Control-driven critical fluctuations across quantum trajectories

We begin by defining the Bernoulli circuit model as shown in Fig. 1 and reviewing its phase diagram [8, 10]. We first describe the classical version of the model before discussing what changes when quantum superpositions are introduced.

We now introduce the main concept of isolating different types of fluctuations to reveal the quantum nature of seemingly-classical transitions in the adaptive monitored circuit. This idea is broadly applicable to various models with a dark or control state, e.g., the CIPT in the competition between chaotic and controlled dynamics [8], and absorbing-state transitions in adaptive monitored circuits [13]. In the following results, we will use the CIPT as a paradigmatic example to illustrate the concept. The application of this idea to another model with an absorbing state transition is discussed in Appendix F.

The aim is to determine the variance of a generic quantum operator $O$. To begin, we define all quantities being averaged. Starting from an initial pure state $\ket{\psi_{0}}$, we have a sequence of stochastic events that either apply a unitary or a measurement with feedback. The combination of the initial state and the sequence of unitaries and measurements defines one circuit realization labeled by $\mathcal{C}$. Each time step $t_{i}$ (e.g., $t=0,1,2\dots$ in Fig. 1(a)) that the circuit performs a measurement, the outcome is randomly sampled from the Born probability distribution with outcome $m^{(i)}$. We denote the full measurement history of a trajectory $\bm{m}$ for circuit realization $\mathcal{C}$ by the vector $\bm{m}_{\mathcal{C}}=\left(m^{(1)},m^{(2)},\dots\right)$; we seek to parse the fluctuations into those due to variations between circuits and those due to variations within a fixed circuit $\mathcal{C}$.

We are now in a position to specify an average over “samples”, where each “sample” is a single trajectory of a single circuit, indexed by $\bm{M}=\{\mathcal{C},{\bm{m}}_{\mathcal{C}}\}$. The average over samples is then

Note that since this is a simple average (no post-selection), it can be experimentally estimated by gathering many samples. We further stress the nested form of the averages (also denoted through the conditional probability $p_{\bm{m}_{\mathcal{C}}|\mathcal{C}}$) that allows us to take the classical limit, which has deterministic measurement outcomes, so for fixed circuit $\mathcal{C}$, all classical trajectories are identical.

If we make projective measurements of operator $O$, giving results $O_{\bm{M}}$ for $\bm{M}$, the variance of these single-shot experimental observations $O_{\bm{M}}$ across $\bm{M}$ can be estimated without post-selection by

It is now straightforward, using Eq. (12), to see that this will involve a combination of fluctuations over quantum trajectories (i.e., measurement outcomes for a fixed circuit realization) and circuit-to-circuit fluctuations that also exist in the classical limit of the model. Therefore, we can now decompose this variance over samples into classical circuit fluctuations and quantum fluctuations:

where $\mathbb{E}_{\mathcal{C}}$ and $\mathbb{E}_{\bm{m}_{\mathcal{C}}}$ denote averages over circuits $\mathcal{C}$, and different trajectories $\bm{m}_{\mathcal{C}}$ given a circuit $\mathcal{C}$, respectively. In going from the first to the second line in Eq. (14), we have added and subtracted the quantity $\mathbb{E}_{\mathcal{C}}\left[\left(\mathbb{E}_{\bm{m}_{\mathcal{C}}}\left[O_{\bm{m}_{\mathcal{C}}}\right]\right)^{2}\right]$, where $O_{\bm{m}_{\mathcal{C}}}$ is for a single shot measurement of $O$ the measurement history $\bm{m}_{\mathcal{C}}$ given a circuit $\mathcal{C}$. The first term, circuit fluctuations, pertains to the classical limit of choosing different circuits to evolve. The second term, quantum fluctuations, is the variance of the outcome of measuring $O$ over many trials with the same circuit $\mathcal{C}$. For the classical model, only the circuit fluctuations $\sigma_{\mathcal{C}}^{2}[O]$ are nonzero because the measurement outcome is deterministic, while for the quantum model, both circuit and quantum fluctuations can be nonzero.

