Quantum measurements are an essential ingredient in numerous protocols and quantum technologies.As a characteristic trait, quantum measurements are typically employed a vast number of times to allow for higher accuracy, better control or just increased performance of quantum devices.In this thesis, we pursue a resource-theoretic approach to quantum measurements, aiming at the quantification of their energetic cost.While there exists a plethora of literature on this topic already, we develop a general framework that allows us to unify several of these previous results and to improve them systematically.We highlight an issue common to most of these previous works, where a notion of quantum measurement is employed which is not detailed enough to be applicable to several realistic and important applications for quantum measurement.More concretely, previous work typically involves a state transformation concept, similar to thermodynamic considerations, where only a specific initial and final state of the measured system are specified to characterise the process.For real applications however, quantum measurement devices are typically required to perform the measurement for all possible initial states of the measured system.In this thesis we analyse this requirement and obtain as a consequence much stronger energy constraints on quantum measurements than previously known.For particular forms of measurement, namely dephasing channels, projective quantum instruments, and projection valued measures, we even determine the exact energy costs in contrast to the common lower bound statements in previous literature.Remarkably, the exact cost of projective quantum instruments constitutes the strongest energy constraint on quantum measurement known so far.We demonstrate the implications of these results in two important applications, where quantum measurements play a significant role, namely quantum Zeno stabilisation and quantum error correction.The exact energetic requirements we compute for these two protocols show that any restriction on the amount of available energy immediately places fundamental constraints on the performance of these protocols.
Quantum measurements are an essential ingredient in numerous protocols and quantum technologies.As a characteristic trait, quantum measurements are typically employed a vast number of times to allow for higher accuracy, better control or just increased performance of quantum devices.In this thesis, we pursue a resource-theoretic approach to quantum measurements, aiming at the quantification of their energetic cost.While there exists a plethora of literature on this topic already, we develop a general framework that allows us to unify several of these previous results and to improve them systematically.We highlight an issue common to most of these previous works, where a notion of quantum measurement is employed which is not detailed enough to be applicable to several realistic and important applications for quantum measurement.More concretely, previous work typically involves a state transformation concept, similar to thermodynamic considerations, where only a specific initial and final state of the measured system are specified to characterise the process.For real applications however, quantum measurement devices are typically required to perform the measurement for all possible initial states of the measured system.In this thesis we analyse this requirement and obtain as a consequence much stronger energy constraints on quantum measurements than previously known.For particular forms of measurement, namely dephasing channels, projective quantum instruments, and projection valued measures, we even determine the exact energy costs in contrast to the common lower bound statements in previous literature.Remarkably, the exact cost of projective quantum instruments constitutes the strongest energy constraint on quantum measurement known so far.We demonstrate the implications of these results in two important applications, where quantum measurements play a significant role, namely quantum Zeno stabilisation and quantum error correction.The exact energetic requirements we compute for these two protocols show that any restriction on the amount of available energy immediately places fundamental constraints on the performance of these protocols.
Measurements and feedback are essential in the control of any device operating at the quantum scale and exploiting the features of quantum physics. As the number of quantum components grows, it becomes imperative to consider the energetic expense of such elementary operations. Here, we derive energy requirements for general quantum measurement, extending previous models and obtaining stronger bounds in relevant situations, and then study two important classes of measurements in detail. One is the projective measurement, where we obtain the exact cost rather than a lower bound, and the other is the so-called inefficient measurement, in which we explicitly show that energy extraction is possible. As applications, we derive the energy-precision trade-off in quantum Zeno stabilisation schemes and the exact energy expense for quantum error correction. Our results constitute fundamental energetic limitations against which to benchmark implementations of future quantum devices as they grow in complexity.
