Feedback and Continuous Measurement

A feedback simulation must evolve both the physical measurement record and the conditional state used by the controller. Reusing the wrong noise increment or allowing the controller access to future data can produce artificial stabilization. The algorithm therefore needs an explicit causal ordering.

Helpful background. Stochastic equations establishes Itô increments; density operators supplies state checks.

Measurement record and conditional filter

Let $c$ be a monitored collapse operator, $\eta\in[0,1]$ the detection efficiency, and $H$ the Hamiltonian in energy units. Include the square roots of decay rates in all collapse operators. The unconditional generator is

$$\mathcal L(\rho)=-\frac{i}{\hbar}[H,\rho] +\mathcal D[c]\rho+\sum_j\mathcal D[L_j]\rho, \qquad \mathcal D[L]\rho=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\},$$

where $L_j$ describes any additional unobserved channels. In a homodyne convention,

$$\begin{aligned} d\rho_c&=\mathcal L(\rho_c)\,dt +\sqrt\eta\,\mathcal H[c]\rho_c\,dW,\\ dY&=\sqrt\eta\,\operatorname{Tr}[(c+c^\dagger)\rho_c]\,dt+dW,\\ \mathcal H[c]\rho&=c\rho+\rho c^\dagger -\operatorname{Tr}[(c+c^\dagger)\rho]\rho. \end{aligned}$$

The dissipator $\mathcal D[c]$ remains in $\mathcal L$ even at $\eta=0$. Turning off observation does not turn off environmental disturbance. The normalized noise innovation is $dW=dY-\sqrt\eta\,\langle c+c^\dagger\rangle_cdt$.

A simulated physical state and an estimator with imperfect parameters may differ. Generate the record from the physical state, then drive the estimator with that record and its own predicted mean. Giving both independent innovations makes the estimator solve a different experiment.

Causal finite-step control

For a bounded controller $u_n=f(\rho_{{\rm est},n})$ computed from the record through time $t_n$, set $H_n=H_0+u_nF$. Then:

1. Propagate the physical state and generate $\Delta Y_n$ under the chosen measurement model.
2. Update the estimator using the same $\Delta Y_n$.
3. Compute $u_{n+1}$ for the next interval, including any physical delay or bandwidth filter.

This state-based feedback is predictable at the start of each interval. Instantaneous feedback proportional to the raw current $\Delta Y/\Delta t$ is singular as $\Delta t\to0$; its stochastic correction must be derived consistently. It cannot be substituted into an ordinary deterministic Hamiltonian update without analysis.

Use a positivity-preserving measurement update when available, or monitor minimum eigenvalues carefully in an Euler scheme. Renormalizing the trace does not guarantee a physical conditional state.

Qubit measurement without control

Set $c=\sqrt\kappa Z$, $H=0$, and start at $|+\rangle$. The record has mean increment $2\sqrt{\eta\kappa}\,z\,dt$. The filter obeys

$$dz=2\sqrt{\eta\kappa}(1-z^2)\,dW.$$

For $\eta=1$, conditional purity stays one in the continuous limit; for $\eta=0$, the state follows deterministic dephasing and loses purity. In both cases the ensemble coherence decays as $e^{-2\kappa t}$. These limits separate detector efficiency from the dephasing rate.

Only after these tests pass should control be introduced. Compare feedback with the same initial ensemble and noise model under zero control, and report control amplitude, delay, and bandwidth. An improvement obtained with unlimited instantaneous control may disappear when realistic delay and amplitude limits are included.

An exact finite-bin QND update

For the undriven qubit above, constant $\kappa$ and $\eta$, and no other channels, the quantum nondemolition (QND) measurement admits an exact finite-step instrument. Let $h$ be the bin duration, $y$ the integrated record, and $p_s=\langle s|\rho|s\rangle$ for $Z|s\rangle=s|s\rangle$, $s=\pm1$. The record density is

$$p(y)=\sum_{s=\pm1}p_s \frac{\exp[-(y-2s\sqrt{\eta\kappa}\,h)^2/(2h)]}{\sqrt{2\pi h}}.$$

To draw $y$, sample an auxiliary label $s$ with probabilities $p_s$, then a Gaussian with mean $2s\sqrt{\eta\kappa}\,h$ and variance $h$. This label is a sampling device for the mixture; the conditional state is computed from the full record likelihood, not projected onto that label.

