Classical Estimation Theory

Sanjoy K. Mitter, Stephen K. Young

Mankei Tsang

Classical and quantum theories of time-symmetric smoothing, which can be used to optimally estimate wave forms in classical and quantum systems, are derived using a discrete-time approach, and the similarities between the two theories are emphasized. Application of the quantum theory to homodyne phase-locked loop design for phase estimation with narrowband squeezed optical beams is studied. The relation between the proposed theory and weak value theory of Aharonov et al. is also explored.

Yu Watanabe

Bakmou Lahcen, Mohammed Daoud A

Quantum estimation theory is a reformulation of random statistical theory with the modern language of quantum mechanics. In fact, the density operator plays a role similar to that of probability distribution functions in classical probability theory and statistics. However, the use of the probability distribution functions in classical theories is founded on premises that seem intuitively clear enough. Whereas in quantum theory, the situation with operators is different due to its non-commutativity nature. By exploiting this difference, quantum estimation theory aims to attain ultra-measurement precision that would otherwise be impossible with classical resources. In this thesis, we reviewed all the fundamental principles of classical estimation theory. Next, we extend our analysis to quantum estimation theory. Due to the non-commutativity of quantum mechanics, we prove the different families of QFIs and the corresponding QCRBs. We compared these bounds and discussed their accessibility in the single-parameter and multiparameter estimation cases. We also introduce HCRB as the most informative alternative bound suitable for multiparameter estimation protocols. Since the quantum state of light is the most accessible in practice, we studied the quantum estimation theory with the formalism of these types of quantum states. We formulate, with complete generality, the quantum estimation theory for Gaussian states in terms of their first and second moments. Furthermore, we address the motivation behind using Gaussian quantum resources and their advantages in reaching the standard quantum limits under realistic noise. In this context, we propose and analyze a measurement scheme that aims to exploit quantum Gaussian entangled states to estimate the displacement parameters under a noisy Gaussian environment.

Sanjoy K. Mitter, Stephen K. Young

Keiji Matsumoto

Two important classes of the quantum statistical model, the locally quasi-classical model and the quasi-classical model, are introduced from the estimation theoretical viewpoint, and they are characterized geometrically by the vanishing conditions of the relative phase factor (RPF), implying the close tie between Uhlmann parallel transport and the quantum estimation theory.