Theoretically and, if we allow postselection, experimentally, the quantum fluctuations in Eq. (14) can be further decomposed into two parts:

where $\expectationvalue{\dots}_{\bm{m}_{c}}$ denotes the quantum state expectation over the wave function $\ket{\psi_{\bm{m}_{c}}}$ of the system immediately before any measurement of $\dots$ is made. (Note that $\mathbb{E}_{\bm{m}_{\mathcal{C}}}\left[\expectationvalue{O^{n}}_{\bm{m}_{\mathcal{C}}}\right]\equiv\mathbb{E}_{\bm{m}_{\mathcal{C}}}\left[O^{n}_{\bm{m}_{\mathcal{C}}}\right]$ with $n=$1 and 2.) The first term, trajectory fluctuations $\sigma_{t}^{2}[O]$, characterizes the uncertainty of different quantum trajectories $\bm{m}_{\mathcal{C}}$, coming from the randomness in the outcomes of the measurements that are made earlier than the measurement of $O$. The second term, state fluctuations $\sigma_{s,\bm{m}_{\mathcal{C}}}^{2}[O]$, characterizes the quantum nature (superposition) of the pure state $\ket{\psi_{\bm{m}_{\mathcal{C}}}}$ immediately before the measurement of $O$ is made. These two terms capture different aspects of the quantum fluctuations. In the MIPT without feedback, the transition is seen only in properties of the state $\ket{\psi_{\bm{m}_{\mathcal{C}}}}$, although this is in entanglement properties rather than simple observables.

For our model with feedback, we will show that the trajectory fluctuations $\sigma_{t}^{2}[O]$ and state fluctuations $\sigma_{s,\bm{m}_{\mathcal{C}}}^{2}[O]$ both become critical at the CIPT, i.e., they obey single-parameter scaling. Whereas both parts of the quantum fluctuations are large for $p<p_{c}^{\text{CIPT}}$, they become small for $p>p_{c}^{\text{CIPT}}$, where only small fluctuations around a classical trajectory remain. One open question about more general CIPT models is whether there exist models where the critical behavior differs between the trajectory fluctuations and the state fluctuations.

Having addressed these preliminaries, we now briefly summarize our main results regarding the quantum nature of the CIPT.

1. 1.

   At the level of ensemble-averaged wave function properties (e.g. the ensemble-averaged distribution of the wave function over computational basis states), the quantum model looks similar to the classical model near the CIPT.

2. 2.

   The CIPT nevertheless manifests a phase transition in a quantum information-theoretic quantity: the ensemble-averaged $l_{1}$-coherence $\mathfrak{C}$ changes from an exponential growth as a function of system size $L$ in the chaotic phase to a saturating $L$-independent value in the controlled phase. At the critical point, the coherence growth remains exponential, indicating the critical wave function is quantum coherent.

3. 3.

   At the level of fluctuations in observables, the circuit-to-circuit fluctuations present in both the classical and quantum limits provide the dominant contribution and manifest the critical properties of the classical model.

4. 4.

   Although the trajectory-to-trajectory fluctuations, which appear only in the quantum limit, are subdominant, their contribution can be separated from the circuit-to-circuit fluctuations and measured experimentally. These intrinsic quantum fluctuations also become critical at the CIPT, and develop a non-zero scaling dimension that governs the algebraic decay of these fluctuations at the critical point.

These results provide a precise framework to demonstrate that the CIPT, which naively appears fully classical in nature, in fact, manifests inherent quantum critical properties that were previously “washed out” (or hidden) by equally averaging over circuits and measurement outcomes. We defer discussion of this point and further consequences to Sec. V.

