Quantum Filtering

Quantum filtering is the recursive estimation of a quantum state from the measurement record available up to the present time. In continuous measurement it is the rule that turns a stream of detector data into the conditional state $\rho_c(t)$ used for prediction, feedback, parameter estimation, and trajectory interpretation.

The shortest definition is:

$$\rho_c(t) = \rho(t\mid\mathcal Y_t),$$

where $\mathcal Y_t$ denotes the classical information generated by the record up to time $t$. The notation is deliberately conditional. A filter is not the unconditional density matrix, and it is not a hidden trajectory independent of the measurement. It is the state assignment justified by a specified measurement model and the record actually retained.

Quantum filtering is the continuous-time analogue of Bayesian filtering in classical probability. The analogy is useful, but incomplete: a quantum measurement changes the state through an instrument, and the measured observables at different times need not be jointly classical random variables.

What the Filter Does

At each time, a filter performs two logically distinct steps:

• prediction: propagate the previous conditional state through known dynamics;
• update: condition on the new measurement increment.

In a laboratory this distinction is practical. Between digitizer samples, the state evolves under Hamiltonian dynamics, damping, dephasing, drives, and unmonitored loss. At a sample time, the new voltage increment, quadrature increment, or count updates the state according to the calibrated measurement model.

The filter therefore connects four objects:

model + previous state + new record increment -> updated conditional state

The same mathematical object appears under several names: conditional state, information state, filtered state, a posteriori state, or quantum trajectory conditioned on the observed record. The preferred term depends on community and context.

For the record conventions themselves, see Measurement Records. For the stochastic differential equations usually used to write the filter, see Stochastic Master Equations.

Discrete Instrument Form

The cleanest starting point is a finite time step. Suppose $\rho_{k|k}$ is the state conditioned on all records through bin $k$. First propagate it through a trace-preserving prediction map $\mathcal P_{\Delta t}$:

$$\rho_{k+1|k} = \mathcal P_{\Delta t}(\rho_{k|k}).$$

If the next outcome is $r_{k+1}$, represented by an instrument operation $\mathcal I_{r_{k+1}}^{\Delta t}$, the unnormalized updated state is

$$\tilde\rho_{k+1|k+1} = \mathcal I_{r_{k+1}}^{\Delta t} \!\left(\rho_{k+1|k}\right).$$

The likelihood of that record increment is its trace:

$$p(r_{k+1}\mid\mathcal Y_k) = \operatorname{Tr} \!\left[ \mathcal I_{r_{k+1}}^{\Delta t} \!\left(\rho_{k+1|k}\right) \right].$$

After normalization,

$$\rho_{k+1|k+1} = \frac{\tilde\rho_{k+1|k+1}} {\operatorname{Tr}(\tilde\rho_{k+1|k+1})}.$$

This is the quantum version of the prediction-update cycle. The probability law of the record and the state update are not separate assumptions; both come from the same instrument.

Relation to Bayesian Updating

For a classical hidden variable $x$ and outcome $r$, Bayes’ rule says

$$p(x\mid r) = \frac{p(r\mid x)p(x)} {\sum_{x'}p(r\mid x')p(x')}.$$

For a quantum measurement with Kraus operators $M_r$, the corresponding selective update is

$$\rho_r = \frac{M_r\rho M_r^\dagger} {\operatorname{Tr}(M_r^\dagger M_r\rho)}, \qquad p(r\mid\rho) = \operatorname{Tr}(M_r^\dagger M_r\rho).$$

The denominator is a likelihood, just as in Bayes’ rule. The numerator, however, is not merely multiplication by a classical likelihood function. It also contains the transformation $M_r\rho M_r^\dagger$, which represents measurement backaction and coherence changes.

This is the essential distinction:

$$\text{classical filtering updates a probability distribution;} \qquad \text{quantum filtering updates a state through an instrument.}$$

When the measured quantities commute and the measurement is nondemolition in the relevant basis, quantum filtering can reduce to a classical Bayesian filter over pointer states. In general it does not.

