Abstract We establish the rigorous operational meaning of dynamical total coherence within the resource non-increasing (RNI) framework for the task of single-parameter quantum channel estimation. By connecting distance-based and maximal-increment-based dynamical coherence measures with the right-logarithmic-derivative (RLD) Fisher information and the Cramér-Rao bound (CRB), we prove that each dynamical coherence measure directly determines the fundamental precision limit of channel parameter estimation. We show that the diamondnorm measure yields a tight error lower bound, the induced trace-norm measure gives a simplified bound for device-limited scenarios, the relative-entropy measure governs the asymptotic error-exponent rate, the l 2 -based measure quantifies the precision gain from optimal input states, and the RLD-Fisher-based measure provides an achievable error upper bound. This work unifies dynamical resource theory with quantum channel estimation and endows dynamical total coherence with clear, task-oriented operational significance.
Abstract We establish the rigorous operational meaning of dynamical total coherence within the resource non-increasing (RNI) framework for the task of single-parameter quantum channel estimation. By connecting distance-based and maximal-increment-based dynamical coherence measures with the right-logarithmic-derivative (RLD) Fisher information and the Cramér-Rao bound (CRB), we prove that each dynamical coherence measure directly determines the fundamental precision limit of channel parameter estimation. We show that the diamondnorm measure yields a tight error lower bound, the induced trace-norm measure gives a simplified bound for device-limited scenarios, the relative-entropy measure governs the asymptotic error-exponent rate, the l 2 -based measure quantifies the precision gain from optimal input states, and the RLD-Fisher-based measure provides an achievable error upper bound. This work unifies dynamical resource theory with quantum channel estimation and endows dynamical total coherence with clear, task-oriented operational significance.
The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel's Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when
Abstract One of the fundamental tasks in quantum metrology is to estimate multiple parameters embedded in a noisy process, i.e. a quantum channel. In this paper, we study fundamental limits to quantum channel estimation via the concept of amortization and the right logarithmic derivative (RLD) Fisher information value. Our key technical result is the proof of a chain-rule inequality for the RLD Fisher information value, which implies that amortization, i.e. access to a catalyst state family, does not increase the RLD Fisher information value of quantum channels. This technical result leads to a fundamental and efficiently computable limitation for multiparameter channel estimation in the sequential setting, in terms of the RLD Fisher information value. As a consequence, we conclude that if the RLD Fisher information value is finite, then Heisenberg scaling is unattainable in the multiparameter setting.
We define the resource non-increasing (RNI) framework to study the dynamical resource theory. With this definition, we propose several potential quantification candidates under various free operation sets. For explicit demonstrations, we quantify the quantum dynamical coherence in the scenarios with and without post-selective measurements. Correspondingly, we show that the maximally incoherent operations (MIO) and the incoherent operations (IO) in the static coherence resource theory are free in the sense of dynamical coherence. We also provide operational meanings for the measures by the quantum discrimination tasks. Moreover, for the dynamical total coherence, we also present convenient measures and give the analytic calculation for the amplitude damping channel.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the k-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most k, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.
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.
Practical quantum single-parameter estimation is rarely a pristine task; it almost invariably involves nuisance parameters, casting it as a one-from-many problem. Existing approaches to this problem rely on multi-parameter metrology, reducing the matrix quantum Cramér-Rao bound to obtain scalar quantum precision limits. However, these methods are often hampered by the demanding inversion of the quantum Fisher information (QFI) matrix and the requisite choice of a weight matrix, and they break down when the QFI matrix becomes singular. Here, we show that for dynamical one-from-many estimation, a previously overlooked sum rule connecting the QFI about all model parameters to the QFI about time necessitates including the latter to consider an augmented QFI matrix while simultaneously rendering it inherently singular--precisely the scenario where conventional approaches fail. To meet this challenge, we derive a tight, matrix-free quantum precision bound that involves only scalar quantities, offers broad applicability, and subsumes existing results as special cases. Validated in both unitary and noisy settings, our findings provide a refined operational framework for practical quantum single-parameter metrology.