# Critical Properties of Weak Measurement Induced Phase Transitions in Random
Quantum Circuits

Kemal Aziz  Department of Physics and Astronomy, Center for Materials Theory,
Rutgers University, Piscataway, NJ 08854, USA  Ahana Chakraborty  Department
of Physics and Astronomy, Center for Materials Theory, Rutgers University,
Piscataway, NJ 08854, USA  J. H. Pixley  Department of Physics and Astronomy,
Center for Materials Theory, Rutgers University, Piscataway, NJ 08854, USA
Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue,
New York, NY 10010

###### Abstract

The effects of different forms of weak measurements on the nature of the
measurement induced phase transition are theoretically studied in hybrid
random quantum circuits of qubits. We use a combination of entanglement
measures, ancilla purification dynamics, and a transfer matrix approach to
compute the critical exponents, the effective central charge, and the
multifractal spectrum of the measurement induced transitions. We compare weak
measurements with an infinite number of discrete outcomes to a protocol with
only a pair of outcomes and find that to within our numerical accuracy the
universal critical properties are unaffected by the weak measurement protocols
and are consistent with the universality class found for strong projective
measurements.

##  I Introduction

Measures of quantum entanglement, such as Reyni entanglement entropies Eisert
_et al._ (2010); Vidal  _et al._ (2003); Horodecki  _et al._ (2009); Abanin
_et al._ (2019); Calabrese and Cardy (2005), provide critical insights into a
variety of equilibrium and non-equilibrium properties in quantum many-body
systems. In hybrid quantum circuits Fisher  _et al._ (2023); Potter and
Vasseur (2022), where generic unitary dynamics competes with disentangling
random local measurements, a measurement induced phase transition (MIPT) takes
place in the structure of the entanglement Skinner  _et al._ (2019); Li  _et
al._ (2018, 2019); Bao  _et al._ (2020); Choi  _et al._ (2020); Chan  _et al._
(2019); Turkeshi  _et al._ (2021). The critical point has been shown to be
Lorentz invariantLi  _et al._ (2021a); Bao  _et al._ (2020); Skinner  _et al._
(2019); Vasseur  _et al._ (2019) and can be described by a logarithmic
conformal field theory (log-CFT)Jian  _et al._ (2020); Zabalo  _et al._ (2022)
in one-dimensional chains (though several perturbations, such as long-range
gates Block  _et al._ (2022) or static measurement profiles that are
disordered Zabalo  _et al._ (2023) or quasiperiodic Shkolnik  _et al._ (2023),
can dramatically modify this critical behavior to no longer be Lorentz
invariant). The properties of the log-CFT depends crucially on the quantum
nature of the problem and can be numerically investigated through a bulk or
boundary transfer matrix approach Zabalo  _et al._ (2022); Kumar  _et al._
(2024). In both stabilizer circuits and qudits of local Hilbert space size
$q\rightarrow\infty$, the MIPT does not have multifractal correlations Zabalo
_et al._ (2022); Li  _et al._ (2021b) that reflect the discrete nature of
Clifford gates and the classical nature of the percolation problem,
respectively. In Haar random gates on qubit chains on the other hand, the
multi-fractal correlations are strongly pronounced providing a qualitative
distinction between these problemsLudwig and Cardy (1987); Zabalo  _et al._
(2022).

A monitored circuit with weak, as opposed to strongly projective, measurements
that extract only partial information from the system provide a much more
versatile implementation of an open quantum system. Microscopically, a generic
example of a means to implement a weak measurement is via an ancilla degree of
freedom (e.g. a qubit) coupled to the system Nielsen and Chuang (2010). Upon
measurement of the ancilla, the state of the system is updated based on the
outcome of the ancilla measurement. Such measurements are generically given by
a measurement strength $J$ which can be used to tune between no observation
($J=0$) and projective measurements ($J\rightarrow\infty$) of some system
observable. At intermediate strengths, only partial information of the systems
observable that is being weakly measured can be obtained. Unlike projective
measurements, which are described by projection-valued measures, weak
measurements are described by positive operator valued measures (POVMs)Nielsen
and Chuang (2010). The size of POVMs may be larger than the dimension of the
Hilbert space of the qubit being measured (because its elements are not
necessarily orthogonal), which leads to the possibility of making simultaneous
measurements of non-commuting observables Ochoa  _et al._ (2018); Aharonov
_et al._ (1988). Putting such weak measurements into a hybrid random quantum
circuit, as depicted in Fig. 1, thus represents an interesting class of models
that have more entanglement than its strongly projective counterpart. Previous
work has shown that the MIPT can remain in the presence of weak measurements
provided the strength of the measurement is not too weakBao  _et al._ (2020);
Szyniszewski  _et al._ (2019); Doggen  _et al._ (2023).

For the generic MIPT it is commonly believed that the nature of the
universality class is independent of the type of random local measurements
being strongly projective or weak Potter and Vasseur (2022). This question is
directly investigated in this work through a detailed numerical study of the
effects of weak measurements on the MIPT in Haar random hybrid quantum
circuits. We compare and contrast two distinct models for a weak measurement;
one representing an infinite number of measurement outcomes, with one that has
a binary outcome. We focus on dual Haar random gates as this reduces the error
in the analysis of the transfer matrix as shown in Ref.Zabalo  _et al._ ,
2022. The universality class of the MIPT in Haar and dual Haar models are
expected to be the same and we further verify this by computing the remaining
(straightforwardly accessible) unknown exponents with strong projective
measurements in Appendix A. Using purification based probes on an auxiliary
ancilla qubit we provide unbiased estimates of the location of the MIPTs as a
function of the measurement strength. Across the MIPT we use finite size
scaling of the entanglement entropy of an ancilla qubit, and the mutual
information between a pair of ancilla qubits to extract several critical
exponents and show that the transition remains Lorentz invariant. This allows
us to apply the bulk transfer matrix construction of Ref. Zabalo  _et al._
(2022) to study the universal properties of the log-CFT.

To formulate the free energy of the log-CFT Zabalo  _et al._ (2022) based on
the entropy of the measurement record and further the Lyapunov spectra of the
transfer matrix, we find that it is essential to have discrete, not
continuous, measurement outcomes. To make this construction explicit we review
a model for weak measurements with continuous outcomes to then show how we can
effectively “bin” the measurement outcomes to a width $\epsilon$ allowing us
to construct a model with a discrete but infinite number of measurement
outcomes. Using this approach we compute the leading Lyapunov spectrum of the
transfer matrix to extract the effective central charge of the log-CFT, the
typical scaling dimension of the order parameter, and the multifractal
spectrum of the order parameter. To summarize, all of the typical critical
exponents agree well with the strong projective case while the leading
multifractal exponent that we have computed does have a slightly larger
deviation. Taken together, our results strongly suggest that the universality
class of the MIPT is unaffected by going from strong to weak measurements and
they are ultimately described by the same log-CFT.

The remainder of this paper is organized as follows: In Sec. II, we introduce
the models of the monitored circuits including the Haar dual unitary
entangling gates and three different protocols for weak measurements. In Sec.
III, we compute the phase diagram of the entanglement transition as a function
of the measurement strength $J$ and measurement rate. We provide numerical
evidence of Lorentz invariance at the transition and compute several critical
exponents. In Sec. IV, we study the log-CFT governing the transition through a
transfer-matrix based approach and calculate the effective central charge, the
typical scaling dimension of the order parameter, and the multifractal
spectrum. We conclude and provide outlook for future works in Sec. V. In
Appendix A we compute critical exponents of the strongly projective dual
unitary Haar model, while in Appendix B and C we provide additional details on
the models with an infinite number of measurement outcomes. Lastly in Appendix
D we show that introducing a discrete measurement outcome does not affect our
estimate of the critical properties.

##  II Models

In the following section, we will describe the models we use for the
entangling unitary gates and the various forms of weak measurements. We
consider a class of hybrid random quantum circuits, depicted in Fig. 1(a),
consisting of a chain of qubits where the unitary dynamics is generated by the
entangling gates (blue squares) that we take to be either randomly drawn from
the Haar random or dual Haar random distribution (defined below). Random local
measurements are applied to each site with probability $p$ and the red circles
in Fig. 1(a) denote where a measurement has taken place. We define one
timestep as one layer of gates, followed by one layer of measurements. With
increasing $p$, for both strong projective measurements Skinner  _et al._
(2019); Zabalo  _et al._ (2020) and (certain strengths of) weak measurements
Szyniszewski  _et al._ (2019), this hybrid circuit exhibits a MIPT in its
entanglement structure. In the following, we explore the effects of different
forms of weak measurements on the nature of the universality class of the
MIPT. In particular, we consider three models that differ in the nature (or
protocol) of the weak measurement we apply. The first two models involve weak
measurements with an infinite number of outcomes that can be continuous or
discrete. The third model involves only two outcomes but a “softened”
projection operator.

###  II.1 Entangling Unitary gates

We aim to study the most generic quantum many body circuit of qubits. At the
level of the unitary gates this is achieved by sampling each gate randomly
from the Haar distribution of random U$(4)$ matrices. Importantly, previous
work Zabalo  _et al._ (2022); Kumar  _et al._ (2024) has shown that we can
restrict this generic gate set to a smaller subset of “dual-unitary” Haar
(HDU) gates Bertini  _et al._ (2019); Gopalakrishnan and Lamacraft (2019);
Claeys  _et al._ (2022), which are unitary along the space and time direction,
and still probe the same transition while obtaining more accurate numerical
results for the free energy of the log-CFT (explained in more detail in Sec.
IV and see Refs. Zabalo  _et al._ (2022); Kumar  _et al._ (2024)). In Appendix
A, we explore this universality class further to provide additional evidence
beyond Ref. Zabalo  _et al._ , 2022 that the strong projective MIPT in random
HDU and random Haar circuits are within the same universality class. As a
result, for the majority of the paper, we use two site HDU qubit gates between
neighboring sites (unless otherwise specified) that are given by,

$U=e^{i\phi}(U_{+}\otimes U_{-})\cdot V[\theta]\cdot(V_{-}\otimes V_{+})$ (1)

where $\phi,\theta\in\mathbb{R}$ are chosen randomly from $[0,\pi)$ and
$U_{\pm},V_{\pm}\in SU(2)$, are randomly chosen from the Haar measure, and

$V[\theta]=\exp\left[-i\left(\frac{\Pi}{4}\sigma_{x}\otimes\sigma_{x}+\frac{\Pi}{4}\sigma_{y}\otimes\sigma_{y}+\frac{\Pi}{4}\theta\sigma_{z}\otimes\sigma_{z}\right)\right].$
(2)

Here, $\sigma^{x},\sigma^{y},\sigma^{z}$ are the Pauli spin-1/2 matrices.

