Establishes a structural isomorphism between Shannon's information channel theorem and the triadic coherence invariant R = P × I × Pr ≠ 0. The sender–encoding–receiver decomposition maps to Intent–Pattern–Presence. Derives: topological channel capacity C ≈ 10.7 bits per anyon pair at SU(2)₃₀; spectral Shannon entropy H ≈ 0.929 nats with efficiency η ≈ 0.845; Landauer floor P_min ≈ 7.42 kT/τ; a self-reference bound unifying Shannon, Gödel, and the Ouroboros fixed-point condition. Proves DPI saturation for topological encoding and provides a channel-theoretic formulation of the measurement problem. Part of the CAT'S Theory corpus (ISBN 979-8160469362).
Establishes a structural isomorphism between Shannon's information channel theorem and the triadic coherence invariant R = P × I × Pr ≠ 0. The sender–encoding–receiver decomposition maps to Intent–Pattern–Presence. Derives: topological channel capacity C ≈ 10.7 bits per anyon pair at SU(2)₃₀; spectral Shannon entropy H ≈ 0.929 nats with efficiency η ≈ 0.845; Landauer floor P_min ≈ 7.42 kT/τ; a self-reference bound unifying Shannon, Gödel, and the Ouroboros fixed-point condition. Proves DPI saturation for topological encoding and provides a channel-theoretic formulation of the measurement problem. Part of the CAT'S Theory corpus (ISBN 979-8160469362).
The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H ( B ) -H ( E )—a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. We proved achievability of the coherent information for quantum data transmission in Corollary 22.2.1, but the roundabout path that we followed to prove achievability there perhaps does not give much insight into the structure of a quantum code that achieves the coherent information. Our approach in this chapter is different and should shed more light on this structure. Specifically, we show how to make coherent versions of the private classical codes from the previous chapter.
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.
We cannot overstate the importance of Shannon's contribution to modern science. His introduction of the field of information theory and his solutions to its two main theorems demonstrate that his ideas on communication were far beyond the other prevailing ideas in this domain around 1948. In this chapter, our aim is to discuss Shannon's two main contributions in a descriptive fashion. The goal of this high-level discussion is to build up the intuition for the problem domain of information theory and to understand the main concepts before we delve into the analogous quantum information-theoretic ideas. We avoid going into deep technical detail in this chapter, leaving such details for later chapters where we formally prove both classical and quantum Shannontheoretic coding theorems. We do use some mathematics from probability theory (namely, the law of large numbers). We will be delving into the technical details of this chapter's material in later chapters (specifically, Chapters 10, 13, and 14). Once you have reached later chapters that develop some more technical details, it might be helpful to turn back to this chapter to get an overall flavor for the motivation of the development. Data Compression We first discuss the problem of data compression. Those who are familiar with the Internet have used several popular data formats such as JPEG, MPEG, ZIP, GIF, etc. All of these file formats have corresponding algorithms for compressing the output of an information source. A first glance at the compression problem might lead one to believe that it is not possible to compress the output of the information source to an arbitrarily small size, and Shannon proved that this is the case. This result is the content of Shannon's first noiseless coding theorem. An Example of Data Compression We begin with a simple example that illustrates the concept of an information source. We then develop a scheme for coding this source so that it requires fewer bits to represent its output faithfully. Suppose that Alice is a sender and Bob is a receiver. Suppose further that a noiseless bit channel connects Alice to Bob—a noiseless bit channel is one that transmits information perfectly from sender to receiver, e.g., Bob receives “0” if Alice transmits “0” and Bob receives “1” if Alice transmits “1.”
This chapter begins our exploration of “dynamic” information-processing tasks in quantum Shannon theory, where the term “dynamic” indicates that a quantum channel connects a sender to a receiver and their goal is to exploit this resource for communication. We specifically consider the scenario where a sender Alice would like to communicate classical information to a receiver Bob, and the capacity theorem that we prove here is one particular generalization of Shannon's noisy channel coding theorem from classical information theory (overviewed in Section 2.2). In later chapters, we will see other generalizations of Shannon's theorem, depending on what resources are available to assist their communication or depending on whether they are trying to communicate classical or quantum information. For this reason and others, quantum Shannon theory is quite a bit richer than classical information theory. The naive approach to communicate classical information over a quantum channel is for Alice and Bob simply to mimic the approach used in Shannon's noisy channel coding theorem. That is, they select a random classical code according to some distribution pX(x) , and Bob performs individual measurements of the outputs of a noisy quantum channel according to some POVM. The POVM at the output induces some conditional probability distribution pY∣X(y∣x) , which we can in turn think of as an induced noisy classical channel. The classical mutual information I(X; Y) of this channel is an achievable rate for communication, and the best strategy for Alice and Bob is to optimize the mutual information over all of Alice's inputs to the channel and over all measurements that Bob could perform at the output.
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.
The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) – H(E) —a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain.