The goal of the covering lemma is perhaps opposite to that of the packing lemma because it applies in a setting where one party wishes to make messages indistinguishable to another party (instead of trying to make them distinguishable as in the packing lemma of the previous chapter). That is, the covering lemma is helpful when one party is trying to simulate a noisy channel to another party, rather than trying to simulate a noiseless channel. One party can accomplish this task by randomly covering the Hilbert space of the other party (this viewpoint gives the covering lemma its name).

One can certainly simulate noise by choosing a quantum state uniformly at random from a large set of quantum states and passing along the chosen quantum state to a third party without telling which state was chosen. But the problem with this approach is that it could potentially be expensive if the set from which we choose a random state is large, and we would really like to use as few resources as possible in order to simulate noise. That is, we would like the set from which we choose a quantum state uniformly at random to be as small as possible when simulating noise. The covering lemma is similar to the packing lemma in the sense that its conditions for application are general (involving bounds on projectors and an ensemble), but it gives an asymptotically efficient scheme for simulating noise when we apply it in an IID setting.

The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information.

The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B)–H(E)—a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain.