Open-System Notebook Index

These computational lessons turn channel and master-equation formulas into small calculations you can inspect. Choose a lesson by the result you want to understand. Begin with a qubit whose analytical solution is known before adding larger Hilbert spaces, stochastic records, or optimization.

Required background. The Model Index helps choose the physical model. Each computational lesson links the specific mathematical preparation it uses.

Channels, generators, and monitored records

Lesson	What you learn to calculate
Simulating Quantum Channels	Apply Kraus maps, build a Choi matrix, and distinguish trace preservation from complete positivity
Bloch Vector Noise Models	Compare dephasing, isotropic mixing, and the shifted ellipsoid of amplitude damping
Solving Lindblad Equations	Evolve density matrices, compare direct and vectorized equations, and identify stationary states
Quantum Jump Simulation	Generate counting records, conditional states, and waiting-time statistics
Diffusive Trajectory Simulation	Evolve continuous measurement records and compare ensemble averages with the master equation

For a first calculation, the Computational Notebooks overview contains a runnable standard-library Python example for amplitude damping. It prints populations, coherence, and purity without requiring a differential-equation solver. Then use the channel and Bloch-vector lessons to compare several inputs rather than testing one state only.

Move to a Lindblad solver once you can distinguish a channel’s finite-time probability from a generator’s rate. For zero-temperature damping, for example, the decay probability is $p(t)=1-e^{-\Gamma t}$, rather than $\Gamma t$ at arbitrary time. The linear expression is only its short-time approximation.

Add trajectories after the unconditional solution is understood. The comparison is

$$\frac1N\sum_{r=1}^N\rho_c^{(r)}(t)\longrightarrow\rho(t)$$

as the number of independent records increases. Individual records are not supposed to agree. Different unravelings may also have different fluctuations while sharing the same mean state.

Explicit environments, noise, control, and work

Lesson	Question to explore
Non-Markovian Toy Models	Can exact joint evolution produce reduced-state revivals, and which memory diagnostic detects them?
Decoherence Timescale Estimation	How do spectral cutoffs and pulse filters change a measured coherence envelope?
Optimal Control Toy Problems	How do bounded controls improve a stated state-transfer or gate objective under a specified noise model?
Quantum Thermodynamics Toy Models	How are work probabilities constructed, and when do fluctuation-relation checks apply?

A two-qubit explicit environment is often a simpler next step than a large dissipative simulation: diagonalize its joint Hamiltonian, evolve unitarily, and take the partial trace. A revival is then something to explain physically, rather than a numerical defect to suppress.

For noise calculations, distinguish a fitted exponential time from an envelope that is Gaussian or has several decay scales. For control, specify which inputs the objective tests: preparing one target state is not the same as implementing a gate on every input. For work distributions, specify the initial state and measurement protocol before interpreting an energy difference as work.

Understanding numerical disagreement

Different sources of error require different changes to the calculation.

Source of error	Useful comparison
Basis or tensor ordering	A hand-calculated matrix element or product state
Time discretization	Smaller timestep or a matrix exponential for the same generator
Oscillator truncation	Larger number-state cutoff and population near its boundary
Finite trajectory sample	More independent records and uncertainty of the estimated observable
Physical approximation	Exact small system–environment evolution or a different controlled approximation

Trace and positivity are essential structural checks, but a wrong channel can pass both. Likewise, agreement between two implementations can miss an assumption they share. Combine a solvable physical example with convergence tests suited to the actual observable.

Keep the parameters, units, tensor order, package versions, and random seed alongside a result you want to reproduce. A seed repeats one random sample; it does not estimate uncertainty or remove sampling bias. The linked lessons provide the model-specific calculations.

Exercises

More trajectories or smaller steps? The mean of $N$ independent trajectory measurements has a statistical standard error proportional to $N^{-1/2}$. How many trajectories reduce it by a factor of four? Does that fix a biased time integrator?

Solution

Use $16N$ trajectories. The statistical error shrinks, but timestep bias remains. Refine the timestep separately and compare averages with uncertainties small enough to reveal that bias.

A misleading physical check. Two channel implementations preserve trace and positivity but give different damping of an initial $|+\rangle$. What simple test helps?

Solution

For amplitude damping with probability $p$, check $\rho_{11}'=(1-p)/2$ and $\rho_{01}'=\sqrt{1-p}/2$. Confirm that both programs use the same ground-state ordering and that $p=1-e^{-\Gamma t}$ when comparing with a rate-based solver. Trace preservation alone cannot detect a wrong coherence factor.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Solvable models and stochastic methods.
• Johansson, J. R., P. D. Nation, and F. Nori. “QuTiP: An open-source Python framework for the dynamics of open quantum systems.” Computer Physics Communications 183, 1760–1772 (2012). doi:10.1016/j.cpc.2012.02.021. Computational examples; consult the documentation matching the installed software version.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Conditional evolution and ensemble comparisons.