Stochastic Master Equations

A stochastic master equation is a differential equation for a quantum state conditioned on a continuously acquired measurement record. It describes a single observer’s best state assignment during monitoring, while the ordinary master equation describes the ensemble average when the record is ignored.

The central contrast is:

unconditional state:  average over all records
conditional state:    state given a particular record

Stochastic master equations are the bridge between generalized measurement, Lindblad dynamics, quantum trajectories, feedback control, and experimentally recorded signals such as photon counts, homodyne currents, heterodyne currents, and qubit readout voltages.

For the conceptual entry point to records, conditioning, measurement strength, and ensemble averaging, see Continuous Monitoring.

Conditional Versus Unconditional Evolution

An unconditional Markovian master equation has the form

$$\frac{d\rho}{dt} = \mathcal L\rho,$$

where, in Lindblad–GKSL form,

$$\mathcal L\rho = - \frac{i}{\hbar}[H,\rho] + \sum_k \mathcal D[L_k]\rho,$$

with

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12 \{L^\dagger L,\rho\}.$$

This equation describes the average state when environmental outputs or measurement records are not retained.

A stochastic master equation describes the conditional state $\rho_c(t)$ given the measurement record up to time $t$:

$$d\rho_c = \text{deterministic drift} + \text{record-dependent stochastic update}.$$

The ensemble average over records must recover the unconditional state:

$$\mathbb E[\rho_c(t)] = \rho(t), \qquad \frac{d\rho}{dt} = \mathcal L\rho.$$

This identity assumes the generator does not depend on the realized record. Under feedback one must average the record-dependent drift as well. For a fixed generator, disagreement beyond sampling and time-step errors indicates a problem in the stochastic equation or its numerical implementation.

Measurement Record

A continuous measurement produces a time series, not just a final outcome. The dedicated Measurement Records page discusses calibration, time bins, detector imperfections, and the main record types. The record may be:

• a counting record $N_t$ for photon detection;
• a noisy current $Y_t$ for homodyne measurement;
• two noisy quadratures for heterodyne measurement;
• a voltage trace for dispersive qubit readout;
• a coarse-grained detector signal in a solid-state measurement.

The conditional state is filtered from this record. It is not an additional hidden ontic variable in the formalism; it is the state assignment conditioned on the information made available to the observer.

Ignoring the record gives the nonselective state:

$$\rho(t) = \mathbb E_{\text{records}}\rho_c(t).$$

This is the continuous-time analogue of the distinction between selective and nonselective measurements.

Itô Increments

Most stochastic master equations in quantum optics and measurement theory are written in Itô form.

For a Wiener increment $dW_t$,

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

For a counting increment $dN_t$,

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

Its conditional mean is set by the instantaneous detection rate. For a monitored collapse operator $c$ with unit efficiency,

$$\mathbb E[dN_t\mid\rho_c(t)] = \operatorname{Tr}(c^\dagger c\rho_c(t))\,dt.$$

These multiplication rules are part of the equation. Treating $dW_t$ like an ordinary small number loses the Itô correction terms that make the ensemble average work.

Diffusive SME

A common normalized diffusive stochastic master equation for monitoring an output associated with a collapse operator $c$ is

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta}\, \mathcal H[c]\rho_c\,dW_t,$$

Here $c$ includes its rate factor and has units of inverse square-root time; $\mathcal L$ includes the full $\mathcal D[c]$, independent of detection efficiency. The efficiency is $0\le\eta\le1$, and

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

The corresponding homodyne-style record may be written, in this convention, as

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

The stochastic term is an innovation: the difference between the observed noisy increment and its conditional expectation. It updates the state because the record contains partial information about the system.

The stochastic coefficient is evaluated using the record before the new increment. Since $\mathbb E[dW_t\mid\mathcal Y_t]=0$, its product with that coefficient has zero mean. This conditional-mean property, together with linearity of the fixed $\mathcal L$, yields the unconditional equation; zero unconditional mean of a noise variable alone would not justify dropping a correlated product.

The dedicated Diffusive Trajectories page treats homodyne, heterodyne, and continuous weak-measurement records in more detail. For the local-oscillator detection model behind the homodyne record, see Homodyne Detection.

Jump SME

For photon counting or other discrete detection events, a normalized jump SME has Poisson increments. For one monitored collapse operator $c$ and no unmonitored channels, a standard form is

$$d\rho_c = - \frac{i}{\hbar}[H,\rho_c]dt - \frac12 \{c^\dagger c,\rho_c\}dt + \operatorname{Tr}(c^\dagger c\rho_c)\rho_c\,dt + \left( \frac{c\rho_c c^\dagger} {\operatorname{Tr}(c^\dagger c\rho_c)} - \rho_c \right)dN_t.$$

The increment satisfies

$$\mathbb E[dN_t\mid\rho_c] = \operatorname{Tr}(c^\dagger c\rho_c)\,dt.$$

If the intensity is zero, the normalized jump fraction need not be evaluated: that event has zero probability. When no count occurs, the state evolves under a no-jump drift. When a count occurs, the state jumps by the normalized update

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

Averaging over $dN_t$ recovers

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \mathcal D[c]\rho.$$

The dedicated Quantum Jump Trajectories page treats waiting times, effective non-Hermitian Hamiltonians, and trajectory simulations in more detail.

