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Measurements on an Anderson Chain

We study the dynamics of a monitored single particle in a one-dimensional, Anderson-localized system. The time evolution is governed by Hamiltonian dynamics for fixed time intervals, interrupted by local, projective measurements. The competition between disorder-induced localization and measurement-induced jumps leads to interesting behaviour of readout-averaged quantities. We find that measurements at random positions delocalize the average position, similar to a classical random walk. Along each quantum trajectory, the particle remains localized, however with a modified localization length. In contrast to measurement-induced delocalization, controlled measurements can be used to introduce transport in the system and localize the particle at a chosen site. In this sense, the measurements provide a controlled environment for the particle.

If disorder is introduced into a one-dimensional non-interacting chain, the system becomes Andersonlocalized [23][24][25]. All eigenstates decay exponentially, and transport from one end of the chain to the other is exponentially suppressed with the size of the system. Because of this special property of the eigenstates, it is natural to ask about measurements in this context. In particular, one may wonder whether the introduction of measurements leads to dephasing, destroying localization and establishing transport. A related issue of noiseinduced dynamics in a localized system is discussed in Ref. [26]. Recently, the effect of measurements on the many-body localized phase has been studied [27]. Considering a single-particle model has the benefit of being more tractable numerically, and intuitively comprehensible. At the same time, we find interesting physics in the simple model.
In the present work, we consider local projective measurements. If such a measurement occurs within the localization length of a localized particle, the particle is often detected at the measured site. In these cases, the center of the wave function shifts to the measured site as a consequence of the projection. Repeated measurements are therefore expected to induce transport in the system, by projecting the particle on successive sites. With this idea in mind, we follow two general directions: On the one hand, we choose measurement locations at random and investigate the consequent dynamics. On the other hand, we try to manipulate the state of the system in a controlled way, by designing "measurement protocols" that aim for spatial steering of the particle. The engineering and manipulation of quantum states through measurements has been explored in Refs. [28][29][30][31].
In the case of random measurement locations, our key observation is that while the particle position is randomized over the entire system, the wave packet typically remains exponentially localized around its center site. Considering averages over measurement outcomes, measurement locations, as well as over disorder realizations, we investigate the spread of the ensemble of wave-packet centers throughout the system and modifications to the localization lengths due to the measurements. Supported by a connection to a classical random walk, we argue that the particle trajectories of different random realizations spread diffusively in the long-time limit. The idea of measurement-induced random walks has been addressed in Ref. [32]. In Ref. [33], the authors use a mapping to a classical random walk to model dynamical properties of quantum systems subject to measurements and disorder.
Regarding steering, we analyze and compare different kinds of protocols, either using the detector readout only at a designated target site, or at every measurement location. Having access to all readouts allows us to induce ballistic transport. In this sense, both types of steering, passive [28] and active [29], are considered. Diffusion due to random measurements allows for polynomial steering times, even if only reading out the target site.
The paper is structured as follows. We introduce the time evolution protocol for random measurements in Sec. II. In Sec. III, we describe the resulting dynamics qualitatively, motivating the discussion of particle trajectories and corresponding observables. With these observables, we investigate "delocalization" due to measurements in Sec. IV. We present the numerical results on steering by non-random measurements in Sec. V. Finally, in Sec. VI, the relation to a classical random walk is formalized, and, based on that, our numerical results are further discussed. We conclude in Sec. VII.

II. HAMILTONIAN AND MEASUREMENT PROTOCOL
We consider an Anderson-localized chain with nearestneighbour hopping described by the Hamiltonian where |i denotes a state perfectly localized at site i and the disordered onsite potentials, {ε i }, are randomly drawn from a Gaussian distribution with zero mean and standard deviation W/ √ 3. We initialise the state at time t = 0 at site i 0 , |ψ(t = 0) = |i 0 , and study the dynamics following a discrete-time-evolution protocol. The protocol consists of two steps. In the first step, the state is subject to unitary evolution for a time interval ∆t with the Hamiltonian (1). In the second step, with probability p, a single projective measurement of the number operator is done at a randomly chosen site. Formally, whereM t encodes the measurement at time t and t − = t − 0 refers to the time just before the time of measurement t. The operator carries an index t to make explicit the fact that it depends on time. In a given realisation of the dynamics, we denote the site that is measured at time t as i t and the sequence of sites i ∆t , i 2∆t , · · · , i N ∆t as the measurement path, where N is the total number of measurements. At any time step, the measurement site is chosen randomly and equiprobably from the entire chain.