Define the diagonal likelihood operator and unobserved dephasing map by

$$D_y=\exp[\sqrt{\eta\kappa}\,Zy-\eta\kappa h\,I], \qquad [\Phi_{\rm unobs}(\rho)]_{01} =e^{-2(1-\eta)\kappa h}\rho_{01},$$

with $\Phi_{\rm unobs}$ leaving populations unchanged. Update

$$\rho'=\frac{\Phi_{\rm unobs}(D_y\rho D_y)} {\operatorname{Tr}(D_y\rho D_y)}.$$

The physical instrument has Kraus density $M_y=p_0(y)^{1/2}D_y$, where $p_0(y)=e^{-y^2/(2h)}/\sqrt{2\pi h}$, followed by the unobserved dephasing channel. Indeed, $\int M_y^\dagger M_y\,dy=I$ and $p(y)=p_0(y)\operatorname{Tr}(D_y\rho D_y)$. The common Gaussian factor cancels only in the normalized update, not in the probability law. Each outcome map is completely positive, so the conditional state stays positive at any bin size.

Integrating over $y$ gives the exact total coherence factor $e^{-2\kappa h}$: observed measurement contributes $e^{-2\eta\kappa h}$ and unobserved dephasing contributes the remainder. At $\eta=1$ the conditional update preserves a pure input; at $\eta=0$ the record is independent noise and the update is deterministic dephasing. For a general input with $|z|<1$, its posterior population difference is $z'=\tanh[\operatorname{atanh}z+2\sqrt{\eta\kappa}\,y]$. These are stringent checks on positivity, efficiency factors, and sampling.

A noncommuting drive breaks this exact QND reduction. Alternating a unitary step under the frozen predictable control with this measurement step gives a positive approximation, but its record law and conditional evolution still require time-step convergence. General positivity-preserving filtering schemes are developed by Rouchon and Ralph (2015).

Statistical and numerical checks

The continuous innovation, obtained by subtracting the time integral of the evolving conditional mean from the record, is a Wiener process for a correct filter. A finite-bin residual formed with the mean frozen at the bin start agrees only to leading order. In the exact QND benchmark,

$$\begin{aligned} \mathbb E[y\mid\rho_n]&=2\sqrt{\eta\kappa}\,z_nh,\\ \operatorname{Var}(y\mid\rho_n)&= h+4\eta\kappa h^2(1-z_n^2). \end{aligned}$$

The $h^2$ correction follows from the variance of the two Gaussian means; subtracting a fixed predicted mean does not remove it. Check zero conditional mean and absence of predictable correlations, and compare variance with the finite-step observation model actually used. Persistent excess correlations can indicate a wrong model, an omitted detector filter, or a timing error.

Refine time steps at fixed control bandwidth and physical delay; do not shrink those physical parameters together with the numerical step. Estimate target infidelity and control expenditure across independent records. A single successful trajectory can conceal a low-probability destabilization.

Exercise

For $\eta=0$, why should feedback based solely on the observed record fail to extract state information?

Solution

The record becomes $dY=dW$ and the stochastic term in the conditional filter vanishes. Any controller driven by that record receives noise independent of the system. It may still enact predetermined or random control, but cannot gain measurement-based state information. A simulator showing record-dependent purification in this limit has coupled noise or efficiency incorrectly.

References

• Jacobs, K., and Steck, D. A., “A straightforward introduction to continuous quantum measurement,” Contemporary Physics 47, 279–303 (2006), arXiv:quant-ph/0611067.
• Rouchon, P., and Ralph, J. F., “Efficient quantum filtering for quantum feedback control,” Physical Review A 91, 012118 (2015), doi:10.1103/PhysRevA.91.012118.
• Wiseman, H. M., and Milburn, G. J., Quantum Measurement and Control, Cambridge University Press (2010), doi:10.1017/CBO9780511813948 — measurement-based feedback and stochastic timing conventions.