Quantum algorithms require quantum computers performing logical operations on a sufficiently large number of entangled qubits. Unfortunately, state-of-the-art quantum computers can only operate on tens of qubits. A solution to this scalability challenge is to employ the distributed paradigm, where a network of small-scale quantum computers are used in a distributed manner. It is the aim of my work to study the fundamental limits of distributed quantum problems and to further enhance their performance by exploiting the structure inherent to these problems employing asymptotically good Algebraic codes. This thesis consists of two parts. The first part studies the task of faithfully simulating a distributed quantum measurement, wherein we provide a protocol for the three parties, Alice, Bob and {Charlie}, to simulate a repeated action of a distributed quantum measurement using a pair of non-product approximating measurements by Alice and Bob, followed by a stochastic mapping at {Charlie}. The objective of the protocol is to utilize minimum resources, in terms of classical bits needed by Alice and Bob to communicate their measurement outcomes to {Charlie}, and the common randomness shared among the three parties, while faithfully simulating independent repeated instances of the original measurement. We characterize a set of sufficient communication and common randomness rates required for asymptotic simulatability in terms of single-letter quantum information quantities. We further improve the results obtained in the above by exploiting the structure present in the Charlie's stochastic bivariate mapping using random structured POVMs based on asymptotically good algebraic codes. The algebraic structure of these codes is matched to that of the bivariate function that models the action of Charlie. This leads to the computation being performed on the fly, thus obviating the need to reconstruct individual measurement outcomes at Charlie. We provide examples to illustrate the information-theoretic gains attained by endowing POVMs with algebraic structure. As an application of the distributed measurement compression problem, we also demonstrate a multi-party purity distillation protocol. Concluding this part, we consider the lossy quantum source coding problem where the objective is to compress a given quantum source below its von Neumann entropy. Inspired by the duality connections between the rate-distortion and channel coding problems in the classical setting, we propose a new formulation for the lossy quantum source coding problem. We require that the reconstruction of the compressed quantum source fulfill a global error constraint and employ the notion of a ``posterior reference map'' to measure the reconstruction error. Using these, we characterize the asymptotic performance limit of this problem in terms of single-letter coherent information of the given posterior reference map. In the second part of this thesis, we study the advantage algebraic structured codes can provide, to the class of the classical-quantum network problems. In particular, we investigate two problems. Firstly, we consider the problem of communicating a general bivariate function of two classical sources observed at the encoders of a classical-quantum multiple access channel. We propose and analyze a coding scheme based on algebraic structured coset codes that enables the decoder to recover the desired function without recovering the sources themselves. We derive a new set of sufficient conditions that are weaker than the current known for identified examples. In addition, we analyze the performance of these algebraic codes toward studying the capacity of a special class of 3-user classical-quantum interference channel.
Quantum algorithms require quantum computers performing logical operations on a sufficiently large number of entangled qubits. Unfortunately, state-of-the-art quantum computers can only operate on tens of qubits. A solution to this scalability challenge is to employ the distributed paradigm, where a network of small-scale quantum computers are used in a distributed manner. It is the aim of my work to study the fundamental limits of distributed quantum problems and to further enhance their performance by exploiting the structure inherent to these problems employing asymptotically good Algebraic codes. This thesis consists of two parts. The first part studies the task of faithfully simulating a distributed quantum measurement, wherein we provide a protocol for the three parties, Alice, Bob and {Charlie}, to simulate a repeated action of a distributed quantum measurement using a pair of non-product approximating measurements by Alice and Bob, followed by a stochastic mapping at {Charlie}. The objective of the protocol is to utilize minimum resources, in terms of classical bits needed by Alice and Bob to communicate their measurement outcomes to {Charlie}, and the common randomness shared among the three parties, while faithfully simulating independent repeated instances of the original measurement. We characterize a set of sufficient communication and common randomness rates required for asymptotic simulatability in terms of single-letter quantum information quantities. We further improve the results obtained in the above by exploiting the structure present in the Charlie's stochastic bivariate mapping using random structured POVMs based on asymptotically good algebraic codes. The algebraic structure of these codes is matched to that of the bivariate function that models the action of Charlie. This leads to the computation being performed on the fly, thus obviating the need to reconstruct individual measurement outcomes at Charlie. We provide examples to illustrate the information-theoretic gains attained by endowing POVMs with algebraic structure. As an application of the distributed measurement compression problem, we also demonstrate a multi-party purity distillation protocol. Concluding this part, we consider the lossy quantum source coding problem where the objective is to compress a given quantum source below its von Neumann entropy. Inspired by the duality connections between the rate-distortion and channel coding problems in the classical setting, we propose a new formulation for the lossy quantum source coding problem. We require that the reconstruction of the compressed quantum source fulfill a global error constraint and employ the notion of a ``posterior reference map'' to measure the reconstruction error. Using these, we characterize the asymptotic performance limit of this problem in terms of single-letter coherent information of the given posterior reference map. In the second part of this thesis, we study the advantage algebraic structured codes can provide, to the class of the classical-quantum network problems. In particular, we investigate two problems. Firstly, we consider the problem of communicating a general bivariate function of two classical sources observed at the encoders of a classical-quantum multiple access channel. We propose and analyze a coding scheme based on algebraic structured coset codes that enables the decoder to recover the desired function without recovering the sources themselves. We derive a new set of sufficient conditions that are weaker than the current known for identified examples. In addition, we analyze the performance of these algebraic codes toward studying the capacity of a special class of 3-user classical-quantum interference channel.