###  II.2 Models for Weak Measurement

Figure 1: Circuit model and different measurement protocols: (a) We consider
models of a hybrid quantum circuit with a bricklayer geometry acting on a
chain of length $L$ of spin-1/2 qubits with periodic boundary conditions. The
two-site unitary gates (blue squares) are interspersed with weak measurements
at every site with probability $p$ (red circles denote the location that a
measurement has taken place). (b) In both the models coupled to the classical
pointer (see Secs. II.2.1 and II.2.2), $p(x)$ denotes the probability
distribution of the location of a classical Gaussian pointer coupled to a
qubit. The outcome of a readout operation on the pointer in the DGPM is shown
by the blue region of width $\epsilon$. The strength of the measurement
depends on the ratio of the separation between the two Gaussian peaks
($\lambda$) and the width of the Gaussian distribution $(\Delta)$. (c) The
measurement set-up for the Softened Projective Measurement Model (SPMM). For a
single qubit oriented along the $|+n\rangle$ direction in the Bloch sphere,
intermediate values of $\Lambda$ interpolate between $|+z\rangle$ and
$|+n\rangle$.

To implement a weak measurement locally in the circuit, we use the generic
description of the von Neumann model Von Neumann (2018), where the system is
first entangled with an ancilla locally, and then a projective measurement is
performed on the ancilla. We consider three weak measurement models. In the
first two models, we consider a measuring device with a “classical pointer”
Szyniszewski  _et al._ (2019) having canonically conjugate position $\hat{x}$
and momentum $\hat{q}$ operators, satisfying $[\hat{x},\hat{q}]=i$ (in units
with $\hbar=1$). The classical pointer interacts with a qubit of the system
for a time $\delta t$ with a coupling strength $\lambda$, which is then
followed by a readout of the pointer position. The readout location $x_{o}$ is
a continuous variable with an infinite number of outcomes. This readout
operation on the pointer updates the state of the qubit in the system either
partially or fully, being controlled by $\lambda$. This implements a weak
measurement on the qubit with varying strength of measurement. We first
provide a review of the continuous outcome model in Sec. II.2.1. However, we
are unable (at present) to form a transfer matrix description of this MIPT due
to the infinitesimal Born probabilities associated with it. To overcome this
limitation, we bin the measurement outcomes $x_{o}$ to a small window
$\epsilon$ around that point to form a discrete weak measurement model, that
is described in detail in Sec. II.2.2.

We also find it interesting to contrast and compare these models with a weak
measurement model with only a pair of outcomes. In subsection II.2.3, we
consider a weak measurement model obtained from softening the projective
measurement of the $z-$component of the qubit, which implements a weak
measurement with two outcomes.

####  II.2.1 Continuous Gaussian pointer

In this section, we review a well known model for a weak measurement model
where the system is coupled to a classical pointer that can be measured
continuously at position $x$Szyniszewski  _et al._ (2019). We dub this the
continuous Gaussian pointer model (CGPM). This will set the stage in the
following section to define a similar model with discrete measurement
outcomes.

The wavefunction of the pointer is initialized in a Gaussian state
$|\phi(x_{c}=0)\rangle$ of width $\Delta$ centered at $x_{c}=0$, which can be
expanded in terms of the position basis states $|x\rangle$ as:

$|\phi(x_{c}=0)\rangle=\frac{1}{\sqrt{\Delta}}\int\limits_{-\infty}^{\infty}a(x)|x\rangle
dx.$ (3)

The corresponding probability amplitudes are chosen from a squared Gaussian
distribution, namely $|a(x)|^{2}=[G_{\Delta}(x)]^{2}$, and

$G_{\Delta}(x)=\frac{e^{\frac{-x^{2}}{2\Delta^{2}}}}{\pi^{1/4}\Delta^{1/4}}$
(4)

is a Gaussian distribution of width $\Delta$ centered at $x=0$. On the other
hand, the system is initialized in a state $|\psi\rangle$ which can be written
in terms of the $2^{L}$ product states $|e_{i}\rangle$ spanning the Hilbert
space of $L$ qubits with corresponding expansion coefficients $c_{i}$s as,

$|\psi\rangle=\sum_{i=1}^{2^{L}}c_{i}|e_{i}\rangle.$ (5)

Hence the system-pointer combined initial state at $t=0$ is given by,

$|\Psi(t=0)\rangle=|\psi\rangle\otimes|\phi(x_{c}=0)\rangle.$ (6)

At the start of each measurement operation on a system qubit ($t=0$), we
couple the system with the measuring device with a tunable coupling strength
$\lambda$ via the Hamiltonian $\hat{H}_{\mathrm{int}}$,

$\hat{H}_{\mathrm{int}}=\lambda\Theta(t)\Theta(\delta
t-t)\sigma_{z}^{(j)}\otimes\hat{q}.$ (7)

Here the pointer interacts with the $j^{th}$ site of the system where
z-component of the spin is to be measured and $\Theta(x)$ denotes the
Heaviside step function. The system-pointer interaction is turned on for an
interval of time $\delta t$ imposed by the theta function. During this time
interval, the system and pointer jointly evolve under the unitary operator,

$\hat{U}_{\mathrm{int}}(\delta t)=e^{-i\hat{H}_{\mathrm{int}}\delta
t}=\Pi_{+}^{(j)}\otimes e^{-i\lambda{\delta t}\hat{q}}+\Pi_{-}^{(j)}\otimes
e^{i\lambda{\delta t}\hat{q}},$ (8)

where

$\Pi_{\pm}^{(j)}=[(\mathbb{I}\pm\sigma_{z}^{(j)})/2]\otimes_{i\not=j}\mathbb{I}^{(i)}$
(9)

projects the $z-$component of the $j^{th}$ spin onto the spin-up, or the spin-
down state. We set $\delta t=1$ without any loss of generality. The unitary
evolution under $\hat{U}_{\mathrm{int}}$ generates translation of the position
space wave-packet of the pointer and entangles them with the spin at the
$j^{th}$ site as,

$\displaystyle|\Psi(\delta t)\rangle$
$\displaystyle=\hat{U}_{\mathrm{int}}|\psi\rangle\otimes|\phi(x_{c}=0)\rangle$
$\displaystyle=$
$\displaystyle\Pi_{+}^{(j)}|\psi\rangle\otimes|\phi(x_{c}=\lambda)\rangle+\Pi_{-}^{(j)}|\psi\rangle\otimes|\phi(x_{c}=-\lambda)\rangle.$

Here, $|\Psi(\delta t)\rangle$ denotes the combined state of the system and
the pointer after the unitary evolution that consists of two Gaussian states
$|\phi(x_{c}=\pm\lambda)\rangle$ of the same width $\Delta$ with their centers
shifted to $x_{c}=\pm\lambda$ corresponding to the spin eigenstates of
$\sigma_{z}^{(j)}$.

The next step is to perform a readout operation on the pointer location by
measuring the operator
$\hat{M}_{C}(x_{o})=\mathbb{I}\otimes|x_{o}\rangle\langle x_{o}|$, where we
use the $C$ subscript to denote continuous measurement outcomes $x_{0}$. These
set of Krauss operators Nielsen and Chuang (2010) satisfy the completeness
relation:
$\int_{-\infty}^{\infty}\hat{M}_{C}^{\dagger}(x_{o})\hat{M}_{C}(x_{o})dx_{o}=1$,
as required for a POVM. The system-pointer state after the readout operation
is given by,

$\displaystyle|\Psi(\delta t)\rangle$ $\displaystyle\rightarrow$
$\displaystyle\frac{\hat{M}_{C}(x_{o})|\Psi\rangle}{||\hat{M}_{C}(x_{o})|\Psi\rangle||}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p(x_{o};\lambda/\Delta)}}\Big{[}\Pi_{+}^{(j)}G_{\Delta}(x_{o}-\lambda)$
$\displaystyle+$
$\displaystyle\Pi_{-}^{(j)}G_{\Delta}(x_{o}+\lambda)\Big{]}|\psi\rangle\otimes|x_{o}\rangle.$

The probability of measuring the pointer at $x_{o}$ is given by

$\displaystyle p(x_{o};\lambda/\Delta)$ $\displaystyle=$
$\displaystyle\langle\Psi|\hat{M}_{C}^{\dagger}(x_{o})\hat{M}_{C}(x_{o})|\Psi\rangle$
$\displaystyle=$ $\displaystyle\langle\psi|\Pi_{+}^{(j)}|\psi\rangle
G^{2}_{\Delta}(x_{o}-\lambda)+\langle\psi|\Pi_{-}^{(j)}|\psi\rangle
G^{2}_{\Delta}(x_{o}+\lambda).$

$p(x_{o})$ is schematically shown in Fig. 1(b) which consists of two
overlapping Gaussians and hence, there is no one to one correspondence between
the pointer readout position and the spin eigenstates of the qubit. As a
result, a readout/projection of pointer location to $x_{o}$ does not project
the state of system to one of eigenstates of $\sigma_{z}^{(j)}$ and the state
of the spin is only weakly measured. We recover projective measurements in the
limit of non-overlapping Gaussians ($\lambda\gg\Delta$), as the pointer will
only have nonzero probability to be in eigenstates $|x\rangle$ near the peaks
of the Gaussian wavepackets, $x_{c}=\pm\lambda$.

To summarize this subsection, we implement a weak measurement protocol on a
qubit by entangling it to a classical pointer whose position can be measured
in a continuous basis. For any arbitrary strength of measurements determined
by the ratio $\lambda/\Delta$, we perform a readout operation on the pointer
with its location $x_{o}$ sampled from the probability distribution
$p(x_{o};\lambda/\Delta)$ and update the system-pointer state governed by
Eq.(II.2.1). The two step weak-measurement operation is denoted by the
operator

$\hat{P}_{\mathrm{CGPM}}=\hat{M}_{C}(x_{o})\hat{U}_{\mathrm{int}}(\delta t).$
(13)

In the continuous model the probability density $p(x_{o};\lambda/\Delta)$ has
the dimension of inverse length and is vanishingly small as it has
infinitesimal support in real space. That is, the Born probability has a
continuous set of outcomes and within an infinitesimal spatial interval $dx$
goes like

$p_{x}^{\mathrm{CGPM}}=p(x;\lambda/\Delta)dx.$ (14)

As a result of the infinitesimal and continuous Born probability, it is not
currently straightforward to define the free energy of the log-CFT based on
the Born probabilities using this measurement model (ellaborated on in Sec.
IV). It is however, natural to do so in the limit of a discrete number of
outcomes with a finite Born probability, and we now therefore turn to
constructing a discrete measurement model based on the continuous pointer
outcomes.

