Notebooks for Channels and Open-System Dynamics

Computational experiments make channels, relaxation, and measurement records tangible. Start with a small model whose answer is known, then change one physical or numerical parameter at a time. This chapter guides calculations in which the state, observable, and numerical error can all be interpreted. A smooth plot alone cannot distinguish correct dynamics from a stable solution of the wrong equation.

Required background. Quantum Channels and Noise supplies finite-time maps, Markovian Master Equations supplies deterministic dynamics, and Continuous Measurement and Quantum Trajectories supplies conditional records.

Start with an exact qubit reference

Take zero-temperature amplitude damping, no drive, and initial state $|+\rangle$. In the ground–excited basis, with decay rate $\gamma$ and $s=e^{-\gamma t}$,

$$\rho(t)= \begin{pmatrix} 1-s/2&\sqrt{s}/2\\ \sqrt{s}/2&s/2 \end{pmatrix}.$$

This gives a reference for populations, coherence, positivity, and late-time behavior. The following standard-library Python code prints it through the corresponding scalar quantities; times are in units of $1/\gamma$.

from math import exp, sqrt

print("gamma*t   excited   coherence   purity")
for scaled_time in (0.0, 0.5, 1.0, 2.0, 5.0):
    survival = exp(-scaled_time)
    excited = survival / 2
    ground = 1 - excited
    coherence = sqrt(survival) / 2
    purity = ground**2 + excited**2 + 2 * coherence**2
    print(f"{scaled_time:7.2f} {excited:9.6f} "
          f"{coherence:11.6f} {purity:9.6f}")

The initial purity is one. It first falls as the atom becomes correlated with its environment, then returns toward one as the ground state is approached. A solver that reproduces the population but decays coherence at the same rate has a factor-of-two error.

Choose the next numerical experiment

Calculation	What to compare
Simulating Quantum Channels	Kraus propagation with Choi reconstruction and the trace-preserving marginal condition
Bloch Vector Noise Models	Matrix-state evolution with the affine action on the Bloch ball
Solving Lindblad Equations	ODE integration, matrix exponentiation, analytic decay, and steady-state equations
Quantum Jump Simulation	Individual records, waiting times, and the ensemble master-equation solution
Diffusive Trajectory Simulation	Conditional states, record increments, and ensemble averages
Non-Markovian Toy Models	Exact enlarged-system dynamics with a reduced approximation and distinguishability revivals
Decoherence Timescale Estimation	Spectral integrals, filter functions, fit windows, and cutoff dependence
Optimal Control Toy Problems	A baseline pulse with a constrained optimized pulse, retested on a finer grid
Quantum Thermodynamics Toy Models	Exact transition-probability sums with sampled work distributions

The pages develop calculations and their checks. Follow any implementation instructions on the chosen page; a description of a calculation alone does not imply a separately downloadable executable notebook. The Notebook Index collects these routes.

Separate the possible sources of error

Representation. Fix basis order, tensor order, frequency units, and vectorization. For column stacking, $\operatorname{vec}(A\rho B)=(B^{\mathsf T}\otimes A)\operatorname{vec}(\rho)$. Compare a matrix Liouvillian with direct operator evaluation before trusting either propagator.

Propagation. Refine the time step or solver tolerances over the full interval of interest. Two solvers using the same wrongly assembled matrix can agree perfectly. Keep an analytic limit or an independently constructed representation in the comparison.

Truncation. Increase oscillator or environment cutoffs and compare the requested observable. Small occupation at the final retained level is a useful warning test, but does not by itself bound the effect of omitted levels.

Sampling. For independent trajectory samples with finite variance, the usual standard error of a mean is $s_X/\sqrt N$. It does not measure integration bias. Rare-event tails and exponentially weighted thermodynamic averages can converge much less conveniently than a mean population.

Physical approximation. A numerically converged Markovian equation can still be inaccurate for a bath with long memory. Numerical convergence answers whether the specified equation was solved accurately; comparison with a more complete model tests whether that equation is adequate.

Check states and records together

Inspect trace, Hermiticity, and the smallest eigenvalue of each density matrix. Renormalizing trace or clipping negative eigenvalues changes the answer and can hide a bad integrator or invalid approximation. Judge small residuals relative to solver precision and convergence, rather than silently repairing them.

For trajectories, also check the probability law of the record. A norm-preserving algorithm may still use incorrect jump probabilities. Compare the ensemble with a direct master equation and test waiting times or innovation means and variances. Save the parameters, units, solver settings, cutoffs, and random seed with the numerical output so that the comparison can be repeated.

Exercises

Minimum purity of the reference. Find the minimum purity of the amplitude-damping state above and the time when it occurs.

Solution

$\operatorname{Tr}\rho^2=1-s/2+s^2/2$. Its derivative with respect to $s$ vanishes at $s=1/2$, giving purity $7/8$. Since $s=e^{-\gamma t}$, the time is $t=\ln2/\gamma$. The state starts pure and ends pure, but is mixed in between.

Which refinement helps? An ensemble estimate has standard error $0.02$ from $2500$ independent trajectories, with negligible integration error. How many are needed for error $0.005$ under ordinary Monte Carlo scaling?

Solution

A fourfold error reduction requires sixteen times as many independent trajectories, or $40000$. Reducing the integration step alone would not remove this sampling uncertainty. The estimate assumes the variance remains comparable and the usual finite-variance scaling applies.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Analytic reduced dynamics and stochastic methods.
• Hairer, E., S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd revised edition, Springer, 1993. Integration error and refinement.
• Higham, N. J. Accuracy and Stability of Numerical Algorithms. 2nd edition, SIAM, 2002. Numerical stability and representation errors.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Conditional states and record statistics.