Postselection in lattice bosons undergoing continuous measurements

Diego Barberena; Matthew P. A. Fisher

REVIEW 3 major objections 4 minor 103 references

Postselection in lattice bosons undergoing continuous measurements

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Continuously measured lattice bosons can be postselected using one or two per-site estimators instead of the full measurement record.

desk verdict Strong core result on postselection in monitored bosons; the near-continuum test has a parameter inconsistency and likely too short integration time. read the letter →

arxiv 2411.14582 v1 pith:PS4QMJH6 submitted 2024-11-21 quant-ph cond-mat.stat-mech

Diego Barberena , Matthew P. A. Fisher This is my paper · ORCID

classification quant-phcond-mat.stat-mech

keywords postselectioncontinuousmeasurementquantumtrajectorieslatticebosonsGaussianstatesestimatorstwo-pointcorrelatorscavityQED

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the postselection problem in a solvable model of lattice bosons under continuous quadrature measurement can be dramatically reduced. Instead of requiring identical full measurement records to build an ensemble of identical quantum states, the late-time conditional state depends on just one or two numbers per lattice site, the estimators. The authors show that connected two-point correlators, which are nonlinear in the conditional density matrix and invisible in the unconditional state, can be recovered by binning trajectories according to these estimators. They further show that the estimators can be constructed from information available in the unconditional dynamics or in the measurement record alone, and they verify the protocol in numerical experiments mimicking experimental conditions. A sympathetic reader would take away that efficient postselection is possible in this Gaussian bosonic model and that the structural features identified may guide mitigation in more general monitored systems.

What carries the argument

The central objects are the per-site estimators defined as spatiotemporal convolutions of the measurement record with filter kernels, $(\hat{x}_i)_{\rm est}(t)=2\sqrt{\Gamma}\sum_j\int_{-\infty}^{t} K_x(i-j,t-s)\,dI_j(s)$ and similarly for $\hat{p}_i$, with $K_x$ and $K_p$ given in Eq. (30) in terms of the momentum-space steady-state covariances $v_q$, $u_q$, $w_q$. These filters encode a memory time $\tau=(2\Gamma v^\infty_x)^{-1}$ and, on the lattice, a correlation length $\xi=\sqrt{J/[2(J_0-dJ)]}$, so that only the measurement record in a spacetime correlation volume around a site matters. The machinery is Gaussian Kalman-Bucy filtering: the conditional state is described by means and covariances, the means are linear functionals of the record at late times, and the estimators convert postselection on the full record into postselection on one or two numbers per site.

What would settle it

Take a one-dimensional lattice at $J_0 = 2.0025\Gamma$, $J=\Gamma$ so that $\xi=20$, evolve from vacuum for a fixed time $T=10\Gamma^{-1}$, and apply the binning protocol with $N_{\rm trial}=3\times 10^4$; compare the recovered $C^P(r)$ against the exact conditional covariance computed by directly integrating Eq. (26). If the recovered profile deviates from the exact profile by more than the statistical error bars, the steady-state-filter assumption has broken down. A sharper test increases $\xi$ (by moving $J_0$ closer to $2J$) at fixed $T$: the protocol should fail once the equilibration time $\sim \tau$ exceeds $T$.

Watch

Extended reading notes

Core claim

The paper establishes that in a lattice of bosons with local continuous measurements of the $\hat{x}_i$ quadrature, the conditional quantum state at late times is fully determined by per-site linear estimators of the measurement record: $(\hat{x}_i)_{\rm est}$ and $(\hat{p}_i)_{\rm est}$ given by Eq. (29) with time-translation-invariant filter kernels $K_x(i-j,t-s)$ and $K_p(i-j,t-s)$ from Eq. (30). Because these estimators capture the full dependence of the state on the record, connected two-point functions $C^X_{ij}$ and $C^P_{ij}$, which are nonlinear in the conditional density matrix, can be obtained by binning trajectories on the estimators rather than on the entire measurement history. The filter kernels can be derived analytically from the steady-state covariances $v_q$, $u_q$, $w_q$, and the paper demonstrates numerically that the same kernels can be inferred from the record-record and system-record correlations of the unconditional dynamics. The protocol recovers the exponentially decaying spatial profile of $C^P(r)$ that is absent in the unconditional correlators, using only experimentally accessible data and a few tens of thousands of repetitions.