####  II.2.2 Discrete Gaussian pointer

We now consider a weak measurement model where the classical pointer can be
measured in discrete positions, $|x_{i}\rangle$ separated by a distance
$\epsilon$. We call this the discrete Gaussian pointer model (DGPM). The DGPM
can be deduced from the continous measurement model, which replaces the basis
states $|x\rangle$ by $|x_{i}\rangle$. The basis states $|x_{i}\rangle$ are
obtained by binning the continuous pointer positions $|x\rangle$ over a width
of $\epsilon$. The measurement region $\epsilon$ is shown schematically by the
blue rectangles in Figure 1(b). This binning gives the orthonormal basis
states of the pointer,

$|x_{i}\rangle=\frac{1}{\sqrt{\epsilon}}\int\limits_{x_{i}-\frac{\epsilon}{2}}^{x_{i}+\frac{\epsilon}{2}}|x\rangle
dx,~{}\forall~{}i\in\mathbb{Z}.$ (15)

The pointer is initialized in a Gaussian state ${|\phi(x_{c}=0)\rangle}$ of
width $\Delta$ centered at $x_{c}=0$, which can be expanded in the discrete
position basis as,

$|\phi(x_{c}=0)\rangle=\frac{1}{\sqrt{\Delta}}\sum\limits_{i=-\infty}^{i=\infty}a_{i}|{x_{i}}\rangle,$
(16)

where the corresponding probabilities, $|a_{i}|^{2}$s are chosen from a
squared Gaussian distribution $G_{\Delta}^{2}(x)$ averaged over a width
$\epsilon$ around $|{x_{i}}\rangle$ as,

$|a_{i}|^{2}=\int\limits_{x_{i}-\frac{\epsilon}{2}}^{x_{i}+\frac{\epsilon}{2}}|a(x^{\prime})|^{2}dx^{\prime}=\int\limits_{x_{i}-\frac{\epsilon}{2}}^{x_{i}+\frac{\epsilon}{2}}G_{\Delta}^{2}(x^{\prime})dx^{\prime}.$
(17)

As previously, the system is coupled to a classical pointer whose Hamiltonian
is given by Eq. (7), but is now measured in a discrete basis.

The readout operation on the pointer location is performed by measuring the
operator $\hat{M}_{\mathrm{D}}(x_{o})=\mathbb{I}\otimes|x_{o}\rangle\langle
x_{o}|$ in discrete basis $|x_{o}\rangle$ given in Eq.15 .The measurement
operation is now denoted by the operator

$\hat{P}_{\mathrm{DGPM}}=\hat{M}_{D}(x_{o})\hat{U}_{\mathrm{int}}(\delta t).$
(18)

Thus, the system-pointer state after the readout operation is analogously
given by,

$\displaystyle|\Psi(\delta t)\rangle$ $\displaystyle\rightarrow$
$\displaystyle\frac{\hat{M}_{\mathrm{D}}(x_{o})|\Psi\rangle}{||\hat{M_{\mathrm{D}}}(x_{o})|\Psi\rangle||}$
(19) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p(x_{o};\lambda/\Delta,\epsilon)}}\Big{[}\Pi_{+}^{(j)}\sqrt{p_{+}(x_{o};\lambda/\Delta,\epsilon)}$
$\displaystyle+$
$\displaystyle\Pi_{-}^{(j)}\sqrt{p_{-}(x_{o};\lambda/\Delta,\epsilon)}\Big{]}|\psi\rangle\otimes|x_{o}\rangle,$

where $x_{o}$ is discrete and $p_{\pm}(x_{o};\lambda/\Delta,\epsilon)$ denotes
the nonzero probability amplitudes from the two Gaussian wave-packets shifted
to $x_{c}=\pm\lambda$ that is given by,

$p_{\pm}(x_{o};\lambda/\Delta,\epsilon)=\int\limits_{x_{o}-\frac{\epsilon}{2}}^{x_{o}+\frac{\epsilon}{2}}G_{\Delta}^{2}(x^{\prime}\mp\lambda)dx^{\prime}.$
(20)

Fig. 1(b) illustrates the probability distribution of the classical pointer
when two Gaussians significantly overlap. In this case the measured spin will
not collapse to an eigenstate of $\sigma_{z}$ (unless it was already in an
eigenstate prior to measurement due to the overlap).

Figure 2:  Phase diagrams for the weak measurement models considered: To
qualitatively see the phase boundary we present the variance of the bipartite
entanglement entropy, $\mathrm{var}(S)$ as color, for a chain of $L=12$ qubits
with varying measurement rate $p$ and varying strength of measurement
$J=\lambda/\Delta$ [introduced in Eq. (7) for the CGPM defined in Sec. II.2.1]
in (a) and $J=\Lambda$ [introduced in Eq. (22) for the SPMM defined in Sec.
II.2.3] in (b). The blue circles track the critical measurement rate
$p_{c}(J)$ obtained from the finite-size scaling collapse of the ancilla
entanglement entropy $S_{\rm{anc}}$ following Eq. (26). For small $p$ and weak
coupling strength the models are in the volume law (VL), entangled phase,
whereas for large coupling strength and measurement rate the models have long
time stead states that are area-law (AL) entangled. In both models, at the
phase boundary between the area and volume law phases, the fluctuations in $S$
are effectively maximized shown by the light yellow color. The white line
denotes the transitions that we focus on by computing their critical exponents
in Sec. III and their log-CFT properties in Sec. IV.

Now, the Born probability of measuring the pointer at $x_{o}$ over width
$\epsilon$ is,

$\displaystyle p(x_{o};\lambda/\Delta,\epsilon)$ $\displaystyle=$
$\displaystyle\langle\psi|\Pi_{+}^{(j)}|\psi\rangle
p_{+}(x_{o};\lambda/\Delta,\epsilon)$ (21) $\displaystyle+$
$\displaystyle\langle\psi|\Pi_{-}^{(j)}|\psi\rangle
p_{-}(x_{o};\lambda/\Delta,\epsilon).$

In the limit of small $\epsilon/\Delta$, we find analytical expressions for
the probabilities $p(x_{o};\lambda/\Delta,\epsilon)$ given in Appendix C. In
our numerical calculation, we use at most $\epsilon/\Delta=10^{-5}$ and thus
use the Born probabilities given in Eq.41. In the discrete measurement model,
we now have a well-defined Born probability,
$p(x_{o};\lambda/\Delta,\epsilon)$ of the measurement outcome measuring the
pointer within a bin centered at $x_{o}$. Therefore, we use the discrete model
in computing quantities involving the probability of the measurement record
(see Sec. IV). However, we find that the average entanglement properties such
as the half-cut and ancilla entanglement entropies are numerically equivalent
between the discrete and continuous outcome models as shown in Appendix B.

In the next subsection, we will discuss a different weak measurement model
which does not involve a system-pointer coupling to implement the measurement
protocol.

####  II.2.3 Softened Projective Measurement Model

Figure 3: Properties of the ancilla entanglement entropy and associated
critical exponents: Late time ancilla entanglement entropy
$S_{\mathrm{anc}}(t=2L;p,J)$ vs $p$ for the for the the CGPM with
$J=\lambda/\Delta=1$ in (a) and the SPMM with $J=\Lambda=0.45$ in (b) show a
crossing at the critical point Gullans and Huse (2020); Zabalo  _et al._
(2020) $p=p_{c}(J)$ in each model. We define $y=(p-p_{c})L^{1/\nu}$. The
location of MIPT $p_{c}(J)$ and the correlation length exponent $\nu(J)$ are
obtained from a finite size scaling collapse using Eq. (26) shown in the
insets. In both models, we perform the collapse with
$\mathrm{L}\in\\{12,16,20\\}$. We find $p_{c}=0.19(1)$ and $\nu=1.3(3)$ for
the CGPM with $J=\lambda/\Delta=1$ and $p_{c}=0.28(2)$ and $\nu=1.6(3)$ for
the SPMM with $J=\Lambda=0.45$. $\nu(J)$ agrees within error-bars with the
strongly projective case having $\nu=1.3(3)$. We comment however that $\nu$
within these proxies is not expected to be sufficiently accurate to make
conclusions about the critical properties. The dynamical critical exponent
$z(J)$ is obtained from the time-dependence of $S_{\mathrm{anc}}(t;p_{c},J)$
at $p=p_{c}(J)$ shown in (c) for the CGPM and (d) for the SPMM. Following Eq.
(26), we collapse $S_{\mathrm{anc}}(t;p_{c},J)$ vs $t/L^{z}$ as shown in the
insets and obtain $z(\lambda/\Delta=1)=0.94(6)$ for the CGPM and
$z(\Lambda=0.45)=0.95(5)$ for the SPMM. In both models, we perform the
collapse with $L\in\\{12,16,20\\}$.

We will now illustrate a weak measurement protocol introduced in Ref.Yang  _et
al._ , 2023 obtained by softening of the projective measurement in the
eigenbasis of $\sigma_{z}$ by a tunable softening parameter $\Lambda$. We call
this model the softened projective measurement model (SPMM). This weak
measurement operator at site $j$ is defined as,

$\hat{P}_{\pm}^{(j)}=\frac{1\pm\Lambda\sigma_{z}^{(j)}}{\sqrt{2(1+\Lambda^{2}})},$
(22)

which also satisfies the completeness condition $P_{+}+P_{-}=1$. Here
$\Lambda$ can be varied between $[0,1]$ to control the strength of the
measurement. Fig. 1 schematically shows the effect of the weak measurement in
the Bloch sphere for a single qubit. In the limit $\Lambda=0$, $\hat{P}_{\pm}$
reduces to the identity (up to a multiplicative constant). At $\Lambda=1$ the
weak measurement operators $\\{\hat{P}_{+}^{(j)},\hat{P}_{-}^{(j)}\\}$ map
onto the strong projector operators $\\{\Pi_{j,+}^{z},\Pi_{j,-}^{z}\\}$
defined in Eq. (9) implementing measurement along $|\pm z\rangle$ axis. For
the intermediate values of $\Lambda$, $\hat{P}_{\pm}$ interpolates between
$|\pm z\rangle$ and $|\pm n\rangle$. To perform the weak measurement, we
calculate the Born probabilities $p_{\pm}(\Lambda)$ corresponding to the
operators $\hat{P}_{\pm}^{(j)}$ from the system wave-function $|\psi\rangle$
as,

$\displaystyle p_{\pm}(\Lambda)$
$\displaystyle=\langle\psi|\hat{P}_{\pm}^{(j)}\hat{P}_{\pm}^{(j)}|\psi\rangle$
(23) $\displaystyle=\frac{1}{2(1+\Lambda^{2})}(1+\Lambda^{2}\pm
2\Lambda\langle\psi|\sigma_{z}^{(j)}|\psi\rangle).$

Based on $p_{\pm}(\Lambda)$, we measure either $\hat{P}_{+}^{(j)}$ or
$\hat{P}_{-}^{(j)}$ and update the state of the system to

$|\psi\rangle\rightarrow\frac{\hat{P}_{\pm}^{(j)}|\psi\rangle}{||\hat{P}_{\pm}^{(j)}|\psi\rangle||}.$
(24)

This implements a soften version of the standard projective measurement
controlled by the parameter $\Lambda$.

