Continuous Measurement and Quantum Trajectories

Continuous quantum measurement is the limit of many weak, time-resolved measurement steps. Instead of one isolated outcome, the apparatus produces a record: a stream of detector clicks, a noisy quadrature current, a voltage trace, or another classical stochastic signal. The state used to predict later events is conditioned on the record observed so far.

The defining distinction is

\begin{gathered} \text{conditional state} \\ \text{depends on the record}, \\ \text{unconditional state} \\ \text{averages over records}. \end{gathered}

A quantum trajectory is one realization of that conditioned state evolution. Averaging trajectories generated by a consistent monitoring model recovers the ordinary master equation. Different monitoring schemes can produce different trajectory ensembles for the same unconditional dynamics.

The Operational Chain

Continuous monitoring connects four objects:

\begin{gathered} \text{instrument for each short time step} \\ \longrightarrow \text{measurement record} \\ \longrightarrow \text{conditional state} \\ \longrightarrow \text{ensemble-averaged state}. \end{gathered}

For a short interval $dt$ with outcome $r$ , an operation $\mathcal I_r^{(dt)}$ gives

p(r|\rho_c) = \operatorname{Tr} \left[ \mathcal I_r^{(dt)}(\rho_c) \right]

and the normalized update

\rho_c' = \frac{\mathcal I_r^{(dt)}(\rho_c)} {p(r|\rho_c)}.

Taking a limit of many weak updates produces a stochastic differential equation. The apparatus model determines whether the record is jump-like, diffusive, or a combination of both.

Reading Path

Read this page	Use it for
Continuous Monitoring	Moving from weak discrete instruments to time-resolved conditional states.
Measurement Records	Modeling click times, currents, voltages, bandwidth, calibration, and coarse graining.
Stochastic Master Equations	Writing normalized and unnormalized conditional equations using Itô calculus.
Quantum Jump Trajectories	Simulating event-conditioned jumps, no-click evolution, and waiting times.
Diffusive Trajectories	Modeling continuous noisy currents and innovation-driven updates.
Homodyne Detection	Monitoring one output-field quadrature with a phase reference.
Heterodyne Detection	Monitoring two quadratures with a complex record and added vacuum noise.
Photon Counting	Connecting direct detection, counting processes, inefficiency, and jump rates.
Unravelings	Understanding why one master equation admits many conditioned ensembles.
Quantum Filtering	Estimating the state recursively from the record available up to the present.
Bayesian Quantum Measurement	Separating probabilistic conditioning from physical measurement backaction.
Feedback from Measurement Records	Using a filtered record causally to choose later controls.

A compact technical route is

\begin{gathered} \text{continuous monitoring} \\ \longrightarrow\text{records}, \\ \text{stochastic master equations} \\ \longrightarrow\text{trajectories}, \\ \text{filtering} \longrightarrow \text{feedback}. \end{gathered}

Conditional and Unconditional States

Let $\mathcal Y_t$ denote the complete measurement record available up to time $t$ . The filtered state is

\rho_c(t) = \rho(t|\mathcal Y_t).

It gives probabilities for future measurements conditioned on the actual record. If the record is ignored, the state is

\rho(t) = \mathbb E \left[ \rho_c(t) \right],

where the expectation is over records generated by the measurement model.

The conditional state can purify because an observer gains information. The unconditional state can decohere because averaging discards that information. These are not contradictory predictions; they answer different conditioning questions.

A detector with efficiency below one leaves some output channels unobserved. Even with a perfect state estimator, the conditional state can remain mixed because information has genuinely escaped into unmonitored degrees of freedom.

Records Are Classical Stochastic Processes

A record is classical data produced by a quantum apparatus. It is not generally the instantaneous value of a pre-existing observable.

Record type	Increment	Typical apparatus
counting	$dN_t\in\{0,1\}$	direct photon or electron detection
homodyne	real current increment $dY_t$	one field quadrature
heterodyne	complex increment $dZ_t$	two noisy quadrature channels
dispersive readout	filtered voltage or phase trace	circuit-QED or cavity probe

Raw laboratory data also include bandwidth, gain, offsets, dark counts, dead time, latency, digitization, and filtering. An ideal Wiener or Poisson process is a calibrated model of those data, not the data acquisition chain itself.