To explicitly visualize how the control map is imprinted upon the quantum state, we plot the ensemble-averaged probability distribution over bit strings as a function of the control rate $p_{\text{ctrl}}$ for both the classical [Fig. 3(a)] and quantum [Fig. 3(b)] Bernoulli circuit models for system size $L=10$. Here, the probability of finding the state in a specific bit string $x$ is defined as

where $\ket{\psi_{\bm{M}}}$ is the steady state of the system for a specific sample $\bm{M}$ including a fixed circuit realization $\mathcal{C}$ and measurement history $m_{\mathcal{C}}$. In the classical model, the system is in a simple product state at all times.

In Fig. 3(a,b), we notice a striking visual resemblance between the classical and quantum models—as the control rate $p_{\text{ctrl}}$ increases from 0, the distribution of the bit string changes from a uniform distribution over all bit strings (corresponding to an ergodic chaotic phase) to a distribution localized at the fixed point of the polarized state $\ket{00\dots 0}$.

To understand the structure in the computational basis in more detail we turn to Figs. 3(c-e), where we present bit string distributions $f(x)$ for three typical values of $p_{\text{ctrl}}$ corresponding to the chaotic phase at $p_{\text{ctrl}}=0.4$ (orange), the critical point at $p_{\text{ctrl}}=0.5$ (red), and the controlled phase at $p_{\text{ctrl}}=0.6$ (green), respectively. We see for $p_{\mathrm{ctrl}}<p_{\mathrm{ctrl}}^{c}$ that $f(x)$ is almost uniformly distributed across all the basis states, namely $f(x)\sim dx=2^{-L}$, where $dx$ is the measure coming from the discretization of the interval $[0,1)$. For $p_{\mathrm{ctrl}}>p_{\mathrm{ctrl}}^{c}$, we find $f(x)\sim\delta_{x}dx$ up to a tail that goes to zero in the thermodynamic limit (see Appendix A for more details on the scaling of the distribution with system size $L$), demonstrating that the wave function “sticks” to the fixed point in the controlled phase. At the CIPT $p_{\mathrm{ctrl}}=p_{\mathrm{ctrl}}^{c}$, the bit string distribution takes the power law form $f(x)\sim L^{-1}dx$.

An interesting feature is the staircase pattern in the distribution of bit strings, $f(x)$, at a given $p_{\text{ctrl}}$, where abrupt changes all happen at bit strings that are inverse of the power of 2, and probability density within each interval remains almost uniform. The positions of abrupt changes motivate us to define the position of FDW measured as the first (leftmost) “1” from the right edge of the bit string $x$. For example, $\ket{0000000010}$ and $\ket{0000000011}$ have $k=2$, and the polarized state $\ket{0000000000}$ has $k=0$. Formally, we can define the FDW position for $x\in[0,1)$ as

where $\lfloor\dots\rfloor$ is the floor function.

Therefore, the state of the system can be effectively described by the FDW location $k$ instead of tracking the entire wave function, hence providing an effective metric to monitor its dynamics. (see Appendix A for more quantitative analysis of the FDW distribution.)

In the previous section, we found that the ensemble-averaged FDW distribution is the same for both the classical and quantum models (see Appendix A for a quantitative analysis of the two distributions), which does not inform us about the quantum nature of the CIPT. However, we can probe quantum effects by directly looking at the essence of a quantum state, i.e., its coherence due to superposition.

The coherence measure we adopt here is the ensemble-averaged $l_{1}$-quantum coherence [21]

where $\rho_{\bm{M}}$ is the density matrix for a specific sample $\bm{M}$, and ${(\rho_{\bm{M}})_{x,x^{\prime}}}$ is the off-diagonal element of the density matrix corresponding to $\outerproduct{x}{x^{\prime}}=\bigotimes_{i=1}^{L}\outerproduct{b_{i}}{b_{i}^{\prime}}$. The summation runs over all the off-diagonal elements of the density matrix, which originate from multiple superposition components within the quantum state. Consequently, the $l_{1}$-coherence is manifestly zero in the classical model. Note that we choose to focus on the average coherence of the state rather than the coherence of the average state as, in the latter case, the sample average washes out the off-diagonal density-matrix elements, leading to zero coherence. This is similar to the reason why ensemble-averaged entanglement measures are considered in studies of MIPTs.