Diffusive Filtering

For a homodyne-like record, a common normalized convention is

$$dY_t = \mu_t\,dt + dW_t,$$

where $dW_t$ is a Wiener increment and the predicted signal is

$$\mu_t = \sqrt{\eta}\, \operatorname{Tr} \!\left[ (c+c^\dagger)\rho_c(t) \right].$$

The innovation is the surprise in the record:

$$dW_t = dY_t-\mu_t\,dt.$$

Written in innovation form, the corresponding filter is

$$\begin{aligned} d\rho_c =& \mathcal L\rho_c\,dt \\ &+ \sqrt{\eta}\, \mathcal H[c]\rho_c \left(dY_t-\mu_t\,dt\right), \end{aligned}$$

with

$$\mathcal H[c]\rho = c\rho+\rho c^\dagger - \operatorname{Tr} \!\left[ (c+c^\dagger)\rho \right]\rho.$$

This is the same physical equation as the usual diffusive stochastic master equation, but the filtering form emphasizes that the state is driven by the mismatch between observed and predicted signal.

In a correctly specified model,

$$\mathbb E[dW_t\mid\mathcal Y_t]=0, \qquad \mathbb E[dW_t^2\mid\mathcal Y_t]=dt.$$

This property is not just formal. Innovation residuals are a practical diagnostic: systematic drift, wrong variance, or residual correlations usually signal an incorrect measurement strength, local-oscillator phase, detector efficiency, bandwidth model, Hamiltonian, or noise calibration.

For homodyne and heterodyne detector models, see Homodyne Detection and Heterodyne Detection. For the corresponding trajectory equations, see Diffusive Trajectories.

Counting Filtering

For photon counting, the record is a counting process $N_t$. In a simple ideal model with monitored collapse operator $c$, the conditional intensity is

$$\lambda_t = \eta\, \operatorname{Tr}(c^\dagger c\rho_c(t)).$$

The filter predicts

$$\mathbb E[dN_t\mid\mathcal Y_t] = \lambda_t\,dt.$$

The counting innovation, often called a compensated counting increment, is

$$dM_t = dN_t-\lambda_t\,dt,$$

with conditional mean zero. A click gives a large update, while a no-click interval also gives information because it says that no event occurred when the current state predicted some rate.

For unit efficiency and no dark counts, the click update is

$$\rho_c \longmapsto \frac{c\rho_c c^\dagger} {\operatorname{Tr}(c^\dagger c\rho_c)}.$$

Dark counts, finite efficiency, dead time, and unresolved detector channels modify the instrument and therefore modify the filter. It is not enough to keep the ideal jump equation and add a detector note in prose; the likelihood and update must match the record actually being used.

For the record model, see Photon Counting. For the conditioned dynamics, see Quantum Jump Trajectories.

Filtering, Prediction, and Smoothing

Three related tasks are often confused:

• filtering estimates the present state using records up to the present;
• prediction uses the present filtered state to estimate future records or future states;
• smoothing uses later records as well, usually to infer a past state, trajectory feature, event time, or parameter.

In notation:

$$\begin{array}{lll} \text{filtering:} & \rho(t\mid\mathcal Y_t), & \text{uses records through }t, \\ \text{prediction:} & \rho(t+s\mid\mathcal Y_t), & \text{uses records through }t, \\ \text{smoothing:} & \rho(t\mid\mathcal Y_T), & \text{uses records through }T,\ T\gt t. \end{array}$$

Smoothing can be valuable for offline data analysis, but it is not available to a real-time controller at time $t$. A feedback law that uses smoothed information is acausal unless implemented with a delay and interpreted accordingly.

Unobserved Channels and Mixed Conditional States

If every relevant environmental channel is monitored with unit efficiency, a pure initial state can often remain pure along a trajectory. Real experiments rarely meet this ideal. Loss, inefficient detectors, unmonitored decay channels, thermal noise, detector bandwidth, and classical technical noise all remove information from the observer.

