Canonical examples of quantum channels

Question

I have a conjecture about quantum channels. On which examples should I test it before I try to prove it, ask it on StackExchange, or write a paper about it?

(Note: This is meant to be a reference question. But whenever I have a conjecture, I do test it on the channels listed, and it has saved me a lot of time trying to prove wrong statements.)

Answer 1

The following is a list of channels you can use to test your conjecture. (Some are special cases of subsequent ones -- it makes more sense to first test the special cases.) Here, $d$ is the dimension of the space.

1. The identity channel: $$ \mathcal E(\rho)=\rho\ . $$

2. The fully depolarizing channel: $$ \mathcal E(\rho) = \mathrm{tr}(\rho)\,\tfrac1d\mathbb I\ . $$

3. The depolarizing channel: $$ \mathcal E(\rho) = \gamma\rho + (1-\gamma)\mathrm{tr}(\rho)\,\tfrac1d\mathbb I $$ for $-\tfrac1d\le\gamma\le1$.

4. The dephasing channel $$ \mathcal E(\rho) = \gamma\rho + (1-\gamma)Z\rho Z $$ for qubits (with $Z$ the Pauli $Z$ matrix), and possibly some suitable generalizations for $d>2$. If your conjecture is not rotationally symmetric, test rotated versions as well.

5. The "Wirf weg und mach neu™" ("throw away and make new") channel: $$ \mathcal E(\rho) = \mathrm{tr}(\rho)\,\sigma $$ with $\sigma$ a density matrix. Obviously a generalization of 2, but test e.g. pure states $\sigma$.

6. The Holevo-Werner channel: $$ \mathcal E(\rho) = \tfrac{1}{d-1}(\mathrm{tr}(\rho)\,\mathbb I -\rho^T)\ , $$ where $\rho^T$ is the transposition.

7. Entanglement breaking channels: $$ \mathcal E(\rho) = \sum_i \mathrm{tr}(\rho F_i) \sigma_i $$ with $\sigma_i$ density matrices and the $F_i$ a POVM (i.e., $F_i\ge0$ and $\sum F_i=\mathbb I$). Obviously a generalization of 5: These are all channels which can be realized by first measuring the input and then preparing a new output conditioned on the measurement outcome. (You might want to test specific instances of these, like $F_i$ projectors onto subspaces, and $\sigma_i$ supported in the same subspace, etc.)

If your conjecture has passed all examples: Congratulations! It is probably true, and you can start proving it!