In our numerical simulations of the coherence, we use both state-vector and MPS evolution algorithms. We use state-vector evolution to compute the coherence throughout the phase diagram (including the volume-law phase) up to $L=18$. Within the area-law phase, which includes the CIPT, the object of our study, we use MPS to simulate system sizes beyond $L=20$. Here, the nontrivial part is to compute the element-wise absolute value of the MPS wave function without fully contracting the internal legs of the MPS, which would lead to an exponential memory cost. We, therefore, use tensor cross interpolation [22, 23, 24, 25] (TCI) to learn the mapping from the physical index of an MPS to the norm of the corresponding wave function amplitude:

where $\ket{b_{1}b_{2}\dots b_{L}}$ corresponds to the computational basis state $\ket{x}$ with $x\in[0,1)$. With TCI, we can directly obtain each element in the density matrix—in complexity $O(\chi^{4})$ where $\chi$ is the maximal bond dimension of the MPS—without fully contracting the MPS. We choose a sufficiently large maximal bond dimension to capture the wave function within the area-law phase. More details on the TCI method are provided in Appendix C.

We start with an initial product state with FDW location $k=1$ and evolve the system to the steady state, which is reached after $2L^{2}$ time steps. In Fig. 4, we plot the steady-state coherence $\log_{2}\mathfrak{C}$ as a function of system size $L$ for different $p_{\text{ctrl}}$ using state-vector evolution for the chaotic phase in Fig. 4(a) and MPS for the critical and controlled phases in Fig. 4(b). In the chaotic phase [Fig. 4(a)], we find that the coherence grows exponentially with $L$, namely $\mathfrak{C}\sim 2^{a_{1}(p)L}$. At $p_{\text{ctrl}}=0$, we analytically find $\mathfrak{C}=\frac{\pi}{4}2^{L}$ as $L\rightarrow\infty$ from the coherence of a Haar random state (see Appendix B), which is the maximal possible coherence. For a finite $p_{\text{ctrl}}$ in the chaotic phase, the coherence continues to grow exponentially, however, with $0<a_{1}(p)<1$. Surprisingly, we find that at the critical point, the coherence remains exponential with $a_{1}(p_{c})\approx 0.85$. In contrast, for $p>p_{c}$, we find that $\mathfrak{C}$ saturates to a system size independent value. This motivates the hypothesis that the coherence can be parameterized as $\mathfrak{C}\sim 2^{h(p,L)}$, with $h(p,L)$ becoming non-analytic at the control transition. To test this, we apply an empirical “proof by contradiction” by assuming $h(p,L)$ is analytic in $p$, so that an asymptotic expansion of the form

holds. We will show that the coefficients of this expansion become singular near the critical point, demonstrating that $h(p,L)$ becomes non-analytic at this dynamical quantum phase transition.

We analyze the fitted parameters $a_{1}$, $a_{0}$, and $a_{-1}$ in Fig. 5 by gradually excluding smaller system sizes from the fitting. In Fig. 5(a), we find that $a_{1}$, which parameterizes the exponential growth of the coherence, remains almost constant throughout the chaotic phase up to the CIPT, where it jumps to zero. As we continue to exclude smaller system sizes, the jump becomes more abrupt, as shown in the inset. In Fig. 5(b), we study $a_{0}$, which parameterizes the system-size independent contribution. Starting with $a_{0}\approx-0.34\approx\log_{2}(\pi/4)$ at zero control rate, we find that $a_{0}$ decreases monotonically with $p_{\text{ctrl}}$ and reaches its minimum at the critical point, which is consistent with the decreasing growth exponent in Fig. 4(a). In the controlled phase at $p_{\text{ctrl}}>0.6$, we find that $a_{0}$ also decreases monotonically but from a large positive value. Here, $a_{0}$ corresponds to the saturation coherence at large $L$, which is consistent with the decreasing saturation coherence for larger $p_{\text{ctrl}}$ in Fig. 4(b). The growth from the minimum value to the large positive value between $p_{\text{ctrl}}=0.5$ and $p_{\text{ctrl}}=0.55$ is a finite-size effect: The data in the inset show that the peak of $a_{0}$ in the controlled phase near $p_{\text{ctrl}}=0.55$ becomes sharper as we exclude smaller system sizes from the fitting as shown in the inset.