In the event that a measurement does happen, say at i t , there are two possibilities. One is that the particle gets detected (the detector "clicks"), in which case the state collapses into a perfectly localized (on a single site) state This happens with a probability ρ it (t) ≡ |ψ it | 2 given by the Born rule, where ψ it = i t |ψ(t − ) . The second possibility, with probability 1 − ρ it (t), is that the particle does not get detected (detector does not click -"no-click" event), in which case the wavefunction develops a hole at i t , such that For a given measurement path specified by the locations and times of the projective measurements, we define the sequence of clicks and no-clicks as the outcome sequence.
The above makes clear the various sources of stochasticity (both spatial and temporal) in the dynamics, Figure 1. Time evolution of the probability density starting from the initial state |51 in a system of 100 sites withξ ≈ 7, ∆t = 10 and p = 1. We show 100 intermediate times for each step of the protocol. We observe both no-click and click measurements: For measurements at N = 84, 179, 207, 258 clicks occur at sites 52, 69, 69, 65 and the particle is thus projected to these positions. The dark spots correspond to the no-click measurements.
namely, the disorder realisation, the measurement path, and the outcome sequence. For a given disorder realisation and i 0 , a particular measurement path r t and a particular outcome sequence n t ∈ [0, 1] together define uniquely the quantum trajectory {t, r t , n t }. It is useful to introduce notation for averages over each of these stochasticity sources as • · · · d ≡ average over disorder realisations, • · · · p ≡ average over measurement paths, • · · · o ≡ average over outcome sequences. These averages are defined by varying the corresponding quantities between runs of the protocol.
In the following, we consider the disorder strengths in the interval W ∈ [2, 10], corresponding to average localization lengths, ξ, in the range 0.9 ξ 7. The system size is chosen to be larger than then the localization length for any of the disorder strengths, L ξ , to avoid finite size effects. Furthermore, we consider long unitary time evolution intervals ∆t/p ξ , avoiding the quantum Zeno effect-the particle has typically enough time to spread over its localization length between two click-measurements.

III. PARTICLE TRAJECTORIES AND OBSERVABLES
To first get a broadbrush view of the phenomenology of the dynamics, we simulate a particular quantum trajectory and plot ρ i (t) = |ψ i (t)| 2 as a heatmap in the (i, t) space, as shown in Fig. 1. The state is initialised at the centre of the chain, and we consider W = 2 and p = 1 as an example. The following features are of note in the dynamics: (i) the wavefunction remains reasonably localized around its localization centre throughout the dynamics; (ii) the profile of the wavefunction is quite robust to a series of several no-click measurements; (iii) no-click measurements are much more likely than click measurements.
Much of these can be argued for based on the fact that the unitary dynamics between the measurements is governed by an Anderson localized Hamiltonian. This mandates that in between measurements the state can only spread within a region of size of order of ξ on average around the localization centre. Note that our wavefunction does not exactly correspond to an eigenfunction at finite disorder, which would feature a localization length distributed according to ξ. This is because we initialize on / project to individual sites, which share overlap with O(ξ) sites. For this reason, the wavefunction does not necessarily have one bright center.
If the consequent measurement is near the localization centre, it is highly likely that the detector clicks and the state gets further localized to a single site. On the other hand, if the measurement is away from the localization centre, then the probability of a click is exponentially suppressed in the distance from the localization centre. This, in turn, leads to a concomitantly small perturbation to the wavefunction in the most likely case of a noclick event and, hence, to the robustness of the state to individual no-click measurements. Finally, note that the probability of a click decreases with increasing system size. This can be understood as follows: the probability of a click, if the measurement is at site i, is simply |ψ i | 2 . However, since every site is equiprobable for it to be measurement site (with probability pL −1 ), the probability that we have a click measurement outcome is pL −1 i |ψ i | 2 = pL −1 , which is extremely small for large L. This also suggests that the average time interval between two click measurements is L∆t/p. In combination with the typical smallness of the perturbation due to one no-click, this implies the robustness of the wavefunction to typical no-click sequences.