We consider the task of faithfully simulating a distributed quantum measurement, wherein we provide a protocol for the three parties, Alice, Bob and Charlie, to simulate a repeated action of a distributed quantum measurement using a pair of non-product approximating measurements by Alice and Bob, followed by a stochastic mapping at Charlie. The objective of the protocol is to utilize minimum resources, in terms of classical bits needed by Alice and Bob to communicate their measurement outcomes to Charlie, and the common randomness shared among the three parties, while faithfully simulating independent repeated instances of the original measurement. To achieve this, we develop a mutual covering lemma and a technique for random binning of distributed quantum measurements, and, in turn, characterize a set of sufficient communication and common randomness rates required for asymptotic simulatability in terms of single-letter quantum information quantities. In the special case, where the Charlie’s action is restricted to a deterministic mapping, we develop a one-shot performance characterization of the distributed faithful simulation problem. Furthermore, using these results we address a distributed quantum rate-distortion problem, where we characterize the achievable rate distortion region through a single-letter inner bound. Finally, via a technique of single-letterization of multi-letter quantum information quantities, we provide an outer bound for the rate-distortion region.
We consider the task of faithfully simulating a quantum measurement, acting on a joint bipartite quantum state, in a distributed manner. In this setup, the constituent sub-systems of the joint quantum state are measured by two agents, Alice and Bob. A third agent, Charlie, receives the measurement outcomes sent by Alice and Bob. Charlie uses local and pairwise shared randomness to compute a bivariate function of the measurement outcomes. The objective of three agents is to faithfully simulate the given distributed quantum measurement acting on the given quantum state while minimizing the communication and shared randomness rates. We demonstrate a new inner bound to the rate region using random structured POVMs based on asymptotically good algebraic codes, and characterize the performance limit using single-letter quantum mutual information quantities. This new bound subsumes the largest known inner bound and improves upon it strictly for identified examples. One of the challenges in analyzing these structured POVMs is that they exhibit only pairwise independence and induce only uniform single-letter distributions. We address these in the non-commutative quantum setting, and provide a two-party distributed faithful simulation and function computation protocol.
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.
In this work, we consider the task of faithfully simulating a quantum measurement, acting on a joint bipartite quantum state, in a distributed manner. In the distributed setup, the constituent sub-systems of the joint quantum state are measured by two agents, Alice and Bob. A third agent, Charlie receives the measurement outcomes sent by Alice and Bob. Charlie uses local and pairwise shared randomness to compute a bivariate function of the measurement outcomes. The objective of three agents is to faithfully simulate the given distributed quantum measurement acting on the given quantum state while minimizing the communication and shared randomness rates. We demonstrate a new achievable information-theoretic rate-region that exploits the bivariate function using random structured POVMs based on asymptotically good algebraic codes. The algebraic structure of these codes is matched to that of the bivariate function that models the action of Charlie. The conventional approach for this class of problems has been to reconstruct individual measurement outcomes corresponding to Alice and Bob, at Charlie, and then compute the bivariate function, achieved using mutually independent approximating POVMs based on random unstructured codes. In the present approach, using algebraic structured POVMs, the computation is performed on the fly, thus obviating the need to reconstruct individual measurement outcomes at Charlie. Using this, we show that a strictly larger rate region can be achieved. One of the challenges in analyzing these structured POVMs is that they exhibit only pairwise independence and induce only uniform single-letter distributions. To address this, we use nesting of algebraic codes and develop a covering lemma applicable to pairwise-independent POVM ensembles. Combining these techniques, we provide a multi-party distributed faithful simulation and function computation protocol.