A common model splits one physical channel into monitored and unmonitored parts:

$$c_{\mathrm{mon}} = \sqrt{\eta}\,c, \qquad c_{\mathrm{loss}} = \sqrt{1-\eta}\,c.$$

The total unread dissipator is unchanged:

$$\mathcal D[c_{\mathrm{mon}}]\rho + \mathcal D[c_{\mathrm{loss}}]\rho = \mathcal D[c]\rho.$$

Only the monitored part contributes to the record. The lost part still decoheres or damps the system, but the observer cannot condition on its specific outcome. The filtered state is therefore generally mixed even if the underlying system began pure.

This is one of the most common experimental mistakes: fitting a pure-state filter to data whose detection efficiency or unmonitored loss requires a mixed-state filter.

Circuit-QED Readout Example

In dispersive circuit QED, a microwave resonator maps qubit information onto an outgoing field. After amplification and demodulation, a simplified calibrated readout model for a qubit measurement of $\sigma_z$ is often written

$$dY_t = 2\sqrt{\eta\Gamma_{\mathrm m}}\, \langle\sigma_z\rangle_c\,dt + dW_t,$$

where $\Gamma_{\mathrm m}$ is a convention-dependent measurement rate and $\eta$ is the total efficiency. A corresponding schematic filter is

$$\begin{aligned} d\rho_c =& - \frac{i}{\hbar}[H,\rho_c]\,dt + \Gamma_{\mathrm m}\mathcal D[\sigma_z]\rho_c\,dt \\ &+ \sqrt{\eta\Gamma_{\mathrm m}}\, \mathcal H[\sigma_z]\rho_c\,dW_t. \end{aligned}$$

The exact coefficients depend on the normalization of the integrated voltage, the resonator elimination, the amplifier chain, and whether additional dephasing is separated from the measured dephasing. The operational point is stable across conventions: the live voltage record changes the observer’s state estimate, and the innovation tests whether the readout model is calibrated.

The broader platform context is Circuit QED.

Filters for Parameters

Quantum filtering usually assumes that the Hamiltonian, decay rates, measurement efficiencies, phases, and detector response are known. In real data analysis, some of these quantities may also be uncertain.

One strategy is to run a family of filters indexed by a classical parameter $\theta$ and update a classical posterior:

$$p(\theta\mid\mathcal Y_t) \propto p(\mathcal Y_t\mid\theta)p(\theta).$$

For each candidate $\theta$, the quantum state $\rho_c^{(\theta)}(t)$ evolves according to its own measurement model, and the record likelihood updates the parameter weights. This is a hybrid classical-quantum filtering problem: the parameter uncertainty is classical, while the system state update remains quantum.

This distinction matters when interpreting fitted trajectories. A single best-fit state trajectory is not the same object as a posterior distribution over models and states.

Connection to Feedback

Measurement-based feedback uses the record, or the filtered state inferred from it, to choose later control actions. A schematic feedback Hamiltonian might be

$$H_{\mathrm{fb}}(t) = H_0 + u_t(\mathcal Y_t)F,$$

or, in state-estimate form,

$$u_t = u\!\left(\rho_c(t)\right).$$

The filter is what makes the feedback causal: at time $t$, the controller may depend only on the information already in $\mathcal Y_t$, along with any allowed delays. Unmonitored information, future records, and postselected outcomes cannot be used for real-time control.

This also explains the conceptual difference between measurement-based feedback and reservoir engineering. In Reservoir Engineering, the environment or dissipation is designed so that the desired dynamics occurs autonomously. In measurement-based feedback, a classical record is processed and fed back through a controller.

Common Mistakes

• Using the unconditional master-equation state as if it were the filtered state for a single record.
• Treating the detector current as the system expectation value rather than as signal plus noise.
• Updating the state with an ideal instrument while comparing to a coarse-grained, inefficient, or bandwidth-limited record.
• Forgetting that no-click intervals are part of the counting record and can be informative.
• Using future data in a calculation and then calling the result a real-time filter.
• Assuming that a conditional state remains pure when efficiency is less than one or unmonitored channels are present.
• Ignoring latency and filtering in a feedback loop.
• Diagnosing unexpected innovations as new physics before checking calibration, phase, efficiency, and model mismatch.

