The packing lemma is a general method for one party to pack classical messages into a Hilbert space so that another party can distinguish the packed messages. The first party can prepare an ensemble of quantum states, and the other party has access to a set of projectors from 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 in many coding theorems in quantum Shannon theory. Examples of coding theorems that we can prove with the packing lemma are the Holevo–Schumacher–Westmoreland (HSW) theorem for 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 21 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.
The packing lemma is a general method for one party to pack classical messages into a Hilbert space so that another party can distinguish the packed messages. The first party can prepare an ensemble of quantum states, and the other party has access to a set of projectors from 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 in many coding theorems in quantum Shannon theory. Examples of coding theorems that we can prove with the packing lemma are the Holevo–Schumacher–Westmoreland (HSW) theorem for 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 21 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.
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.
The goal of the covering lemma is perhaps opposite to that of the packing lemma because it applies to a setting in which 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 indicating 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), and it gives an effective scheme for simulating noise when we apply it in an i.i.d. setting. One application of the covering lemma in quantum Shannon theory is in the construction of a code for transmitting private classical information over a quantum channel (discussed in Chapter 23). The method of proof for private classical transmission involves a clever combination of packing messages so that Bob can distinguish them, while covering Eve's space in such a way that Eve cannot distinguish the messages intended for Bob. A few other applications of the covering lemma are in secret key distillation, determining the amount of noise needed to destroy correlations in a bipartite state, and compressing the outcomes of an i.i.d. measurement on an i.i.d. quantum state.
Optimally encoding classical information in a quantum system is one of the oldest and most fundamental challenges of quantum information theory. Holevo's bound places a hard upper limit on such encodings, while the Holevo-Schumacher-Westmoreland (HSW) theorem addresses the question of how many classical messages can be “packed” into a given quantum system. In this article, we use Sen's recent quantum joint typicality results to prove a one-shot multiparty quantum packing lemma generalizing the HSW theorem. The lemma is designed to be easily applicable in many network communication scenarios. As an illustration, we use it to straightforwardly obtain quantum generalizations of well-known classical coding schemes for the relay channel: multihop, coherent multihop, decode-forward, and partial decode-forward. We provide both finite blocklength and asymptotic results, the latter matching existing classical formulas. Given the key role of the classical packing lemma in network information theory, our packing lemma should help open the field to direct quantum generalization.
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.
In this correspondence, we give an alternative proof of the direct part of the classical-quantum channel coding theorem (the Holevo-Schumacher-Westmoreland (HSW) theorem), using ideas of quantum hypothesis testing. In order to show the existence of good codes, we invoke a limit theorem, relevant to the quantum Stein's lemma, in quantum hypothesis testing as the law of large numbers used in the classical case. We also apply a greedy construction of good codes using a packing procedure of noncommutative operators. Consequently we derive an upper bound on the coding error probability, which is used to give an alternative proof of the HSW theorem. This approach elucidates how the Holevo information applies to the classical-quantum channel coding problems
Most coding theorems in quantum Shannon theory can be proven using the decoupling technique. To send data through a channel, one guarantees that the environment gets no information about it. Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method, sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a dequantizing theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland theorem to the one-shot scenario, where a quantum channel can be applied only once.