Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper, we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional, and memoryless.
Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper, we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional, and memoryless.
Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entangle- ment distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper, we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional, and memoryless. We formalize two principles that have long been tacitly under- stood. First, we describe how the Church of the larger Hilbert space allows us to move flexibly between states, channels, en- sembles, and their purifications. Second, we introduce finite and asymptotic (quantum) information processing resources as the basic objects of quantum Shannon theory and recast the protocols used in direct coding theorems as inequalities between resources. We develop the rules of a resource calculus which allows us to manipulate and combine resource inequalities. This framework simplifies many coding theorem proofs and provides structural insights into the logical dependencies among coding theorems. We review the above-mentioned basic coding results and show howasubsetofthemcanbeunifiedintoafamilyofrelatedresource inequalities. Finally, we use this family to find optimal tradeoff curves for all protocols involving one noisy quantum resource and two noiseless ones.
These notes (in French) provide a concise introduction to information theory, spanning its classical foundations: Shannon entropy, joint and conditional entropies, mutual information, statistical correlations, and Markov chains; as well as Shannon's source coding and channel capacity theorems, which govern the optimal compression and transmission of information through noisy channels. They also cover the quantum extensions of these concepts, including the density operator formalism, distances between quantum states, von Neumann entropy, coherent information, conditional entropy, entanglement theory, quantum discord, and quantum resource theory.
Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.
One of the fundamental tasks in classical information theory is the compression of information. Given access to many uses of a noiseless classical channel, what is the best that a sender and receiver can make of this resource for compressed data transmission? Shannon's compression theorem demonstrates that the Shannon entropy is the fundamental limit for the compression rate in the IID setting (recall the development in Section 13.4). That is, if one compresses at a rate above the Shannon entropy, then it is possible to recover the compressed data perfectly in the asymptotic limit, and otherwise, it is not possible to do so. This theorem establishes the prominent role of the entropy in Shannon's theory of information. In the quantum world, it very well could be that one day a sender and a receiver would have many uses of a noiseless quantum channel available, and the sender could use this resource to transmit compressed quantum information. But what exactly does this mean in the quantum setting? A simple model of a quantum information source is an ensemble of quantum states { pX(x) , ∣ψ x ⟩}, i.e., the source outputs the state ∣ψ x ⟩ with probability pX(x) , and the states {∣ψ x ⟩} do not necessarily have to form an orthonormal basis. Let us suppose for the moment that the classical data x is available as well, even though this might not necessarily be the case in practice. A naive strategy for compressing this quantum information source would be to ignore the quantum states coming out, handle the classical data instead, and exploit Shannon's compression protocol from Section 13.4.
We present the unification of many previously disparate results in noisy quantum Shannon theory and the unification of all of noiseless quantum Shannon theory. More specifically we deal here with bipartite, unidirectional, and memoryless quantum Shannon theory. We find all the optimal protocols and quantify the relationship between the resources used, both for the one-shot and for the ensemble case, for what is arguably the most fundamental task in quantum information theory: sharing entangled states between a sender and a receiver. We find that all of these protocols are derived from our one-shot superdense coding protocol and relate nicely to each other. We then move on to noisy quantum information theory and give a simple, direct proof of the "mother" protocol, or rather her generalization to the Fully Quantum Slepian-Wolf protocol(FQSW). FQSW simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation, and state transfer from the sender to the receiver. As a result, in addition to her other "children," the mother protocol generates the state merging primitive of Horodecki, Oppenheim, and Winter as well as a new class of distributed compression protocols for correlated quantum sources, which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called "father" protocol, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother.