Disordered monitored free fermions

Scrambling of quantum information in unitary evolution can be hindered due to measurements and localization, which pin quantum mechanical wavefunctions in real space suppressing entanglement in the steady state. In monitored free-fermionic models, the steady state undergoes an entanglement transition from a logarithmically entangled critical state to area-law. However, disorder can lead to Anderson localization. We investigate free fermions in a random potential with continuous monitoring, which enables us to probe the interplay between measurement-induced and localized phases. We show that the critical phase is stable up to a finite disorder and the criticality is consistent with the Berezinskii-Kosterlitz-Thouless universality. Furthermore, monitoring destroys localization, and the area-law phase at weak dissipation exhibits power-law decay of single-particle wave functions. Our work opens the avenue to probe this novel phase transition in electronic systems of quantum dot arrays and nanowires, and allow quantum control of entangled states.


I. INTRODUCTION
The preservation of information in many-body quantum systems poses a substantial challenge in quantum computing.Generically, as quantum systems evolve in time, any initial quantum information is scrambled throughout the system becoming inaccessible through local measurements, leading to thermalization.In recent years it has become clear that there are quantum systems that can fail to thermalize, the most prominent example being the phenomenon of many-body localization (MBL) [1][2][3][4][5].In such systems, quantum information remains accessible via local measurements even at long times and preserves correlations in the initial state.The MBL phase transition separating localized and chaotic phases of matter, is characterised by a singular change in the entanglement properties of the system.
Entanglement phase transitions can also occur in quantum trajectories of open quantum systems [6][7][8].In particular, the transition occurs due to a competition between measurements and unitary evolution, hence the name measurement-induced entanglement transition (MIET).This novel type of phase transition has been of interest in many recent studies .Typically, one considers a quantum circuit with unitary gates interspersed with local measurements at random locations.The transition between the volume-law and the area-law phase occurs at a finite measurement probability, and is known to occur in a wide variety of systems: unitaries can be randomly drawn either from the Haar measure or the Clifford group [6][7][8][9], or a Hamiltonian evolution of interacting systems [45][46][47][48][49], while the measurements can be chosen to be projective or weak.The universal properties of the MIET in random unitary circuits have similarities with those of percolation, though there appear to be some differences in surface critical behavior [8,11,[16][17][18][19]21].
Intriguingly, the phase diagram changes significantly for a free-fermionic system [32][33][34][35][36][37][38][39][40][41][42][43][44].The volume-law entangled steady state for non-zero measurement probability is fragile due to its lack of multipartite entanglement in free-fermion systems [32,33].For a range of measurement probabilities, an extended critical phase with logarithmic growth of entanglement and conformal symmetry emerges [33].Beyond a critical measurement probability the systems transition into an area-law phase.A substantial amount of evidence implies that this MIET is within the Berezinskii-Kosterlitz-Thouless (BKT) universality class [33,34], which puts it in a distinct class from random unitary circuits.Recent developments also suggest that the transition happens due to pinning of the wavefunction trajectory to the eigenstates of the measurement operator [34].
However, several important questions relating to the robustness of the critical logarithmic phase remain unanswered.For example, the logarithmic phase remaining stable against breaking of the continuous U (1) symmetry, for particle number conservation, to a discrete Z 2 fermion parity symmetry is associated with continuous replica symmetry breaking which doesn't appear to have a physical analog [31,34,38,60].For free-fermionic systems in one dimension, it is particularly interesting to ask about robustness to quenched disorder.For a non-interacting Hamiltonian, arbitrarily weak disorder localizes the singleparticle modes in 1D, a phenomenon known as Anderson localization [128][129][130].The role of measurements can destroy the localized phase at intermediate couplings while facilitating localization into product states at strong coupling.The competition between measurements and quenched disorder can result in a rich phase structure for an entanglement transition and is also relevant for observing the critical to area-law phase transition in an experimental setting.Disorder plays an essential role in a system of electrons in quantum dot arrays and nanowires where this phenomena can be explored.Motivated by this question, in this article we investigate the impact of quenched disorder on the measurementinduced transition in a one dimensional free-fermion system.A careful analysis of the entanglement entropy and central charge leads us to a phase diagram in terms of measurement strength γ and disorder W [see Fig. 1(b)] which exemplifies the robustness of the logarithmic phase and the relationship between Anderson localized and measurement induced area-law phases of non-interacting electrons.