Itô Increments

Diffusive equations use a Wiener increment $dW_t$ satisfying

\mathbb E[dW_t]=0, \qquad dW_t^2=dt.

Terms such as $dW_t^2$ must be retained because they are order $dt$ in Itô calculus. Higher products obey the Itô table.

Counting equations use an increment

dN_t\in\{0,1\}, \qquad dN_t^2=dN_t,

with conditional mean

\mathbb E[dN_t|\mathcal Y_t] = \lambda_c(t)dt.

The conditional intensity $\lambda_c(t)$ depends on the current filtered state and detector model.

Itô and Stratonovich equations require different drift terms. Converting between them is not a typographical change; the noise-dependent correction must be included.

A Diffusive Stochastic Master Equation

For one monitored output operator $c$ and homodyne phase zero, a common normalized Itô equation is

\begin{aligned} d\rho_c ={}& \mathcal L(\rho_c)dt \\ &+ \sqrt{\eta}\, \mathcal H[c](\rho_c)dW_t, \end{aligned}

where $\mathcal L$ includes the unconditional dynamics and

\begin{aligned} \mathcal H[c](\rho) ={}& c\rho+\rho c^\dagger \\ &- \operatorname{Tr} \left[ (c+c^\dagger)\rho \right] \rho. \end{aligned}

The idealized record is

dY_t = \sqrt{\eta}\, \operatorname{Tr} \left[ (c+c^\dagger)\rho_c \right]dt + dW_t.

The innovations process is the observed record minus its conditional prediction:

dW_t = dY_t - \mathbb E[dY_t|\mathcal Y_t].

With the correct filter, innovations have zero conditional mean. A systematic drift in inferred innovations is a practical sign of model mismatch, calibration error, or an omitted signal.

Quantum Jump Trajectories

For a monitored channel with operator $c$ , a click has conditional rate

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

When a click occurs, the ideal conditional update is

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

Between clicks, the absence of a detection also carries information. For a perfectly monitored decay channel, the no-click pure-state evolution is generated before normalization by

H_{\mathrm{eff}} = H - \frac{i\hbar}{2}c^\dagger c.

The decreasing norm gives the survival probability for no click. Thus a trajectory is not “ordinary evolution interrupted by occasional measurements.” Both clicks and no-click intervals update the conditioned state.

Detector inefficiency splits the output into observed and unobserved components. Unobserved jumps still contribute to the unconditional dissipator and generally reduce conditional purity.

Photon Counting

Direct detection produces time-tagged events or counts in finite bins. In an ideal Markovian output channel,

\mathbb E[dN_t|\mathcal Y_t] = \eta \langle c^\dagger c\rangle_c dt + \nu_{\mathrm{dark}}dt,

where $\eta$ is efficiency and $\nu_{\mathrm{dark}}$ is a dark-count rate in a simple model.

Real detectors can have dead time, afterpulsing, finite timing resolution, missed counts, and background events. These effects alter the likelihood and filter. Treating every click as a system quantum jump can be wrong when the record includes detector noise or multiple physical channels.

Photon Counting owns the record model; Quantum Jump Trajectories owns the conditioned state dynamics.

Homodyne Detection

Balanced homodyne detection mixes an output field with a strong local oscillator and subtracts detector currents. The local-oscillator phase $\theta$ selects the measured quadrature. A schematic record is

\begin{aligned} dY_t ={}& \sqrt{\eta} \left\langle e^{-i\theta}c + e^{i\theta}c^\dagger \right\rangle_c dt \\ &+dW_t. \end{aligned}

The signal is buried in white-noise fluctuations over an infinitesimal interval. Information appears through integration and filtering, not through reading an exact instantaneous quadrature value.

Changing $\theta$ changes both the measured signal and the conditional backaction. The unconditional master equation can remain the same.

Heterodyne Detection

Heterodyne detection produces two quadrature channels, often combined into a complex increment

dZ_t = \sqrt{\eta}\, \langle c\rangle_cdt + d\zeta_t,

with complex noise conventionally satisfying

d\zeta_t^2=0, \qquad d\zeta_t d\zeta_t^*=dt.

Heterodyne detection does not perform two noiseless simultaneous measurements of conjugate quadratures. Splitting the signal or using a frequency-offset local oscillator introduces the extra vacuum noise required by quantum mechanics.

Homodyne and heterodyne records are different unravelings of output-field monitoring. Their trajectories can look very different even when their ensemble average obeys the same Lindblad equation.