Load-bearing premise

The protocol assumes that the covariance matrices reach their unique Gaussian steady state within the observation time, so the time-translation-invariant filter kernels derived from steady-state covariances are valid over the entire postselected evolution; this requires $h_q = J_0 - J\sum_\mu \cos(q_\mu) > 0$ for all $q$, i.e. $J_0 > dJ$.

Editorial extensions

If this is right

Connected two-point functions of monitored quantum trajectories become experimentally accessible with roughly $10^4$ repetitions, instead of the exponentially many repetitions required to reproduce a full measurement record.
The filters needed for postselection can be designed from the unconditional dynamics, specifically from record-record and system-record correlation functions, without solving the conditional evolution.
The same set of measured trajectories can be reused to recover all two-point correlators $C^P_{ij}$ by rebinning according to the estimators at different site pairs.
Locality in time and space emerges dynamically: only the measurement record within a memory time and correlation length of a site is relevant, which reduces the postselection overhead in extended systems.
In the continuum limit the filter kernels exhibit ballistic light-cone-like structure with power-law tails, providing a concrete classical postprocessing rule for near-critical parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The estimator framework suggests that postselection can be reformulated as a classical regression problem: if good filters can be learned from unconditional response functions, the same binning idea may apply to non-Gaussian monitored systems where the state-to-estimator map is not linear but still low-dimensional.
Near the critical point $J_0 = dJ$, the diverging correlation length and memory time imply that the steady-state filter assumption fails at fixed observation time; a testable prediction is that the recovered correlators will be biased when $\xi$ exceeds the system size or the observation window.
The cavity-QED and circuit-QED implementations proposed in the paper could be used to probe the robustness of the estimator protocol against finite detection efficiency and nonlinearities, with the expectation that imperfect filters produce systematically biased estimates of $C^P$.
If this method extends to area-law phases of more general monitored circuits, it would convert a fundamental obstruction into a practical signal-processing task, with the measurement record itself acting as a classical shadow of the conditional state.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read it. The central result is real: for a Gaussian boson lattice with continuous quadrature measurement, they derive explicit space-time filter kernels and show that conditional correlators can be recovered by binning trajectories on one or two estimators per site instead of the full record. The numerical experiment in Fig. 8, with 3e4 runs, reproduces the analytic profile at J0=3Γ. That is a genuine step toward mitigating postselection in a solvable, experimentally relevant class.

The construction of the filters from unconditional dynamics or record correlations alone (Section III B) is the most valuable part; it separates the protocol from the exactly-solvable model. The experimental sections are thoughtful, with plausible cavity-QED and circuit-QED routes.

Now the soft spots. The near-continuum test in Fig. 10(b) is not as clean as the text claims. The paper states J0=2.0025Γ, J=Γ, and ξ=20, but Eq. (34) gives ξ≈0.7 for those parameters. One of those numbers is wrong. If ξ=20 is intended, then the memory time is τ≈28/Γ, and the numerical run at T=10/Γ has not equilibrated; the steady-state filter kernels are not yet valid, and the gray comparison curve is from the J0=3Γ run, not the analytic profile at the new parameters. So the recovery of the decaying profile in that regime is not established. This is fixable by extending T and/or providing the direct comparison, and it does not affect the core J0=3Γ result.

The unproved spatial-averaging assumption in Section IV C and the heuristics in Appendix C are flagged by the authors; they are real but secondary. The paper's own limitations are mostly stated honestly.

Who is this for: people working on measurement-induced dynamics, quantum trajectories, and postselection in bosonic/continuous-measurement systems. It deserves serious refereeing. I would suggest the referee focus on the parameter inconsistency and the equilibration condition, and ask for a direct analytic comparison in the continuum regime.