##  III Phase Diagrams of weak MIPTs

In this section, we will locate the entanglement phase transition as a
function of measurement strength ($J$) and rate ($p$). To get a qualitative
sense of the phase diagram, we use the variance of the bipartite entanglement
entropy as a function of measurement strength and probability in Sec. III.1.
To provide an accurate, unbiased estimate of the location of the MIPT we use
finite size scaling of an ancilla based local order parameter Gullans and Huse
(2020); Zabalo  _et al._ (2020) in Sec. III.2. To compute all the Reyni
entropies (e.g. the half-cut bipartite Renyi entropy and the ancilla Renyi
entropies defined below) in the models with a classical pointer coupled to the
system, we use the CGPM. As we show in Appendix B, both models yield the same
numerical value for the Reyni entropies, up to negligible numerical
fluctuations of the order $10^{-3}$ times smaller than the value of the Reyni
entropies.

Figure 4: Mutual information between two ancillas at the critical point and
exponent $\eta$: (a) Mutual information between two ancillas entangled at
$r=1$ and $r^{\prime}=\frac{L}{2}$ in the steady state of the circuit evolving
with the CGPM at $\lambda/\Delta=1$ and $p=p_{c}=0.19(3)$ shown in (a) and
that for SPMM at $\Lambda=0.45$ and $p=p_{c}=0.28(2)$ shown in (b). We use
three different system sizes $L=12,16,20$ for the scaling collapse following
Eq. (27) shown in the inset. We find the bulk exponent $\eta=0.19(3)$ in SPMM
and $\eta=0.21(2)$ in CGPM. We average over $10^{4}$ trajectories for each
system size.

###  III.1 Bi-partite entanglement entropy

We begin by estimating the phase boundary from fluctuations in bi-partite the
entanglement entropy. To compute the bipartite entanglement entropy, we divide
the system into two halves (denoted A and B) and the reduced density matrix
$\rho_{A}=\rm{Tr}_{B}[|\psi(t)\rangle\langle\psi(t)|]$ is used to compute the
half-cut von Neumann entanglement entropy

$S_{n=1}=-\rm{Tr}_{A}[\rho_{A}\ln\rho_{A}].$ (25)

Following Ref. Szyniszewski  _et al._ , 2019, we compute the variance of the
bi-partite entanglement entropy $\mathrm{var}(S)$ as a qualitative proxy of
the phase diagram. We compute the variance from an ensemble of entanglement
entropy values constituted from (i) the steady state $S(t=2L)$ values from
different quantum trajectories Zabalo  _et al._ (2022) and (ii) $S(t)$ values
from a single trajectory choosing $t$ from a quasi-stationary regime, $t=L/2$
to $100$. This steady state regime is chosen such that the initial growth of
the entropy with time has saturated. Fig. 2 shows $\rm{var}(S)$ in the two-
dimensional parameter space spanned by $p$ and $J$ with a color plot where the
lighter color indicates larger variance. Fig. 2(a) corresponds to the CGPM
while Fig. 2(b) is for the SPMM. We average the bi-partite entanglement
entropy over 3000 trajectories for CGPM and 2000 trajectories for SPMM for
each values of $p$ and $J$. The variance is expected to be maximal at the
critical measurement rate $p=p_{c}(J)$ (where $J=\lambda/\Delta$ for CGPM/DGPM
and $J=\Lambda$ for SPMM), and as shown in the data we see a clear maximum in
the parameter space for a fixed system size $L=12$. However, the location of
this maximum is known to drift with increasing system size Szyniszewski  _et
al._ (2019) and therefore in the next section we consider a separate unbiased
estimate of $p_{c}(J)$. In particular, we use the ancilla entanglement entropy
defined below in the next subsection, and its crossing for various system
sizes as a function of $p$ to locate $p_{c}(J)$.

###  III.2 Ancilla Qubit Order Parameter

To construct a local order parameter of the MIPT Gullans and Huse (2020), we
couple an ancilla qubit locally by putting it in a Bell pair with a spin in
the system. We then apply an encoding step to scramble the locally entangled
ancilla by running the circuit without measurements and only unitary gates out
to $t_{o}=2L$, which prepares the ancilla in a maximally entangled states with
the system. We then run the hybrid measurement and unitary dynamics and call
this time $t=0$ in the results presented.

As a function of time we calculate the ancilla von Neumann entanglement
entropy $S_{\rm{anc}}$ of the reduced density matrix of the ancilla after
integrating out all of the spins in the circuit. In the monitored dynamics,
$S_{\rm{anc}}(t;p,J)$ decreases monotonically with time from its maximum value
$1$ as time increases. The ancilla entanglement entropy $S_{\rm{anc}}(t;p,J)$
serves as an order parameter for the MIPT in the steady state ($t\approx 2L$)
Gullans and Huse (2020); Zabalo  _et al._ (2020). To estimate the critical
point $p_{c}(J)$, we perform a finite-size scaling with the following ansatz

$S_{\rm{anc}}(t;p,L)\sim Q((p-p_{c})L^{1/\nu},t/L^{z}),$ (26)

where $Q(x,y)$ is an arbitrary scaling function, $\nu(J)$ and $z(J)$ are the
correlation length exponent and the dynamical exponents of the MIPT,
respectively.

We show the data for $S_{\rm{anc}}(t=2L;p,J)$ vs $p$ for the for the CGPM
model with $J=\lambda/\Delta=1$ in Fig. 3(a) and SPMM model with
$J=\Lambda=0.45$ in Fig. 3(b), in which both display a clear crossing at
$p_{c}(J)$ for various systems sizes, consistent with the scaling ansatz in
Eq. (26). Fixing the aspect ratio of time and the system-size ($t=2L$), we
collapse the ancilla order parameter for different system sizes $L=12,16,20$
to estimate $p_{c}(J)$ and $\nu(J)$ as shown in the insets of Fig. 3(a) and
(b). This yields $p_{c}(\lambda/\Delta=1)=0.19(1)$ and
$\nu(\lambda/\Delta=1)=1.3(3)$ for the CGPM and $p_{c}(\Lambda=0.45)=0.28(2)$
and $\nu(\Lambda=0.45)=1.6(3)$ for the SPMM. These estimates of $\nu$ agree
within their numerical accuracy with the value for the projective measurement
case $\nu\approx 1.2(3)$ Zabalo  _et al._ (2020). We average
$S_{\rm{anc}}(t=2L;p,J)$ over 3,000 trajectories for CGPM and 2,000
trajectories in SPMM for each values of $p,L$ and $J$.

Varying the strength of the measurements $J$, we calculate
$p_{c}(\lambda/\Delta)$ for the CGPM and $p_{c}(\Lambda)$ for the SPMM shown
in Fig.2 (a) and (b) respectively as solid blue circles. The critical
transition rate $p_{c}(J)$ increases with decreasing strength of measurement
$J(=\lambda/\Delta~{}\mathrm{or}~{}\Lambda)$ in both the models as expected.
This estimation of $p_{c}$ from the ancilla order parameter ($S_{\rm{anc}}$)
clearly follows (though not precisely) the locus of the maximum of var$(S)$
(at fixed system size $L=12$) that is shown by bright yellow color.
Interestingly, in both models we find that if the measurement strength is too
weak its not possible to drive a phase transition out of the volume law phase
with measurements.

To estimate the dynamical exponent $z$, we study the time-dependence of the
ancilla order-parameter at the critical measurement $p=p_{c}$. In Fig. 3 we
show $S_{\rm{anc}}(t;p_{c},J)$ as a function of $t$ for the CPGM with
$\lambda/\Delta=1$ in (c) and for the SPMM with $\Lambda=0.45$ in (d) for
three different system-sizes $L=12,16,20$. We average the time-dependent
$S_{\rm{anc}}(t;p_{c},J)$ over 10,000 trajectories for both CGPM and and SPMM
for each values of $L$ and $J$. We perform a finite size scaling collapse of
$S_{\rm{anc}}(t;p_{c},J)$ as a function of $t/L^{z(J)}$ using Eq.26 and the
collapsed curves are shown in the insets of Fig. 3 (c) and (d). We find
$z(\lambda/\Delta=1)=0.94(6)$ and $z(\Lambda=0.45)=0.95(5)$. Thus our results
provide strong numerical evidence of conformal invariance at the weak
measurement induced transition.

