Quantum Estimation utilizing Bayesian Techniques and Quantum Error Correction

Abstract

Quantum estimation explores the fundamental limits of precision imposed by quantum mechanics, offering a quadratic advantage (in terms of the probe's energy budget) by leveraging entanglement and non-classical states. Among various estimation frameworks, the Bayesian approach is particularly useful when the Fisherian methods, such as those based on the Cramér-Rao bound (CRB), are not well defined, such as in uncertainty regimes or when limited measurements are available. Bayesian quantum estimation allows for adaptive strategies that refine precision as more data is collected, making it well-suited for practical quantum sensing applications. However, quantum estimation remains susceptible to decoherence and noise, which degrade measurement accuracy. To address these challenges, quantum error correction (QEC) has been explored as a means to preserve quantum information and enhance the robustness of estimation. The first part of this thesis establishes the mathematical framework for quantum estimation theory, beginning with classical estimation and moving to quantum estimation, with the two basic approaches: Fisherian and Bayesian. The second part focuses on the application of quantum estimation to transmissivity sensing. By employing a Bayesian framework, we derive the optimal probe states for estimating the transmissivity of a quantum channel under different prior distributions. The study demonstrates how quantum resources can enhance estimation performance and explores the trade-offs between different measurement strategies. The third part examines phase estimation, a key task in quantum metrology and quantum computing. This section analyzes various quantum states, including NOON states and general photon-number states, under the Bayesian framework. Additionally, adaptive Bayesian methods are explored, demonstrating how iterative updates improve phase estimation precision. The fourth part investigates the estimation of the spatial separation between two incoherent point sources, a problem relevant to quantum imaging and microscopy. We analyze the performance of direct imaging (DI) and spatial-mode demultiplexing (SPADE) under different prior assumptions and extend the study to multi-source scenarios. The findings illustrate the conditions under which quantum measurements provide a resolution advantage over classical methods. Finally, this thesis explores the role of QEC in quantum estimation and sensing. By integrating Gottesman-Kitaev-Preskill (GKP) codes and two-mode squeezing techniques, we develop noise-resilient estimation protocols for distributed quantum sensing (DQS). Different QEC concatenation schemes are analyzed, and their impact on hypothesis testing and machine learning-assisted sensing is examined. The results presented in this thesis contribute to the advancement of quantum estimation theory and its practical applications. By addressing fundamental challenges and exploring novel strategies, this work provides a pathway toward more robust and enhanced quantum sensing, and quantum estimation theory.