Referee Report

3 major / 4 minor

Summary. The paper studies a one-dimensional lattice of bosons with local continuous quadrature measurements, a Gaussian model that is exactly solvable. It shows that, after the covariance matrices reach their steady state, the conditional means at each site are determined by linear filtered versions of the measurement record (Eq. (29) with kernels Eq. (30)). The authors exploit this to reduce postselection: instead of conditioning on the full measurement record, one can bin trajectories by one or two scalar estimators per site and still recover connected two-point correlators C^X and C^P that are inaccessible from the unconditional state. The paper derives the filters analytically, shows how the same filters can be estimated from the unconditional dynamics or from record correlations, verifies the protocol numerically in a way that mimics experimental access, discusses the continuum or large-correlation-length regime, and proposes cavity-QED and circuit-QED implementations. Several limitations are acknowledged in the text, including the unproved spatial-averaging assumption in Section IV C and the heuristic cancellation of divergent terms in Appendix C.

Significance. If the claims hold, this is a valuable exactly solvable example in which the postselection barrier for local observables is explicitly broken: the conditional state depends on the record only through a few estimators, and those estimators can be constructed from experimentally accessible data. The analytic derivation is careful, the numerical protocol in Figure 8 matches the analytic steady-state profile at J0=3Γ, and the paper is unusually explicit about what is assumed and what is demonstrated. The authors correctly distinguish between observables that are linear in the conditional state and genuine nonlinear correlators, and they provide concrete experimental implementations. The central result at J0=3Γ is convincing. However, the near-continuum numerical demonstration and two admitted gaps—the divergent-term cancellation in Appendix C and the spatial-averaging assumption in Section IV C—prevent the paper from being fully self-contained in its broader claims.

major comments (3)

[§IV B, final paragraph; Fig. 10(b)] The claim that the postselection protocol "can still recover the decaying spatial profile" for ξ=20 is not substantiated by the data shown. At J0=2.0025Γ and J=Γ, the observation time T=10Γ^{-1} is shorter than the memory/equilibration time: from Section III C the slowest covariance relaxation is τ/2, and with τ≈ξ√(2/(ΓJd)) one obtains τ≈28Γ^{-1} at ξ=20, so equilibration requires about 14Γ^{-1}. The steady-state kernels in Eq. (30) therefore need not be valid at T=10Γ^{-1}. Moreover, the gray line in Fig. 10(b) is the J0=3Γ profile from Fig. 8(b), not the analytic steady-state profile Eq. (32) for the simulated parameters, so the red points are not compared to the correct curve. Please rerun the near-continuum test with T substantially larger than τ, or explicitly benchmark the time-dependent conditional covariance at the observation time, and compare against Eq. (32) at the same J0, J, and Γ.
[Appendix C, Eqs. (C4)-(C11)] The derivation of the filter from the record-record correlator assumes, without proof, that the time-non-invariant and ∝T terms in the record-record correlator cancel against analogous contributions on the right-hand side of Eq. (15); the text says "hoping that the diverging terms will cancel against a similar contribution on the right-hand side of Eq. (15)". Because this is one of the routes used to establish that the filter can be obtained from the record correlation alone, this step should be made rigorous or explicitly regularized. Please provide a well-defined T→∞ limit or prove that the decaying solution of Eq. (C11) with Eq. (C8) is the unique filter following from the minimization problem in Section III B.
[§IV C, step 2] The single-sample protocol rests on replacing measurement-realization averages by spatial averages over regions separated by more than a few correlation lengths. The manuscript states in step 2 that "Although we do not present a formal proof, we justify this assumption by arguing that spatial regions separated by more than few correlations lengths are uncorrelated in practical terms." Since this assumption is load-bearing for the "application to single samples" discussion and for the claimed phase-of-matter framing, please either provide numerical evidence of convergence at the system sizes used, or explicitly demote this to a conjecture with the required separation length identified as an open condition.

minor comments (4)

