Diffusive Trajectory Simulation

This notebook guide specifies a reproducible simulation of diffusive quantum trajectories for continuous weak qubit measurement. The baseline model is a monitored $\sigma_z$ measurement with noisy records, stochastic state updates, state purification under monitoring, and ensemble averages that recover the unconditional master equation.

As of this review, no executable notebook under notebooks/density-open-systems/diffusive-trajectory-simulation/ is promoted as a reproduced artifact. This page is the admission contract for that notebook: it states the stochastic equation, record convention, numerical scheme, validation tests, convergence checks, and accepted outputs required before trajectories or records should be cited.

Purpose

The notebook should demonstrate how to:

• generate Wiener increments with a declared convention;
• simulate continuous weak measurement records;
• update a conditioned qubit density matrix with a diffusive stochastic master equation;
• distinguish conditional trajectories from unconditional master-equation dynamics;
• show state purification under efficient monitoring;
• average many trajectories to recover dephasing;
• test time-step, trajectory-number, and random-seed sensitivity;
• diagnose trace, positivity, and Itō-convention errors.

The first version should use a single qubit and one monitored observable. More elaborate homodyne, heterodyne, and feedback examples should wait until this baseline is validated.

Directory Plan

Use a dedicated directory:

notebooks/density-open-systems/diffusive-trajectory-simulation/
  diffusive-trajectory-simulation.ipynb
  README.md

The opening notebook cell or README.md should state:

• Python and package versions;
• basis ordering and Pauli convention;
• stochastic calculus convention;
• measurement strength and detection efficiency;
• time step and total time;
• number of trajectories;
• random-number generator and seed;
• normalization and positivity tolerances;
• date and commit identifier when the notebook is promoted.

Baseline Model

Use a continuous measurement of $\sigma_z$ with collapse operator

$$c = \sqrt{\kappa}\,\sigma_z,$$

where $\kappa$ is the measurement-induced dephasing scale in this convention. With no Hamiltonian and no other dissipative channel, the unconditional generator is

$$\mathcal L\rho = \kappa\mathcal D[\sigma_z]\rho,$$

where

$$\mathcal D[\sigma_z]\rho = \sigma_z\rho\sigma_z-\rho.$$

The normalized diffusive stochastic master equation is

$$d\rho_c = \kappa\mathcal D[\sigma_z]\rho_c\,dt + \sqrt{\eta\kappa}\, \mathcal H[\sigma_z]\rho_c\,dW_t,$$

with efficiency $0\le\eta\le1$ and

$$\mathcal H[\sigma_z]\rho = \sigma_z\rho+\rho\sigma_z - 2\langle\sigma_z\rangle\rho.$$

The corresponding record convention is

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

The notebook must use the same convention for the SME and the record. Mixing record normalizations is one of the easiest ways to produce plausible-looking but wrong trajectories.

Itō Increment Checks

Generate Wiener increments as

$$dW_n = \sqrt{dt}\,\xi_n, \qquad \xi_n\sim\mathcal N(0,1).$$

The notebook should check, for the generated increments,

$$\frac{1}{N_{\mathrm{steps}}} \sum_n dW_n \approx 0, \qquad \frac{1}{N_{\mathrm{steps}}} \sum_n dW_n^2 \approx dt.$$

These checks are not decorative. If the increments have variance $1$ instead of $dt$, the trajectory will be wrong even if the code runs without errors.

Numerical Update

For a first implementation, Euler-Maruyama is acceptable:

$$\rho_{n+1} = \rho_n + \kappa\mathcal D[\sigma_z]\rho_n\,dt + \sqrt{\eta\kappa}\, \mathcal H[\sigma_z]\rho_n\,dW_n.$$

After each step, the notebook should measure:

• trace error;
• Hermiticity error;
• smallest eigenvalue;
• purity $\operatorname{Tr}\rho^2$;
• Bloch vector length;
• current conditional expectation $\langle\sigma_z\rangle_c$.

Small numerical symmetrization may be used for plotting, but raw errors should be recorded before any repair. If positivity failures become visible at normal tolerances, reduce $dt$ or use a more stable integrator.

Measurement Records

For each trajectory, store the time series

$$\Delta Y_n = 2\sqrt{\eta\kappa}\, \langle\sigma_z\rangle_n\,dt + dW_n.$$

Also store the innovation estimate

$$\Delta W_n = \Delta Y_n - 2\sqrt{\eta\kappa}\, \langle\sigma_z\rangle_n\,dt.$$

The innovation should have zero mean and variance $dt$ within sampling error. This is a central validation test: the state update must be driven by unpredictable record increments, not by the predicted signal that was already encoded in $\rho_c$.

Ensemble Average

The unconditional master equation is pure dephasing. If

$$\rho(0) = \begin{pmatrix} \rho_{00}(0)&\rho_{01}(0)\\ \rho_{10}(0)&\rho_{11}(0) \end{pmatrix},$$

then, under the convention above,

$$\rho_{01}(t) = e^{-2\kappa t}\rho_{01}(0), \qquad \rho_{00}(t)=\rho_{00}(0).$$

The notebook should average many conditioned trajectories,

$$\bar\rho_N(t) = \frac{1}{N} \sum_{j=1}^N \rho_c^{(j)}(t),$$

and compare $\bar\rho_N(t)$ with the analytic unconditional solution. The maximum error should decrease as the number of trajectories increases, with the expected Monte Carlo scaling visible over a reasonable range.

