and the noisy
vector}\qquad\boldsymbol{j}_{t}=\gamma^{-1}\begin{pmatrix}a^{-1}\\\
b^{-1}\end{pmatrix}I_{t}.$ (87)

Note that $Q$ is deterministic while only $\boldsymbol{j}_{t}$ depends on the
stochastic measurement outcomes. The operator $E_{I_{t}}$ is decomposed in
terms of an imaginary time evolution $U_{{\rm d}t}=e^{-iJQ{\rm d}t}$ and the
displacement operator
$\mathcal{D}(\boldsymbol{v})=e^{i\boldsymbol{v}^{\top}J\boldsymbol{s}}$ as

$E_{I_{t}}\sim\mathcal{D}(-\boldsymbol{j}_{t})\mathcal{D}(U_{{\rm
d}t}^{-1}\boldsymbol{j}_{t})e^{-\frac{1}{2}\boldsymbol{s}^{\top}Q\boldsymbol{s}\,{\rm
d}t}.$ (88)

Applying the measurement operator the state at time $t$ will give the updated
first moments and covariance matrix for one time step ${\rm d}t$:

$\displaystyle\langle\boldsymbol{s}\rangle_{c,t}$ $\displaystyle\rightarrow$
$\displaystyle\langle\boldsymbol{s}\rangle_{c,t+{\rm
d}t}\sim\langle\boldsymbol{s}\rangle_{c,t}+(U_{{\rm
d}t}^{-1}-\mathbb{1}_{2})\boldsymbol{j}_{t},$ (89)
$\displaystyle\boldsymbol{\sigma}_{c,t}$ $\displaystyle\rightarrow$
$\displaystyle\boldsymbol{\sigma}_{c,t+{\rm d}t}\sim U_{{\rm
d}t}\,\boldsymbol{\sigma}_{c,t}\,U_{{\rm d}t}^{\top},$ (90)

where we neglected normalization constants. The first moments are updated in a
stochastic way according to the outcomes of $I_{t}$, while the update of the
covariance matrix is deterministic. This is not too surprising since we know
from unitary time evolution that Hamiltonian terms linear in the fields will
only affect the dynamics of the first moments.

###  C.3 Review of the Classical Kalman Filter

In Sec. 3.1 we introduced the notion of the free boson Kalman filter to denote
the conditioned equations of motion for the moments. In the following we will
motivate this notation by first reviewing the classical Kalman filter and then
making the connection to the quantum problem below.

In classical control theory, or more specifically in classical estimation
theory, one frequently encounters the following problem: The state of a system
of interest is described by the real vector
$\boldsymbol{x}_{t}=(x_{1,t},\ldots,x_{N,t})^{\top}$ and obeys the following
linear equation of motion

$\displaystyle\dot{\boldsymbol{x}}_{t}$ $\displaystyle=$ $\displaystyle
A\boldsymbol{x}_{t}+V_{x}\boldsymbol{\zeta}_{t}.$ (91)

Its deterministic part is described by the $N\times N$-matrix $A$, but the
system is also affected by the process noise characterized by
$\mathbb{E}\zeta_{i,t}=0$ and
$\mathbb{E}\zeta_{i,t}\zeta_{j,s}=(V_{x}V_{x}^{\top})_{ij}\delta(t-s)$. The
state is continuously monitored through a noisy measurement device with
outcome

$\displaystyle\boldsymbol{y}_{t}$ $\displaystyle=$ $\displaystyle
C\boldsymbol{x}_{t}+V_{y}\boldsymbol{\xi}_{t}.$ (92)

Here, $C$ is an $N_{m}\times N$ matrix, where $N_{m}$ is the number of
independent measurements, and the measurement noise is characterized by
$\mathbb{E}\xi_{i,t}=0$ and
$\mathbb{E}\xi_{i,t}\xi_{j,s}=(V_{y}V_{y}^{\top})_{ij}\delta(t-s)$. There are
no cross-correlations, $\mathbb{E}\xi_{i,t}\zeta_{j,s}=0$.

The problem of filtering in classical estimation theory [99] refers to the
problem of finding an estimator $\boldsymbol{\hat{x}}_{t}$ provided that the
measurement results $\\{\boldsymbol{y}_{t}\\}$ and some aspects of the system
dynamics are known. This estimator is chosen such that it converges in some
sense to the true unknown system state $\boldsymbol{x}_{t}$. One prominent
example, where it is assumed that the system dynamics $A$, as well as the
strengths of the process and measurement noises ($V_{x}$ and $V_{y}$) are
known, is the Kalman filter. It is the prescription for constructing an
optimal estimator such that

$J_{t}=\mathbb{E}\,\boldsymbol{\epsilon}_{t}^{\top}\boldsymbol{\epsilon}_{t}\longrightarrow{\rm
min.}\qquad\text{
with}\qquad\boldsymbol{\epsilon}_{t}\equiv\boldsymbol{x}_{t}-\hat{\boldsymbol{x}}_{t}.$
(93)

This means that the fluctuations in $\boldsymbol{\epsilon}_{t}$, which is the
deviation of the estimator from the true state, are minimized. In this case
the Kalman-filter can be treated analytically and the equation of motion for
the estimated state $\hat{\boldsymbol{x}}_{t}$ evolves according to the
equation of motion

$\displaystyle\dot{\hat{\boldsymbol{x}}}_{t}$ $\displaystyle=$ $\displaystyle
A\hat{\boldsymbol{x}}_{t}+K_{F}(\boldsymbol{y}_{t}-\hat{\boldsymbol{y}}_{t}),$
(94)

where $\hat{\boldsymbol{y}}_{t}=C\hat{\boldsymbol{x}}_{t}$ are the estimated
measurement results, consistent with the estimator dynamics. The usual
procedure in filtering is the following; an experimentalist who continuously
receives a stream of measurement results $\boldsymbol{y}_{t}$ uses Eq. (94) to
solve for the estimator $\hat{\boldsymbol{x}}_{t}$ and thereby approximates
the true system state $\boldsymbol{x}_{t}$. The Kalman filter refers to the
choice of the gain, $K_{F}$, such that the cost function $J_{t}$ is minimized.
The simplicity of the Kalman filter permits that the gain $K_{F}$ can be
obtained from a simple variational computation [99] and has the explicit form

$K_{F}=\boldsymbol{S}_{t}C^{\top}(V_{I}V_{I}^{\top})^{-1}\qquad\text{with}\qquad\boldsymbol{S}_{t}=\mathbb{E}\,\epsilon_{t}\epsilon_{t}^{\top},$
(95)