# Advances in Quantum Channel Capacity

I have been reading about the Quantum Channel Capacity and it seems to be an open problem to find such capacity in general. Quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to a receiver.

Known results on the field are the Hashing bound, which is a lower bound on such quantum capacity and which is given by the LSD (Lloyd-Shor-Devetak) theorem; or the HSW (Holevo-Schumacher-Westmoreland) theorem for classical capacity over quantum channels.

I was wondering if there have been any advances in what a general expression for the quantum capacity since the release of those theorems. A glimpse on the advance in such field is enough for me, or references to papers where such task is developed.

Let's recap a bit:

In classical information theory, the analogous formula is the Shannon noisy channel coding theorem. It's charming, because it is basically just a very simple optimization of the mutual information.

The quantum channel capacity is that it is given by

$$\lim\limits_{n\to\infty} \frac{1}{n}Q(T^{\otimes n})$$

where $$T$$ is the quantum channel in question and $$Q$$ is the coherent information.

Obviously, we'd want a formula that doesn't depend on $$n$$ just like in the classical case. The problem is: It's known that such an expression cannot exist (see https://arxiv.org/abs/quant-ph/9706061). It gets worse: You could hope that there is a maximal $$n$$ after which you at least know that the capacity is zero. But that's false (published recently: Unbounded number of channel uses may be required to detect quantum capacity (Cubitt et al., 2015).