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.

Predicting and Updating the Conditional State

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,

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

The prediction map must contain only evolution not already included in the instrument. If the instrument describes the whole bin, including Hamiltonian evolution and damping, set the extra prediction map to the identity. Otherwise those dynamics would be counted twice. 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.

Classical filtering updates a probability distribution; 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

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

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 Later Data

Filtering uses records through $t$ to estimate the present state. Prediction propagates that estimate to a later time without using new data. Smoothing uses records after $t$ to infer an earlier event or hidden variable. Only the first two are available to a controller at time $t$.

Quantum inference about the past needs more care than a notation such as $\rho(t\mid\mathcal Y_T)$ suggests. In general, a single density matrix cannot reproduce all probabilities for measurements at $t$ conditioned on both earlier and later records. One useful representation is a pair: the forward filtered state $\rho_c(t)$ and a backward effect $E(t)$ describing the likelihood of the later record.

For a specified measurement at $t$ with one Kraus operator $M_m$ per outcome, the retrodictive probability is

$$p(m\mid\text{earlier and later records}) =\frac{\operatorname{Tr}[E(t)M_m\rho_c(t)M_m^\dagger]} {\sum_n\operatorname{Tr}[E(t)M_n\rho_c(t)M_n^\dagger]}.$$

It depends on the intermediate instrument. The product $\rho_cE$ need not be Hermitian or a state. This is the past quantum state formulation of Gammelmark, Julsgaard, and Mølmer. Specific quantum-state smoothing constructions can define positive state estimates with additional assumptions, but they should not be confused with a universally valid replacement for the forward filtered state.

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

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

For these displayed equations, $c=\sqrt{\Gamma_{\mathrm m}}\sigma_z$ and the unread coherence-decay rate is $2\Gamma_{\mathrm m}$. Other voltage conventions require changing the record and state-update coefficients together. Additional unobserved dephasing belongs in $\mathcal L$, not in the information term. 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:

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

Each candidate filter is driven by the same observed $dY_t$, with its own predicted drift $\mu_\theta(t)$; drawing a new noise history for each candidate would compare different data sets. Relative to a Wiener reference record, the log-likelihood obeys

$$d\log\mathcal L_\theta=\mu_\theta\,dY_t-\frac12\mu_\theta^2dt.$$

Accumulating these log-likelihoods avoids underflow when multiplying many small likelihood densities. The quantum state $\rho_c^{(\theta)}$ and the classical parameter weight are updated together. 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.

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 records through $T>t$ to infer the system at $t$ and choose a control at $t$. Identify the conceptual error.

Solution

The inference uses records after time $t$, so it is retrospective rather than a filtered estimate available at $t$. 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.

References

• A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer, 2009.
• V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” Journal of Multivariate Analysis 42, 171–201, 1992.
• A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Reviews of Modern Physics 93, 025005, 2021.
• L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM Journal on Control and Optimization 46, 2199–2241 (2007), doi:10.1137/060651239.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• S. Gammelmark, B. Julsgaard, and K. Mølmer, “Past Quantum States of a Monitored System,” Physical Review Letters 111, 160401 (2013), doi:10.1103/PhysRevLett.111.160401.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press, 2014.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.