[§IV, first paragraph] The condition "J0 > J2" should be written as "J0 > dJ", and the later phrase "J0 → J + 2" should be replaced by the correct d-dimensional expression (or "J0→J+2" only after specifying d=1).
[Fig. 9(b) caption] The notation "fixed p ΓJ d/2 T* = 10" is unclear; please write √(ΓJd/2) T* = 10 or equivalent, matching the exponent in Eq. (35).
[Fig. 7(c) and surrounding text] The statement that the correlators decay with "a correlation length of the order of the lattice size" would be clearer if the value of ξ from Eq. (34) for J0=3Γ, J=Γ were quoted explicitly, since the text later defines ξ and discusses the continuum limit.
[§III B, Eq. (19)] The sentence "This is enough to determine f(t)" relies on an implicit boundary-condition argument for the fourth-order differential equation; please spell out why the exponentially growing solutions are discarded and why the two normalization conditions in Eq. (20) give a unique filter.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the analytic derivation is self-contained and the numerical recovery tests use independently derived analytic profiles as benchmarks.

full rationale

The derivation chain is self-contained. Section III solves the single-site stochastic Schrödinger equation and derives the estimator filter Eq. (10) directly from the exact conditional dynamics; Eqs. (12)-(20) then show that the same filter can be obtained by minimizing a cost functional built only from record-record and system-record correlations, not from the target variance. Section IV generalizes this machinery: Eq. (26) gives the deterministic covariance evolution, Eq. (30) is obtained analytically from the steady-state covariance parameters (v_q, u_q, w_q), and Eq. (32) independently defines the target C^X and C^P profiles. The binning protocol in Section IV A uses only sampled p-values and record-derived estimators, and Fig. 8 compares the recovered profile against Eq. (32), so the target correlator is not fed into the filter construction. The paper's self-citations (e.g., Refs. [22,25,42,47,57]) are contextual and non-load-bearing; the filtering methodology rests on standard external Kalman-Bucy and quantum filtering references. The skeptical concern about the near-continuum test in Fig. 10(b), where T = 10/Gamma may be shorter than the memory time tau ~ 28/Gamma at xi = 20, is a parameter-regime/correctness issue rather than a circularity: the steady-state-kernel assumption may or may not be satisfied there, but the claimed equivalence between estimator-based postselection and the conditional correlator is not an identity of inputs and outputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Gaussian-state filtering theory, the stability of the covariance fixed point, and two domain assumptions (spatial averaging as ensemble averaging, and the continuum small-q expansion). No new particles or fields are introduced. The model parameters J0, J, Gamma, h0 are physical inputs, not fitted constants.

assumptions (5)

standard math Quadratic bosonic Hamiltonians with linear continuous measurements preserve Gaussian states; all moments beyond second order vanish.
Used throughout Sections III and IV to close the equations of motion for means and covariances; valid because the generator is quadratic and initial states, including non-Gaussian ones, become Gaussian at late times (Section III E).
standard math The optimal estimator for a Gaussian linear system is a linear filter minimizing a quadratic cost, with the filter determined by record-record and record-state correlations.
Used in Section III B to derive the filter equations (15)-(20), citing Refs. [58,69,70].
domain assumption The covariance equations (7) or (26) have a unique stable fixed point and equilibration is exponential with a rate set by the memory time.
Used to treat the covariances as constant at late times and to write time-translation-invariant filters. The existence of the fixed point is shown in Section III C; reaching it within the observation time is assumed for the finite-time protocol.
domain assumption In a single sample, spatial averages over regions separated by more than the correlation length can replace averages over measurement realizations.
Stated in Section IV C step 2 without formal proof: 'we do not present a formal proof, we justify this assumption by arguing that spatial regions separated by more than few correlations lengths xi are uncorrelated in practical terms.'
domain assumption For xi >> 1, the q-integral for the filter is dominated by small q, giving u_q approximately 1/2 and the expansions in Eq. (33).
Used to derive the continuum form of K_p (Eq. (35)) and the power-law profile; valid in the limit J0 approaching dJ.