State Purification

Start from the maximally mixed state,

$$\rho(0)=\frac12I.$$

For efficient monitoring, individual conditioned states should tend toward $\sigma_z$ eigenstates as the measurement record accumulates information. A useful diagnostic is the purity

$$P(t) = \operatorname{Tr}\rho_c(t)^2.$$

The ensemble average can remain mixed even while individual trajectories purify. This is the main conceptual point the notebook should make: conditioning on a record changes the state assignment, while ignoring the record gives the nonselective state.

The notebook should compare at least two efficiencies, such as $\eta=1$ and $\eta=0.3$, to show how lost information weakens purification.

Plots and Outputs

The accepted notebook should produce:

• several sample measurement records $\Delta Y_n/dt$ or binned currents;
• sample trajectories for $\langle\sigma_z\rangle_c(t)$ and purity;
• an ensemble-average density-matrix comparison with the analytic dephasing solution;
• innovation mean and variance diagnostics;
• convergence plots versus $dt$ and number of trajectories.

Save accepted outputs under:

notebooks/density-open-systems/diffusive-trajectory-simulation/outputs/
  sample-measurement-records.svg
  diffusive-trajectories-purity.svg
  ensemble-average-dephasing.svg
  innovation-diagnostics.svg
  diffusive-trajectory-validation.json

The JSON validation file should record:

• package versions;
• random seed;
• $\kappa$, $\eta$, $dt$, total time, and trajectory count;
• maximum trace error;
• minimum eigenvalue observed;
• innovation mean and variance;
• maximum ensemble-average error against the analytic solution;
• whether convergence checks passed.

Validation Tests

The notebook should include automated checks:

Check	Required behavior
Wiener scaling	sample variance of $dW_n$ near $dt$
trace	$\operatorname{Tr}\rho_c$ stays near $1$
Hermiticity	$\rho_c-\rho_c^\dagger$ stays small
positivity	eigenvalues stay nonnegative within tolerance
record convention	innovations reconstruct generated $dW_n$
ensemble average	trajectories recover $e^{-2\kappa t}$ coherence decay
efficiency	lower $\eta$ reduces conditioned purification
convergence	errors decrease when $dt$ is reduced

The notebook should deliberately include one bad-step demonstration in an appendix or disabled cell: for example, using $dW_n=\xi_n$ instead of $\sqrt{dt}\xi_n$ should fail the validation tests. This is a good way to prove the tests can catch a common stochastic-simulation bug.

Common Mistakes

• Treating $dW_t$ as an ordinary deterministic small number.
• Generating Gaussian increments with variance $1$ instead of $dt$.
• Updating the state with one measurement-record convention and plotting another.
• Forgetting that ensemble averages must recover the unconditional master equation.
• Interpreting a conditioned trajectory as a hidden classical path of the system.
• Assuming purification occurs when the measurement record is discarded.
• Comparing one noisy trajectory to the master equation instead of comparing the ensemble average.
• Ignoring small positivity violations until they dominate the result.

Exercises

Wiener Scaling

If $dW=\sqrt{dt}\xi$ with $\xi\sim\mathcal N(0,1)$, compute $\mathbb E[dW]$ and $\mathbb E[dW^2]$.

Solution

Since $\mathbb E[\xi]=0$,

$$\mathbb E[dW] = \sqrt{dt}\,\mathbb E[\xi] = 0.$$

Since $\mathbb E[\xi^2]=1$,

$$\mathbb E[dW^2] = dt\,\mathbb E[\xi^2] = dt.$$

This is the numerical version of the Itō rule $dW_t^2=dt$.

Unconditional Coherence Decay

For $\mathcal L\rho=\kappa(\sigma_z\rho\sigma_z-\rho)$, show that $\dot\rho_{01}=-2\kappa\rho_{01}$.

Solution

In the $\sigma_z$ basis,

$$\sigma_z\rho\sigma_z = \begin{pmatrix} \rho_{00}&-\rho_{01}\\ -\rho_{10}&\rho_{11} \end{pmatrix}.$$

Therefore

$$\sigma_z\rho\sigma_z-\rho = \begin{pmatrix} 0&-2\rho_{01}\\ -2\rho_{10}&0 \end{pmatrix}.$$

Thus

$$\dot\rho_{01} = -2\kappa\rho_{01},$$

so

$$\rho_{01}(t) = e^{-2\kappa t}\rho_{01}(0).$$

Innovation Reconstruction

Given the record increment

$$\Delta Y_n = 2\sqrt{\eta\kappa}\, \langle\sigma_z\rangle_n dt + dW_n,$$

write the innovation increment.

Solution

Subtract the predicted conditional signal:

$$\Delta W_n = \Delta Y_n - 2\sqrt{\eta\kappa}\, \langle\sigma_z\rangle_n dt.$$

Using the record equation, this gives

$$\Delta W_n=dW_n.$$

Thus the innovation is the unpredictable part of the observed record.

Conditional Versus Unconditional Purity

Why can individual monitored trajectories become more pure while the ensemble average remains mixed?

Solution

The monitored trajectory is conditioned on a specific record, so the observer gains information about which $\sigma_z$ eigenstate is favored. That information updates the conditional density matrix and can increase its purity. If the record is ignored, different possible records are averaged together. The average state then represents a mixture over incompatible records and can remain mixed even though each conditioned state became more definite.

Cross-Links

• Stochastic Master Equations
• Diffusive Trajectories
• Measurement Backaction
• Pure Dephasing Master Equation
• Solving Lindblad Equations
• Quantum Jump Simulation
• Circuit QED
• Cavity QED

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.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.