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.)

• Is the conjecture to be found somewhere now? Oct 1 '20 at 11:08
• @DescheleSchilder What do you mean? Oct 1 '20 at 11:35
• I mean the conjecture you concocted. Has it been published? Oct 1 '20 at 12:05
• @DescheleSchilder Have you read the post fully, including the part in the brackets? Oct 1 '20 at 15:35
• @DescheleSchilder Precisely. There were a few questions here which made me realize that such a catalogue would be useful (since I could shoot down the conjectures with one of those examples). In fact, Emilio Pisanty lured me into writing the reference question by promising a bounty which he never awarded ;) As you can see from the linked question, I have also used those examples since. Oct 1 '20 at 17:40

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!

• Nice list. Is it useful to consider only depolarization along a specific subspace? Nov 9 '16 at 19:12
• @EmilioPisanty Probably. Guess I usually encounter qubit channels ... To be honest, in my experience 1, 2, 5 with pure $\sigma$, and 6 are enough to shoot down almost all wrong conjectures. Nov 9 '16 at 19:35