Quantum Estimation Theory

Yu Watanabe

Sanjoy K. Mitter, Stephen K. Young

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.

Keiji Matsumoto

We give a rigorous treatment on the foundation of the first order asymptotic theory of quantum estimation, with tractable and reasonable regularity conditions. Different from past works, we do not use Fisher information nor MLE, and an optimal estimator is constructed based on locally unbiased estimators. Also, we treat state estimation by local operations and classical communications (LOCC), and estimation of quantum operations.

François Chapeau‐Blondeau

Zishi Chen, Xueyuan Hu

We present a resource theory to investigate the power of a multqubit system as a probe in the task of dephasing estimation. Our approach employs the quantum Fisher information about the dephasing parameter as the resource measure. Based on the monotonicity of quantum Fisher information, we propose two sets of free operations in our resource theory: the Hamming distance preserving operations and the selectively Hamming distance preserving operations. We derive a necessary condition for the state transformation under these free operations and demonstrate that uniform superposition states are the golden states in our resource theory. We further compare our resource theory with the resource theory of coherence and thoroughly investigate the relation between their free operations in both single-qubit and multiqubit cases. Additionally, for multiqubit systems, we discover the incompatibility between the resource theory of dephasing estimation and that of $\text{U}(1)$ asymmetry, which is responsible for phase estimation. The condition for enhancing the performance of a probe state in phase estimation while preserving its ability in dephasing estimation is also discussed. Our results provide insights into quantum parameter estimation by the resource-theoretic approach.