II. MODEL
We consider spinless fermions hopping in a onedimensional lattice with a random potential, subject to continuous measurements [see Fig. 1(a)], where the random potential is distributed uniformly h i ∈ [−W, W ] with disorder strength W .The system is initially set to a separable Néel state.The evolution is implemented using the stochastic Schrödinger equation, which describes the continuous monitoring of particle number operator n i on each site, with measurement strength γ [32].The Itô increments dη t i have zero mean and variance of γ dt (see Appendix A for details).
We monitor each quantum trajectory, characterized by a set of measurement outcomes for a single realization of the random potential; the results are then averaged over multiple trajectories.Importantly, this provides access to averages of non-linear functions of the reduced density matrix, which in turn allow us to capture the entanglement phase transition.Specifically, we use the von Neumann entropy, a measure of entanglement between subsystem A and its complement, defined as S = −tr(ρ A ln ρ A ), where ρ A is the reduced density matrix of A. S is initially zero for a separable state, and grows in time, saturating near a fixed point S ∞ at long times, estimated as time average after saturation, S ∞ = lim ∆T →∞ tsat+∆T tsat S(t)dt/∆T .Finally, S ∞ is averaged over trajectories, giving S.
Entanglement phase transitions can be directly observed by monitoring how S changes with the system size L.However, even in free fermion circuits, where we can access larger system sizes, finite size effects are significant and impede our analysis.Special care needs to be taken for the critical phase, where both S and the correlation length ξ diverge logarithmically with L -extraction of the critical point is difficult for phase transitions with slowly diverging length scales [33].This critical phase is expected to be described by a 1+1D non-unitary conformal field theory (CFT) with periodic boundaries, with where l is the length of the subsystem A, c is the effective central charge of the non-unitary CFT, and s 0 is the residual entropy.For large enough systems, c is expected to be zero in the area law phase and finite in the log phase, and thus can be used as a transition diagnostic.

A. Phase diagram
Using the results for S(L/2, L) as a function of L, we perform a fit to Eq. 3 and obtain a central charge estimate, c L/2 .This allows us to draw the dependence of c L/2 on the measurement strength γ and the disordered field strength W -see Fig. 1(b).The central charge remains non-zero at low values of γ and W , implying the existence of the critical phase.However, at large values of either γ or W , c L/2 stays close to zero, a signature of the area law.This suggests that the logarithmic phase survives the introduction of the random disordered field, and only when the field is strong enough (W ≳ 3.5), the phase breaks down.
Estimation of the precise phase boundary is, however, difficult, as c L/2 does not decay sharply to zero.Large finite size effects necessitate a scaling analysis, which we perform in the next sections.Nonetheless, two peculiar features can immediately be seen in the density plot in Fig. 1(b): for small W , there is a non-monotonic behavior of the phase boundary [see Fig. 2(a)]; and for small γ, the density plot shows a rapid change of c L/2 for γ = 0 (Anderson localized) versus when γ is finite [see Fig. 2(b)].

