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Quantum estimation theory is a reformulation of random statistical theorywith the modern language of quantum mechanics. In fact, the density operatorplays a role similar to that of probability distribution functions in classicalprobability theory and statistics. However, the use of the probabilitydistribution functions in classical theories is founded on premises that seemintuitively clear enough. Whereas in quantum theory, the situation withoperators is different due to its non-commutativity nature. By exploiting thisdifference, quantum estimation theory aims to attain ultra-measurementprecision that would otherwise be impossible with classical resources. In thisthesis, we reviewed all the fundamental principles of classical estimationtheory. Next, we extend our analysis to quantum estimation theory. Due to thenon-commutativity of quantum mechanics, we prove the different families of QFIsand the corresponding QCRBs. We compared these bounds and discussed theiraccessibility in the single-parameter and multiparameter estimation cases. Wealso introduce HCRB as the most informative alternative bound suitable formultiparameter estimation protocols. Since the quantum state of light is themost accessible in practice, we studied the quantum estimation theory with theformalism of these types of quantum states. We formulate, with completegenerality, the quantum estimation theory for Gaussian states in terms of theirfirst and second moments. Furthermore, we address the motivation behind usingGaussian quantum resources and their advantages in reaching the standardquantum limits under realistic noise. In this context, we propose and analyze ameasurement scheme that aims to exploit quantum Gaussian entangled states toestimate the displacement parameters under a noisy Gaussian environment.

Advantages of quantum mechanics in the estimation theory