The upshot of all of the above is that, since the wavefunction behaves like a semiclassical wavepacket at all times, its dynamics can be effectively characterised quantitatively by the moments of the wavepacket which we define as For instance, r q=1 (t) denotes the expectation value of the position of the wavepacket (which we refer to as particle trajectory), r 2 (t) − r 2 1 (t) denotes the width of the wavepacket for a given realisation and so on.
Before analysing different averages of r q (t) in the following section and motivating their physical meaning, we explain some of the features of r 1 (t) for a typical trajectory shown in Fig. 2. For all data therein we consider W = 2, p = 1, ∆t = 100, and the disorder realisation and measurement path are held fixed. The upper panel shows r 1 (t) for different outcome sequences (represented by different colours). Note that for large intervals of time, O(L/p), r 1 (t) only fluctuates weakly around some value -this is a manifestation of the fact that each of the measurements are no-click measurements or click measurements very close / at the center site of the wavefunction. Importantly, the probability of such "zero-distance" jumps depends on the disorder strength, which thus also impacts the typical time scale between two visible jumps. The weak fluctuations are brought about by the unitary quantum evolution of the state with the hole punctured by the no-click measurements. However, every O(L/p) time instance on average, the detector clicks far from the previous center, and r 1 (t) jumps to the new location.
This can be made quantitative further by considering the probability distribution of the wavepacket's expected position over all outcome sequences. This can be formally expressed as where r The key point that should be noted from the data is that the distribution of Π i (t) keeps getting broader in i as time progresses. This can be attributed to the click measurements, albeit rare. This is most noticeable at early times in the figure, where a click measurement creates a new stream of finite Π i (t), which is spatially separated from other regions of finite Π i (t). A click measurement displaces the localization centre of the wavepacket. The probability of having a given distance between the two localization centres naturally decreases exponentially with the distance; nevertheless, averaged over outcome sequences, these rare click measurements can delocalise the state in an average sense.
In order to quantify the spread of the wavepacket's expected position, we introduce the "classical" displacement where µ can be "d", "o", or "p", or a combination of them, and r q (t) is defined in Eq. (6). If the trajectories spread equally over the entire chain, the spread is given by ∆(t) = L/ √ 12 for L 1. While ∆(t) defined in Eq. 8 encodes the classical spreading of the wavepacket due to the stochasticity in measurement paths or outcome sequences, we show in Sec. III that quantum-mechanically, the wavefunction remains reasonably localized within each trajectory.
To quantify this, we introduce as a second observable the "effective localization length" where the notation . . . ψ(t) denotes the quantummechanical average with the wavefunction ψ at time t. This quantity was also studied in Ref. [34] to find the distribution of localization lengths of eigenstates in Anderson Hamiltonians. We use it as a dynamical definition of the localization length, sensitive to changes to the "effective localization length"ξ characterizing the interplay between disorder-induced localization and measurements. For point-like densities ξ eff = 0, for a localized wavefunction with localization length ξ it holds that ξ eff = ξ/ √ 2 (if L ξ and ξ 1). Note that ξ eff (t) = ∆(t), if no average over the measurement runs is performed. In general,  before the measurements. With this convention, a clickmeasurement, for example, leads to a contribution of O(ξ) to ξ eff , instead of zero, which we would get immediately after the measurement.

IV. MEASUREMENT-INDUCED DELOCALIZATION
In the previous section, we introduced the observables ∆(t)-quantifying the spread of the particle trajectories-and the effective localization length ξ eff (t). In the absence of measurements, these observables assume their final values within O(ξ) hopping times, as a consequence of eigenfunction's localization in the Anderson chain. In this case, both observables measure the usual disorder localization length, as the particle remains confined to its initial position. In the following, we quantify the impact of uniformly distributed measurements on the dynamics of the system through these observables.
A. Different averages Figure 3 shows the observables ∆, ξ eff and r o obtained from an outcome average over 4 · 10 3 runs in a system of L = 1600 sites, with fixedξ ≈ 7 disorder realization and fixed random measurement path with p = 1 (blue lines) and ∆t = 100. Grey, dashed lines show the same observables in the non-measured case p = 0 for comparison.
