Notebook Index

This index organizes the computational notebook contracts for measurement, decoherence, and open quantum systems. Each linked page states what a notebook should compute, which conventions it must declare, and which validation tests are required before numerical output should be cited.

As of this review, the linked pages are admission contracts rather than promoted reproduced artifacts. They are still useful: they define the models, tests, tolerances, and expected outputs needed to turn exploratory notebooks into trustworthy scientific objects.

For site-wide notebook status and validation policy, see Notebooks, Validation Tests, and Reproducibility Status.

Skill-Level Map

Skill Level	Notebook Contract	Core Skill	Best Starting Pages
beginner	Simulating Quantum Channels	Apply finite-dimensional channels and validate Kraus, Choi, trace, and positivity checks.	Completely Positive Maps, Kraus Representation, Choi Matrix
beginner	Bloch Vector Noise Models	Visualize qubit dephasing, depolarizing, and amplitude-damping noise in Bloch-vector form.	Common Noise Channels, Dephasing Channel, Amplitude Damping Channel
intermediate	Solving Lindblad Equations	Build Liouvillians, integrate finite-dimensional master equations, compute steady states, and inspect spectra.	Lindblad–GKSL Equation, Lindblad Operators
intermediate	Quantum Jump Simulation	Simulate conditioned jump trajectories, ensemble averages, and waiting-time histograms.	Quantum-Jump Trajectories, Quantum Optical Master Equation
intermediate	Diffusive Trajectory Simulation	Simulate continuous weak-measurement records and validate diffusive stochastic master equations.	Stochastic Master Equations, Diffusive Trajectories
advanced	Non-Markovian Toy Models	Evolve small system-environment models exactly, trace out the environment, and diagnose revivals.	Non-Markovian Dynamics, Information Backflow, Reduced Dynamics
advanced	Decoherence Timescale Estimation	Estimate coherence envelopes and $T_2$, $T_2^*$, and $T_\phi$ from noise spectra and filters.	Pure Dephasing Master Equation, Noise Spectra, Dynamical Decoupling
advanced	Optimal Control Toy Problems	Optimize small driven-qubit pulses under pure dephasing and validate state-transfer or gate-style fidelities.	Lindblad–GKSL Equation, Pure Dephasing Master Equation
advanced	Quantum Thermodynamics Toy Models	Compute TPM work distributions for driven two-level systems and verify Jarzynski–Crooks checks.	Two-Point Measurement Scheme, Work Distributions, Jarzynski Equality and Crooks Relation

Recommended Path

1. Start with Simulating Quantum Channels. It fixes finite-dimensional state conventions, Choi conventions, and sanity checks for trace preservation and complete positivity.

2. Move to Solving Lindblad Equations. This introduces continuous-time generators, vectorized Liouvillians, steady states, and finite-time channel validation.

3. Add Quantum Jump Simulation after the unconditional Lindblad solver works. Trajectories should average back to the master equation before being interpreted as measurement records.

4. Use Non-Markovian Toy Models once exact joint evolution, partial trace, and Markovian comparison dynamics are familiar.

5. Add specialized advanced notebooks according to the question: Decoherence Timescale Estimation for filter-function and spectrum calculations, Optimal Control Toy Problems for driven-qubit pulse checks, and Quantum Thermodynamics Toy Models for finite work distributions and fluctuation-relation tests.

This path mirrors the conceptual escalation:

channels -> generators -> conditioned trajectories -> explicit environments, control, and thermodynamic records

Beginner: Channels and Bloch-Sphere Diagnostics

The channel notebook should stay small enough that every result can be checked analytically. Qubit dephasing, depolarizing, and amplitude-damping channels are enough to expose the main issues:

• basis ordering,
• Kraus normalization,
• Choi convention,
• Bloch-vector action,
• trace preservation,
• complete positivity,
• limiting cases.

The notebook should never treat a visually plausible noisy state as validation. It must compute numerical residuals and compare with known formulas.

Intermediate: Lindblad Solvers and Trajectories

The Lindblad solver notebook owns deterministic evolution:

$$\dot\rho = \mathcal L(\rho).$$

It should compare at least two implementations of the same generator, such as direct density-matrix right-hand sides and matrix-vectorized Liouvillians. It should also compare numerical integration with analytic qubit solutions or matrix exponentials when available.

The jump notebook owns conditioned stochastic evolution. Its most important validation is

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

where the left side averages many trajectories and the right side is the unconditional master-equation solution. Without that agreement, trajectory plots are not trustworthy.

Advanced: Memory, Control, and Thermodynamic Records

The advanced notebook contracts are still admission contracts, not promoted reproduced artifacts. They now cover several distinct research skills:

Notebook Contract	Nearest Conceptual Pages	What It Should Validate
Non-Markovian Toy Models	Non-Markovian Dynamics and Information Backflow	exact joint evolution, partial trace, Markovian comparison, and revival diagnostics
Decoherence Timescale Estimation	Noise Spectra and Dynamical Decoupling	spectral conventions, filter functions, cutoff sensitivity, and fitted timescale definitions
Optimal Control Toy Problems	Reservoir Engineering and Dynamical Decoupling	objective definitions, optimizer reproducibility, amplitude constraints, and robustness checks
Diffusive Trajectory Simulation	Stochastic Master Equations and Diffusive Trajectories	innovation statistics, ensemble averages, detection-efficiency conventions, and positivity checks
Quantum Thermodynamics Toy Models	Quantum Thermodynamics and Fluctuation Theorems	protocol definition, work convention, branch-probability normalization, and fluctuation-relation residuals

Hierarchical equations of motion examples remain planned. They should be added only when hierarchy truncation, bath-correlation expansion, and tier-depth convergence checks can be stated precisely.

Research-Tool Expectations

Research-facing notebooks need more than a working plot. A notebook should declare:

• package versions and random seeds,
• basis and tensor-product ordering,
• vectorization convention,
• units and rate conventions,
• solver tolerances,
• accepted residual thresholds,
• cutoff and convergence checks,
• date and commit identifier when promoted.

It should also contain at least one independent check: analytic solution, conservation law, trace/positivity residual, finite-time channel test, convergence with timestep, or comparison to a simpler limiting model.

Common Mistakes

• Treating an exploratory notebook as a reproduced result.
• Comparing trajectories to each other instead of checking their ensemble average.
• Mixing row-stacking and column-stacking vectorization.
• Forgetting that oscillator truncations require convergence checks.
• Plotting a density matrix without checking Hermiticity, trace, and positivity.
• Reporting a stochastic result without random seed, trajectory count, and confidence estimate.

Self-Checks

1. Which notebook should come first if you cannot yet build and validate a Choi matrix?

Solution

Start with Simulating Quantum Channels. It establishes the finite-dimensional channel and Choi conventions needed by later notebooks.

2. A quantum-jump simulation produces beautiful individual trajectories, but their average does not match the Lindblad solution. Can the trajectories be cited?

Solution

No. Agreement of the trajectory ensemble average with the unconditional master equation is a core validation requirement.

3. A non-Markovian toy notebook uses an oscillator truncation. What check is mandatory?

Solution

The notebook must check convergence with respect to the oscillator cutoff or otherwise justify why the chosen finite subspace is exact for the initial condition and dynamics.

References

• J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open quantum systems,” Computer Physics Communications 183, 1760, 2012.
• J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP 2: A Python framework for the dynamics of open quantum systems,” Computer Physics Communications 184, 1234, 2013.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer, 1999.