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.

Entanglement is one of the most useful resources in quantum information processing. If a sender and receiver share noiseless entanglement in the form of maximally entangled states, then Chapter 6 showed how they can teleport quantum bits between each other with the help of classical communication, or they can double the capacity of a noiseless qubit channel for transmitting classical information. We will see further applications in Chapter 20 where they can exploit noiseless entanglement to assist in the transmission of classical or quantum data over a noisy quantum channel.

Given the utility of maximal entanglement, a reasonable question is to ask what a sender and receiver can accomplish if they share pure entangled states that are not maximally entangled. In the quantum Shannon-theoretic setting, we make the further assumption that the sender and receiver can share many copies of these pure entangled states. We find out in this chapter that they can “concentrate” these non-maximally entangled states to maximally entangled ebits, and the optimal rate at which they can do so in the asymptotic limit is equal to the “entropy of entanglement” (the von Neumann entropy of half of one copy of the original state). Entanglement concentration is thus another fundamental task in noiseless quantum Shannon theory, and it gives a different operational interpretation to the von Neumann entropy.

Entanglement concentration is perhaps complementary to Schumacher compression in the sense that it gives a firm quantum information-theoretic interpretation of the term “ebit” (just as Schumacher compression did for the term “qubit”), and it plays a part in demonstrating how the entropy of entanglement is the unique measure of entanglement for pure bipartite states.

We have now seen in Chapters 19–21 how Alice can communicate classical or quantum information to Bob, perhaps even with the help of shared entanglement. One might argue that these communication tasks are the most fundamental tasks in quantum Shannon theory, given that they have furthered our understanding of the nature of information transmission over quantum channels. Though, when discussing the communication of classical information, we made no stipulation as to whether this classical information should be public, so that any third party might have partial or full access to it, or private, so that any third party does not have access.

This chapter introduces the private classical capacity theorem, which gives the maximum rate at which Alice can communicate classical information privately to Bob without anyone else in the universe knowing what she sent to him. The information-processing task corresponding to this theorem was one of the earliest studied in quantum information theory, with the Bennett–Brassard-84 quantum key distribution protocol being the first proposed protocol for exploiting quantum mechanics to establish a shared secret key between two parties. The private classical capacity theorem is important for quantum key distribution because it establishes the maximum rate at which two parties can generate a shared secret key.

This chapter marks the beginning of our study of the asymptotic theory of quantum information, where we develop the technical tools underpinning this theory. The intuition for it is similar to the intuition we developed in the previous chapter on typical sequences, but we will find some important differences between the classical and quantum cases.

So far, there is not a single known information-processing task in quantum Shannon theory where the tools from this chapter are not helpful in proving the achievability part of a coding theorem. For the most part, we can straight-forwardly import many of the ideas from the previous chapter about typical sequences for use in the asymptotic theory of quantum information. Though, one might initially think that there are some obstacles to doing so. For example, what is the analogy of a quantum information source? Once we have established this notion, how would we determine if a state emitted from a quantum information source is a typical state? In the classical case, a simple way of determining typicality is to inspect all of the bits in the sequence. But there is a problem with this approach in the quantum domain—“looking at quantum bits” is equivalent to performing a measurement and doing so destroys delicate superpositions that we would want to preserve in any subsequent quantum information-processing task.

So how can we get around the aforementioned problem and construct a useful notion of quantum typicality? Well, we should not be so destructive in determining the answer to a question when it has only two possible answers.

This chapter unifies all of the channel coding theorems that we have studied in this book. One of the most general information-processing tasks that a sender and receiver can accomplish is to transmit classical and quantum information and generate entanglement with many independent uses of a quantum channel and with the assistance of classical communication, quantum communication, and shared entanglement. The resulting rates for communication are net rates that give the generation rate of a resource less its consumption rate. Since we have three resources, all achievable rates are rate triples (C, Q, E) that lie in a three-dimensional capacity region, where C is the net rate of classical communication, Q is the net rate of quantum communication, and E is the net rate of entanglement consumption/generation. The capacity theorem for this general scenario is known as the quantum dynamic capacity theorem, and it is the main theorem that we prove in this chapter. All of the rates given in the channel coding theorems of previous chapters are special points in this three-dimensional capacity region.

The proof of the quantum dynamic capacity theorem comes in two parts: the direct coding theorem and the converse theorem. The direct coding theorem demonstrates that the strategy for achieving any point in the three-dimensional capacity region is remarkably simple: we just combine the protocol from Corollary 21.5.3 for entanglement-assisted classical and quantum communication with the three unit protocols of teleportation, super-dense coding, and entanglement distribution.