All the currently known examples of quantum channels with zero quantum capacity are either PPT or anti-degradable. These notions can be conveniently defined in terms of the Choi matrix of the given channel:
- A channel is said to be PPT if its Choi matrix is positive under partial transposition. In other words, the partial action of such a channel on any bipartite state results in a state which stays positive under partial transposition. Maximally entangled states cannot be distilled from such states at any positive rate, which is why these states are also called bound entangled. This fact can be used to show that quantum information cannot be reliably transmitted through these channels and hence these channels have zero quantum capacity.
- A channel is said to be anti-degradable if its Choi matrix admits a two-copy symmetric extension. Such a channel operates on any input state to yield a combined output-environment state which is symmetric under exchange. One can use this fact to show that if these channels can be used to reliably transmit quantum information, it would lead to a violation of the no-cloning theorem. Hence, anti-degradable channels have zero quantum capacity.
The aforementioned classes of channels are really distinct in the sense that one class is not a subset of the other or vice versa. The two classes have a non-trivial intersection which is known to contain the well-know class of entanglement-breaking channels. Interested readers should refer to this wonderful paper for a more thorough discussion on these concepts.
I am interested in examples of such zero capacity channels $\Phi : M_d \rightarrow M_d$ which have Choi ranks strictly less than $d$. Recall that the Choi rank of a channel is the rank of its Choi matrix which equals the minimal number of Kraus operators required in the channel's Kraus representation. For $d=2$, the rank constraint forces us to look at only unitary conjugations of the form $\Phi (\rho) = U\rho U^\dagger$, which have strictly positive capacity. For the next dimension $d=3$, all examples of zero capacity channels that I know of have Choi ranks $\geq 3$ (such as the PPT channels in the depolarizing/dephasing family or the anti-degradable amplitude damping channels). Hence the question: Are there known examples of zero capacity qutrit channels with Choi rank $=2$?