Quantum measurements lie at the core of quantum information processing, enabling tasks ranging from state reconstruction and parameter estimation to sensing and computation. As quantum technologies scale and become increasingly precise, a central question emerges: what does it mean for a quantum measurement to be good? This thesis addresses this question by developing quantitative, operational, and physically grounded frameworks for assessing the quality and informational power of quantum measurements.The first part of the thesis focuses on the intrinsic quality of quantum measurements independent of any particular input state. Adopting the formalism of quantum resource theories, measurements are treated as resources ordered under classical post-processing, which cannot increase informational content. Restricting attention to informationally complete measurements, we employ frame theory to introduce a novel resource monotone termed completeness stability, defined as the minimum eigenvalue of a suitably scaled frame operator. We establish its operational significance by showing that the average error in quantum state tomography and the average variance in shadow tomography are tightly bounded by the inverse of the completeness stability. Maximizing this monotone identifies weighted complex projective 2-designs, including symmetric informationally complete POVMs and mutually unbiased bases, as optimal measurement configurations.The second part of the thesis extends these ideas to quantum parameter estimation, where measurement performance necessarily depends on correlations between the measurement and the encoded quantum state. By introducing a state-dependent inner product on operator space, we develop an operator-theoretic formulation in which both the classical and quantum Fisher information arise naturally from a single frame operator associated with the measurement and the reference state. Within this framework, the spectral properties of the frame operator provide a complete characterization of estimation performance. The maximum and minimum eigenvalues identify the best- and worst-case parameter-encoding directions, respectively, yielding sharp bounds on the achievable ratio of classical and quantum Fisher information. This establishes a direct connection between informational completeness, estimation robustness, and resource-theoretic monotonicity.The final part of the thesis returns to foundational questions concerning quantum measurement and arrival time. Using Bohmian mechanics, we develop a microscopic, first-principles account of the Hong–Ou–Mandel effect, a paradigmatic two-particle interference phenomenon. By analyzing particle trajectories guided by the wave function, we show how bosonic bunching and fermionic anti-bunching emerge dynamically from the guiding equation without invoking wave-function collapse. Beyond reproducing the Hong–Ou–Mandel dip, we demonstrate how experimental parameters such as barrier strength, phase velocity, and path asymmetry influence both coincidence probabilities and temporal correlations.Taken together, this thesis provides a unified treatment of quantum measurement quality across operational, metrological, and dynamical regimes. By bridging resource theory, frame theory, Fisher information, and trajectory-based quantum dynamics, it offers new insights into the structure, limitations, and physical interpretation of quantum measurements in both applied and foundational contexts.
Quantum measurements lie at the core of quantum information processing, enabling tasks ranging from state reconstruction and parameter estimation to sensing and computation. As quantum technologies scale and become increasingly precise, a central question emerges: what does it mean for a quantum measurement to be good? This thesis addresses this question by developing quantitative, operational, and physically grounded frameworks for assessing the quality and informational power of quantum measurements.The first part of the thesis focuses on the intrinsic quality of quantum measurements independent of any particular input state. Adopting the formalism of quantum resource theories, measurements are treated as resources ordered under classical post-processing, which cannot increase informational content. Restricting attention to informationally complete measurements, we employ frame theory to introduce a novel resource monotone termed completeness stability, defined as the minimum eigenvalue of a suitably scaled frame operator. We establish its operational significance by showing that the average error in quantum state tomography and the average variance in shadow tomography are tightly bounded by the inverse of the completeness stability. Maximizing this monotone identifies weighted complex projective 2-designs, including symmetric informationally complete POVMs and mutually unbiased bases, as optimal measurement configurations.The second part of the thesis extends these ideas to quantum parameter estimation, where measurement performance necessarily depends on correlations between the measurement and the encoded quantum state. By introducing a state-dependent inner product on operator space, we develop an operator-theoretic formulation in which both the classical and quantum Fisher information arise naturally from a single frame operator associated with the measurement and the reference state. Within this framework, the spectral properties of the frame operator provide a complete characterization of estimation performance. The maximum and minimum eigenvalues identify the best- and worst-case parameter-encoding directions, respectively, yielding sharp bounds on the achievable ratio of classical and quantum Fisher information. This establishes a direct connection between informational completeness, estimation robustness, and resource-theoretic monotonicity.The final part of the thesis returns to foundational questions concerning quantum measurement and arrival time. Using Bohmian mechanics, we develop a microscopic, first-principles account of the Hong–Ou–Mandel effect, a paradigmatic two-particle interference phenomenon. By analyzing particle trajectories guided by the wave function, we show how bosonic bunching and fermionic anti-bunching emerge dynamically from the guiding equation without invoking wave-function collapse. Beyond reproducing the Hong–Ou–Mandel dip, we demonstrate how experimental parameters such as barrier strength, phase velocity, and path asymmetry influence both coincidence probabilities and temporal correlations.Taken together, this thesis provides a unified treatment of quantum measurement quality across operational, metrological, and dynamical regimes. By bridging resource theory, frame theory, Fisher information, and trajectory-based quantum dynamics, it offers new insights into the structure, limitations, and physical interpretation of quantum measurements in both applied and foundational contexts.
