Chapter 8. The classical capacity of quantum channel

Alexander S. 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

Before quantum information became an established discipline, John R. Pierce issued the following quip at the end of his 1973 retrospective article on the history of information theory (Pierce, 1973): “I think that I have never met a physicist who understood information theory. I wish that physicists would stop talking about reformulating information theory and would give us a general expression for the capacity of a channel with quantum effects taken into account rather than a number of special cases.” Since the publication of Pierce's article, we have learned much more about quantum mechanics and information theory than he might have imagined at the time, but we have also realized that there is much more to discover. In spite of all that we have learned, we still unfortunately have not been able to address Pierce's concern in the above quote in full generality. The most basic question that we could ask in quantum Shannon theory (and the one with which Pierce was concerned) is how much classical information a sender can transmit to a receiver by exploiting a quantum channel. We have determined many special cases of quantum channels for which we do know their classical capacities, but we also now know that this most basic question is still wide open in the general case. What Pierce might not have imagined at the time is that a quantum channel has a much larger variety of capacities than does a classical channel. For example, we might wish to determine the classical capacity of a quantum channel assisted by entanglement shared between the sender and receiver. We have seen that in the simplest of cases, such as the noiseless qubit channel, shared entanglement boosts the classical capacity up to two bits, and we now refer to this phenomenon as the super-dense coding effect (see Chapter 6). Interestingly, the entanglementassisted capacity of a quantum channel is one of the few scenarios where we can claim to have a complete understanding of the channel's transmission capabilities. From the results regarding the entanglement-assisted capacity, we have learned that shared entanglement is often a “friend” because it tends to simplify results in both quantum Shannon theory and other subfields of quantum information science.

Masanori Ohya, Dėnes Petz, Noboru Watanabe