We consider a distributed quantum hypothesis testing problem with communication constraints, in which the two hypotheses correspond to two different states of a bipartite quantum system, multiple identical copies of which are shared between Alice and Bob. They are allowed to perform local operations on their respective systems and send quantum information to Charlie at limited rates. By doing measurements on the systems that he receives, Charlie needs to infer which of the two different states the original bipartite state was in, that is, which of the two hypotheses is true. We prove that the Stein exponent for this problem is given by a regularized quantum relative entropy. The latter reduces to a single letter formula when the alternative hypothesis consists of the products of the marginals of the null hypothesis, and there is no rate constraint imposed on Bob. Our proof relies on certain properties of the so-called quantum information bottleneck function. The second part of this paper concerns the general problem of finding finite blocklength strong converse bounds in quantum network information theory. In the classical case, the analogue of this problem has been reformulated in terms of the so-called image size characterization problem. Here, we extend this problem to the classical-quantum setting and prove a second order strong converse bound for it. As a by-product, we obtain a similar bound for the Stein exponent for distributed hypothesis testing in the special case in which the bipartite system is a classical-quantum system, as well as for the task of quantum source coding with compressed classical side information. Our proofs use a recently developed tool from quantum functional inequalities, namely, the tensorization property of reverse hypercontractivity for the quantum depolarizing semigroup.
The novel ways in which correlations may be encoded non-locally in quantum mechanics have led to the promise of new technologies. This, in turn, has led to the first attempts at the development of quantum networks. Nonetheless, our understanding of quantum correlations and the limits of how we may process them over a quantum network remains nascent. This thesis studies the limits of processing quantum correlations between spatially separated parties in various settings and the mathematical tools necessary to determine such limits. To do so, this thesis studies three general problems in quantum network theory. The first problem is the minimum resources to prepare a quantum correlation between non-communicative receivers, which may be viewed as a star network. This study is separated into three pieces. The first piece is a near-tight one-shot quantum generalization of a problem introduced by Wyner, distributed source simulation. Establishing our one-shot bounds builds on recent work on soft-covering without smoothing and relies on establishing a variety of properties for one-shot entropic quantities, as well as improving our understanding of the smooth entropy framework in quantum network information theory. As distributed source simulation cannot be achieved to arbitrary error for sufficiently entangled states, the second piece characterizes the minimal error of converting one entangled state to another using zero communication. This relies on establishing new properties of the fidelity measure. The third piece establishes an entanglement-assisted version of the distributed source simulation task that can be achieved to arbitrary error for arbitrary quantum systems. This result builds on recent work on multipartite state splitting and establishing new properties of one-shot multipartite max mutual information quantities. The second problem is the extraction of perfect classical correlation from a multipartite quantum source where the receivers do not communicate. This may be seen as the reverse process of distributed source simulation. We establish that this is asymptotically possible in quantum mechanics for the first time and establish the asymptotic rate for classical-quantum states. The technical barrier in establishing the asymptotic rate is establishing a limit on how much noiseless correlation can be extracted without communication. Our major technical contribution is establishing such a limit by generalizing a technical lemma of Gács and Körner to classical-quantum states. The final problem is distributed state discrimination of quantum product states. This is motivated by its relevance to both understanding non-locality in quantum mechanics and developing quantum cryptographic schemes. Our study again breaks into three key pieces. First, we study the framework of local operations and simultaneous classical or quantum communication (LOSCC and LOSQC respectively) for product states, which is relevant to quantum cryptography. We establish the first separation between quantum and classical communication in the simultaneous communication model using only product state inputs. We also establish a new uncertainty relation that implies error bounds for discriminating product states using LOSQC. Second, we study state discrimination under local operations and classical communication (LOCC). We develop a new framework for zero-error state discrimination under LOCC. Our framework provides a simple proof of orthogonal product states that cannot be distinguished using LOCC--- a property introduced by Bennett et al. Finally, we study the thermodynamics of zero-error distributed state discrimination in the framework of Faist et al. This allows us to resolve an open problem from the aforementioned paper of Bennett et al.