Fisher information is a statistical metric whose inverse represents a lower bound on the variance of an unbiased estimator. Naturally, in order to make the best estimates of a given unknown parameter, one would desire to find ways to minimize the value of this lower bound. Thus, the Fisher information can also be used to compare the quality of different estimators for a given parameter: whichever estimator has a lower value for its Fisher information must also provide better estimates. The same holds true in the framework of quantum information theory, where an analog to the Fisher information, dubbed quantum Fisher information, performs the same role as its classical counterpart. This dissertation concerns itself with both types of Fisher information, and so I present two projects in which Fisher information was a helpful tool for determining the quality of our work. In the first project, we examined potential solutions to the problem of optical loss in long-baseline interferometers, which creates a bottleneck in the resolution these arrays of telescopes can achieve. What we suggested was a novel entanglement-assisted scheme that interferes a two-mode squeezed vacuum state with the target astronomical state, and then measures the resulting state by means of homodyne detection. Ultimately, we showed that when the Fisher information is observed, our proposed scheme did not outperform the traditional schemes used in long-baseline interferometry. In the second project, we sought to write code that could determine how to best use dynamical decoupling pulses to correct dephasing error in a nitrogen-vacancy center realization of a memory and spectator qubit system. Through the use of the Python coding language, we were able to write programs that could determine the characteristic time and model the change in fidelity of the system when XY4 pulses are applied. Ultimately, we found that the quantum fidelity of our model closely mirrored the fidelity decay function our system was expected to follow in the case where XY4 pulses are applied.
Fisher information is a statistical metric whose inverse represents a lower bound on the variance of an unbiased estimator. Naturally, in order to make the best estimates of a given unknown parameter, one would desire to find ways to minimize the value of this lower bound. Thus, the Fisher information can also be used to compare the quality of different estimators for a given parameter: whichever estimator has a lower value for its Fisher information must also provide better estimates. The same holds true in the framework of quantum information theory, where an analog to the Fisher information, dubbed quantum Fisher information, performs the same role as its classical counterpart. This dissertation concerns itself with both types of Fisher information, and so I present two projects in which Fisher information was a helpful tool for determining the quality of our work. In the first project, we examined potential solutions to the problem of optical loss in long-baseline interferometers, which creates a bottleneck in the resolution these arrays of telescopes can achieve. What we suggested was a novel entanglement-assisted scheme that interferes a two-mode squeezed vacuum state with the target astronomical state, and then measures the resulting state by means of homodyne detection. Ultimately, we showed that when the Fisher information is observed, our proposed scheme did not outperform the traditional schemes used in long-baseline interferometry. In the second project, we sought to write code that could determine how to best use dynamical decoupling pulses to correct dephasing error in a nitrogen-vacancy center realization of a memory and spectator qubit system. Through the use of the Python coding language, we were able to write programs that could determine the characteristic time and model the change in fidelity of the system when XY4 pulses are applied. Ultimately, we found that the quantum fidelity of our model closely mirrored the fidelity decay function our system was expected to follow in the case where XY4 pulses are applied.
The first half of this thesis deals with the problem of parameter estimation in a quantum system. In quantum parameter estimation theory, the quantum Fisher information is usually considered as the ultimate precision limit, beyond which no further improvement is possible due to the inherent stochasticity of quantum measurements. On the other hand, in this thesis we will show that a better precision is achievable than predicted by a quantum Fisher information analysis. This is true if some regularity assumptions about the underlying quantum parametric model are relaxed. In such situations, the quantum Fisher information does not completely capture the best possible performance of quantum measurements, and a different approach must be followed. In the second part of the thesis, we will focus on some applications of orthogonal array theory to two notable quantum information problems: the problem of multipartite entanglement classification and the quantum marginal problem. Introduced by Rao in 1947, orthogonal arrays have been usefully applied to different fields, from cryptography and coding theory to the statistical design of experiments, software testing and quality control. Remarkably, orthogonal arrays have also found application in quantum information and, in particular, in the study of quantum entanglement. We will employ tools from orthogonal array theory to study a toy version of the multipartite entanglement classification problem. Finally, we will show how orthogonal arrays can be employed to build constructive solutions to low-dimensional quantum marginal problems.
As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.