Inspired by environmental sciences, we develop a framework to quantify the energy needed to generate quantum entanglement via noisy quantum channels, focusing on the hardware-independent, i.e. fundamental cost. Within this framework, we define a measure of the minimal fundamental energy consumption rate per distributed entanglement (expressed in Joule per ebit). We then derive a lower bound on the energy cost of distributing a maximally entangled state via a quantum channel, which yields a quantitative estimate of energy investment per entangled bit for future quantum networks. We thereby show that irreversibility in entanglement theory implies a non-zero energy cost in standard entanglement distribution protocols. We further establish an upper bound on the fundamental energy consumption rate of entanglement distribution by determining the minimal energy required to implement quantum operations via classical control. To this end, we formulate the axioms for an energy cost measure and introduce a Hamiltonian model for classically-controlled quantum operations. The fundamental cost is then defined as the infimum energy over all such Hamiltonian protocols, with or without specific hardware constraints. The study of the energy cost of a quantum operation is general enough to be naturally applicable to quantum computing and is of independent interest. Finally, we evaluate the energy demands of three entanglement distillation protocols for photonic polarization qubits, finding that, due to entanglement irreversibility, their required energy exceeds the fundamental lower bound by many orders of magnitude. The introduced paradigm can be applied to other quantum resources, with appropriate changes depending on their nature.
Quantifying the energetic cost of implementing quantum operations is essential for assessing the efficiency and scalability of quantum sensing and information-processing technologies. Here, we introduce a framework for analysing the interplay between complexity and energy cost of quantum processes. In particular, we apply our framework to a sequential quantum phase estimation protocol, where a phase of physical significance is encoded in a quantum channel. The channel is applied to a probe state repeatedly until the probe is measured and the outcome leads to an estimate on the phase. We establish a trade-off relation between the total energy cost of the protocol and the number of times the channel is applied (complexity), while reaching a desired estimation precision. A sweet spot is located where the two quantities are co-optimised. The principles of our analysis can be adapted to benchmark the energetic requirements in other quantum protocols and devices.
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.
We show that it is impossible to perform ideal projective measurements on quantum systems using finite resources. We identify three fundamental features of ideal projective measurements and show that when limited by finite resources only one of these features can be salvaged. Our framework is general enough to accommodate any system and measuring device (pointer) models, but for illustration we use an explicit model of an N -particle pointer. For a pointer that perfectly reproduces the statistics of the system, we provide tight analytic expressions for the energy cost of performing the measurement. This cost may be broken down into two parts. First, the cost of preparing the pointer in a suitable state, and second, the cost of a global interaction between the system and pointer in order to correlate them. Our results show that, even under the assumption that the interaction can be controlled perfectly, achieving perfect correlation is infinitely expensive. We provide protocols for achieving optimal correlation given finite resources for the most general system and pointer Hamiltonians, phrasing our results as fundamental bounds in terms of the dimensions of these systems.
The foundational theory of quantum enhanced metrology for parameter estimation is of fundamental importance to the progression of science and technology as the scientific method is built upon empirical evidence, the acquisition of which is entirely reliant on measurement. Quantum mechanical properties can be exploited to yield measurement results to a greater precision (lesser uncertainty) than that which is permitted by classical methods. This has been mathematically demonstrated by the derivation of theoretical bounds which place a fundamental limit on the uncertainty of a measurement. Furthermore, quantum metrology is of immediate interest in the application of quantum technologies since measurement plays a central role. This thesis focuses on the role of quantum correlations and uncertainty relations which govern the precision bounds. We show how correlations can be distributed amongst limited resources in realistic scenarios, as permitted by current experimental capabilities, to achieve higher precision measurements than current approaches. This is extended to the setting of multiparameter estimation in which we demonstrate a more technologically feasible method of correlation distribution than those previously posited which perform as well as, or worse than, our scheme. Furthermore, a quantum metrology protocol is typically comprised of three stages: probe state preparation, sensing and then readout, where the time required for the first and last stages is usually neglected. We consider the more realistic sensing scenario of time being a limited resource which is divided amongst the three stages and demonstrate the most efficient use of this resource. Additionally, we take an information theoretic approach to quantum mechanical uncertainty relations and derive a one-parameter class of uncertainty relations which supplies more information about the quantum mechanical system of interest than conventional uncertainty relations. Finally, we demonstrate how we can use this class of uncertainty relations to reconstruct information of the state of the quantum mechanical system.