Bloch Vector Noise Models

This notebook guide specifies a reproducible qubit calculation for visualizing noise models in Bloch-vector form. The goal is to simulate dephasing, depolarizing, and amplitude-damping channels, plot Bloch-vector trajectories, and compare discrete channels with continuous-time master-equation limits.

As of this review, no executable notebook under notebooks/density-open-systems/bloch-vector-noise-models/ is promoted as a reproduced artifact. This page is the admission contract for that notebook: it states the models, basis conventions, analytic checks, plotting requirements, and validation tests required before its numerical outputs should be cited.

Purpose

The notebook should demonstrate how to:

• convert between density matrices and Bloch vectors;
• apply standard qubit noise channels to fixed test states;
• visualize contractions, translations, and fixed points inside the Bloch ball;
• compare channel parameters with continuous-time rates;
• validate trace preservation, Hermiticity, positivity, and Bloch-ball constraints;
• diagnose convention mistakes in basis ordering and Pauli-matrix signs;
• keep exploratory plots separate from accepted reproducible outputs.

The first version should stay deliberately small. A single qubit is enough to expose the relationship between channel formulas, master-equation rates, and geometric intuition.

Directory Plan

Use a dedicated directory:

notebooks/density-open-systems/bloch-vector-noise-models/
  bloch-vector-noise-models.ipynb
  README.md

The opening notebook cell or README.md should state:

• Python and package versions;
• basis ordering and Pauli-matrix convention;
• whether $|0\rangle$ is the $+z$ or $-z$ Bloch-sphere pole;
• channel parameter definitions;
• rate conventions for continuous-time comparisons;
• numerical tolerance for positivity and trace checks;
• plotting backend and saved-output paths;
• date and commit identifier when the notebook is promoted.

Bloch Representation

Use

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right),$$

with

$$r_j = \operatorname{Tr}(\rho\sigma_j), \qquad j=x,y,z.$$

The notebook should use the standard matrices

$$\sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

With this convention, the state $|0\rangle=(1,0)^{\mathsf T}$ is the $+z$ pole and $|1\rangle=(0,1)^{\mathsf T}$ is the $-z$ pole.

A physical qubit state satisfies

$$\left\|\mathbf r\right\|_2 \le 1.$$

The notebook should verify this inequality after every channel application. A norm slightly above $1$ at the level of roundoff should be reported separately from a true positivity failure.

Affine Channel Form

Every qubit channel acts on Bloch vectors as an affine map

$$\mathbf r' = M\mathbf r+\mathbf c,$$

where $M$ is a real $3\times3$ matrix and $\mathbf c$ is a real translation vector. Unital channels have $\mathbf c=0$ and send the maximally mixed state to itself. Amplitude damping is not unital; it translates the Bloch ball toward the ground state.

The notebook should compute $M$ and $\mathbf c$ numerically from channel action on the four states

$$\frac{I}{2}, \qquad \frac{I+\sigma_x}{2}, \qquad \frac{I+\sigma_y}{2}, \qquad \frac{I+\sigma_z}{2},$$

then compare against analytic formulas. This catches transposition, basis, and sign mistakes that may not show up in one trajectory.

Baseline Channels

Implement at least three noise models.

Dephasing

Use the dephasing-channel convention

$$(r_x,r_y,r_z) \mapsto (\lambda r_x,\lambda r_y,r_z), \qquad -1\le\lambda\le1.$$

For a continuous pure-dephasing model with coherence decay rate $\Gamma_\phi$,

$$\lambda(t) = e^{-\Gamma_\phi t}.$$

The notebook should plot a trajectory whose transverse components shrink while $r_z$ is unchanged. It should also test a state on the equator, where the visual contraction is easiest to see.

Depolarizing

Use the qubit depolarizing-channel convention

$$\mathbf r \mapsto \lambda\mathbf r.$$

Complete positivity requires

$$- \frac13 \le \lambda \le 1.$$

The notebook should test at least one valid negative $\lambda$ and one invalid value below $-1/3$. The invalid value should fail a Choi-positivity check in the companion channel-simulation helpers, not merely look odd in a plot.

