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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.

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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.

Now let's try to answer your question:

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).

In addition, if you use two different channels, their capacity can both be zero while the capacity when used together is larger than zero (see arXiv:0807.4935), which makes it even more difficult to imagine simple formulas for channels.

This implies that the best we can hope for is formulas for particular channels or particular subclasses of channels. There are a few results scattered throughout the literature. For instance, the capacities of certain Gaussian bosonic channels are known (arXiv:quant-ph/0606132).


However, please note that the quantum capacity is only one of many capacities defined and it's not necessarily the most interesting one. A different capacity which is often discussed in the literature is the classical capacity of a quantum channel (i.e. how much classical information can I send over a quantum channel?).

Let me point you to a recent review providing a lot of pointers to the literature: arXiv:1801.02019.

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  • $\begingroup$ Thanks for the answer, I will go through the literature you sent to see if I can get some more insight about the field! $\endgroup$ – Josu Etxezarreta Martinez Jul 2 '18 at 8:04

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