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.