Quantum state estimation is a fundamental problem in quantum information theory with applications in quantum computing and communication. To determine the state of a quantum system, researchers often perform measurements on a set of identically prepared quantum states, which are indexed by a parameter. These measurements provide information not only about the parameter itself but also about the quantum states. While questions related to optimal quantum measurements can be elegantly formulated in the language of mathematical statistics, the underlying non-commutative structure yields inferential results that are distinctly non-trivial compared to their counterparts in classical statistics. Furthermore, in contrast to classical models, where estimates are constructed solely based on measurement outcomes, quantum models introduce an additional layer of complexity because the optimal estimator depends on the choice of measurement as well. In classical statistics, a fundamental paradigm involves approximating complex models with simpler ones. One commonly establishes asymptotic equivalence between i.i.d. models, characterized by a local parameter, and a Gaussian shift model. This approximation, known as local asymptotic normality (LAN), facilitates the construction of an estimator based on a procedure in the Gaussian model, offering comparable risk bounds. Notably, local asymptotic equivalence can be extended to quantum scenarios, linking quantum i.i.d. models with quantum Gaussian models. In this context, we obtain optimal estimators in the complex former models based on optimal estimators in the simpler latter models.
Samriddha Lahiry is a postdoctoral fellow at Harvard University in the Department of Statistics. He received his Ph.D. in Statistics from Cornell University in 2022. His research focuses on asymptotic methods in quantum statistical inference and high-dimensional statistics.