Let us first consider the outcome-averaged position expectation value: At p = 0 this value fluctuates by O(ξ) sites around the initial position. These fluctuations are due to the unitary time evolution, mediating between |i 0 and O(ξ) neighboring sites through the localized eigenfunctions. In contrast, at p = 1, the fluctuations are less pronounced, as they average out over different quantum trajectories. At N ∼ L = 1600, however, the average position starts to slowly drift away from the initial position, reaching i ≈ 810 at N ≈ 10 5 , to then continue fluctuate in this region. The drift velocity is very small, v ≈ 1/10 4 1. Since p = 0 corresponds to a single quantum trajectory, only the spread of the wavefunction contributes to the displacement ∆. Accordingly, ξ eff and ∆ are equal in this case, and ∆ = ξ eff ∼ξ. This is similar for the measured case up to N L/p = 1600, where again both quantities behave similarly (not shown), since all quantum trajectories away from the initial position are very unlikely-few click measurements occur up to this point and there are also few impactful no-click events up to N ∼ L/(pξ) ≈ 250. In consequence, there is no classical contribution to ∆ from the displacement of the wave packet center, and the shape of the wave packet is largely determined by the unitary time evolution. At N ∼ L, however, ξ eff at p = 1 is increased compared to non-measured ξ eff , while ∆ continues to grow as an approximate power law ∝ t γ with γ ≈ 0.59 (close to the diffusive exponent γ diff = 1/2).
At N = L for p = 1, we expect O(1) clickmeasurements for each trajectory, shifting the positions of the wave packets; and no-click measurements close to the center of the wave packets, redistributing the weight which is projected away from the measured site. Importantly, however, ξ eff L, validating the picture of an effective localization length that is still defined in the presence of measurements. Considering the entire time interval, we notice slow fluctuations in the local power law exponent of ∆; ξ eff reaches a local minimum at t ∼ 10 4 and slightly increases at later times.
If we redraw the measurement path between runs, in addition to considering random outcome sequences for every run, the features of the plots are largely similar to the exclusively outcome averaged case (not shown). This is expected even for N L, when measurements play an important role. The outcome-average alone spatially separates the particle trajectories over time, and at this point different trajectories experience independent measurement locations in their vicinities anyway. There may be subtle differences at intermediate time scales, where many trajectories are still overlapping. However, we are mostly interested in the long-time behaviour.
Finally, we consider the observables ∆ and ξ eff and r o,p,d with averages over outcomes, paths and disorder at the same parameters W = 2, L = 1600, p = 1, ∆t = 100 (Fig. 4). In addition, ∆ and ξ are plotted together in the lower right panel, to demonstrate the difference ∆ 2 − ξ 2 eff = r 2 − r 2 . Averaging over disor- der realizations and paths, we remove all spatial inhomogeneities from the averaged quantities. As a result, r o,p,d is constant up to fluctuations, also in the measured case. Again, the obtained exponent for the spread γ = 0.49 is close to the diffusive value. Contrary to the previous cases, ξ eff saturates after O(10 3 ) measurements to a value that does not further change with time. This is because the introduction of an average over disorder realizations removes, right from the outset, correlations between different runs of the simulation. The average number of measurements required to displace the wave packet center from a specific site is determined by the localization lengths of the eigenfunctions peaked close to this site, which are, in turn, determined by the disorder realization. If disorder is not averaged over, a region of small localization lengths ("traps") slows down all trajectories passing through that region. Such traps can lead to drifting of r o,p (t) (partially blocking transport on one side of the system), as well as to fluctuations in ∆(t) (traps slow down the average spread) and ξ eff (t) (the trap corresponds to small ξ and, thus, small ξ eff ). On the other hand, regions of large ξ can speed up the spread and lead to upwards fluctuations in ξ eff . In the thermodynamic limit L, N → ∞, we expect these effects to vanish, even if the observables are only averaged over outcomes, as the trajectories become increasingly spatially distributed and thus less correlated. However, our numerics do not probe the corresponding time scales. Upon disorder averaging, there is an immediate average over different ξ at every point in time, since the observables are averaged with wave functions Based on the approximation that the trajectories spread diffusively due to click-measurements, we can estimate the number of measurements required to achieve a sufficient effective average over different localization lengths. From Fig. 4, we conclude that ∼ 1000 independent sites are sufficient for the effect of traps to be averaged out. Therefore, having ∆ ∼ 10 3ξ without disorder averaging should facilitate a similar average. This corresponds to t ∼ 10 3ξ /ξ, where t = N p/L is the expected number of click measurements. We find N ∼ 10 6 L/p, where we have to set L 10 3ξ to avoid finite-size effects. We find N ∼ 10 9 ξ/p with an additional numericalξ-dependent factor taking into account zero-distance jumps. From this estimate, it is clear that we would have to go to much larger numbers of measurements to find diffusive behaviour without averaging over disorder realizations.