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Cite this review

Pith. "Pith review of Postselection in lattice bosons undergoing continuous measurements." pith.science (2026). https://pith.science/paper/PS4QMJH6

@misc{pith2026241114582,
  author       = {Pith},
  title        = {Pith review of: Postselection in lattice bosons undergoing continuous measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PS4QMJH6}},
  note         = {Machine review of arXiv:2411.14582}
}

read the original abstract

We study in detail the postselection problem in a specific model: bosons hopping on a lattice subjected to continuous local measurements of quadrature observables. We solve the model analytically and show that the postselection overhead can be reduced by postprocessing the entire measurement record into one or two numbers for each trajectory and then postselecting based only on these numbers. We then provide a step-by-step protocol designed to recover connected two-point functions of the quantum trajectories, which display an exponentially decaying profile that is not observable in the unconditional, trajectory averaged, state. With the analytical solution in hand, we analyse the features of this postprocessing stage with the intention of abstracting away the properties that make postselection feasible in this model and may help in mitigating postselection in more general settings. We also test the protocol numerically in a way that utilizes only experimentally accessible information, showing that various quantum trajectory observables can be recovered with a few repetitions of the numerical experiment, even after including inevitable coarse-graining procedures expected under realistic experimental conditions. Furthermore, all the information required to design the postprocessing stage is independently present both in the unconditional dynamics and also in the measurement record, thus bypassing the need to solve for the conditional evolution of the model. We finalize by providing experimental implementations of these models in cavity-QED and circuit-QED.

Figures

Figures reproduced from arXiv: 2411.14582 by the authors.

**Figure 2.** FIG. 2. Husimi probability distributions [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Variance and (b) mean of ˆx [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Husimi quasiprobability distributions [ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Flowchart of the protocol designed to recover the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: We used Ntrial = 104 realizations, beginning from the quantum state |0⟩+|5⟩ and evolving the system from t = 0 to t = 10Γ−1 at Γ = h0 using Eq. (4). We sample xmeas directly from the quantum state at the end of each run and discard the state, leaving us only with xmeas…

**Figure 6.** Figure 6: FIG. 6. Results of the numerical implementation of the postselection procedure as outlined in section III D after evolving the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) Variance of ˆp [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8. (a) Binning used to recover [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9. (a) Filter [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: FIG. 10. (a) Schematic depiction of the correlation volume [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: FIG. 11. Schematics of the proposed implementations. (a) [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

103 extracted references · 38 canonical work pages

[1]

Run the experiment once to obtain a single copy of the quantum state ˆρ and a single realization of the record {dI}
[2]

filtered

Blue lines are directly calculated from the time-evolved quantum state [using Eq. (4)], black lines or dots are the analytical formulas given in Eq. (9) and Eq. (10), and orange lines represent the unconditional results of Eq. (11). where we have used Eq. (5) to eliminate dW in favor of dI because, as mentioned before, experiments only have access to dI. ...
[3]

Sample a single value xmeas by measuring the ob- servable ˆx in the quantum state ˆρ, with associated probability distribution P (x) = ⟨x|ˆρ|x⟩ (|x⟩ is an eigenstate of ˆx)
[4]

The output of the last two steps is the pair of numbers ( xmeas, xest)

Construct xest using the specific {dI} obtained in this iteration of the experiment. The output of the last two steps is the pair of numbers ( xmeas, xest)
[5]

Repeat the experiment Ntrial times to obtain Ntrial pairs (xr meas, xr est) for r = 1, ...Ntrial
[6]

Since ˆx is a continuous variable there will necessarily be some coarse-graining

Bin the data according to the values of xr est. Since ˆx is a continuous variable there will necessarily be some coarse-graining. The bin number and size should be chosen such that in most bins there are enough data points to do statistical averages with low sampling error
[7]

This gives us vbin x

Calculate ⟨ˆx⟩ and ⟨ˆx2⟩ in each bin using the experi- mentally measured xmeas and doing statistics. This gives us vbin x . Because vbin x depends on the measure- ment record only through ⟨ˆx⟩ ∼xest, this binning procedure allows us to access the quantum trajec- tory value of vx. Referring back to Fig. 4, each bin corresponds to a postselected distributio...
[8]

For this gaussian system vbin x is bin independent and equal to vx, but this need not be the case for more general models

Finally, we average vbin x over all bins to obtain vx. For this gaussian system vbin x is bin independent and equal to vx, but this need not be the case for more general models. We depict a flow chart of this procedure in Fig. 5. We also test this protocol numerically and show the results in Fig. 6. We used Ntrial = 104 realizations, beginning from the qu...

