Quantum computation and information has become a central research area in theoretical computer science. It studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. On one hand, the ability of a quantum system to be in a superposition of states provides a way to encode an exponential amount of information in a quantum state. However, this information is accessible only indirectly via measurements. The goal of this thesis is precisely to investigate the fundamental question of how information is encoded and processed in quantum mechanical systems. (1) We investigate their power relative to classical encodings in the model of one-way communication complexity and show that they can be exponentially more efficient, answering a long standing open question in the area of quantum communication complexity. (2) We show how the theory developed for the study of quantum information can be employed in order to answer questions about classical coding theory and complexity. We resolve a main open question about the efficiency of Locally Decodable Codes by reducing the problem to one about quantum codes and solving the latter using tools from quantum information theory. This is the first example where quantum techniques are essential in resolving a purely classical question. (3) We investigate the power of quantum encodings in the area of cryptography. We study certain cryptographic primitives, such as Private Information Retrieval, Symmetrically-Private Information Retrieval and Coin Flipping.
Quantum computation and information has become a central research area in theoretical computer science. It studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. On one hand, the ability of a quantum system to be in a superposition of states provides a way to encode an exponential amount of information in a quantum state. However, this information is accessible only indirectly via measurements. The goal of this thesis is precisely to investigate the fundamental question of how information is encoded and processed in quantum mechanical systems. (1) We investigate their power relative to classical encodings in the model of one-way communication complexity and show that they can be exponentially more efficient, answering a long standing open question in the area of quantum communication complexity. (2) We show how the theory developed for the study of quantum information can be employed in order to answer questions about classical coding theory and complexity. We resolve a main open question about the efficiency of Locally Decodable Codes by reducing the problem to one about quantum codes and solving the latter using tools from quantum information theory. This is the first example where quantum techniques are essential in resolving a purely classical question. (3) We investigate the power of quantum encodings in the area of cryptography. We study certain cryptographic primitives, such as Private Information Retrieval, Symmetrically-Private Information Retrieval and Coin Flipping.
The description of the state of an n-bit quantum system requires 2n-1 complex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing both exponential speed-up, and exponential savings in communication costs in solving certain problems using quantum computers. In this dissertation, we establish limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed. More specifically, we give tight bounds for random access codes, which allow us to encode classical information using quantum bits such that only a small portion of the encoded information may be recovered via a measurement. This also sheds light on the power of computing with a finite number of quantum bits—using these techniques, we show an exponential size lower bound for quantum finite automata for a problem which can be solved on a linear size classical automaton. We then consider the complexity of solving certain problems in the quantum black-box model, an information theoretic model that has been a rich source of insights into the nature of quantum computation. We derive nearly optimal lower bounds for several problems in this model, including that of approximating the median. We also give new, optimal algorithms for approximate medians and other order statistics.
One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet there are ways of conveying information with exponentially fewer qubits than possible classically. Moreover, these methods have a very simple structure---they involve little interaction between the communicating parties. We look more closely at the ways in which information encoded in quantum states may be manipulated, and consider the question as to whether every classical protocol may be transformed to a ``simpler'' quantum protocol of similar efficiency. By a simpler protocol, we mean a protocol that uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity, thus answering the above question in the negative. Our result builds on two primitives, local transitions in bi-partite states (based on previous work) and average encoding which may be of significance in other applications as well.
Quantum information processing (QIP) studies the representation, manipulation, and transmission of information using quantum systems. By leveraging counterintuitive quantum phenomena such as superposition and entanglement, QIP enables new paradigms for computation, communication, and learning that can surpass classical limits in certain regimes. Foundational tasks in QIP include quantum data compression, quantum communication, and quantum learning and optimization, which together form the basis for scalable quantum technologies. This thesis develops new frameworks and algorithms at the intersection of quantum information theory, quantum learning, and quantum-inspired modeling. The thesis is organized into three parts. The first part studies quantum compression problems; the second addresses quantum communication; and the third develops quantum learning and optimization frameworks with applications to complex systems. In the first part, a new formulation for lossy quantum-classical source coding, termed rate-channel theory, is introduced, replacing the conventional distortion-based formulation with a posterior-channel-based framework under a block-error criterion. The framework is further generalized to incorporate quantum side information at the decoder. A connection between rate-channel theory and classical rate-distortion theory is also established, showing that optimal rate-distortion performance can be recovered under suitable conditions. These results provide a unified and flexible framework for data compression, which is essential for scalable quantum computing infrastructure, enabling efficient storage of quantum data, such as outputs from quantum sensors (e.g., MRI) and information encoded in photonic systems for satellite and optical communication. In the second part, the problem of computing functions of distributed sources over