Unraveling Freedom

An unraveling is a representation of an unconditional state as an average over conditioned trajectories:

\rho(t) = \mathbb E_{\mathcal U} \left[ \rho_c^{(\mathcal U)}(t) \right].

The label $\mathcal U$ specifies the monitoring scheme or stochastic representation. Direct detection, homodyne detection at different phases, heterodyne detection, and purely numerical stochastic decompositions can all reproduce one master equation.

Consequently:

a master equation does not identify a unique measurement record;
a Lindblad operator is not automatically a physically detected jump;
trajectories corresponding to different compatible measurements are not competing predictions for the same record;
a numerical unraveling need not correspond to an implemented detector;
trajectory-dependent quantities must be interpreted relative to the chosen monitoring scheme.

The unconditional density operator is invariant under the unraveling choice. Conditional states and record statistics are not.

Filtering, Prediction, and Smoothing

Quantum filtering estimates the present state from the record up to the present:

\rho_c(t)=\rho(t|\mathcal Y_t).

Prediction propagates that filtered state into the future before later data arrive. Smoothing uses data obtained after time $t$ to retrodict an earlier state or event. These are distinct inference tasks and should not be labeled interchangeably.

A practical filter must specify:

the process model and Hamiltonian;
measured and unmeasured channels;
detector efficiency, gain, bandwidth, and noise;
the record convention and time discretization;
unknown parameters included in the state estimate;
initialization and model-mismatch diagnostics.

The filtered state is an operational state assignment for prediction. Whether one interprets it as an observer’s information, a physical conditional state, or part of a broader ontology is an additional foundational question.

Bayesian Updating and Quantum Backaction

Classical Bayes rule updates probabilities using a likelihood. Quantum measurement also updates a state, but the physical operation contains more information than the likelihood alone.

For an outcome $r$ with effect $F_r$ , the likelihood is

p(r|\rho) = \operatorname{Tr}(F_r\rho).

The conditional output requires an instrument:

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

Two instruments can have the same likelihoods and different backaction. Bayesian language captures conditioning on data; it does not replace the quantum operation that physically transforms the state.

Continuous filters are recursive quantum-instrument updates in differential form. Bayesian Quantum Measurement develops the analogy and its limits.

Feedback from Records

Measurement-based feedback uses only information available causally up to the current time. A typical loop is

record -> filter -> control law -> actuator -> new system dynamics

The controller may change a Hamiltonian, trigger an event-based pulse, adjust a drive amplitude, or modify a dissipative coupling. Latency, finite bandwidth, estimation error, and actuation noise can destabilize the loop or limit performance.

Feedback also changes energy and entropy bookkeeping. Measurement backaction, information acquisition, control work, and dissipated heat should not be combined into one unnamed “feedback cost.”

Feedback from Measurement Records owns the trajectory-to-control bridge. Measurement-Based Feedback owns controller design and control limits.

Numerical Simulation

Trajectory simulation can replace one density-matrix evolution with an ensemble of stochastic pure or mixed states. It is especially useful for sparse jump records, conditional observables, waiting-time distributions, and large Hilbert spaces where state vectors are cheaper than density matrices.

A trustworthy simulation should:

state the Itô or Stratonovich convention;
preserve normalization or track unnormalized weights consistently;
use time steps small relative to all rates and detector bandwidths;
ensure jump probabilities remain much smaller than one per step;
compare the ensemble average with a direct master-equation solution;
report trajectory count and statistical uncertainty;
test detector efficiency and unmonitored-channel limits;
validate waiting-time or innovation statistics;
document random seeds and convergence criteria;
distinguish a physical unraveling from a numerical convenience.

One trajectory is not an ensemble estimate. Rare events can require variance reduction or many more realizations than smooth mean observables.

Canonical Boundaries

This chapter owns time-resolved measurement records and the quantum states conditioned on them.

Generalized Measurements and Instruments owns discrete outcome-resolved operations and POVMs.
Markovian Master Equations owns unconditional GKSL generators and Lindblad-operator structure.
Quantum Noise, Dissipation, and Baths owns input-output relations and the field correlations that feed detector models.
Quantum Control and Feedback owns controller synthesis, optimal control, coherent feedback, and reservoir engineering.
Quantum Thermodynamics owns systematic work, heat, entropy-production, and information-cost accounting.
Quantum Information owns large-scale tomography, channel discrimination, error correction, and information-processing tasks.