Finally, in Fig. 5(c), we show data for $a_{-1}$, which parameterizes the subleading correction of order $L^{-1}$. We notice that it remains zero in both the chaotic and controlled phases but diverges at the critical point, consistent with the function $h(p,L)$ becoming non-analytic at the transition in the thermodynamic limit $(L\rightarrow\infty)$. It is striking that the coherence remains exponentially growing at the transition, demonstrating that the critical wave function is inherently quantum mechanical.

We first consider the dominant contribution to the unresolved fluctuations in the quantum model, namely the classical circuit-to-circuit fluctuations

where $\mathbb{E}_{\bm{m}_{\mathcal{C}}}\left[{k}_{\bm{m}_{\mathcal{C}}}\right]$ is the trajectory-averaged steady-state FDW expected value given a specific circuit $\mathcal{C}$.

To probe the classical fluctuations, we choose an initial state with the FDW at $k=L/2$, and allow the circuit to reach the steady state (the dynamics will be discussed in a sequel [28]). The classical nature of the circuit-to-circuit fluctuation can be seen in the steady-state value $\sigma_{\mathcal{C}}^{2}[k]$ as a function of $p_{\rm ctrl}$ for different system sizes $L$, as shown in Fig. 6. Here, we find that the steady-state fluctuations peak at the critical point, with the peak value scaling as $O(L^{2})$, consistent with the dynamical exponent of $z=2$ in the unbiased random walk. With the finite-size scaling ansatz

we find data collapse as shown in the inset of Fig. 6 with the estimated critical exponents $\nu=1.007(2)$ and $p_{c}^{\text{CIPT}}=0.500(1)$ consistent with the random walk universality reported in Ref. [8].

The steady-state trajectory fluctuation $\sigma_{t}^{2}[k]$ in Eq. (13), which measures fluctuations across different quantum trajectories for a fixed circuit realization, is unique to the quantum setting as it is only nonzero when the measurement outcomes are random:

where $\expectationvalue{k}_{\bm{m}_{\mathcal{C}}}=\expectationvalue{k}{\psi_{\bm{m}_{\mathcal{C}}}}$ is the steady state FDW expected value for a specific quantum trajectory $\bm{m}_{\mathcal{C}}$ given a specific circuit $\mathcal{C}$. It is subdominant [$\sigma_{t}^{2}[k]\sim O(1)$] compared to the classical circuit-to-circuit fluctuations [$\sigma_{\mathcal{C}}^{2}[k]\sim O(L^{2})$], so its contribution is not visible in the full average over samples. However, as discussed in Sec. II.2, it can be measured independently in experiments. We also examine the trajectory fluctuation using the magnetization density $M_{z}$ in Appendix D, finding results consistent with those for the FDW.

As an illustrative example, we first visualize the temporal spreading of the trajectories of the averaged FDW $k(t)=\mathbb{E}_{\bm{m}_{\mathcal{C}}}\left[\expectationvalue{k}{\psi(t)_{\bm{m}_{\mathcal{C}}}}\right]$ in Fig. 1(e) starting from $\expectationvalue{k(0)}=1$ for the three representative control rates $p_{\text{ctrl}}=0.4$, $0.5$, and $0.6$ in the quantum model. Within each panel, we plot a collection of quantum trajectories given the same circuit realization $\mathcal{C}$ (i.e., the same realization of the unitary scrambler $U$ in the Bernoulli map and the same sequence of chaotic and control time steps). The visible fluctuations come from the different measurement outcomes $\bm{m}_{\mathcal{C}}$. In the chaotic phase (left panel), the trajectories start to diverge immediately from the initial state at $k=1$ and quickly saturate to the left end of the chain at $k=L$. The large variance over different trajectories reflects the nature of the quantum dynamics due to the chaos in the Bernoulli map. However, in the controlled phase (right panel), the trajectories remain close to the fixed point, overlapping with each other, and the spreading of the trajectories only happens when the chaotic map is applied multiple times in a row (which is a rare event inside the controlled phase). This demonstrates the scenario where the quantum dynamics with a large spreading of trajectories is reduced to an effectively classical dynamics with a common trajectory. At the critical point in the middle panel, we see critical behavior where the FDW undergoes an unbiased random walk, and the spreading of the trajectories remains $O(1)$.