In Fig. 6, we compare (o, p, d)-averaged ∆ and ξ eff at different measurement probabilities in order to demonstrate that the measurement frequency only rescales the time axis, as long as ∆t ξ. For this purpose, we plot each quantity as a function of the expected number of measurements N · p on top of each other. Indeed, after an initial phase corresponding to few measurements, the curves lie on top of each other.

B. Length and time scales
In order to separate the influence of measurements on the average quantities from the non-measured time evolution, we consider small measurement probabilities p ∈ {0.025, 0.05} in a system with L = 100 andξ ≈ 7 (p = 0 is shown for reference), averaging over O(10 3 ) runs, see Fig. 5. We choose ∆t = 1, which does not come with the Zeno effect, since ∆t/p ξ . In order to avoid the related intermediate-scale effects, we average not only over outcomes and measurement paths, but also over disorder realizations. Because of the small measurement probability, it takes many steps of the time evolu-  Figure 6. Displacement (left) and effective localization length (right) for a system of L = 1600 sites, atξ ≈ 7 for different measurement probabilities. Each curve is averaged over 4 · 10 3 runs, with random measurement outcomes, paths, and disorder realizations. The data is plotted over the expected number of measurements N · p.
tion protocol for the measurements to show a pronounced effect on the system, leading to a separation of the initial time scale, where the observables basically behave as in the absence of measurements (gray, dashed lines), from the time scale, where the effect of measurements sets in. Specifically, for these parameters of the protocol, N = L/(ξp) ∼ 300 steps are required until one measurement within the localization radius has taken place on average, and about N = L/p ∼ 2000 steps (black dotted lines) until one click is encountered. Consequently, a difference between the p = 0 and p = 1 curves becomes apparent between these time steps.
As we observed before, ξ eff (t) increases to its saturated valueξ < ξ ∞ eff L while ∆ grows according to a power law. The spread of the trajectories is limited by the system size, which shows as a second plateau with ∆ ≈ L/ √ 12 ≈ 30-corresponding to uniform spreading over the system. Assuming diffusive spreading of trajectories, driven by click measurements, the corresponding scale can be estimated as t 1/2 ∼ L/ξ, where the time is counted in expected clicks t = N p/L, giving N ∼ L 3 /(pξ) 2 ∼ 6 · 10 5 , in agreement with the actual time of saturation to the second plateau. Since ξ ∞ eff is related to no-click measurements, the corresponding plateau sets in when there is averaging over contributions of all relevant numbers of successive no-click measurements. As a rough estimate, the probability of a sequence of exclusively no-click measurements decreases exponentially with the length of this sequence. We thus estimate N ∼ L/(pξ) ∼ 300 with the numerical factor depending on the details of effective localization length's origin.
In the upper panel of Fig. 7, we demonstrate the scaling of ∆ ∞ (left) and ξ ∞ eff (right) with the size of the system for parameters p = 1, W = 2, ∆t = 100. In agreement with uniform spread over the entire system, we find √ 12∆ ∞ (L) ≈ L (lower left, dashed line). On the contrary, ξ ∞ eff (L) (lower right) saturates to ξ eff (L → ∞) ≈ 13 for L 250. This further validates that there are still localized wave functions in the presence of measurements, despite the spread of trajectories. In the lower panels, we  plot the number of measurements required to reach the plateaus in ∆ (left) and ξ eff (right); confirming the cubic (linear) dependence on the system size.

C. Effective localization lengths
The value of ξ ∞ eff depends non-trivially on W , p, ∆t and L. In particular, W determinesξ and, thus, ξ ∞ eff (p = 0). We already observed that finite p leads to an increase in ξ ∞ eff in the limit ∆t/p ξ . We do not consider the Zeno limit ∆t/p → 0, where ξ ∞ eff → 0; however this limit implies that ξ ∞ eff (∆t/p) has a maximum at finite ∆t/p. In the upper right panel of Fig. 7, we see the effect of L acting as an upper cutoff on ξ ∞ eff .
