Choose amplitude damping with \(\gamma=9/25\):
The channel \(\Phi(\rho)=\sum_aK_a\rho K_a^*\) is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational.
Reading the drawing
The left matrix acts on the ordered matrix entries shown by its row and column labels. The middle annotations identify the damping parameter and coherence factor. The smaller matrix at right is the output from the displayed input, with its trace and determinant below.
Drawing labels and data 33 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
- Superoperator on matrix entries
| Row / column | 00 | 01 | 10 | 11 |
|---|---|---|---|---|
| 00 | \(1\) | \(0\) | \(0\) | \(\frac{9}{25}\) |
| 01 | \(0\) | \(\frac{4}{5}\) | \(0\) | \(0\) |
| 10 | \(0\) | \(0\) | \(\frac{4}{5}\) | \(0\) |
| 11 | \(0\) | \(0\) | \(0\) | \(\frac{16}{25}\) |
- Output from \({\left|+\right\rangle}\)
| Row / column | Column 1 | Column 2 |
|---|---|---|
| Row 1 | \(\frac{17}{25}\) | \(\frac{2}{5}\) |
| Row 2 | \(\frac{2}{5}\) | \(\frac{8}{25}\) |
- \(\gamma=9/25\)
coherence factor
\(4/5\)
- \({\operatorname{Tr}}\rho=1,\quad\det\rho=36/625>0\)
The superoperator uses the entry order \((\rho_{00},\rho_{01},\rho_{10},\rho_{11})\). Population leaves the excited state with factor \(9/25\), while coherences contract by \(4/5\). These are not the same rate or the same coordinate.
For \(\rho_+=\frac12{\begin{pmatrix}1&1\\1&1\end{pmatrix}}\), the exact output is
Its trace is one and determinant is \(36/625>0\), so it is a mixed positive state. Arrows that show population transfer alone would leave its off-diagonal entries unspecified.
How complete positivity is certified
The Choi matrix is an explicit sum of two outer products of vectorized Kraus operators. Its eigenvalues are \(41/25,9/25,0,0\), all nonnegative. This gives an independent finite certificate in addition to Kraus normalization. The Choi matrix has trace two under the convention used by the check, which leaves the maximally entangled vector unnormalized.
Read this as
A packet channel acts on pairs of packet addresses, because density matrices have two indices. Population transfer alone does not reconstruct a quantum channel.