###  III.3 Mutual information between two ancilla qubits

We next compute the anomalous dimension exponent $\eta(J)$ following the
protocol prescribed in Ref.Gullans and Huse, 2020; Zabalo  _et al._ , 2020. We
initialize the system in a random product state and then evolve the circuit
with monitored dynamics to reach a steady state at time $t_{o}=20L$. We note
that in HDU circuit this wait time is much longer to reach the steady state
for the mutual information between two ancillas compared to that with Haar
random gates (where $t_{o}\approx 2L$), though we find this is important to
reach a stable (i.e. wait time independent) estimate of $\eta$. We introduce
two ancilla qubits $\tilde{A}$ and $\tilde{B}$, and maximally entangle them
with the circuit at the spacetime points $(r,t_{o})$ and $(r^{\prime},t_{o})$.
We define the connected order-parameter correlation function $C(t-t_{o})$ as
the mutual information between the ancillas $\tilde{A}$ and $\tilde{B}$. In
both the volume law phase, $p<p_{c}$, and area law phase, $p>p_{c}$,
$C(t-t_{o})\sim\mathrm{exp}(-L/\xi)$ for $L\gg\xi$ where
$\xi\sim(p-p_{c})^{\nu}$ is the finite correlation length at generic $p\neq
p_{c}$. At MIPT $p=p_{c}(J)$, $C(t-t_{o})\sim 1/L^{\eta}$. To compute the bulk
exponent $\eta$, we entangle the two ancillas to antipodal sites
$(r-r^{\prime}=L/2)$ with periodic boundary conditions in the circuit and
perform a finite-size scaling with the following ansatz (assuming $z=1$)

$C(t-t_{o})\sim L^{-\eta}\tilde{Q}((t-t_{o})/L).$ (27)

where $\tilde{Q}(x)$ is an arbitrary scaling function. In Fig.4(a) and (b) we
show $C(t-t_{o})$ vs $t-t_{o}$ for the CGPM model at $\lambda/\Delta=1$ and
$p=p_{c}=0.19(1)$ and the SPMM model at $\Lambda=0.45$ and $p=p_{c}=0.28(2)$
for three different system sizes $L=12,16,20$. In the data presented, we
average over $10^{4}$ trajectories for each system size. The scaling collapse
is shown in the inset yielding $\eta=0.21(3)$ for CGPM and $\eta=0.19(2)$ for
the SPMM. We also compute the bulk exponent $\eta$ for the projective
measurement case in the HDU circuit in Appendix A and find $\eta=0.23(2)$.
Hence, based on numerical estimation of the exponents $\eta$ and $\nu$, we
observe that weakening the strength of measurements does not change the bulk
exponents of the MIPT.

Figure 5: Properties of log-CFT in DGPM model: (a) shows the decay of the free
energy density $f(L)$ defined in Eqs. (31) and (30) vs $1/L^{2}$ by the black
circles with its slope related to the effective central charge,
$c_{\mathrm{eff}}$. We perform the double fitting procedure from Zabalo  _et
al._ (2022) by successively removing the smallest system size from the fit,
and then fitting the data with $L\geq L_{\mathrm{min}}$, for
$L_{\mathrm{min}}\in$ {8,10,12} shown by the dashed lines where darker color
denotes larger $L_{\mathrm{min}}$. The inset shows
$c_{\mathrm{eff}}(L_{\mathrm{min}})$ and we use
$c_{\mathrm{eff}}(L_{\mathrm{min}})=c_{\mathrm{eff}}(L_{\mathrm{min}}=\infty)+b/L^{2}_{\mathrm{min}}$
to obtain $c_{\mathrm{eff}}(L_{\mathrm{min}}=\infty)=0.25(3)$ quoted in the
main text. (b) shows the decay of $k_{i}/Lt$ vs $1/L^{2}$ where $k_{1}$ and
$k_{2}$ are the first two cumulants of the correlation functions
($C^{\vec{m}}$) defined in Eq. (37). The slope of $k_{1}/Lt$ gives the typical
anomalous scaling dimension $x_{1}^{\mathrm{typ}}=0.14(2)$ while that of
$k_{2}/Lt$ gives the leading multifractal exponent $x_{1}^{(2)}=0.19(2)$. The
presence of multifractality is further confirmed in (c) by showing the scaling
collapse of the distribution of $Y(t)=-\ln C^{\vec{m}}(t)$ onto a universal
scaling function $H(s)$, following a multifractal scaling given in Eq. (39).
The values are shown for different system sizes from $L=8$ to $L=18$, and
using data from $t=5L$ to $t=32L$, in the solid yellow lines. The darker color
denotes a larger system size. The black dashed line shows the $H(s)$ curve
extracted from Ref. Zabalo  _et al._ , 2022. The x-axis is plotted as
$s-s_{o}$, where $H(s)$ attains a minimum at $s_{o}=0.14\approx
x_{1}^{\mathrm{typ}}$. We use $\epsilon=0.1$, $\lambda=\Delta=10^{4}$,
$p=p_{c}=0.19$ and periodic boundary conditions for all the plots. We use
$3\times 10^{6}$ quantum trajectories for statistical averaging of the $f$ and
$1.5\times 10^{5}$ quantum trajectories for averaging of the cumulants.

##  IV Transfer Matrix Approach and Properties of the log-CFT

At the critical point $p=p_{c}$ of the MIPT we have shown numerical evidence
of Lorentz invariance at the transition in Figs. 3 and 4. Through mappings to
classical statistical mechanics models via an infinite onsite Hilbert space
dimension Vasseur  _et al._ (2019); Jian  _et al._ (2020); Bao  _et al._
(2020) or the zeroth Renyi entropy Skinner  _et al._ (2019) a firm connection
to percolation in $1+1$-dimensions, which is a well known example of a log-CFT
Ludwig and Cardy (1987); Cardy (2013); Gurarie and Ludwig (2005); Gurarie
(1993), has been established. Moving away from these tractable limits
numerical evidence for the nature of the log-CFT in qubit chains with strong
projective measurements was unveiled through a transfer matrix description
that probe the bulk Zabalo  _et al._ (2022) and the boundary Kumar  _et al._
(2024) critical exponents. Focusing on the bulk Lyapunov spectrum of this non-
unitary transfer matrix, the effective central charge of the log-CFT, the
leading typical scaling dimension of the order parameter, and the potential
multifractal nature of the transition can be computed numerically by utilizing
numerical techniques from percolation applied to the MIPT Cardy and Jacobsen
(1997); Jacobsen and Cardy (1998); Zabalo  _et al._ (2022). In the following
section, motivated by this past work and the numerically observed Lorenz
invariance of the transition in Sec. III, we utilize this transfer matrix
description to study the nature of the log-CFT governing the MIPT with weak
measurements.

###  IV.1 Lyapunov Spectrum of the Transfer Matrix

We adopt the transfer matrix method introduced in Ref.Zabalo  _et al._ , 2022
to describe the non-unitary evolution of the hybrid circuit. We summarise the
key ingredients of the method here for both the sake of completeness and to
understand the reason why we construct the DGPM. The time-evolution of a
quantum circuit is represented as an ensemble of quantum trajectories where
each trajectory is defined by a fixed set of unitary gates and the location
and time of measurement operations. Each time-step of a trajectory is defined
by a Krauss operator $K_{t}^{\vec{m}}=P_{t}^{\vec{m}}U_{t}$ where $U_{t}$ are
the unitary gates acting at each time step and $P_{t}^{\vec{m}}$ is the weak
measurement operator defined in Sec. II. The Krauss operators at time $t$
depends on the history of measurement outcomes $\vec{m}$ in that trajectory.
In the DGPM $\vec{m}=(x_{o}^{1},x_{o}^{2},x_{o}^{3},\dots)$ where $x_{o}^{i}$
refers to the measured pointer position at the $ith$ measurement event and
take an infinite number of possibilities. Whereas, for the SPMM
$\vec{m}=(m^{1},m^{2},m^{3},\dots)$ and $m^{i}=\pm 1$ for the two possible
outcomes given in Eq. (23). The time evolution of the density matrix in a
trajectory is represented as,

$\rho(t)=\frac{1}{Z_{\vec{m}}}K_{\vec{m}}\rho(t=0)K^{\dagger}_{\vec{m}},$ (28)

where $K_{\vec{m}}=\prod_{t^{\prime}=0}^{t}K_{t^{\prime}}^{\vec{m}}$ and
$Z_{\vec{m}}$ is the partition function of the statistical mechanics model
describing the trajectory Jian  _et al._ (2020); Zabalo  _et al._ (2022) and
is given by,

$Z_{\vec{m}}(t)=\mathrm{Tr}\left[K_{\vec{m}}\rho(t=0)K^{\dagger}_{\vec{m}}\right]=\sum_{i}e^{\lambda_{i}^{\vec{m}}t}.$
(29)

Here $\lambda_{i}^{\vec{m}}$ are the Lyapunov exponents governing the
exponential decays in the partition function in the long time limit
($\lambda_{i}^{\vec{m}}<0)$. The average Lyapunov exponents
$\lambda_{0},\lambda_{1},...$ are obtained by averaging
$\lambda_{i}^{\vec{m}}$ over trajectories $\vec{m}$ weighing by their
corresponding Born probability $p_{\vec{m}}$, i.e.
$\lambda_{i}=\sum_{\vec{m}}p_{\vec{m}}\lambda_{i}^{\vec{m}}$. In the next
subsections, we will study the average first two leading Lyapunov exponents
$\lambda_{0}$ and $\lambda_{1}$ to extract critical exponents governing the
log-CFT.

###  IV.2 Free Energy

The leading Lyapunov exponent is related to the free energy $F=-\lambda_{0}t$
which can be calculated as the Shannon entropy of the measurement record
$\vec{m}$ averaging over quantum trajectories,

$F=-\sum_{\vec{m}}p_{\vec{m}}\ln p_{\vec{m}}=-\langle\ln p_{\vec{m}}\rangle.$
(30)

Here, we point out that a discrete measurement outcome is required for the
average of $\ln p_{\vec{m}}$ to be well defined. At the critical point of the
MIPT $p=p_{c}$, the conformal invariance of the system dictates that the free
energy density $f(L,t)=F(L,t)/A$ where the (implicitly defined) space-time
area is given by $A=\alpha Lt$. The space time anisotropy factor
$\alpha_{\mathrm{HDU}}=1$ for the HDU gates we consider here Zabalo  _et al._
(2022), and this exact knowledge of $\alpha$ is why we are considering these
HDU gates. If we consider Haar unitary gates on the other hand, a direct
computation of $\alpha$ through mutual information in space and time Zabalo
_et al._ (2022), yields $\alpha_{\mathrm{HU}}=0.81(9)$ that introduces an
additional error in the free energy calculation. The free energy density after
long times ($t\gg L$) for spatial periodic boundary conditions decays with the
system size as,

$f(L)=f(L=\infty)-\frac{\pi c_{\rm{eff}}}{6L^{2}}+O(L^{3}),$ (31)

where $c_{\mathrm{eff}}$ is the effective central charge of the log-CFT.