classical-quantum multiple access channels is studied. A new coding framework is developed that combines algebraic structured codes with unstructured random coding techniques, enabling efficient communication of functions without reconstructing individual sources. The resulting achievable conditions, expressed in terms of single-letter quantum information quantities, strictly enlarge the class of sources for which reliable computation is possible. These results contribute to the theoretical foundations of quantum network communication and are directly relevant to deep-space satellite-based quantum communication, distributed quantum computing, and the design of scalable quantum network infrastructures. In the final part, new approaches to quantum learning and optimization are developed. A geometry-aware quantum optimization framework is proposed based on a novel quantum information metric and an unbiased estimator constructed from single-shot measurements. This leads to an optimization method with constant sample complexity per iteration and provable exponential convergence under mild conditions, while avoiding saddle points and local minima. Such advances are particularly relevant for challenging optimization tasks in quantum sensing and quantum computational modeling of many-body systems. Building on this optimization framework, a Hamiltonian learning approach is introduced to infer interaction structures from time-resolved measurement data, with finite-sample guarantees and polynomial scaling with system size. This framework is further extended to develop a quantum-inspired generative model for gene regulatory network inference, where interactions are encoded as a parameterized Hamiltonian evolving over pseudotime. The proposed methods demonstrate strong performance on synthetic benchmarks and reveal biologically meaningful structures in real-world data, with applications to multi-omics integration, cancer genomics, and systems biology. Overall, this thesis advances the theoretical and algorithmic foundations of quantum information processing, with contributions spanning compression, communication, and learning. These developments support the design of resource-efficient and scalable quantum technologies, enabling applications in quantum sensing, quantum networks, distributed quantum computation, and the modeling of complex physical and biological systems.
This thesis is a study of how information theory can be used to analyse and interpret quantum mechanical systems, specifically, systems that are being used for information processing and communication tasks. Quantum information science is a relatively new field, and it has only been recently that we have started to look at using quantum mechanical systems to accomplish information processing tasks that are not possible using classical systems. As this method of information processing is a relatively new paradigm, there are many possibilities for new algorithms and discoveries in this area of computing. The application of information theory towards the analysis and description of quantum mechanical systems yields new interesting results - it can even be argued that information theory provides a natural and intuitive way to intepret quantum mechanics, as correlations and measurement outcomes in quantum processes can be described using information theory. Chapter 1 is an introduction to the strange new world of quantum mechanics and takes a brief look at one of the applications of quantum information science in quantum cryptography. A mathematical overview of the postulates and framework that quantum mechanics is built on is presented in Chapter 2. Chapter 3 is a brief study of classical information theory and some of the fundamental concepts and ideas that form the crux of the subject. In Chapter 4, we present the main results of the thesis by taking a look at an optimal measurement scheme that improves on previous schemes by allowing us to extract the maximal amount of information from states in a quantum channel of specific geometry. The findings have an impact on various other areas of the field, including security analysis of quantum cryptographic protocols and building better quantum cloning machines.
This thesis establishes a number of connections between foundational issues in quantum theory, and some quantum information applications. It starts with a review of quantum contextuality and non-locality, multipartite entanglement characterisation, and of a few quantum information protocols. Quantum non-locality and contextuality are shown to be essential for different implementations of quantum information protocols known as quantum random access codes and quantum communication complexity protocols. I derive sufficient experimental conditions for tests of these quantum properties. I also discuss how the distribution of quantum information through quantum cloning processes can be useful in quantum computing. Regarding entanglement characterisation, some results are obtained relating two problems, that of additivity of the relative entropy of entanglement, and that of identifying different types of tripartite entanglement in the asymptotic regime of manipulations of many copies of a given state. The thesis ends with a description of an information processing task in which a single qubit substitutes for an arbitrarily large amount of classical communication. This result is interpreted in different ways: as a gap between quantum and classical computation space complexity; as a bound on the amount of classical communication necessary to simulate entanglement; and as a basic result on hidden-variable theories for quantum mechanics. I also show that the advantage of quantum over classical communication can be established in a feasible experiment.
This paper investigates the core methods of information encoding and decoding within quantum networks, aiming to enhance both the efficiency and security of quantum information transmission. We explore various encoding schemes, including amplitude encoding, phase encoding, and polarization encoding, alongside their corresponding decoding strategies. The core mechanism relies on manipulating quantum states to represent and transmit information securely. Our research focuses on optimizing these processes to mitigate the effects of noise and loss, critical challenges in long-distance quantum communication. We delve into error correction techniques specifically tailored for quantum systems, such as quantum error correction codes (QECCs) and their application in decoding. Furthermore, we examine the role of quantum key distribution (QKD) protocols as a fundamental decoding process, highlighting the security aspects derived from the laws of quantum mechanics. The ultimate goal is to provide a comprehensive overview of the current state-of-the-art in quantum network information encoding and decoding, paving the way for future advancements in quantum communication technologies.