For a continuous-time depolarizing semigroup, one common parametrization is

$$\lambda(t) = e^{-\Gamma_{\mathrm{dep}}t}.$$

The notebook must state which generator convention produces this decay.

Amplitude Damping

Use the zero-temperature amplitude-damping Kraus operators

$$K_0 = \begin{pmatrix} 1&0\\ 0&\sqrt{1-p} \end{pmatrix}, \qquad K_1 = \begin{pmatrix} 0&\sqrt p\\ 0&0 \end{pmatrix}, \qquad 0\le p\le1.$$

With the $|0\rangle$ as $+z$ convention, the Bloch-vector action is

$$\begin{aligned} r_x'&=\sqrt{1-p}\,r_x,\\ r_y'&=\sqrt{1-p}\,r_y,\\ r_z'&=p+(1-p)r_z. \end{aligned}$$

For the continuous spontaneous-emission master equation,

$$p(t) = 1-e^{-\gamma t}.$$

The notebook should show that all initial states flow toward the ground-state fixed point

$$\mathbf r_\infty = (0,0,1).$$

Channel Versus Master-Equation Limits

The notebook should compare finite-time channels with master-equation solutions in at least two ways:

1. Use analytic finite-time parameters such as $\lambda(t)=e^{-\Gamma_\phi t}$ and $p(t)=1-e^{-\gamma t}$.
2. Use the finite-time map $\Phi_t=e^{t\mathcal L}$ from the vectorized Liouvillian in Solving Lindblad Equations.

The comparison should report a numerical error such as

$$\epsilon(t) = \max_{\rho\in\mathcal S} \left\| \Phi_t^{\mathrm{analytic}}(\rho) - \Phi_t^{\mathrm{numeric}}(\rho) \right\|_F,$$

where $\mathcal S$ is a declared finite test set and $\|\cdot\|_F$ is the Frobenius norm.

This page does not require a full proof that the sampled set detects every possible error. It requires enough fixed tests that a convention mistake is likely to fail loudly.

Test States

Use a deterministic set of initial states:

$$\begin{aligned} \rho_{+x}&=\frac12(I+\sigma_x),& \rho_{+y}&=\frac12(I+\sigma_y),\\ \rho_{+z}&=\frac12(I+\sigma_z),& \rho_{-z}&=\frac12(I-\sigma_z),\\ \rho_{\mathrm{mix}}&=\frac12I,& \rho_{\mathrm{off}}&= \frac12 \left( I+ \frac{\sigma_x+\sigma_y+\sigma_z}{\sqrt3} \right). \end{aligned}$$

Random states may be added after the deterministic tests pass, but they should not replace them.

Plot Requirements

The accepted notebook should produce:

• a Bloch-ball or Bloch-disk plot for dephasing showing transverse contraction;
• a depolarizing plot showing isotropic shrinkage toward the origin;
• an amplitude-damping plot showing translation toward $+z$;
• time traces of $r_x(t)$, $r_y(t)$, and $r_z(t)$ for at least one initial state;
• an error-versus-time or error-versus-step-size diagnostic for a continuous-time comparison.

Plots should label the basis convention and channel parameter. A beautiful Bloch sphere is less important than a plot whose coordinates can be checked from the saved data.

Validation Tests

The notebook should include automated checks:

Check	Required condition
Hermiticity	$\|\rho-\rho^\dagger\|_F$ below tolerance
trace	$
positivity	smallest eigenvalue above negative tolerance
Bloch norm	$\|\mathbf r\|_2\le1$ within tolerance
Kraus completeness	$\|\sum_rK_r^\dagger K_r-I\|_F$ below tolerance
analytic Bloch map	matrix-vector result agrees with formula
finite-time limit	channel and Liouvillian propagation agree

At least one deliberately invalid depolarizing parameter should fail positivity. A validation notebook that never tests a bad input may be less trustworthy than one that proves it can reject one.