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(c) Equilibrium profile of the conditional correlators C X,P r ≡ C X,P i,i+r as a func- tion of separation between the operators

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Run the experiment once to obtain a single copy of the quantum state ˆρ and a single realization of the record {dIi}
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They all commute, so this is allowed

Sample a single value ( pi)meas for each lattice site i by measuring all the observables ˆpi in the quantum state ˆρ. They all commute, so this is allowed
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[16]

Repeat the experiment Ntrial times to obtain Ntrial collections of {(pi)r meas, (pi)r est} for r = 1, ...Ntrial
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This results in a two dimensional binning procedure (except for the onsite variance), which is slightly more intensive than the procedure for a single site, but not unmanageable

If we want to recover C P 1,2, for example, we then bin the data according to the values of ( p1)r est and (p2)r est. This results in a two dimensional binning procedure (except for the onsite variance), which is slightly more intensive than the procedure for a single site, bu...
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S. Sang and T. H. Hsieh, Measurement-protected quan- tum phases, Phys. Rev. Res. 3, 023200 (2021). Appendix A: Single site gaussian dynamics In this Appendix we begin from the stochastic Schr¨ odinger equation dˆρ = −i h0ˆa†ˆa, ˆρ dt + Γ ˆxˆρ ˆx − 1 2 {ˆx2, ˆρ} dt + √ Γ ˆxˆρ +...

2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Run the experiment once to obtain a single copy of the quantum state ˆρ and a single realization of the record {dI}

[2] [2]

filtered

Blue lines are directly calculated from the time-evolved quantum state [using Eq. (4)], black lines or dots are the analytical formulas given in Eq. (9) and Eq. (10), and orange lines represent the unconditional results of Eq. (11). where we have used Eq. (5) to eliminate dW in favor of dI because, as mentioned before, experiments only have access to dI. ...

[3] [3]

Sample a single value xmeas by measuring the ob- servable ˆx in the quantum state ˆρ, with associated probability distribution P (x) = ⟨x|ˆρ|x⟩ (|x⟩ is an eigenstate of ˆx)

[4] [4]

The output of the last two steps is the pair of numbers ( xmeas, xest)

Construct xest using the specific {dI} obtained in this iteration of the experiment. The output of the last two steps is the pair of numbers ( xmeas, xest)

[5] [5]

Repeat the experiment Ntrial times to obtain Ntrial pairs (xr meas, xr est) for r = 1, ...Ntrial

[6] [6]

Since ˆx is a continuous variable there will necessarily be some coarse-graining

Bin the data according to the values of xr est. Since ˆx is a continuous variable there will necessarily be some coarse-graining. The bin number and size should be chosen such that in most bins there are enough data points to do statistical averages with low sampling error

[7] [7]

This gives us vbin x

Calculate ⟨ˆx⟩ and ⟨ˆx2⟩ in each bin using the experi- mentally measured xmeas and doing statistics. This gives us vbin x . Because vbin x depends on the measure- ment record only through ⟨ˆx⟩ ∼xest, this binning procedure allows us to access the quantum trajec- tory value of vx. Referring back to Fig. 4, each bin corresponds to a postselected distributio...

[8] [8]

For this gaussian system vbin x is bin independent and equal to vx, but this need not be the case for more general models

Finally, we average vbin x over all bins to obtain vx. For this gaussian system vbin x is bin independent and equal to vx, but this need not be the case for more general models. We depict a flow chart of this procedure in Fig. 5. We also test this protocol numerically and show the results in Fig. 6. We used Ntrial = 104 realizations, beginning from the qu...