Figure 6: Properties of log-CFT in SPMM model: (a) shows the decay of the free
energy density $f(L)$ Eqs. (31) and (30) vs $1/L^{2}$ by the black circles
with its slope related to the effective central charge, $c_{\mathrm{eff}}$. We
perform the double fitting procedureZabalo  _et al._ (2022) by successively
removing the smallest system size from the fit, and then fitting the data with
$L\geq L_{\mathrm{min}}$, for $L_{\mathrm{min}}\in$ {8,10,12,14} shown by the
dashed lines where darker color denotes larger $L_{\mathrm{min}}$. The inset
shows $c_{\mathrm{eff}}(L_{\mathrm{min}})$ and we use
$c_{\mathrm{eff}}(L_{\mathrm{min}})=c_{\mathrm{eff}}(L_{\mathrm{min}}=\infty)+b/L^{2}_{\mathrm{min}}$
to obtain $c_{\mathrm{eff}}(L_{\mathrm{min}}=\infty)=0.26(2)$ quoted in the
main text. (b) shows the decay of $k_{i}/Lt$ vs $1/L^{2}$ where $k_{1}$ and
$k_{2}$ are the first two cumulants of the correlation functions
($C^{\vec{m}}$) defined in Eq. (37). The slope of $k_{1}/Lt$ gives the typical
anomalous scaling dimension $x_{1}^{\mathrm{typ}}=0.12(2)$ while that of
$k_{2}/Lt$ gives the leading multifractal exponent $x_{1}^{(2)}=0.14(2)$.The
presence of multifractality is further confirmed in (c) by showing the scaling
collapse of the distribution of $Y(t)=-\ln C^{\vec{m}}(t)$ onto a universal
scaling function $H(s)$, following a multifractal scaling given in Eq. (39).
The values are shown for different system sizes from $L=8$ to $L=18$, using
data from $t=5L$ to $t=32L$, in the solid yellow lines. The darker color
denotes a larger system size. The black dashed line shows the $H(s)$ curve for
the projective measurement case extracted from Ref. Zabalo  _et al._ , 2022.
The x-axis is plotted as $s-s_{o}$, where $H(s)$ attains a minimum at
$s_{o}=0.12\approx x_{1}^{\mathrm{typ}}$. We use $\Lambda=0.45$,
$p=p_{c}=0.28$ and periodic boundary conditions for all the plots. We use
$2.5\times 10^{4}$ quantum trajectories for statistical averaging.

We first calculate the free energy for the weak measurement model with the
DGPM illustrated in Sec.II.2.2. In this model, the Born probabilities of the
measurement record is calculated from,

$p_{\vec{m}}=\prod_{i}p(x_{o}^{i};\lambda/\Delta,\epsilon),$ (32)

where $p(x^{i}_{o};\lambda/\Delta,\epsilon)$ given in Eq. (21) [and in the
limit $\epsilon\ll\Delta$ by Eq. (45)] is the probability of measuring the
classical pointer in a region of width $\epsilon$ around $x_{o}$ in the $i$th
measurement operation in the trajectory $\vec{m}$. We note that $f(L)$ in this
case explicitly depends on the choice of $\epsilon$ which is the smallest
length scale we keep in our calculation to avoid the vanishingly small
probabilities in the continuous measurement case (see Sec.II.2.1 for the
limiting case $\epsilon\rightarrow 0$). However, in the limit
$\epsilon\ll\Delta$, a change in $\epsilon$ leads to a change in the free
energy density as $\Delta f=p_{c}\ln\Delta\epsilon$ leaving our estimate of
$c_{\mathrm{eff}}$ unaffected in this limit (see Appendix D for more details).
On the other hand in the strong projective limit,
$\lambda/\Delta\rightarrow\infty$, the probabilities
$p(x^{i}_{o};\lambda/\Delta,\epsilon)$ reduce to measuring the spin in the up
or down state up to a multiplicative constant. As we show in detail in
Appendix D, the effective central charge also remains invariant w.r.t. changes
in $\epsilon$ in the projective measurement limit. We compute the free energy
$F(L,t)$ starting from an initial product state of the circuit. To eliminate
the effects of initial conditions in the free energy calculated from the
cumulative Born probabilities $p_{\vec{m}}(t)$, we waited till $t=5L$ before
starting to record $p_{\vec{m}}(t)$. In the long time limit ($t>5L$), $F(L,t)$
grows linearly with time and we extract $f(L)$ from its slope between $t=5L$
and $t=32L$. Fig.5(a) shows the free energy density $f(L)$ vs $1/L^{2}$ for
different system sizes $L=8$ to $18$ with black circles at the strength of
measurement $\lambda/\Delta=1$.

The finite size scaling form in Eq. (31) provides an essential guide to
obtaining the correct numerical estimate of the free energy. An important
point to note is that the DGPM requires us to average over a much larger
number of trajectories ($\sim 2\times 10^{6}$) compared to the strong
projective case ($\sim 2.5\times 10^{4}$) to see the $1/L^{2}$ behaviour
dictated by the log-CFT. To estimate $c_{\rm{eff}}$ from its slope, we use a
double fitting procedure systematically eliminating the effects of the smaller
systems sizes Zabalo  _et al._ (2022); Cardy and Jacobsen (1997). We fit the
data from $L=L_{\rm{min}}$ to $L=18$ shown by dashed lines and estimate
$c_{\rm{eff}}(L_{\mathrm{min}})$ shown in the inset. Extrapolating
$L_{\mathrm{min}}\rightarrow\infty$, we obtain the effective central charge
$c_{\mathrm{eff}}=0.25(3)$ in a weakly measured DGPM at
$p_{c}(\lambda=\Delta)=0.19(1)$, which matches quite well with the case of
strong projective measurement that finds $c_{\mathrm{eff}}=0.25(3)$ in Zabalo
_et al._ (2022).

We next calculate the free energy for the softened projective measurement
model illustrated in Sec. II.2.3. In this case the Born probabilities
$p_{\vec{m}}$ are calculated from,

$p_{\vec{m}}=\prod_{i}p_{m^{i}}(\Lambda)$ (33)

where $p_{m^{i}}(\Lambda)$ are Born probabilities of individual measurement
event (denoted by $m^{i}=\pm$) defined in Eq. (23). Fig. 6(a) shows $f(L)$ vs
$1/L^{2}$ at $\Lambda=0.45$ having $p_{c}=0.28(2)$. We averaged over $25000$
quantum trajectories initialized in both Haar random and product states and
waited till $t=4L$ before starting to record Born probabilities. We extract
the slope of $f(L)$ vs $1/L^{2}$ using the above-explained double fitting
procedure to obtain $c_{\mathrm{eff}}(L_{\rm{min}})$ shown in the inset. This
gives $c_{\mathrm{eff}}=0.26(2)$ in the softened projective measurement case
which also closely matches with the projective measurement case. Thus
weakening strength of measurements does not affect the effective central
charge of the log-CFT governing the MIPT.

###  IV.3 Leading Scaling Dimension

In this section we calculate the next leading Lyapunov exponent $\lambda_{1}$
and the associated typical and multifractal critical exponents following Ref.
Zabalo  _et al._ (2022). To calculate $\lambda_{1}$, we construct the two
orthogonal states of the system $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$.
These states are then evolved in time where at each time step they are
subjected to identical Krauss operators $K_{t}^{\vec{m}}$ with the measurement
operators, $P_{t}^{\vec{m}}$s solely determined by the Born probabilities
calculated from $|\psi_{1}\rangle$. After each timestep, we orthogonalize
$|\psi_{1}\rangle$ to $|\psi_{2}\rangle$ with a Gram-Schmidt projector
$P^{GS}_{t}$, which is akin to an additional measurement operation. The Born
probability of the measurement records in the trajectory $\vec{m}$
corresponding the state $|\psi_{2}\rangle$ is written as,

$p^{\prime}_{\vec{m}}(t)=||\Pi_{t^{\prime}=0}^{t}P^{GS}_{t^{\prime}}K_{t^{\prime}}^{\vec{m}}|\psi_{2}\rangle||^{2}.$
(34)

This gives the first generalized free energy
$F_{1}(L,t)=-\lambda_{1}t=-\sum_{\vec{m}}p^{\prime}_{\vec{m}}\ln
p^{\prime}_{\vec{m}}$. We initialize the system either in Haar state or
product state and wait till $t=5L$ before starting to record
$p^{\prime}_{\vec{m}}(t)$s to eliminate the effects of the choice of initial
conditions. In the long time limit ($t>5L$), $F_{1}(L,t)$ is expected to
increase linearly in time and we compute the first generalized free energy
density, $f_{1}(L)=F_{1}(L,t)/(Lt)$ from its slope between $t=5L$ and $t=32L$.

The log-CFT governing the MIPT suggests that difference between the first two
generalized free energy densities decay with $L$ as (taking $\alpha=1$),

$\frac{k_{1}}{Lt}=f_{1}(L)-f(L)=\frac{2\pi x_{1}^{\rm{typ}}}{L^{2}},$ (35)

with a slope related to the typical scaling dimension of the order parameter,
$x_{1}^{\rm{typ}}$. The exponent $x_{1}^{\rm{typ}}$ is related to the bulk
exponent $\eta$ (computed in Sec. III through Eq. (27), and shown in Fig. 4)
as $x_{1}^{\rm{typ}}=\eta/2$.

We show the first cumulant $k_{1}/Lt$ given in Eq.(35) and (taking $\alpha=1$)

vs $1/L^{2}$ in Fig.5(b) for the DGPM at $\lambda=\Delta$ and in Fig.6(b) for
the SPMM at $\Lambda=0.45$. From the slope of $k_{1}/Lt$, we find
$x_{1}^{\mathrm{typ}}(\lambda=\Delta)=0.14(2)$ for the DGPM and
$x_{1}^{\mathrm{typ}}(\Lambda=0.45)=0.12(2)$ for the SPMM, which closely
agrees with the strong projective case having $x_{1}^{\mathrm{typ}}=0.122(1)$
obtained in Ref Zabalo  _et al._ , 2022. Moreover, our results are consistent
with $x_{1}^{\mathrm{typ}}=\eta/2$ in both the models: In the SPMM,
$\eta=0.19(3)$ and $x_{1}^{\mathrm{typ}}=0.12(2)$, and in the CGPM/DGPM
$\eta=0.21(2)$ and $x_{1}^{\mathrm{typ}}=0.14(2)$, meaning that both results
are almost agree within statistical error bars.

