This chapter demonstrates the power of both coherent communication from Chapter 7 and the particular protocol for entanglement-assisted classical coding from the previous chapter. Recall that coherent dense coding is a version of the dense coding protocol in which the sender and receiver perform all of its steps coherently. Since our protocol for entanglement-assisted classical coding from the previous chapter is really just a glorified dense coding protocol, the sender and receiver can perform each of its steps coherently, generating a protocol for entanglement-assisted coherent coding. Then, by exploiting the fact that two coherent bits are equivalent to a qubit and an ebit, we obtain a protocol for entanglement-assisted quantum coding that consumes far less entanglement than a naive strategy would in order to accomplish this task. We next combine this entanglement-assisted quantum coding protocol with entanglement distribution (Section 6.2.1) and obtain a protocol for which the channel's coherent information (Section 12.5) is an achievable rate for quantum communication. This sequence of steps demonstrates an alternate proof of the direct part of the quantum channel coding theorem stated in Chapter 23. Entanglement-assisted classical communication is one generalization of superdense coding, in which the noiseless qubit channel becomes an arbitrary noisy quantum channel while the noiseless ebits remain noiseless. Another generalization of super-dense coding is a protocol named noisy super-dense coding , in which the shared entanglement becomes a shared noisy state ρ AB and the noiseless qubit channels remain noiseless.
This chapter demonstrates the power of both coherent communication from Chapter 7 and the particular protocol for entanglement-assisted classical coding from the previous chapter. Recall that coherent dense coding is a version of the dense coding protocol in which the sender and receiver perform all of its steps coherently. Since our protocol for entanglement-assisted classical coding from the previous chapter is really just a glorified dense coding protocol, the sender and receiver can perform each of its steps coherently, generating a protocol for entanglement-assisted coherent coding. Then, by exploiting the fact that two coherent bits are equivalent to a qubit and an ebit, we obtain a protocol for entanglement-assisted quantum coding that consumes far less entanglement than a naive strategy would in order to accomplish this task. We next combine this entanglement-assisted quantum coding protocol with entanglement distribution (Section 6.2.1) and obtain a protocol for which the channel's coherent information (Section 12.5) is an achievable rate for quantum communication. This sequence of steps demonstrates an alternate proof of the direct part of the quantum channel coding theorem stated in Chapter 23. Entanglement-assisted classical communication is one generalization of superdense coding, in which the noiseless qubit channel becomes an arbitrary noisy quantum channel while the noiseless ebits remain noiseless. Another generalization of super-dense coding is a protocol named noisy super-dense coding , in which the shared entanglement becomes a shared noisy state ρ AB and the noiseless qubit channels remain noiseless.
This chapter demonstrates the power of both coherent communication from Chapter 7 and the particular protocol for entanglement-assisted classical coding from the previous chapter. Recall that coherent dense coding is a version of the dense coding protocol in which the sender and receiver perform all of its steps coherently.1 Since our protocol for entanglement-assisted classical coding from the previous chapter is really just a glorified dense coding protocol, the sender and receiver can perform each of its steps coherently, generating a protocol for entanglement-assisted coherent coding. Then, by exploiting the fact that two coherent bits are equivalent to a qubit and an ebit, we obtain a protocol for entanglement-assisted quantum coding that consumes far less entanglement than a naive strategy would in order to accomplish this task. We next combine this entanglement-assisted quantum coding protocol with entanglement distribution (Section 6.2.1) and obtain a protocol for quantum communication at a rate equal to the channel's coherent information (Section 13.5). This sequence of steps demonstrates an alternate proof of the direct part of the quantum capacity theorem stated in Chapter 24. Entanglement-assisted classical communication is one generalization of superdense coding, in which the noiseless qubit channel becomes an arbitrary noisy quantum channel while the noiseless ebits remain noiseless. Another generalization of super-dense coding is a protocol named noisy super-dense coding , in which the shared entanglement becomes a shared noisy state ρ AB and the noiseless qubit channels remain noiseless. Interestingly, the protocol that we employ in this chapter for noisy super-dense coding is essentially equivalent to the protocol from the previous chapter for entanglement-assisted classical communication, with some slight modifications to account for the different setting. We can also construct a coherent version of noisy super-dense coding, leading to a protocol that we name coherent state transfer . Coherent state transfer accomplishes not only the task of generating coherent communication between Alice and Bob, but it also allows Alice to transfer her share of the state ρ AB to Bob.