B. Survival of the BKT universality class
We firstly discuss the case of small disorder strength.For the clean system (W = 0), we can fully reproduce existing results [32,33].Importantly, Ref. [33] provides numerical evidence of the BKT universality class, for which the half-chain entropy can be collapsed using [131], W = 1; this implies that the non-monotonicity of the phase boundary near small W is a physical phenomenon, and that introduction of weak disorder shifts the transition point γ c to higher values.This behavior seems similar to the one observed in interacting models [89], where a small amount of noise facilitated entanglement spreading and extended the volume law.Here, a small amount of disorder stabilizes the logarithmic phase, as the observed values of c(L) appear to be higher [see Fig. 2(a)].Alternatively, a weak disordered field slightly impedes the ability of measurements to pin the wave function trajectory to the eigenstates of the measurement operator, so that the area law occurs at higher γ c .Since the entanglement transition is a direct result of the competition between the unitary evolution and measurements, we believe that this non-monotonic behavior is tightly connected to the speed of entanglement spreading dictated by the hopping term.For this model, this speed seems low enough that the introduction of a small amount of disorder scrambles the information more                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              efficiently, pushing γ c to higher values.We test this hypothesis by adding next-nearest neighbor interactions to the Hamiltonian, which should increase the entanglement speed induced by the hopping terms.We find that the non-monotonicity in the phase diagram is absent (see Fig. 5), as expected.This suggests that the n.n.n.interactions increase the entanglement speed enough, so that it is no longer impacted by a small disorder.We would also like to comment on the recent results of Ref. [134], which put into question the existence of the critical phase in the clean model.The non-monotonic behavior observed in the disordered free-fermion system studied here implies that the disorder may stabilize the critical phase, even if it is absent in the clean case, signifying the presence of the measurement-induced transition for a finite disorder strength.

C. Destruction of Anderson localization
We now discuss the topic of small measurement strengths.For γ = 0, the system becomes an Anderson insulator and exhibits area law for any finite W . Below W ≲ 1.1, finite size effects cause finite c L/2 : localization length ξ in the Anderson model is inversely proportional to W 2 [135], and ξ becomes comparable to the considered system sizes when c L/2 becomes non-zero.This should, however, not be an issue for larger γ, as the characteristic length ξ is affected by both the disorder and the measurements.
At very small but nonzero values of γ = 0.02, 0.04 [133], we observe an abrupt change to the localized behavior.S(L/2) results suggest that a logarithmic dependence on the system size is present for small W [Fig. 3(b)], with the crossover to the area-law scaling at around 1.5 ≲ W ≲ 2.5.We pinpoint the transition, extracting W S c ≈ 2.1 and W c(L) c ≈ 1.9 for γ = 0.02.Importantly, W c is large enough not to be impacted by the characteristic length ξ being comparable to L, and therefore we believe the observed transition to be physical.Our results suggest the BKT universality class is preserved for the whole transition boundary in Fig. 1(b).Furthermore, the connected correlators change their behavior between γ = 0 and 0.02 [see Fig. 4(c-d)] from faster-than-algebraic to 1/r 2 decay for W < W c , indicating emergence of the conformal phase.
We thus conclude that Anderson localization is immediately broken for any finite value of γ, and the critical phase reappears in the phase diagram.This phenomenon most likely occurs due to measurements impeding interference through impurity scattering-even very weak measurements change the scattered fermionic modes and destructive interference is not possible, rendering the mechanism behind Anderson localization disrupted.A similar mechanism occurs when inelastic scattering is introduced to an Anderson-localized medium, where the phase coherence between outgoing and ingoing modes is disrupted [136].Whether the measurements force the system into the critical phase or the area law depends on the shape of localized single-particle orbitals |ψ i | 2 at γ = 0 [see Fig. 6(a,b)], which can be read off from the Slater determinant.At large disorder, the orbitals decay rapidly and the overlap between their envelopes is negligible.The measurements have little impact, only sharpening the orbitals at their localization centers, and the area law is preserved.At small disorder, the orbitals are broad and their envelopes substantially overlap with each other.The measurements effectively introduce scrambling between them, which leads to a delocalized behavior and the critical phase.This very simple picture would suggest that the transition happens approximately when ξ ∼ 1, while our numerical results reveal a slightly larger critical localization length of ξ ∼ 24/W 2 c ≈ 6 [135].Furthermore, we find a clear distinction between the Anderson-localized area law and the measurementinduced area law for γ > 0. The former is characterized by exponential decay of the orbitals, while the latter exhibits power-law localization, [133].Autocorrelation functions C(τ ) [Fig.6(c,d)] also showcase this difference.For γ = 0, C(τ ) quickly saturates to a constant and does not decay.However, for γ > 0, C(τ ) plateaus for a long period of time (τ ∼ 100), and eventually decays due to the disruption from measurements to a minimum value of 1/(2L).The plateau size depends on γ, where for large γ the decay begins earlier.The decay itself seems to be approximately power law, with larger systems taking more time to reach the minimum value.

