Canonical examples of quantum channels 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.) 
 A: 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. 


*

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

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

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

*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.

*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 $\sigma$.

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

*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!