A trajectory page should link to those canonical homes instead of repeating their complete derivations.

Common Mistakes

Confusing a conditional trajectory with the unconditional density operator.
Averaging nonlinear normalized updates without the correct record probabilities.
Treating a measurement record as a noiseless instantaneous observable value.
Forgetting that no-click intervals carry information.
Interpreting Lindblad operators as unique detected events.
Treating one unraveling as the unique physical history.
Dropping Itô correction terms such as $dW_t^2=dt$ .
Mixing Itô and Stratonovich drifts.
Ignoring detector inefficiency and unobserved channels.
Treating heterodyne detection as two perfect quadrature measurements.
Calling every Bayesian update purely epistemic while ignoring physical backaction.
Using future data in a filter intended for real-time feedback.
Comparing trajectory simulations without convergence or uncertainty estimates.

Exercises

Ensemble average of a diffusive SME

Suppose

d\rho_c = \mathcal L(\rho_c)dt + \mathcal H(\rho_c)dW_t.

Show formally that averaging recovers the unconditional equation when $dW_t$ is an innovations increment and the required regularity conditions hold.

Solution

Take the conditional expectation over the next increment and then the ensemble expectation. Since innovations have zero conditional mean,

\mathbb E \left[ \mathcal H(\rho_c)dW_t \right] =0.

Linearity of $\mathcal L$ gives

d\mathbb E[\rho_c] = \mathcal L (\mathbb E[\rho_c])dt.

Therefore $\rho=\mathbb E[\rho_c]$ satisfies

\frac{d\rho}{dt}=\mathcal L(\rho).

Click probability and update

A two-level atom is monitored with $c=\sqrt\gamma\,\sigma_-$ . For conditional state $\rho_c$ , find the click probability in $dt$ and the state immediately after an ideal click.

Solution

The conditional rate is

\lambda_c = \operatorname{Tr}(c^\dagger c\rho_c) = \gamma\rho_{ee}.

Thus

\Pr(dN_t=1|\mathcal Y_t) = \gamma\rho_{ee}dt.

The jump update is

\rho_c \longrightarrow \frac{\sigma_-\rho_c\sigma_+} {\operatorname{Tr}(\sigma_+\sigma_-\rho_c)} = |g\rangle\langle g|,

provided the click has nonzero probability.

No-click information

Why does observing no photon for a finite interval change the conditional state of an atom initially in a superposition of ground and excited states?

Solution

The excited component could emit, whereas the ground component cannot. Survival without a click is therefore more likely when the atom is in the ground state. Before normalization, the no-click amplitudes evolve under

H_{\mathrm{eff}} = H - \frac{i\hbar\gamma}{2} |e\rangle\langle e|.

The excited amplitude decays relative to the ground amplitude. Renormalizing after conditioning on no click shifts the state toward the ground state even though no detector event occurred.

Innovations test

A homodyne model predicts conditional signal $m_c(t)dt$ , while the observed increment is $dY_t$ . Define the innovations increment and state its expected conditional mean and variance.

Solution

Define

dW_t=dY_t-m_c(t)dt.

For a calibrated ideal Wiener model,

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

Persistent nonzero mean or incorrect variance indicates a mismatch among the physical model, detector calibration, or noise convention.

Inefficient monitoring

A decay channel has total rate $\gamma$ , but the detector efficiency is $\eta$ . Explain how to divide the channel into observed and unobserved parts and what happens as $\eta\to0$ .

Solution

Use observed and unobserved operators

\begin{aligned} c_{\mathrm{obs}} &= \sqrt{\eta\gamma}\,\sigma_-, \\ c_{\mathrm{unobs}} &= \sqrt{(1-\eta)\gamma}\,\sigma_-. \end{aligned}

Their unconditional dissipators add to $\gamma\mathcal D[\sigma_-]$ . Only the observed channel contributes to the record-conditioned innovation or click update. As $\eta\to0$ , the record carries no information and the conditional evolution reduces to the unconditional master equation.

Cross-Links

References

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).
C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).
J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Physical Review Letters 68, 580–583 (1992).
R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Physical Review A 45, 4879–4887 (1992).
N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” Journal of Physics A 25, 5677–5691 (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).