We introduce a positive Hermitian operator, the Fisher operator, and use it to examine a measurement process incorporating unitary dynamics and complete measurements. We develop the idea of information complement, the minimization of which establishes the optimal precision for a fixed input. The formalism demonstrates that, in general, the classical Fisher Information has the Hamiltonian semi-norm as an upper bound. This is achievable with a qubit probe and only projective measurements, and is independent of the true value of the estimated parameter. In an interferometry context, we show that an optimal measurement scheme can be constructed from linear optics and photon counting, without recourse to generalised measurements or exotic unitaries outside of SU(2).
This thesis presents a collection of projects spanning different sub-fields of quantum science, undertaken with the goal of understanding and advancing quantum technologies. Broadly, it can be divided into two thematic parts — parameter estimation and open system dynamics. Each project addresses inverse and ill-posed problems in Physics, which lie at the heart of developing robust and scalable quantum devices.A central theme in quantum technology is the challenge of inference of microscopic system parameters from limited, indirect observations. The first part of the thesis investigates this through the problem of Hamiltonian reconstruction via local measurements. Indirect estimation of couplings in quantum systems relies on the measurement of the spectrum and the overlap of eigenvectors with some reference state. For a one-dimensional chain, this data can be obtained by local measurement on one end of the chain, thereby reducing the need for full Hamiltonian tomography. Information about couplings is extracted from the temporal data obtained by measuring the expectation values of the time evolution 𝑒−𝑖𝐻𝑡 for a reference state. In Chapter 3, we analyze the robustness of this estimation scheme under perturbations arising from beyond nearest neighbour interactions. We show that the error scales linearly with the strength of such interactions and on average, the error increases with the distance of interaction between sites.Chapter 4 examines whether coupling information can be extracted from local thermal measurements, using expectation values of the Gibbs state at varying temperatures rather than temporal data. We develop a procedure to obtain the coefficients of the Taylor expansion of these expectation values and relate them to the Hamiltonian parameters. For very small chains, this mapping can be carried out analytically, while for larger systems it reveals the algebraic structure that links local measurements to the underlying couplings and highlights the trade-offs between measurement resources, numerical stability, and identifiability. The analysis also clarifies how additional local fields or self-interaction terms enhance the information content of thermal data, suggesting a role for local thermal probes as practical tools for Hamiltonian verification and parameter estimation.Building on the ideas of controlling and inferring information about a quantum system, we now turn to active intervention through adaptive measurement. In Chapter 5, we develop a process tensor driven procedure for cooling the spin bath that is coupled to a nitrogen vacancy (NV) center in diamond. By modeling the sequence of intermediate quantum instruments acting on a correlated system-environment process, we optimize a pair of completely positive (CP) maps to maximize the population in a lower-energy state. Unlike existing approaches that rely solely on positive operator valued measures (POVMs), our method encompasses a broader class of CP maps, including adaptive feed-forward and mixed operations. We explicitly account for initial correlation between system and environment and distinguish between the controllable and uncontrollable parts of the dynamics. Our framework supports non-Markovian operations and benchmarks them against optimal Markovian strategies. The optimization is performed via Semi-definite Programming (SDP), ensuring global optimality. We also assess robustness to Hamiltonian noise and provide a proposal for implementation using NV center in diamonds.Finally, Chapter 6 explore the relaxation dynamics of topological qubits, focusing on a tetron qubit formed from two uncoupled Kitaev chains coupled to a finite fermionic bath. Using the correlation matrix formalism for purely quadratic couplings, we show that the relaxation is fundamentally independent of temperature when either the bath is large, or the zero-mode energy of the system is close to zero. This challenges a key assumption in the field that cooling enhances coherence. Interestingly, we find that introducing glassy disorder into the system slows down relaxation, offering a potential route to prolong qubit lifetimes.Collectively, these studies form a coherent narrative around extraction, control and utilization of quantum information under fundamental constraints. By tackling such inverse and often ill-posed problems, this thesis contributes new tools and insights essential for the realization of future quantum technologies.
The formalism of quantum estimation theory, focusing on the quantum and classical Fisher information, is applied to the estimation of the coupling strength in an optomechanical system. In order to estimate the optomechanical coupling, we have considered a cavity optomechanical model with non-Markovian Brownian motion of the mirror and employed input-output formalism to obtain the cavity output field. Our estimation scenario is based on balanced homodyne photodetection of the cavity output field. We have explored the difference between the associated measurement-dependent classical Fisher information and the quantum Fisher information, thus addressing the question of whether it is possible to reach the lower bound of the mean-squared error of an unbiased estimator by means of balanced homodyne detection. We have found that the phase of the local oscillator in the homodyne detection is crucial; certain quadrature measurements allow very accurate estimation.
We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.