We introduce a resource monotone, the completeness stability, to quantify the quality of quantum measurements within a resource-theoretic framework. By viewing a quantum measurement as a frame, the minimum eigenvalue of a frame operator emerges as a significant monotone. It captures bounds on estimation errors and the numerical stability of inverting the frame operator to calculate the optimal dual for state reconstruction. Maximizing this monotone identifies a well-characterized class of quantum measurements forming weighted complex projective 2-designs, which includes well-known examples such as SIC-POVMs. Our results provide a principled framework for comparing and optimizing quantum measurements for practical applications.
Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.
Quantum states and measurements are two central physical objects in the theory of quantum mechanics. The preparation, control and measurement of quantum states lies at the heart of modern quantum information processing protocols ranging from secure communication to quantum computation and precision sensing. And yet, the reliable operation of these advanced protocols ultimately rests on a far more essential prerequisite - our ability to calibrate, certify and benchmark the quantum states we prepare and the quantum measurements we perform. Quantum information theory provides a natural framework for these tasks, meaningfully connecting the resources we expend for characterisation with the accuracy and precision we can expect in return. While decades of advances in quantum information have refined our understanding of quantum characterisation and sharpened our practical ability to calibrate with enhanced precision, important questions about the fundamental limits of quantum characterisation remain unanswered. Although incorporating entanglement at the measurement stage is known to enhance the estimation of unknown quantum states, the extent of this improvement remains poorly understood. This leaves open a fundamental question on the utility of complex entangling measurement strategies, impeding progress towards bridging the significant gap between practically achievable precision and the ultimate theoretical quantum limit. Complementary to quantum state estimation, which is relatively well-established in theory and practice, the characterisation of quantum measurements lacks a rigorous information-theoretic foundation, leading to a dearth of principled benchmarks and provably optimal strategies despite growing experimental interest. This thesis develops and applies information-theoretic tools to answer questions in quantum characterisation, with an eye to enabling practical improvements beyond current techniques in the field. First, we introduce a computable measure of entanglement enhancement for quantum state estimation and study its variation across general multi-parameter estimation problems for higher-dimensional quantum states. We show that the entanglement enhancement is maximised for full qudit state tomography but grows only linearly with system dimension even when allowing an asymptotically large number of entangling operations. Our proposed tight upper bound to entanglement enhancement therefore constitutes a strong limitation on the utility of such entangling measurements. Next, we formulate the dual problem of quantum detector estimation, developing a comprehensive framework for maximally precise calibration centred around a newly introduced quantity - the detector quantum Fisher information. This information-theoretic measure stipulates the ultimate precision attainable for measurement calibration and identifies the optimal input states that achieve this precision. Further, our framework enables the first provably optimal detector estimation experiment and reveals the scope for entanglement-based enhancements, thereby placing measurement characterisation on equal footing with state characterisation. Considered together, these results suggest two contrasting conclusions about quantum characterisation. On the one hand, they show that for quantum state estimation, increasingly powerful entangling measurements offer only diminishing returns at the ultimate precision limit. On the other hand, they highlight practical quantum advantages available for detector estimation by providing a framework that can improve calibration performance and guide future experimental design. More broadly, these results complete and advance the triad of state, process, and detector characterisation, positioning quantum information measures as not merely abstract bounds, but practical guides for deciding when complex quantum resources are genuinely worth deploying and where near-term quantum advantages are most likely to be realised.
A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence \& purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures.
Quantum physics currently provides our deepest description of nature. To understand precisely how it differs from classical physics is not only of foundational interest, but also important to advance technology: the advent of quantum information has led to the insight that certain quantum phenomena, e.g., entanglement and coherence, are resources which enable us to solve practical tasks that are unsolvable within the realm of classical physics. But how can we quantify quantum resources and how do we decide which resource is relevant for a specific operational advantage in quantum cryptography, quantum computation, or quantum metrology? How can we employ these resources optimally? Addressing these questions is the motivation for the development of quantum resource theories, a mathematically rigorous framework that allows us to investigate a specific quantum phenomenon as a resource. Formally, such theories emerge from restrictions which are imposed in addition to the laws of quantum mechanics. A prominent example is the restriction to local operations and classical communication, which singles out entanglement as a resource. Typically, such theories describe the value of quantum states, i.e., static resources. An important question in such theories concerns their structure as determined by the restriction under consideration, which is analyzed for coherence theory on qubits in this thesis. Only recently, so-called dynamical resource theories that quantify the value of operations were proposed, which are advantageous for several reasons: they include resource theories of states as special cases, allow us to quantify additional quantum properties such as the detection or usage of static resources, which will be a main topic of this thesis, and are more natural from an operational perspective. But are different types of dynamical resources present in quantum operations related? For the case of dynamical coherence and entanglement, it will be shown that there is not only a quantitative, but also a qualitative connection.
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.