Concerning the effect of measurements on the effective localization length, we know that every click measurement resets ξ eff → ξ. A no-click measurement outside of the effective localization length has an exponentially small effect. A no-click event within the effective localization length can lead to enhancement of the wave function tails and, thus, to a slight growth in ξ eff , resulting in ξ ∞ eff >ξ. Figure 8 compares the distributions in the measured and non-measured case forξ ≈ 7, ∆t = 100, L = 300. As expected, the measurements enhance the distribution towards larger localization lengths.
Since in the fully averaged case the effective localization length is still larger than in the free system, we conclude that this increase in the presence of measurements has to be connected to no-clicks, since clicks reset the contribution of a wave function to the non-measured value.
To summarize the above, we have found that randomly distributed measurements lead to delocalization of particle trajectories. Each projective measurement transfers the particle to the measured site. Following this process for a sufficient time span, the probability to find a particle at a particular site is equal for all sites. During this process, the wave functions are still well described by an effective localization length, as opposed to spreading over the size of the system.

V. STEERING WITH MEASUREMENTS
In the preceding section, we concluded that random measurements all over the system lead to delocalization of quantum trajectories, while almost all individual trajectories correspond to localized particles. This raises the question, if the well-defined location of the particle can be efficiently manipulated, inducing controlled transport in a "localized" system by performing measurements according to an appropriate steering protocol.
The concept of having localized trajectories with the location governed by click measurements can be applied to steer the particle from its initial site to a specified target, with the goal of having a click at the target. The average number of measurements required to achieve this goal defines efficiency of the measurement protocol, which dictates the measurement path. In a localized system, where only the target site is measured, the expected number of measurements would increase exponentially with the system size. Contrarily, if the wavefunction is completely delocalized, the expected number of measurements would scale linearly with system size, when measuring again only the target site. In our localized system, where all sites may be measured, we expect to find efficient (sub-exponential), non-trivial measuring strategies, since, on the one hand, quantum trajectories seem to spread over the system on a non-exponential time scale, while, on the other hand, still corresponding to localized wavefunctions.
In order to investigate spatial steering, we consider the following setup. In a system of L sites we initialize the particle at site i 0 = 1 and specify a target site i target = L (using open boundary conditions). Again, we consider large times between two measurements ∆t ξ (W ), in order to avoid confining or repelling the particle through the Zeno effect. The goal is to design a measurement protocol that leads to a click measurement at the target site after as few measurements as possible. The time of arrival in a quantum lattice has also been studied in Refs. [18,35]. Hereby, we differentiate between two types of protocols: Adaptive protocols, which may use the readouts from every performed measurement in determining the position of the successive measurements; and blind protocols, which may only use the readout at the target site.
The simplest protocol one may think of is to just repeatedly measure the target site, until the particle is detected. Since this protocol requires readout only at the target site, it is a blind protocol with the termination policy employed. Based on the localized nature of the system, this protocol typically terminates after an exponentially large number of measurements N ∈ O(exp(L/ξ)), upon averaging over disorder realizations. As this protocol scales exponentially with the system size, it becomes impractical (also for numerical simulations), if L is of the order of a couple of localization lengths.
At the same time, there is a simple adaptive protocol, which is optimal in the sense that N o,p,d ∝ L. This protocol works as follows: 1. Place the detector at i 1 = i 0 + 1.
2. Measure this site, until the particle is detected.
3. Shift the detector by one site towards the target.

Repeat steps 2 -4.
Given that the expected number of measurements until the next click is finite, the total number of measurements scales linearly with the system size. For numerical simulations, we use a slightly improved version of this protocol: Instead of always measuring at a distance 1 from the site where the last click measurement took place, we randomly measure sites within one localization length of this site in the direction of the target site. This has two benefits: Local dips of the wave function as well as effects of preceding no-click measurements are avoided. Indeed, the left panel of Fig. 9 demonstrates that the simulated expected number of measurements for this protocol (blue dots) scales approximately linearly with the system size (black fit line). In this sense, ballistic transport is realized by this protocol.
Evidently, efficient steering is possible, if the readout is always accessible. In an experiment, however, this may not be the case. Therefore, we try to find a blind protocol with an expected number of measurements that behaves polynomially with the system size. As an attempt to improve the runtime of the blind protocol, we perform blind measurements along the chain at random locations until the particle is detected at the target site. As we already observed, this leads to approximately diffusive spread of the trajectories and is thus much more efficient than only measuring the target site. In simulations, we are able to steer the particle to a target site at a distance of several hundred localization lengths, see the right panel of We have thus demonstrated a possibility of efficiently manipulating (dragging) a particle subject to a random potential in a one-dimensional chain by means of measurement-induced steering, using both passive (blind) and active (adaptive) protocols. This type of steering can be further generalized to more sophisticated scenarios, as compared to simply moving the particle through the chain from one end to the other. In particular, one may envision manipulating several particles in the disordered background (not necessarily in a one-dimensional system) to exchange their positions and braid them by measurements.