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.1 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 22.5.2 for entanglement-assisted classical and quantum communication with the three unit protocols of teleportation, super-dense coding, and entanglement distribution. The interpretation of the achievable rate region is that it is the unit resource capacity region from Chapter 8 translated along the points achievable with the protocol from Corollary 22.5.2. In the proof of the converse theorem, we analyze the most general protocol that can consume and generate classical communication, quantum communication, and entanglement along with the consumption of many independent uses of a quantum channel, and we show that the net rates for such a protocol are bounded by the regularization of the achievable rate region.
This chapter begins our first exciting application of the postulates of the quantum theory to quantum communication. We study the fundamental, unit quantum communication protocols. These protocols involve a single sender, whom we name Alice, and a single receiver, whom we name Bob. The protocols are ideal and noiseless because we assume that Alice and Bob can exploit perfect classical communication, perfect quantum communication, and perfect entanglement. At the end of this chapter, we suggest how to incorporate imperfections into these protocols for later study. Alice and Bob may wish to perform one of several quantum information-processing tasks, such as the transmission of classical information, quantum information, or entanglement. Several fundamental protocols make use of these resources: 1. We will see that noiseless entanglement is an important resource in quantum Shannon theory because it enables Alice and Bob to perform other protocols that are not possible with classical resources only. We will present a simple, idealized protocol for generating entanglement, named entanglement distribution . 2. Alice may wish to communicate classical information to Bob. A trivial method, named elementary coding , is a simple way for doing so and we discuss it briefly. 3. A more interesting technique for transmitting classical information is superdense coding . It exploits a noiseless qubit channel and shared entanglement to transmit more classical information than would be possible with a noiseless qubit channel alone. […]
This chapter begins our first exciting application of the postulates of the quantum theory to quantum communication. We study the fundamental, unit quantum communication protocols. These protocols involve a single sender Alice and a single receiver Bob. The protocols are ideal and noiseless because we assume that Alice and Bob can exploit perfect classical communication, perfect quantum communication, and perfect entanglement. At the end of this chapter, we suggest how to incorporate imperfections into these protocols for later study. Alice and Bob may wish to perform one of several quantum informationprocessing tasks, such as the transmission of classical information, quantum information, or entanglement. Several fundamental protocols make use of these resources: 1. We will see that noiseless entanglement is an important resource in quantum Shannon theory because it enables Alice and Bob to perform other protocols that are not possible with classical resources only. We will present a simple, idealized protocol for generating entanglement, named entanglement distribution . 2. Alice may wish to communicate classical information to Bob. A trivial method, named elementary coding , is a simple way of doing so and we discuss it briefly. 3. A more interesting technique for transmitting classical information is superdense coding . It exploits a noiseless qubit channel and shared entanglement to transmit more classical information than would be possible with a noiseless qubit channel alone. 4. Finally, Alice may wish to transmit quantum information to Bob. A trivial method for her to do so is to exploit a noiseless qubit channel. However, it is useful to have other ways for transmitting quantum information because such a resource is difficult to engineer in practice. An alternative, surprising method for transmitting quantum information is quantum teleportation . The teleportation protocol exploits classical communication and shared entanglement to transmit quantum information. Each of these protocols is a fundamental unit protocol and provides a foundation for asking further questions in quantum Shannon theory. In fact, the discovery of these latter two protocols was the stimulus for much of the original research in quantum Shannon theory. One could take each of these protocols and ask about its performance if one or more of the resources involved is noisy rather than noiseless. Later chapters of this book explore many of these possibilities.
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.
We discussed the major noiseless quantum communication protocols such as teleportation, super-dense coding, their coherent versions, and entanglement distribution in detail in Chapters 6, 7,and 8. Each of these protocols relies on the assumption that noiseless resources are available. For example, the entanglement distribution protocol assumes that a noiseless qubit channel is available to generate a noiseless ebit. This idealization allowed us to develop the main principles of the protocols without having to think about more complicated issues, but in practice, the protocols do not work as expected under the presence of noise. Given that quantum systems suffer noise in practice, we would like to have a way to determine how well a protocol is performing. The simplest way to do so is to compare the output of an ideal protocol to the output of the actual protocol using a distance measure of the two respective output quantum states. That is, suppose that a quantum information-processing protocol should ideally output some quantum state ∣ψ⟩, but the actual output of the protocol is a quantum state with density operator ρ. Then a performance measure P (∣ψ⟩,ρ) should indicate how close the ideal output is to the actual output. Figure 9.1 depicts the comparison of an ideal protocol with another protocol that is noisy. This chapter introduces two distance measures that allow us to determine how close two quantum states are to each other.