IV. CONCLUSIONS
The results presented in this work show the nontrivial interplay between Anderson localization and continuous measurements.We convincingly demonstrate that the entanglement phase transition from the critical phase with conformal symmetry to the area-law phase survives the introduction of quenched disorder.Moreover, the universality class of this transition also seems to be preserved, which strongly suggests that the logarithmic phase is stable to weak perturbations.We also find that a small amount of disorder can help stabilizing the critical phase.Gathering all our data from the collapse of entanglement entropy and effective central charge, we estimate the true transition boundary between the logarithmic and area-law phases [solid line in Fig. 1(b)].In general, our results convincingly suggest the conformal phase and free-fermion MIETs are viable for experimental probing in systems such as nanowires and quantum dot arrays, which host Anderson localization along with implementation of local measurements [137].
We find that an introduction of monitoring in the Anderson-localized model results in an instant destruction of the localization for weak disorder.The delocalization results from the destruction of the coherent processes leading to a liquid state.Although, at sufficiently large disorder, the system transitions into an area-law state which is markedly distinct from Anderson localization as the orbitals exhibit a power-law decay in space instead of exponential.The temporal behavior of the autocorrelations exhibits parametrically longer decay time scales compared to Anderson localization.There are several interesting directions for future work emerging from our results.The role of interactions in the logarithmic phase and its relationship to many-body localization remains a challenging open problem.The fate of the critical phase for integrable models which do not map to free fermions could also provide new classes of measurement-induced criticality.
All relevant data present in this publication can be accessed at [138].The measurements and time evolution are implemented using the stochastic Schrödinger equation, where a monitoring of an operator O is done by evolving the wave function according to , with η t a Wiener process and γ the measurement strength/rate.We will measure operator n i = c † i c i on every site.This evolution can be approximated by trotterisation, |ψ(t + dt)⟩ ≈ e M e −iHdt |ψ(t)⟩.
Importantly, this corresponds to an evolution of the matrix U that fully describes the Gaussian state, where M is a matrix with elements M ij = δ ij (η i +(2⟨n i ⟩− 1)γdt), and h corresponds to the free-fermion Hamiltonian H = i c † i c i+1 + h.c., and has elements h ij = δ i,j+1 + δ i,j−1 + h i δ i,j .After each time step dt, the wave function needs to be properly normalized, which can be done by a QR decomposition of matrix U (t + dt) = QR, and setting the new matrix U to be Q.Fig. 7 shows that setting dt = 0.05 is enough to describe the continuous-time regime, and we find that lowering dt does not change our results within the statistical error bars.

Observables
Using the matrix U , one can define the correlation matrix D = U U † with elements D ij = ⟨c † i c j ⟩, giving us direct access to expectation values.Furthermore, to calculate entanglement entropy S of a bipartition of the system into subsystem A and its compliment B, we restrict D to indices associated with the subsystem A, and then diagonalize the restricted matrix to obtain its eigenvalues λ i .S is then simply given by The connected correlation functions C(r) can be deter- mined from the correlation matrix, Similarly, the autocorrelation function C(τ ) can be calculated in the same manner, where Finally, one can easily extract the fermion orbitals by taking the columns of U , i.e. |ψ i (r)| 2 = |U i,r | 2 .We move the orbitals spatially so that they are centered around the maximum value, and then average them over many realizations.