Example: Continuous Qubit Readout

For a continuous measurement of $\sigma_z$, take

$$c=\sqrt{\gamma}\,\sigma_z.$$

The unconditional dissipator is pure dephasing:

$$\mathcal D[c]\rho = \gamma(\sigma_z\rho\sigma_z-\rho).$$

The diffusive SME is

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

The record is

$$dY_t = 2\sqrt{\eta\gamma}\, \langle\sigma_z\rangle_c\,dt + dW_t.$$

The coefficient $\gamma$ gives coherence decay at rate $2\gamma$, since $\sigma_z\rho\sigma_z-\rho$ has off-diagonal entries $-2\rho_{01}$. The deterministic part dephases the ensemble. For $H=0$ or a Hamiltonian commuting with $\sigma_z$, the stochastic part gradually resolves a $\sigma_z$ eigenstate. A competing transverse Hamiltonian can continually rotate the state instead; localization is then a dynamical competition.

Purity and Conditioning

Conditioning can make a state purer even while the ensemble state decoheres. For efficient monitoring, the observer learns information about the system from the record. For inefficient monitoring, some information is lost into unobserved channels, so the conditional state may remain mixed.

This distinction is essential:

decoherence of the average state
  can coexist with
purification of the conditioned state

There is no contradiction. The average state mixes together different possible records; the conditional state follows one record.

Relation to Instruments

An SME can be viewed as the continuous-time limit of a sequence of weak quantum instruments. Over a small interval $dt$, one has outcome-dependent completely positive maps:

$$\rho \longrightarrow \frac{\mathcal I_{dy}(\rho)} {\operatorname{Tr}\mathcal I_{dy}(\rho)}$$

conditioned on the infinitesimal outcome $dy$. Expanding these maps to first order in $dt$ and to the appropriate stochastic order gives the SME.

This connects stochastic master equations to Quantum Instruments rather than making them a separate postulate.

Normalized and Unnormalized Forms

SMEs appear in normalized and unnormalized forms.

Normalized equations keep

$$\operatorname{Tr}\rho_c(t)=1$$

on every trajectory, but the equation is nonlinear because probabilities depend on $\rho_c$.

Unnormalized equations are often linear and easier for likelihood calculations. If $\tilde\rho_c$ is unnormalized, then

$$\rho_c = \frac{\tilde\rho_c} {\operatorname{Tr}\tilde\rho_c}.$$

The trace of $\tilde\rho_c$ carries the likelihood, or likelihood ratio, relative to the reference record measure used to define the linear filter. For diffusive records this is often a Wiener reference measure. One must specify that measure before interpreting the normalization as a probability density. Mixing normalized and unnormalized conventions is a common source of wrong factors and wrong record probabilities.

Itô Versus Stratonovich

Itô and Stratonovich forms are not interchangeable by notation alone. In Itô calculus, increments are nonanticipating and satisfy rules such as $dW_t^2=dt$. In Stratonovich calculus, the chain rule looks more like ordinary calculus, but drift terms are shifted.

Most filtering and trajectory formulas in this volume use Itô form unless stated otherwise. If a source uses Stratonovich form, convert the drift before comparing equations.

Exercises

Diffusive average

For the diffusive SME

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta}\mathcal H[c]\rho_c\,dW_t,$$

show why the stochastic term has no direct contribution to the ensemble mean at order $dt$.

Solution

In Itô form, the increment $dW_t$ has conditional mean zero:

$$\mathbb E[dW_t\mid\rho_c(t)]=0.$$

Therefore

$$\mathbb E[ \sqrt{\eta}\mathcal H[c]\rho_c\,dW_t ] = 0$$

at order $dt$. The ensemble mean obeys

$$\frac{d}{dt}\mathbb E[\rho_c] = \mathbb E[\mathcal L\rho_c] = \mathcal L\,\mathbb E[\rho_c],$$

using linearity of $\mathcal L$.

Counting average

Use the jump SME in the page to show that the ensemble drift contains $c\rho c^\dagger$.

Solution

Let

$$r=\operatorname{Tr}(c^\dagger c\rho_c).$$

The jump term has conditional mean

$$\mathbb E\left[ \left( \frac{c\rho_c c^\dagger}{r} - \rho_c \right)dN_t \middle|\rho_c \right] = \left( \frac{c\rho_c c^\dagger}{r} - \rho_c \right)r\,dt.$$

This equals

$$c\rho_c c^\dagger dt-r\rho_c dt.$$

Adding the no-count drift term $r\rho_c dt$ cancels the last term, leaving $c\rho_c c^\dagger dt$ together with the anticommutator and Hamiltonian pieces. The result is the Lindblad dissipator.

Qubit readout record

For the continuous $\sigma_z$ measurement record

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

what is the conditional mean of $dY_t$?

Solution

Since $\mathbb E[dW_t\mid\rho_c]=0$,

$$\mathbb E[dY_t\mid\rho_c] = 2\sqrt{\eta\gamma}\, \langle\sigma_z\rangle_c\,dt.$$

The innovation is the difference between the observed $dY_t$ and this conditional mean.

Selective versus nonselective

Why does ignoring the measurement record turn an SME into an ordinary master equation?

Solution

The SME updates the state conditioned on each particular record. Ignoring the record means averaging over all possible records with their correct probabilities:

$$\rho(t)=\mathbb E[\rho_c(t)].$$

The stochastic innovation has zero mean in the diffusive case, and jump probabilities average the normalized jumps into the Lindblad jump term. Thus the record-conditioned dynamics reduces to the nonselective Lindblad master equation.

References

• A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer (2009).
• L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM Journal on Control and Optimization 46, 2199–2241 (2007).
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• K. Jacobs and D. A. Steck, “A straightforward introduction to continuous quantum measurement,” Contemporary Physics 47, 279–303 (2006), doi:10.1080/00107510601101934.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).
• A. N. Korotkov, “Continuous quantum measurement of a double dot,” Physical Review B 60, 5737–5742 (1999).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).