Accepted Outputs

The notebook should save:

notebooks/density-open-systems/bloch-vector-noise-models/outputs/
  dephasing-bloch-trajectory.svg
  depolarizing-bloch-trajectory.svg
  amplitude-damping-bloch-trajectory.svg
  bloch-noise-validation.json

The validation file should include:

• package versions;
• declared basis convention;
• channel parameters;
• numerical tolerances;
• maximum trace error;
• minimum eigenvalue observed;
• maximum analytic-comparison error;
• whether the invalid-channel test failed as expected.

No page in the documentation should cite these plots as authoritative until the validation file exists and the notebook has a reproducibility status entry.

Common Mistakes

• Using $|0\rangle$ as the north pole in one cell and the south pole in another.
• Confusing the dephasing shrink factor $\lambda$ with a probability parameter.
• Applying amplitude damping with the ground and excited states swapped.
• Assuming every affine map that keeps a few plotted points inside the Bloch ball is completely positive.
• Comparing a channel parameter $p$ directly with a rate $\gamma$ instead of using $p(t)=1-e^{-\gamma t}$.
• Letting a numerical ODE step produce a slightly unphysical state without measuring the error.
• Drawing Bloch trajectories without saving the numeric coordinates used to make them.

Exercises

Bloch Vector of a Density Matrix

For

$$\rho = \begin{pmatrix} 3/4&1/4\\ 1/4&1/4 \end{pmatrix},$$

compute the Bloch vector in the convention of this page.

Solution

Use $r_j=\operatorname{Tr}(\rho\sigma_j)$. The diagonal entries give

$$r_z = \rho_{00}-\rho_{11} = \frac34-\frac14 = \frac12.$$

The real off-diagonal part gives

$$r_x = \rho_{01}+\rho_{10} = \frac12,$$

and the imaginary off-diagonal part gives

$$r_y = -i\rho_{01}+i\rho_{10} = 0.$$

Thus

$$\mathbf r = \left( \frac12,0,\frac12 \right).$$

Dephasing Trajectory

An initial state has Bloch vector $\mathbf r(0)=(1,0,0)$. Under pure dephasing with $\lambda(t)=e^{-\Gamma_\phi t}$, what is $\mathbf r(t)$?

Solution

Dephasing maps

$$(r_x,r_y,r_z) \mapsto (\lambda r_x,\lambda r_y,r_z).$$

Therefore

$$\mathbf r(t) = \left( e^{-\Gamma_\phi t},0,0 \right).$$

The state moves inward along the $x$ axis toward the maximally mixed state.

Amplitude-Damping Fixed Point

Show that the amplitude-damping affine map on this page has fixed point $(0,0,1)$ for any $0\lt p\le1$.

Solution

The map is

$$r_x'=\sqrt{1-p}\,r_x, \qquad r_y'=\sqrt{1-p}\,r_y, \qquad r_z'=p+(1-p)r_z.$$

A fixed point satisfies $r_x'=r_x$ and $r_y'=r_y$. For $p\gt0$, $\sqrt{1-p}\ne1$, so $r_x=r_y=0$.

For $r_z$,

$$r_z = p+(1-p)r_z.$$

Thus

$$pr_z=p,$$

and since $p\gt0$,

$$r_z=1.$$

The fixed point is therefore $(0,0,1)$, the ground state in this convention.

Invalid Depolarizing Parameter

Why should a notebook deliberately test a depolarizing shrink factor $\lambda\lt-1/3$?

Solution

For a qubit depolarizing channel, complete positivity requires $-1/3\le\lambda\le1$. A value below $-1/3$ can still send some individual Bloch vectors to points inside the Bloch ball, so plotting a few states is not enough. Testing an invalid value verifies that the Choi-positivity check can reject maps that are positive-looking in limited samples but not valid quantum channels.

Cross-Links

• Bloch Sphere
• Common Noise Channels
• Dephasing Channel
• Depolarizing Channel
• Amplitude-Damping Channel
• Pure Dephasing Master Equation
• Amplitude Damping Master Equation
• Simulating Quantum Channels
• Solving Lindblad Equations

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• I. Bengtsson and K. Życzkowski, Geometry of Quantum States, Cambridge University Press, 2006.
• J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation, California Institute of Technology.