VI. RELATION TO A CLASSICAL RANDOM WALK
In this section, we describe how dynamics in our measured system is related to a classical random walk model. The random walk picture is useful for several reasons. For a wide variety of random walks, asymptotic properties are known, allowing us to explain the long-time behaviour of our system. Furthermore, the random walk language offers a simplified description of the most important dynamical features of the system, which are much more difficult to calculate analytically when taking its full quantum nature into account. We discuss some of the implied simplifications after introducing the model.
On the level of the particle trajectories, the measuremen-induced dynamics indeed bears similarity to a classical random walk. Consider a set of states {i} with i ∈ [1, L], representing the sites of the system. Approximating the position of a trajectory by the nearest site and limiting our consideration to the discrete set of time points immediately after a measurement, every trajectory is described by transitions i → j with i, j ∈ [1, L]. A natural approach is to describe the ensemble of different particle trajectories in terms of transition matrices M (n) with n ∈ [1, N ], acting on a state ρ with ρ i (n) corresponding to the probability to find a particle on site i at time step n, and For the random walk description, we assume that the transitions are mediated by click events. Since on average every L-th measurement produces a click, one time step in Eq. (10) thus implies L steps of the measurement protocol.
At this point, we need to make some assumptions about the elements of M (n), incorporating localization into the classical picture through the statistics of these matrix elements. Localization implies that the transition probability should decrease exponentially with the distance. Therefore, we keep only distance-one transitions (which represent unit distance O(ξ) jumps) and consider a symmetrical nearest-neighbor transition matrix where the terms at the boundary should be written according to the respective boundary condition. In this model, the particle either remains on a given site i with waiting probability p i , or jumps with equal probabilitiesp i /2 to one of the two adjacent sites. The waiting probabilities are assumed to be time dependent and uncorrelated in time. The next step is to derive a distribution function P p (p) for the p i , based on the statistics of Anderson localized wave functions-this completes the incorporation of the localizing properties of the Hamiltonian time evolution into the model.
The probability distribution of the inverse localization length x := 1/ξ at a given energy E and disorder strenth W is given by a Gaussian distribution [25,34], with the normalization constant N x , variance σ 2 x , and mean µ x . For the toy model, we assume that the waiting probability p i is given by the center-site maximum of the localized probability density ρ i = N exp(−2|i−i 0 |/ξ) (the probability to have a zero-distance jump). This assumption provides a mapping between p and x: From x ∈ [0, ∞), it follows that waiting probabilities between 0 and 1 can be found: p ∈ [0, 1). The probability distribution for the p i at a given energy takes the following form: Since our time evolution protocol includes all eigenstates, we average over the band to obtain the waiting probability distribution with the density of states ν.
At this point, we briefly discuss some key simplifications involved in the development of the toy model: • We approximate all jumps over distances greater than zero by a unit-distance jump instead of incorporating the full distribution of possible distances or connecting the jump size to the jump probability via the corresponding localization length. Given the localization length of a wave function ξ, the probability of jumps over a distance x decreases exponentially with x/ξ. For this reason, approximating all jump distances belonging to one localization length by a unit distance is not expected to change the qualitative behavior. Considering jump of distance one for all localization lengths can only underestimate the spread of the random walker, since a non-zero jump implies a jump of distance one at least.
• The model is based on the statistics of the eigenstates of the Anderson Hamiltonian. However, the wavefunctions in the actual system result from time evolution of one-site states. The approximation is again justified by the exponential decay of the eigenstates, which implies that only O(ξ) sites overlap significantly with the original site. We replace these by the eigenfunction with the largest overlap. The correspondence becomes exact in the largedisorder limit. Note, however, that this correspondence can only be assumed if the time between two measurements is chosen, as in our simulations, to be ∆t ≥ξ.