###  IV.4 Multifractility of the log-CFT

Going beyond the typical scaling exponent, it is also interesting to probe the
multifractal properties of the correlation function. At the critical point,
the system is scale-invariant and all the moments of the correlation functions
of the log-CFT Zabalo  _et al._ (2022); Ludwig (1990); Chatelain and Berche
(2000),
$C^{\vec{m}}=\mathrm{exp}\\{t(\lambda_{1}^{\vec{m}}-\lambda_{0}^{\vec{m}})\\}$
after averaging over trajectories (denoted by $\varepsilon$ ) decay as a power
law in distance ($r$) in the long time limit,

$\varepsilon[\\{{C^{\vec{m}}}\\}^{n}]\sim\frac{B_{n}}{r^{2x_{1}(n)}}.$ (36)

A cumulant expansion of the correlation function yields,

$\ln[\varepsilon[\\{{C^{\vec{m}}}\\}^{n}]]=n\varepsilon[\ln{C^{\vec{m}}}]+\frac{n^{2}}{2!}\varepsilon[\\{\ln
C^{\vec{m}}-\varepsilon[\ln{C^{\vec{m}}}]\\}^{2}]+\dots$ (37)

This gives,
$x_{1}(n)=nx_{1}^{\rm{typ}}+\frac{n^{2}}{2!}x_{1}^{(2)}+\mathcal{O}(n^{3})$
for sufficiently small $n$. We identify the presence of multi-fractality in
the spectrum of correlation functions by the a non-zero estimate of
$x_{1}^{(n)}$ where $n\geq 2$.

From the scaling of the second moment $k_{2}$ (taking $\alpha=1$) defined as

$\frac{k_{2}}{Lt}=\frac{\varepsilon\\{\ln
C^{\vec{m}}-\varepsilon[\ln{C^{\vec{m}}}]\\}^{2}}{Lt}$ (38)

and its slope vs $1/L^{2}$ we find a finite
$x_{1}^{(2)}(\lambda=\Delta)=0.19(2)$ for the DGPM and
$x_{1}^{(2)}(\Lambda=0.45)=0.14(2)$ for the SPMM. Thus our numerical results
present evidence for multifractal scaling in MIPTs with weak measurements.
These estimates are similar to that in the projective measurement case having
$x_{1}^{(2)}=0.145(2)$ computed in Ref.Zabalo  _et al._ (2022).

We further confirm the multifractal scaling by calculating the probability
distribution function $P[Y(t)]$ of the correlation functions, where
$Y(t)\equiv-\ln C^{\vec{m}}(t)$. In presence of multi-fractality, this is
expected to follow the universal scaling form Ludwig (1990); Chatelain and
Berche (2000),

$P[Y(t)]\sim\left(\frac{2\pi t}{L}\right)^{\frac{1}{2}}\exp\left[\frac{-2\pi
t}{L}H\left(\frac{Y(t)}{2\pi t/L}\right)-b\right],$ (39)

collapsing onto a single curve $H(s)$ having a minimum at
$s_{o}=x_{1}^{\mathrm{typ}}$, where we have choosen $b$ in Eq. (39) to bring
$H(s_{o})=0$.

We compute the distribution function $P[Y(t)]$ using data from systems sizes
$L=8$ to $18$ and times $t=5L$ and $32L$ to find the universal scaling
function $H(s)$ for the weak measurement models through the scaling form given
in Eq. 39. $H(s)$ vs $s-s_{o}$ are shown in Fig.5(c) for the DGPM and in
Fig.6(c) for the SPMM. Darker colors denote larger system sizes. We also plot
the curve for projective measurement case by black dashed lines extracting the
data points from Ref. Zabalo  _et al._ (2022) and shifting the origin of the
$s$ axis to $s_{o}=x_{1}^{\mathrm{typ}}$. In both the models of weak
measurements, the curves corresponding to weak measurements have similar shape
compared to that of the projective measurement case, though not precisely
overlapping. This may possibly arise from differences in higher $x_{1}^{(n)}$s
in the multifractal spectra.

| $\frac{\lambda}{\Delta}=\infty$ | $\frac{\lambda}{\Delta}=1$ | $\Lambda=0.45$
---|---|---|---
$p_{c}$ | 0.14(1) | 0.19(1) | 0.28(2)
$\nu$ | 1.3(3) | 1.3(3) | 1.6(3)
$z$ | 0.98(8) | 0.94(6) | 0.95(5)
$\eta$ | 0.23(2) | 0.21(2) | 0.19(3)
$c_{\mathrm{eff}}$ | 0.24(2) | 0.25(3) | 0.26(2)
$x_{1}^{\mathrm{typ}}$ | 0.12(2) | 0.14(2) | 0.12(2)
$x_{1}^{(2)}$ | 0.14(2) | 0.19(2) | 0.14(2)
Table 1:  Critical data of universal parameters for the various models:
central effective charge $c_{\mathrm{eff}}$, scaling dimension of the order
paramter $x_{1}^{\mathrm{typ}}$, the multifractal critical exponent
$x_{1}^{(2)}$, and the dynamical exponent $z$. These exponents were extracted
at the critical probability $p_{c}$. At $\frac{\lambda}{\Delta}=1$, the
quantities involving the free energy ($c_{\mathrm{eff}}$,
$x_{1}^{\mathrm{typ}},x_{1}^{(2)}$) were obtained from the DGPM, whereas the
quantities involving the ancilla entanglement entropy ($\nu$,$z$,$\eta$) were
obtained from the CGPM.

##  V Conclusion

In this work, we have studied the effect of weak measurements on the critical
properties of MIPTs. The summary of our findings are provided in Table 1.
Based on scaling collapse we estimate $p_{c}$, $\nu$, $\eta$, and $z$, which
reveals a Lorentz invariant transition with exponents that are consistent with
their values in the limit of strong projective measurements. As prior work
found it challenging to determine the universality class from these estimates
alone Zabalo  _et al._ (2020) we turned to a more accurate transfer matrix
approach. This is also based on the Lorentz invariance of the transition and
the log-CFT nature of the field theory, allowing us to utilize universal
finite size scaling corrections to estimate the effective central charge
$c_{\mathrm{eff}}$ and the scaling dimension of the order parameter
$x_{1}^{\mathrm{typ}}=\eta/2$, which both agree quite well across each model
considered. We therefore take the agreement in $c_{\mathrm{eff}}$ and
$x_{1}^{\mathrm{typ}}$ across the different measurement protocols as the
strongest evidence that each of these models belong to the same universality
class, namely the Haar random MIPT Zabalo  _et al._ (2022). We find clear
evidence of multifractal scaling, which is indicitave of strong quantum
fluctuations at the MIPT, in both measurement models. As these moments are
higher statistical quantities they also carry much more numerical uncertainty.
Therefore the larger discrepancies across models that we have observed in the
multifractal properties, namely $x_{1}^{(2)}$ and the shape of $H(s)$, are not
sufficient for us to make any firm conclusions regarding their higher moments
differing. However, we leave this possibility open for future work, which will
require orders of magnitude more statistical samples than we have considered
here. An interesting and open question we leave for later is a construction of
the log-CFT for the continuous Gaussian pointer model. To summarize, taking
all of numerical results together lead us to conclude that the nature of the
universality class of the Haar MIPT and its underlying log-CFT description is
unaffected by weak or strong measurement protocols.

###### Acknowledgements.

We thank Piers Coleman, Haining Pan, and Romain Vasseur for insightful
discussions. This work was partially supported by the Office of Naval Research
grant No. N00014-23-1-2357 (K.A., J.H.P.), the Army Research Office Grant No.
79849-PE-H (K.A., J.H.P.), a Sloan Research Fellowship (J.H.P.), and the
Abrahams Postdoctoral Fellowship at the Center for Materials Theory Rutgers
(A.C.). We acknowledge the Office of Advanced Research Computing (OARC) at
Rutgers, The State University of New Jersey for providing access to the Amarel
cluster and associated research computing resources that have contributed to
the results reported here as well as the services provided by the OSG
Consortium, which is supported by the National Science Foundation awards No.
2030508 and No. 1836650.

Figure 7: Critical Properties of Strong Projective Measurements with HDU
gates: (a) Data is taken for 2,000 trajectories per system size. We find a
crossing near $p=p_{c}=0.14(1)$ consistent with previous work Zabalo  _et al._
(2022), and a scaling collapse of the data in the inset yielding
$p_{c}=0.14(1)$ and $\nu=1.3(3)$. (b) Data is taken for 10,000 trajectories
per system size. We find $z=0.98(8)$ from a scaling collapse of the data, as
shown in the inset. (c) Mutual Information between two ancilla qubits
entangled at $r=1$ and $r^{\prime}=\frac{L}{2}$ in the steady state of the
circuit evolving with projective measurements. We use three different system
sizes $L=12,16,20$ for the scaling collapse following Eq. (27) shown in the
inset. We find the bulk exponent $\eta=0.23(2)$. We average over $10^{4}$
trajectories for each system size.

##  Appendix A Critical properties of the Haar-dual-unitary Strong Projective
Hybrid Quantum Circuit

In this Appendix we compute the remaining (easily accessible) critical
exponents of the strongly projective HDU MIPT. In Ref. Zabalo  _et al._ (2022)
$p_{c}$ and $x_{1}^{\mathrm{typ}}$ were computed. Here we compute $\nu$ and
$\eta$ from the ancilla probes in Sec. III in order to further verify that the
HDU gates do not affect the universal nature of the MIPT. As shown in Fig.
7(a) and (c) we find that, for the von Neumman entropies,
$\eta^{\rm{HDU}}=0.23(2)$, and $\nu^{\rm{HDU}}=1.3(3)$ for HDU gates with
strong measurement, which are close to the values $\eta^{\rm{Haar}}=0.19(1)$
and $\nu^{\rm{Haar}}=1.2(2)$ for Haar random gates with strong measurement as
computed in Ref.Zabalo  _et al._ , 2020. Fig. 7(b) also shows signatures of
Lorentz invariance in HDU circuit. In conclusion, we find good agreement
between HDU and Haar random gates as expected.

##  Appendix B Comparison of Discrete and Continuous Reyni Entropies

In this section, we compare the Reyni entanglement entropies in the discrete
and continuous Gaussian weak measurement models. We compute both the ancilla
and bipartite entanglement entropies following the protocols explained in
Secs. III.2 and III.1. We find that for $L=12$ and all measurement
probabilities centered around the critical point $p_{c}\sim 0.19$, the Reyni
entropies in both models are virtually indistinguishable. This allows us to
use either measurement model to compute the Reyni entropies. In Fig. 8, we
compute the bipartite entanglement entropies in (a) and the ancilla
entanglement entropies in (b) for both the CGPM and DGPM. The difference in
entanglement entropies is orders of magnitude smaller than the mean value in
both instances: $\frac{\Delta{S_{\mathrm{L/2}}}}{S_{\mathrm{L/2}}}\sim
10^{-2}$ and $\frac{\Delta{S_{\mathrm{anc}}}}{S_{\mathrm{anc}}}\sim 10^{-3}$.
The difference is also of the same order of magnitude as the statistical error
bars in both models, which is smaller than the size of the points.