Cross-Links

• Measurement Records for calibrated records, finite time bins, and innovation diagnostics.
• Bayesian Quantum Measurement for the relation between likelihoods, posterior inference, and quantum backaction.
• Feedback from Measurement Records for using filtered estimates in causal control loops.
• Measurement-Based Feedback for the control-design view of record-dependent interventions.
• Stochastic Master Equations for the main continuous-time filter equations.
• Diffusive Trajectories for homodyne, heterodyne, and weak-readout filters.
• Quantum Jump Trajectories for counting filters and no-click evolution.
• Unravelings for why different records lead to different conditioned ensembles.
• Quantum Instruments for the outcome-resolved operations behind finite-step filtering.
• Circuit QED for a platform where filtering of continuous voltage records is central.
• Approximation Checklist for assumptions that must be checked before trusting a filter.

References

• V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” Journal of Multivariate Analysis 42, 171–201, 1992.
• L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM Journal on Control and Optimization 46, 2199–2241, 2007.
• A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer, 2009.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press, 2014.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Reviews of Modern Physics 93, 025005, 2021.

Exercises

1. Let $\{\mathcal I_r\}$ be a measurement instrument with $\sum_r\mathcal I_r$ trace preserving. Show that the normalized update

$$\rho_r = \frac{\mathcal I_r(\rho)} {\operatorname{Tr}[\mathcal I_r(\rho)]}$$

is a valid density matrix whenever $p(r)=\operatorname{Tr}[\mathcal I_r(\rho)]\gt0$.

Solution

Because an instrument operation is completely positive, $\mathcal I_r(\rho)$ is positive whenever $\rho$ is positive. Its trace is $p(r)$, which is positive by assumption. Dividing by $p(r)$ preserves positivity and gives unit trace:

$$\operatorname{Tr}(\rho_r) = \frac{\operatorname{Tr}[\mathcal I_r(\rho)]} {\operatorname{Tr}[\mathcal I_r(\rho)]} = 1.$$

Thus $\rho_r$ is positive and normalized, so it is a valid density matrix.

2. In a diffusive filter, suppose

$$dY_t=\mu_t\,dt+dW_t,$$

with $\mu_t$ determined by the present conditional state. Show that the innovation $dY_t-\mu_tdt$ has conditional mean zero and conditional variance $dt$.

Solution

By definition,

$$dY_t-\mu_tdt=dW_t.$$

For a Wiener increment in Itô convention,

$$\mathbb E[dW_t\mid\mathcal Y_t]=0, \qquad \mathbb E[dW_t^2\mid\mathcal Y_t]=dt.$$

Therefore the innovation has conditional mean zero and conditional variance $dt$. If experimental residuals fail this test, the model or calibration is suspect.

3. For a counting filter with conditional intensity $\lambda_t$, define

$$dM_t=dN_t-\lambda_tdt.$$

Show that $dM_t$ has conditional mean zero.

Solution

The defining property of the conditional intensity is

$$\mathbb E[dN_t\mid\mathcal Y_t] = \lambda_tdt.$$

Therefore

$$\mathbb E[dM_t\mid\mathcal Y_t] = \mathbb E[dN_t\mid\mathcal Y_t] - \lambda_tdt = 0.$$

This makes $dM_t$ the counting analogue of an innovation increment.

4. A real-time controller uses the estimate $\rho(t\mid\mathcal Y_T)$ with $T\gt t$ to choose a control at time $t$. Identify the conceptual error.

Solution

The estimate $\rho(t\mid\mathcal Y_T)$ uses records after time $t$, so it is a smoothed estimate, not a filtered estimate available at time $t$. A real-time controller can use only past and present information, possibly with known delays. Using future records makes the control law acausal unless the calculation is reinterpreted as offline analysis or delayed feedback.