From Classical to Quantum Shannon Theory

TL;DR

The book maps classical Shannon theory onto the quantum domain, building a unified framework for quantum information processing. It develops both noiseless and noisy quantum theories, introducing qubits and qudits, entanglement, and fundamental protocols (teleportation, super-dense coding) while formalizing resources with the density-operator and Kraus formalisms. Core results include the Holevo bound, Schumacher compression, the quantum channel capacity theorem, and entanglement-assisted capacities, all tied together through the resource-inequality paradigm. Through CHSH game analyses, Tsirelson bounds, and a detailed treatment of entanglement as a resource, the text provides a cohesive, operational foundation for quantum communication and network information theory.

Abstract

The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and post-millenium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quantum information theory (Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution (Part III), and we develop many of the tools necessary for understanding information transmission or compression (Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory.

Figures (82)

• Figure 1: This figure depicts Shannon's idea for a classical source code. The information source emits a long sequence $x^{n}$ to Alice. She encodes this sequence as a block with an encoder $\mathcal{E}$ and produces a codeword whose length is less than that of the original sequence $x^{n}$ (indicated by fewer lines coming out of the encoder $\mathcal{E}$). She transmits the codeword over noiseless bit channels (each indicated by "id" which stands for the identity bit channel) and Bob receives it. Bob decodes the transmitted codeword with a decoder $\mathcal{D}$ and produces the original sequence that Alice transmitted, only if their chosen code is good, in the sense that the code has a small probability of error.
• Figure 2: This figure indicates that the typical set is much smaller (exponentially smaller) than the set of all sequences. The typical set is roughly the same size as the set of all sequences only when the entropy $H( X)$ of the random variable $X$ is equal to $\log\left\vert \mathcal{X}\right\vert$---implying that the distribution of random variable $X$ is uniform.
• Figure 3: This figure depicts the action of the bit-flip channel. It preserves the input bit with probability $1-p$ and flips it with probability $p$.
• Figure 4: This figure depicts Shannon's idea for a classical channel code. Alice chooses a message $m$ from a message set $\left[ M\right] \equiv\left\{ 1,\ldots,M\right\}$. She encodes the message $m$ with an encoding operation $\mathcal{E}$. This encoding operation assigns a codeword $x^{n}$ to the message $m$ and inputs the codeword $x^{n}$ to a large number of i.i.d. uses of a noisy channel $\mathcal{N}$. The noisy channel randomly corrupts the codeword $x^{n}$ to a sequence $y^{n}$. Bob receives the corrupted sequence $y^{n}$ and performs a decoding operation $\mathcal{D}$ to estimate the codeword $x^{n}$. This estimate of the codeword $x^{n}$ then produces an estimate $\hat{m}$ of the message that Alice transmitted. A reliable code has the property that Bob can decode each message $m\in\left[ M\right]$ with a vanishing probability of error when the block length $n$ becomes large.
• Figure 5: This figure depicts the notion of a conditionally typical set. Associated to every input sequence $x^{n}$ is a conditionally typical set consisting of the likely output sequences. The size of this conditionally typical set is $\approx2^{nH\left( Y|X\right) }$. It is exponentially smaller than the set of all output sequences whenever the conditional random variable is not uniform.

Theorems & Definitions (289)

• Definition 3.3.1: Function of a Hermitian operator
• Definition 3.6.1: Pure-State Entanglement
• Theorem 3.8.1: Schmidt decomposition
• Remark 3.8.1
• Definition 4.1.1: Trace
• Definition 4.1.2: Density Operator
• Definition 4.1.3: Density Operator as the State
• Definition 4.1.4: Maximally Mixed State
• Definition 4.1.5: Purity
• Definition 4.2.1: POVM