We survey the field of quantum information theory. In particular, we discuss the fundamentals of the field, source coding, quantum error-correcting codes, capacities of quantum channels, measures of entanglement and quantum cryptography.
Quantum information theory represents a revolutionary paradigm that merges quantum mechanics with information science, fundamentally transforming our understanding of computation, communication, and information processing. This chapter provides a comprehensive examination of quantum information theory, exploring its theoretical foundations, practical applications, and transformative potential across multiple domains. We investigate the core principles including quantum bits (qubits), superposition, entanglement, and quantum interference, while analyzing their implications for quantum computing, cryptography, and communication protocols. Through systematic literature review and methodological analysis, we present current research trends, breakthrough developments, and emerging applications. The chapter addresses significant challenges including decoherence, error correction, and scalability issues that currently limit practical implementation. Our discussion encompasses quantum algorithms, quantum error correction codes, and the evolving landscape of quantum technologies. We conclude by identifying future research directions and the potential societal impact of quantum information systems, emphasizing both the unprecedented opportunities and the technical obstacles that define this rapidly advancing field.
Quantum information theory is the study of the achievable limits of information processing within quantum mechanics. Many different types of information can be accommodated within quantum mechanics, including classical information, coherent quantum information, and entanglement. Exploring the rich variety of capabilities allowed by these types of information is the subject of quantum information theory, and of this Dissertation. In particular, I demonstrate several novel limits to the information processing ability of quantum mechanics. Results of especial interest include: the demonstration of limitations to the class of measurements which may be performed in quantum mechanics; a capacity theorem giving achievable limits to the transmission of classical information through a two-way noiseless quantum channel; resource bounds on distributed quantum computation; a new proof of the quantum noiseless channel coding theorem; an information-theoretic characterization of the conditions under which quantum error-correction may be achieved; an analysis of the thermodynamic limits to quantum error-correction, and new bounds on channel capacity for noisy quantum channels.
Quantum information processing exploits the quantum nature of information. It offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems. For quantum communication channels, many new capacity definitions were developed in comparison to classical counterparts. A quantum channel can be used to realize classical information transmission or to deliver quantum information, such as quantum entanglement. Here we review the properties of the quantum communication channel, the various capacity measures and the fundamental differences between the classical and quantum channels.
Quantum information processing exploits the quantum nature of information. It offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems. For quantum communication channels, many new capacity definitions were developed in comparison to classical counterparts. A quantum channel can be used to realize classical information transmission or to deliver quantum information, such as quantum entanglement. Here we review the properties of the quantum communication channel, the various capacity measures and the fundamental differences between the classical and quantum channels.
The paper is intended to be a survey of all the important aspects and results that have shaped the field of quantum computation and quantum information. The reader is first familiarized with those features and principles of quantum mechanics providing a more efficient and secure information processing. Their applications to the general theory of information, cryptography, algorithms, computational complexity and error-correction are then discussed. Prospects for building a practical quantum computer are also analyzed.
Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.