Figure 1(e) provides an intuitive picture of the trajectory fluctuations. To quantify these fluctuations, we plot in Figs. 7(a-c) the probability distribution of $\sigma_{t}^{2}[k]$ in the steady state for the three representative values $p_{\text{ctrl}}=0.4$, 0.5, and 0.6 over different circuit realizations $\mathcal{C}$. ¹¹1We evolve the circuit up to $t=2L^{2}$, and confirm that the dynamics reaches a steady state for $t>L^{2}$. Therefore, all states within $t\in[L^{2},2L^{2}]$ can be treated as separate samples for constructing the histogram in Fig. 7 to enhance the smoothness of the statistics with minimal additional computational cost. We also confirm the qualitatively same results using only the data at final time step $t=2L^{2}$. We find that in the chaotic phase [Fig. 7(a)], most of the trajectory fluctuations are finite (around $0.1$) and it is unlikely to have zero fluctuations in the thermodynamic limit—the probability of zero fluctuations is exponentially small in system size. At the critical point [Fig. 7(b)], the probability of zero fluctuations instead decays algebraically as a function of system size. In the controlled phase [Fig. 7(c)], most of the trajectory fluctuations are zero regardless of system size with an exponential tail of finite fluctuations, indicating classical dynamics that converge around a single trajectory with deterministic measurement outcomes.

This change in the distribution of the trajectory fluctuations $\sigma_{t}^{2}[k]$ as a function of $p_{\text{ctrl}}$ motivates us to define an order parameter $\mathcal{O}_{t}[k]$ as the probability of zero fluctuations,

where $\epsilon=10^{-5}$ is a small numerical cutoff. This order parameter is shown in Fig. 8 as a function of $p_{\rm ctrl}$ for different system sizes $L$. It vanishes in the chaotic phase, decays algebraically at the critical point, and saturates to a finite value in the controlled phase.

The algebraic decay of the order parameter $\mathcal{O}_{t}[k]$ at the critical point exposes another scaling dimension defined as

in the $t\rightarrow\infty$ and $L\rightarrow\infty$ limit, characterizing the onset of classicality. Here, we denote the scaling dimension by $\beta$ to make an analogy to the order-parameter critical exponent in the conventional context of continuous phase transitions. In the classical model, $\beta_{t}$ does not exist since the trajectory fluctuations $\sigma_{t}^{2}[k]$ are manifestly zero, and therefore $\mathcal{O}_{t}[k]$ will be always 1. In the quantum model, to extract the scaling dimension $\beta_{t}$, we perform finite-size scaling of the order parameter $\mathcal{O}_{t}[k]$ following the scaling form

In the inset of Fig. 8, we estimate the new scaling dimension as $\beta_{t}=0.98(8)$, along with $\nu=0.99(3)$, $p_{c}^{\text{CIPT}}=0.50(2)$ consistent with the classical results. This unveils another inherently quantum aspect of the CIPT which was previously overlooked due to the subleading nature of the quantum contribution to the sample-to-sample fluctuations. These universal quantum fluctuations should be generic to other adaptive quantum dynamics as well. In Appendix F, we show that the absorbing-state transition studied in Ref. [13] exhibits a similar behavior where the quantum trajectory fluctuations become critical and are then suppressed once the system enters the absorbing phase (analogous to the controlled phase here). The only difference lies in the vanishing scaling dimension $\beta$ in the trajectory fluctuation.