Equilibration to the steady state
The time it takes to reach the steady state is nontrivially dependent on two variables: measurement strength γ and disorder W .In the absence of the disorder, for large γ we find that the equilibration takes O(1) time, while for small γ it takes at most O(L) time.Introducing the disorder prolongs the equilibration time roughly proportionally to W (see Fig. 8).
We also note that near W ≈ 2, the time dependence of the trajectory-averaged half-chain entropy seems to be collapsing into one curve [see Fig. 9], with the initial behavior scaling as S(L/2) ∼ ln(time/L).This suggests an emergence of z = 1 conformal symmetry near this point.

Finite size scaling
The data collapse for the finite-size scaling analysis is performed by minimizing the cost function ϵ, which measures how well the data collapses into a single curve given the parameters.First, the data is rescaled using the finite-size scaling ansatze from Eqs. ( 4) and ( 5) to produce a set of triples x i , y i , d i representing the rescaled x coordinate, rescaled y coordinate, and the error in the y coordinate.For example, for Eq. ( 4), ), and d is the error of the half-chain entropy.Then, the triples are sorted by their x-values, and one can calculate the cost function, After obtaining the minimum ϵ min , one can estimate the error in the collapse parameters by investigating the region where ϵ = 2ϵ min .
In Table I we report the estimates for the parameters from data collapses in Figs. 3, 10, and 11.The scaling function g(L) from Eq. ( 5) has the following form, g(L) = [1 + 1/(2 ln L − β)] −1 , and can be determined from a superfluid stiffness scaling analogy for the BKT transition [33,131,132].
Supporting data for W = 0.25 and γ = 0.04 is shown in Fig. 10.

Decay of correlation functions
Although we find that single-particle wave functions are power-law localized, the issue is that these orbitals are not uniquely defined, as the matrix U can be multiplied on the right by any unitary, while not changing the physical state.However, we also find that for γ > 0, correlation functions do not seem to exhibit exponential decay (see Fig. 12, where we show the data of Fig. 4, but on a linear-log plot), which would be in agreement with our findings for the orbitals.Perhaps the reason why we do not find "scrambled" orbitals, is due to the uniqueness of the method: both unitary evolution and measurements uniquely transform matrix U (during normalization, QR decomposition is unique as well).

FIG. 1 .
FIG. 1.(a) Sketch of disordered monitored free fermions.(b) Phase diagram.The density plot shows the effective central charge estimate, c L/2 .Data collapses of half-chain entanglement entropy (green circles) and central charge (blue squares) are used to estimate the transition boundary (solid line).

8 FIG. 2 .
FIG. 2. Effective central charge estimate, c L/2 , calculated using half-chain entanglement entropy (a) for small values of W , and (b) for small values of γ.

FIG. 3 .
FIG. 3. Behavior for small W (left plots) and small γ (right plots).(a,b) Half-chain entropy S(L/2) for different values of the measurement strength γ or disorder strength W (see labels on the right).(c,d) Central charge c(L) as a function of γ (or W ) and system size L. Data collapses for (e,f) c(L), and (g,h) S(L/2).Legend from (c,d) applies in (e-h).

FIG. 5 .
FIG.5.Central charge as a function of measurement strength γ and disorder W for a system with additional next-nearest neighbor interactions.

FIG. 8 .
FIG. 8. Time dependence of averaged entanglement entropy S for (a) γ = 0.02, L = 256, (b) γ = 0.02, L = 385, and (c) W = 0.5, L = 256.The time after which the value reaches saturation value is roughly proportional to the system size L, but is also impacted by the disorder strength W and the measurement rate γ.

FIG. 10 .
FIG. 10.Behavior of (a,b) half-chain entanglement entropy S(L/2) for different values of the measurement strength γ (see labels on the right), and (c,d) central charge c(L).Data collapse for (e,f) S(L/2), and (g,h) c(L); legend from (c,d) applies in (e-h).Left plots are for W = 0.25 and the right plots are for W = 1.0.