The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it allows to derive the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In Quantum Phase Estimation protocols, this leads to identify coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e., the Symmetric Logarithmic Derivative, in many cases allows to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows: to discuss the role of coherence vs correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behaviour. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behaviour proper of a large class of quantum phase transitions.
The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it allows to derive the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In Quantum Phase Estimation protocols, this leads to identify coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e., the Symmetric Logarithmic Derivative, in many cases allows to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows: to discuss the role of coherence vs correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behaviour. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behaviour proper of a large class of quantum phase transitions.
Abstract The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it establishes the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In quantum phase estimation protocols, this leads to propose coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e. the symmetric logarithmic derivative, in many cases allows one to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows one to discuss the role of coherence versus correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behavior. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behavior proper of a large class of quantum phase transitions.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N2 with the number of particles involved---substantially surpassing the 1/N-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic N limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large N the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.
Coherent ensembles of N qubits present an advantage in quantum phase estimation over separable mixtures, but coherence decay due to classical phase diffusion reduces overall precision. In some contexts, the strength of diffusion may be the parameter of interest. We examine estimation of both phase and diffusion in large spin systems using a novel mathematical formulation. For the first time, we show a closed form expression for the quantum Fisher information for estimation of a unitary parameter in a noisy environment. The optimal probe state has a non-Gaussian profile and differs also from the canonical phase state; it saturates a new tight precision bound. For noise below a critical threshold, entanglement always leads to enhanced precision, but the shot-noise limit is beaten only by a constant factor, independent of N. We provide upper and lower bounds to this factor, valid in low and high noise regimes. Unlike other noise types, it is shown for N≫1 that phase and diffusion can be measured simultaneously and optimally.
It is suggested that computers exploiting quantum phenomena can solve tasks with an efficiency surpassing that achievable by classical computers. One example of this quantum advantage arises in the task of phase estimation. Here, an unknown eigenvalue of a unitary evolution is estimated, a critical subroutine for many quantum algorithms and quantum metrology protocols. For a fixed amount of resources, the utilisation of quantum coherence and entanglement can lead to a quadratic increase in precision of the final estimate compared to classical statistical limits. In systems affected by depolarising noise, including current quantum hardware, this improvement becomes a constant factor. While numerous phase estimation algorithms have been proposed claiming to achieve these theoretical limits, the lack of well-defined criteria for both precision and resource utilisation casts doubt on these results. Further issues arise due to a lack of point identification (the existence of multiple equally likely estimates) that some algorithms have. I aim to address these issues in this thesis. I employ statistical inference, the theoretical foundation for interpreting any quantum measurement, in order to find rigorous definitions for precision and resource use. By combining these definitions with previous work in the field of quantum metrology, I am able to derive theoretical precision limits that one can hope to achieve. I then outline several phase estimation algorithms that can achieve these limits in different contexts. These algorithms that I describe are split into three major groups. Algorithms in the first group focus on the process of point identification; one set of measurements provides a precise estimate of the unknown phase before an additional set of measurements removes any ambiguities present. This second set of measurements can be either decided before any measurement is taken or during measurements in an iterative fashion. I show that the balancing of resources used in each set of measurements can lead to optimally scaling precision both with and without depolarising noise. These algorithms are based upon two of my peer-reviewed articles, [Phys. Rev. A 106, 062615 (2022)] and [Phys. Rev. A 109, 042412 (2024)]. The second group of algorithms that I introduce use the principles of Bayesian experimental design. Here, previous measurement outcomes are used to select future measurements recursively. These algorithms require large classical computing times to select each measurement but boast high precision in certain metrics. The final group of algorithms exploit the oscillating nature of phases by using a discrete Fourier transform (DFT). The DFT can be implemented using either classical or quantum hardware, the latter of which leads to the famous quantum phase estimation algorithm (QPEA). In this thesis, I show that the QPEA performs worse than some classical precision limits. However, I also suggest some modifications to the QPEA circuit that can lead to optimal precision scaling. These results and modifications are described in one of my articles awaiting publication [arXiv:2505.00764].
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.
Quantum multiparameter estimation offers a framework for the simultaneous estimation of multiple parameters, pertaining to possibly noncommutating observables. While the optimal probe for estimating a single unitary phase is well understood - being a pure state that is an equal superposition of the eigenvectors of the encoding Hamiltonian corresponding to its maximum and minimum eigenvalues - the structure of optimal probes in the multiparameter setting remains more intricate. We investigate the simultaneous estimation of two phases, each encoded through arbitrary qubit Hamiltonians, using arbitrary weight matrices, and considering single-qubit probes. We also consider single-qutrit probes, for which the encoding Hamiltonians are chosen as SU(2) generators. We find that in both the qubit and qutrit scenarios, the optimal probe is a coherent superposition of the eigenstates corresponding to the largest and smallest eigenvalues of the total encoding Hamiltonian, with a fixed - and not arbitrary - relative phase. Remarkably, this optimal probe is independent of the specific choice of weight matrix, making it a universally optimal input state for the estimation of any pair of SU(2) parameters, which can be reparameterized to the phase estimation problem. Furthermore, we show that this probe also maximizes the determinant of the quantum Fisher information matrix, providing a handy tool for identifying the optimal probe state. We also examine the role of commutativity between the generators of the unitary encodings. Our results demonstrate that a high degree of commutativity degrades the achievable precision, rendering estimation infeasible in the extreme case. However, maximal noncommutativity does not necessarily yield optimal precision.