Chapter 5. The classical-quantum channel

Alexander Semenovich Holevo

John A. Cortese

This thesis studies classical communication over quantum channels. Chapter 1 describes an algebraic technique which extends several previously known qubit channel capacity results to the qudit quantum channel case. Chapter 2 derives a formula for the relative entropy function of two qubit density matrices in terms of their Bloch vectors. The application of the Bloch vector relative entropy formula to the determination of Holevo-Schumacher-Westmoreland (HSW) capacities for qubit quantum channels is discussed. Chapter 3 outlines several numerical simulation results which support theoretical conclusions and conjectures discussed in Chapters 1 and 2. Chapter 4 closes the thesis with comments, examples and discussion on the additivity of Holevo Chi and the HSW channel capacity.

E. Desurvire

This chapter introduces the notion of noisy quantum channels , and the different types of “quantum noise” that affect qubit messages passed through such channels. The main types of noisy channel reviewed here are the depolarizing , bit-flip , phase-flip , and bit-phase-flip channels. Then the quantum channel capacity χ is defined through the Holevo–Schumacher–Westmoreland (HSW) theorem. Such a theorem can conceptually be viewed as the elegant quantum counterpart of Shannon's (noisy) channel coding theorem , which was described in Chapter 13. Here, I shall not venture into the complex proof of the HSW theorem but only provide a background illustrating the similarity with its classical counterpart. The resemblance with the channel capacity χ and the Holevo bound , as described in Chapter 21, and with the classical mutual information H ( X ; Y ), as described in Chapter 5, are both discussed. For advanced reference, a hint is provided as to the meaning of the still not fully explored concept of quantum coherent information . Several examples of quantum channel capacity, derived from direct applications of the HSW theorem, along with the solution of the maximization problem, are provided. Noisy quantum channels The notion of “noisiness” in a classical communication channel was first introduced in Chapter 12, when describing channel entropy . Such a channel can be viewed schematically as a probabilistic relation between two random sources, X for the originator, and Y for the recipient.

Mark M. Wilde

The packing lemma is a general method for one party to “pack” or encode classical messages into a Hilbert space so that another party can distinguish the encoded messages. The first party can prepare an ensemble of quantum states, and the other party has access to a set of projectors using which he can form a quantum measurement. If the ensemble and the projectors satisfy the conditions of the packing lemma, then it guarantees the existence of a scheme by which the second party can distinguish the classical messages that the first party prepares. The statement of the packing lemma is quite general, and this approach has a great advantage because we can use it as a primitive for many coding theorems. Examples of coding theorems that we can prove using the packing lemma are the Holevo–Schumacher–Westmoreland (HSW) theorem for the transmission of classical information over a quantum channel and the entanglement-assisted classical capacity theorem for the transmission of classical information over an entanglement-assisted quantum channel (furthermore, Chapter 22 shows that these two protocols are sufficient to generate most known protocols in quantum Shannon theory). Combined with the covering lemma of the next chapter, the packing lemma gives a method for transmitting private classical information over a quantum channel, and this technique in turn gives a way to communicate quantum information over a quantum channel. As long as we can determine an ensemble and a set of projectors satisfying the conditions of the packing lemma, we can apply it in a straightforward way. For example, we prove the HSW coding theorem in Chapter 20 largely by relying on the properties of typical and conditionally typical subspaces that we proved in the previous chapter, and some of these properties are equivalent to the conditions of the packing lemma. The packing lemma is a “one-shot” lemma because it applies to a general scenario that is not limited only to i.i.d. uses of a quantum channel. This “oneshot” approach is part of the reason that we can apply it to a variety of situations. The technique of proving a “one-shot” result and applying it to the i.i.d.