• The effect of no-click measurements on the shape of the wave function is not taken into account by the model. Again, for the time between two measurements it holds that ∆t ≥ξ in our simulations. This means that no-click holes in the wavefunction have time to relax before the next measurement. We can incorporate corrections to this simplification on a phenomenological level through a modification of the localization length.
• Our model suggests uncorrelated onsite probabilities p i (n). In contrast, in our system, there is time correlation between the probabilities, since p i (n+1) can be calculated from p i (n). We assume these correlations to not change the qualitative behaviour, since the resulting random walk behaves ergodically.
Using the random-walk picture, we can now address the question of diffusion in ∆ at asymptotic times. The asymptotic behaviour of the random walk is governed by the behaviour of P p for p → 1, as sites with a waiting probability of almost one provide bottlenecks for the dynamics in the system. It is known that the spatial randomness in the potential can lead to anomalous transport [36], and it was shown by the authors of Ref. [37], that the corresponding random walk behaves diffusively, if lim p→1 P p (p)/(1 − p) is finite. For Eq. (15) and thus Eq. (16), we find that it falls off faster than any power law for p → 1, Thus, we expect ∆ ∝ t 1/2 in the asymptotic limit.
As explained in Sec. IV, this expectation is independent of the performed averages and holds true even if only the average over outcomes is performed. On intermediate time scales (N L, for ∆ ∈ O(ξ)), few sites have a large impact on the dynamics. This can lead to apparent sub-or even superdiffusive dynamics on these time scales. If a fraction of trajectories reaches a site with ξ eff >ξ (ξ eff <ξ), there is an increase (decrease) with time in the effective localization length (see Fig. 3 for an example). In the random-walk picture, this may be understood as a time-dependent jump distance distribution, which can raise or lower the local exponent of the power law in ∆.
Finally, we consider the measurement steering protocols in terms of random walk. Turning first to the blind protocol with random measurements, we can identify the average runtime of the protocol with the maximum expected hitting time of a random walk on a connected graph [38]. This is the number of steps which the random walker needs to take on average, to first arrive on the most distant site, and it scales as L 2 for a simple chain [38]. Since we need ∝ L measurements to induce one step, we find N ∝ L 3 , as seen in the numerics. For the adaptive protocol, the asymptotic behaviour of the waiting probability distribution confirms a linear relation between the system size and the expected number of measurements, since the moments of the waiting times on a site do not diverge. This also implies that there is a Gaussian probability distribution of the steering times for sufficiently large system sizes, in accordance with the central limit theorem.

VII. DISCUSSION AND CONCLUSIONS
We have investigated the dynamics of a single particle in a one-dimensional Anderson-localized system, subject to projective on-site measurements. Performing measurements at random locations, we have found that particle trajectories, driven by click measurements, spread approximately diffusively over the system. At the same time, the wavefunctions remain localized along individual quantum trajectories, but with a modified effective localization length IV. These findings suggest that efficient steering protocols can be formulated, moving the particle through the system to a predefined target site within a number of measurements, which is polynomial in the system size. We have demonstrated this in Sec. V. The average spread of localized entities suggests a random walk picture, which we have utilized in Sec. VI to confirm the long time diffusive nature of the trajectory spread, and to explain the relation between steering time and system size in the steering protocols.
For future studies, it would be interesting to consider a generalization to a many-body system in a disordered chain, which is time evolved with a similar projectivemeasurement protocol. First, quantum statistics of measured particles is expected to influence the dynamics of the system. Second, the effect of measurements may induce correlations between the particles, which could mimic a genuine interparticle interaction that leads to dephasing and many-body delocalization transition. In this respect, it is interesting to explore various types of measurement operators, in particular, those involving simultaneous measurement of two or more particles in a given state. The key question here can be formulated as follows: Is it possible to "measure" a manybody localization transition in a non-interacting disordered system that would have been Anderson-localized without measurement? Finally, while the problem of the measurement-induced entanglement transition was widely discussed for free fermionic chains (see Introduction), understanding the effect of Anderson localization on such a transition remains a challenging problem.
Note added: before submitting our manuscript for publication, we became aware of a preprint, [39] addressing monitored disordered free fermions.

VIII. ACKNOWLEDGEMENTS
Parts of this work were done in close interaction with Sthitadhi Roy. We are grateful to him for useful discussions and insights. We thank J. F. Karcher, A. D. Mirlin, I. V. Poboiko, and O. Yevtushenko for useful discussions.