Having explored trajectory fluctuations in detail we now turn to a state fluctuation $\sigma_{s,\bm{m}_{\mathcal{C}}}^{2}[k]$ in Eq. (14). Similarly, we study the FDW location $k$ as

where $\expectationvalue{k^{2}}_{\bm{m}_{\mathcal{C}}}=\expectationvalue{k^{2}}{\psi_{\bm{m}_{\mathcal{C}}}}$ is the second moment of $k$ for a specific quantum trajectory ${\bm{m}_{\mathcal{C}}}$ and a fixed circuit $\mathcal{C}$. This quantity intuitively estimates the average spreading of the FDW wave packet within a single quantum trajectory. For the classical model, Eq. (27) is also by definition zero because the “wave function” $\ket{\psi_{\bm{m}_{\mathcal{C}}}}$ is a product state with a sharp FDW location, as opposed to a quantum wave packet. For the quantum model, the Bernoulli map generates quantum coherence as it scrambles a product state into a superposition of product states.

Similar to the trajectory fluctuations in Eq. (24), the distribution of the state fluctuations at the steady state can also be used to unveil the quantum nature of the CIPT. Here, we can define the probability of zero fluctuations in $\sigma_{s,\bm{m}_{\mathcal{C}}}^{2}[k]$ for different circuits $\mathcal{C}$ and trajectories $\bm{m}_{\mathcal{C}}$ as an order parameter $\mathcal{O}_{s}[k]$ similar to Eq. (24). The order parameter $\mathcal{O}_{s}[k]$ also changes from zero in the chaotic phase to a finite value in the controlled phase with an algebraic decay at the critical point, as shown in the lower inset in Fig. 9, similar to Eq. (24). To extract the corresponding scaling dimension $\beta_{s}$, we plot in Fig. 9 the order parameter $\mathcal{O}_{s}[k]$ as a function of $p_{\text{ctrl}}$ for $L=10$ to $L=40$. The upper inset in Fig. 9 shows the finite-size scaling of the order parameter $\mathcal{O}_{s}[k]$ following the scaling form in Eq. (26) to find the scaling dimension $\beta_{s}=0.99(3)$, with critical exponent $\nu=0.99(1)$, and critical measurement rate $p_{c}^{\text{CIPT}}=0.500(1)$, which is consistent with the critical exponents extracted from the trajectory fluctuation in Sec. IV.2.

Although both the trajectory and state fluctuations in the Bernoulli map model exhibit the same critical behavior at the CIPT, we notice that they are quantitatively different. In the controlled phase, the trajectory fluctuations (Fig. 8) have a larger order parameter than the state fluctuations in Fig. 9, indicating that the states with zero state fluctuations are a subset of those with zero trajectory fluctuations, given the ferromagnetic state as the dark state. This suggests that the state fluctuations being zero is a stronger condition than the trajectory fluctuations being zero. For example, when the states across all trajectories are reset to the fixed point $\ket{0\dots 0}$ (e.g., after many consecutive applications of the control map), a fixed Bernoulli map in the next time step drives all states to the same state $z_{0}\ket{0\dots 00}+z_{1}\ket{0\dots 01}+z_{2}\ket{0\dots 10}+z_{3}\ket{0\dots 11}$, leading to a zero trajectory fluctuation, while finite state fluctuation due to the coherence.

The opposite scenario with finite trajectory fluctuations but zero state fluctuations can also exist. For example, in the conventional Haar-random MIPT bricklayer model without any feedback, the trajectory fluctuations are always finite regardless of the measurement rate, while the state fluctuations are zero beyond the connectivity transition. Adaptive circuits with other feedback operations targeting other dark states can lead to different qualitative behaviors for the two types of quantum fluctuations, which is an interesting direction for future research.