| Summary: | This thesis presents several optimal estimation strategies for quantum parameter estimation, mostly applied to quantum Markov chains (QMCs) - discrete-time quantum open systems. Chapter 4, based on [78], solves the problem of optimal estimation of a QMC using adaptive measurements. This culminates in an algorithm that determines the optimal measurement basis for the next step of the QMC by tracking an object known as the measurement filter. Crucially, this involves only local measurements on the environment of the QMC; a considerable simplification in comparison to other optimal strategies. Additionally, this estimation scheme utilizes a coherent absorber. This is a secondary system that we use to post-process the quantum Markov chain before measurement, which we include to purify the stationary state of the QMC.
Chapter 5, based on [77], then examines another measurement scheme - null measurements. A null measurement involves measuring the quantum system in a basis that contains the system’s state. It has been claimed that this is an optimal measurement for parameter estimation, but we demonstrate that this is not the case through an independent identically distributed pure state model. This is due to an identifiability issue that occurs when one attempts the naive implementation the null measurement. We then present a solution to this problem, where we introduce some displacement into the null measurement. This removes the identification issue and provides a proper statistical foundation to the estimation scheme.
In chapter 6, based on [75], we apply this displaced null measurement to a QMC; this provides a second optimal estimation scheme for QMCs. In particular, the displaced null measurement is implemented through the coherent absorber and a simple fixed measurement basis on the environment of the QMC. Therefore, this estimation scheme can be implemented with considerably reduced measurement complexity. The coherent absorber plays an essential, but very different, role in this estimation scheme; it effectively reverses the evolution in each step of the QMC when an initial estimate θ_0 is equal to the true value of the parameters θ. The environment of the QMC can then be modelled as a vacuum state, but introducing some displacement results in ’excitation’ patterns in this vacuum state. The chapter develops the mathematical theory of these patterns and the culminates in an optimal estimator that utilizes the observed pattern counts.
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