##  Appendix C Krauss Operators for Weak Measurement

For DGPM, we showed that $p(x_{o};\lambda/\Delta,\epsilon)$ could be defined
in terms of the integrals $p_{\pm}(x_{o},\lambda/\Delta,\epsilon)$, as
expressed in Eq. (20). In this appendix we describe how we numerically
evaluate the Born probabilities $p_{\pm}(x_{o};\lambda/\Delta,\epsilon)$
defined in Eq. (20). To numerically compute these integrals we consider the
limit $\epsilon\rightarrow 0$. In this limit, the integral over position can
be efficiently calculated by numerical evaluation of the error function:

$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt.$ (40)

In terms of the error function the probabilities become,

$\begin{split}p_{\pm}(x_{o};\lambda/\Delta,\epsilon)=&\frac{\sqrt{\Delta}}{2}\Big{[}\mathrm{erf}\left(\frac{\mp
2\lambda-2x_{o}+\epsilon}{2\Delta}\right)\\\ &-\mathrm{erf}\left(\frac{\mp
2\lambda-2x_{o}-\epsilon}{2\Delta}\right)\Big{]}\end{split}$ (41)

Therefore, the Krauss operator which updates the state, as represented in
Eq.(19), becomes

$\begin{split}\hat{M}_{D}(x_{o})=&\frac{1}{2\sqrt{\Delta}}\Big{\\{}\Big{[}\mathrm{erf}\left(\frac{-2x_{o}-2\lambda+\epsilon}{2\Delta}\right)\\\
&-\mathrm{erf}\left(\frac{-2x_{o}-2\lambda-\epsilon}{2\Delta}\right)\Big{]}\Pi_{+}^{(j)}\\\
&+\Big{[}\mathrm{erf}\left(\frac{-2x_{o}+2\lambda+\epsilon}{2\Delta}\right)\\\
&-\mathrm{erf}\left(\frac{-2x_{o}+2\lambda-\epsilon}{2\Delta}\right)\Big{]}\Pi_{-}^{(j)}\Big{\\}}.\end{split}$
(42)

Figure 8: Comparison of Reyni Entropies. Data is taken for $L=12$ and 2,000
trajectories per data point, at measurement strength
$\frac{\lambda}{\Delta}=1$, $\epsilon=0.1$ and $\Delta=10,000$ for the
discrete model, and $\frac{\lambda}{\Delta}=1$ for the continous measurement
model. (a) Comparison of the bipartite entanglement entropies in the
continuous and discrete Gaussian measurement models. The inset is the
difference in the computed entangled entropies $\Delta
S_{L/2}=S_{\mathrm{L/2}}^{(\mathrm{Discrete})}-S_{\mathrm{L/2}}^{(\mathrm{Continous})}$(b)
Comparison of the ancilla entanglement entropies in the continuous and
discrete Gaussian measurement models. The inset presents $\Delta
S_{\mathrm{anc}}=S_{\mathrm{anc}}^{(\mathrm{Discrete})}-S_{\mathrm{anc}}^{(\mathrm{Continous})}$.

##  Appendix D Changing $\epsilon$ does not affect $c_{\mathrm{eff}}$

In this appendix we show that for DGPM, the effective central charge we
calculated is independent of the choice of the binning width $\epsilon$. In
the limit $\epsilon\ll\Delta$ (there are no constraints on $\lambda$, except
noting that for a fixed strength $\lambda/\Delta$, $\lambda$ depends on
$\Delta$) leading us to show in this appendix that the change in the free
energy as a function of the width of the measurement region is given by

$\Delta
f=f(\epsilon^{\prime})-f(\epsilon)=-p_{c}\mathrm{log}\left(\frac{\epsilon}{\epsilon^{\prime}}\right),$
(43)

where $\epsilon$ and $\epsilon^{\prime}$ are two different bin widths.
Consequently, altering $\epsilon^{\prime}$ in this limit leaves
$c_{\mathrm{eff}}$ invariant, as an $L$-independent shift of $f$ only changes
$f(L=\infty)$.

We now provide a derivation of Eq. (43).In the limit $\epsilon\ll\Delta$, the
integrals in Eq.(20) simplify to become

$p_{\pm}(x_{o};\lambda/\Delta,\epsilon)=G_{\Delta}^{2}(x^{\prime}\mp\lambda)\epsilon.$
(44)

Therefore the probability in Eq. (21) becomes

$\displaystyle p(x_{o};\lambda/\Delta,\epsilon)$ $\displaystyle=$
$\displaystyle[\langle\psi|\Pi_{+}^{(j)}|\psi\rangle
G_{\Delta}^{2}(x_{o}-\lambda)$ (45) $\displaystyle+$
$\displaystyle\langle\psi|\Pi_{-}^{(j)}|\psi\rangle
G_{\Delta}^{2}(x_{o}+\lambda)]\epsilon.$

We use $p(x_{o}^{(i)};\lambda/\Delta,\epsilon)$ to denote the probability of
the $i^{th}$ measurement outcome in the DGPM, and
$p(x_{o}^{(i)};\lambda/\Delta)$ to denote the corresponding probability
density in the CGPM. Using $N_{\mathrm{meas}}$ to indicate the total number of
measurements in the circuit, we can now express the probability of the
measurement record in the trajectory $p_{\vec{m}}$ as:

$\displaystyle p_{\vec{m}_{\epsilon}}$
$\displaystyle=\prod_{i=1}^{N_{\text{meas}}}p(x_{o}^{(i)};\lambda/\Delta,\epsilon)$
(46)
$\displaystyle=\epsilon^{N_{\text{meas}}}\prod_{i=1}^{N_{\text{meas}}}p(x_{o}^{(i)};\lambda/\Delta).$

We now use $\vec{m}_{\epsilon}$ to denote each possible measurement trajectory
at the corresponding bin width $\epsilon$. We express

$\vec{m}_{\epsilon}=(x_{o}^{1}(\epsilon),x_{o}^{2}(\epsilon),x_{o}^{3}(\epsilon),\dots)$
(47)

where $x_{o}^{i}(\epsilon$) refers to the measured pointer position at the
$i^{th}$ measurement event, which can depend on $\epsilon$. For each
measurement we use $N_{\mathrm{out}}(\epsilon)$ to denote the number of
measurement outcomes for $x_{o}^{i}(\epsilon)$ in a finite spatial region of
width $W$. We see that the number of outcomes $N_{\mathrm{out}}$ depends on
the bin width $\epsilon$ as

$N_{\mathrm{out}}(\epsilon)=\frac{W}{\epsilon}.$ (48)

For example, halving the bin width doubles the number of measurement outcomes
over any region. Therefore the number of measurement trajectories
$\vec{m}_{\epsilon}$ will scale as

$N_{\mathrm{traj}}(\epsilon)=\left(\frac{W}{\epsilon}\right)^{N_{\mathrm{meas}}}.$
(49)

This leads to the observation that in the limit $\epsilon\ll\Delta$, the sum
over measurement trajectories depends on the width of the measurement region
as:

$\sum_{\vec{m}_{\epsilon}}=\left(\frac{\epsilon^{\prime}}{\epsilon}\right)^{N_{\text{meas}}}\sum_{\vec{m}_{\epsilon^{\prime}}}.$
(50)

Now, recall that we can write the free energy as

$F_{\epsilon}=-\sum_{\vec{m}_{\epsilon}}p_{\vec{m}_{\epsilon}}\mathrm{ln}p_{\vec{m}_{\epsilon}}.$
(51)

$F_{\epsilon}$ represents the free energy using a bin width $\epsilon$. Using
Eq. (46) to relate $p_{\vec{m}_{\epsilon}}$ and
$p_{\vec{m}_{\epsilon^{\prime}}}$, we have that

$\displaystyle F_{\epsilon}$
$\displaystyle=-\sum_{\vec{m}_{\epsilon}}p_{\vec{m}_{\epsilon}}\mathrm{ln}\left[p_{\vec{m}_{\epsilon^{\prime}}}\left(\frac{\epsilon}{\epsilon^{\prime}}\right)^{N_{\text{meas}}}\right]$
(52)
$\displaystyle=-\sum_{\vec{m}_{\epsilon^{\prime}}}p_{\vec{m}_{\epsilon^{\prime}}}\mathrm{ln}p_{\vec{m}_{\epsilon^{\prime}}}-\sum_{\vec{m}_{\epsilon}}p_{\vec{m}_{\epsilon}}N_{\mathrm{meas}}\mathrm{ln}\left(\frac{\epsilon}{\epsilon^{\prime}}\right)$
$\displaystyle=F_{\epsilon^{\prime}}-N_{\text{meas}}\ln\left(\frac{\epsilon}{\epsilon^{\prime}}\right).$

We now take $\varepsilon[...]$ to denote an average over circuits, so that at
$p=p_{c}$ we expect that $\varepsilon[N_{meas}]\approx tLp_{c}$. Thus,

$\displaystyle\Delta F$
$\displaystyle=\varepsilon[F_{\epsilon}]-\varepsilon[F_{\epsilon^{\prime}}]$
(53)
$\displaystyle=-Ltp_{c}\mathrm{ln}\left(\frac{\epsilon}{\epsilon^{\prime}}\right)$
(54)

and

$\Delta f=\frac{\Delta
F}{Lt}=-p_{c}\mathrm{ln}\left(\frac{\epsilon}{\epsilon^{\prime}}\right).$ (55)

$\Delta f$ represents the change in $f(L=\infty)$ as $\epsilon$ is varied,
which is an $L$-independent shift to the free energies at all system sizes at
a fixed measurement strength. As shown in Fig. 9, we perform a calculation of
the free energy for various values of $\epsilon$, at the measurement strength
$\frac{\lambda}{\Delta}=1$. We verify that our analytical calculation for the
dependence of $f$ on $\epsilon$ matches the numerics.

Figure 9: Free Energy vs. $\epsilon$. (a) Dependence of $f$ on $\epsilon$ in
the Gaussian weak measurement model at $p\sim p_{c}=0.19$ and
$\frac{\lambda}{\Delta}=1$ with $\Delta=1,000,000$ and $\epsilon=0.1$ and
$\epsilon^{\prime}=1$.

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