Quantum entanglement and coherence often allow for protocols that outperform classical ones in estimating a system’s parameter. When using infinite-dimensional probes (such as a bosonic mode), one could, in principle, obtain infinite precision in a finite time for both classical and quantum protocols, which makes it hard to quantify potential quantum advantage. However, such a situation is unphysical, as it would require infinite resources, so one needs to impose some additional constraint: typically the average energy employed by the probe is finite. Here we treat both energy and time as a resource, showing that, in the presence of noise, there is a nontrivial interplay between the average energy and the time devoted to the estimation. Our results are valid for the most general metrological schemes (e.g., adaptive schemes, which may involve entanglement with external ancillae or any kind of continuous measurement). We apply recently derived precision bounds for all parameters characterizing the paradigmatic case of a bosonic mode, subject to Lindbladian noise. We show how the time employed in the estimation should be partitioned in order to achieve the best possible precision. In most cases, the optimal performance may be obtained without the necessity of adaptivity or entanglement with ancilla. We compare results with classical strategies. Interestingly, for temperature estimation, applying a fast-prepare-and-measure protocol with Fock states provides better scaling with the number of photons than any classical strategy.
Quantum entanglement and coherence often allow for protocols that outperform classical ones in estimating a system’s parameter. When using infinite-dimensional probes (such as a bosonic mode), one could, in principle, obtain infinite precision in a finite time for both classical and quantum protocols, which makes it hard to quantify potential quantum advantage. However, such a situation is unphysical, as it would require infinite resources, so one needs to impose some additional constraint: typically the average energy employed by the probe is finite. Here we treat both energy and time as a resource, showing that, in the presence of noise, there is a nontrivial interplay between the average energy and the time devoted to the estimation. Our results are valid for the most general metrological schemes (e.g., adaptive schemes, which may involve entanglement with external ancillae or any kind of continuous measurement). We apply recently derived precision bounds for all parameters characterizing the paradigmatic case of a bosonic mode, subject to Lindbladian noise. We show how the time employed in the estimation should be partitioned in order to achieve the best possible precision. In most cases, the optimal performance may be obtained without the necessity of adaptivity or entanglement with ancilla. We compare results with classical strategies. Interestingly, for temperature estimation, applying a fast-prepare-and-measure protocol with Fock states provides better scaling with the number of photons than any classical strategy.
Modern quantum systems have two facets: they are resources we would like to harness, and they contain physics we would like to probe. This thesis studies both through several case studies in quantum metrology and computing, unified by the interplay between entanglement, noise, and dissipation. We first probe entanglement in an optomechanical setting, where a mechanical oscillator is continuously driven and monitored by a light field. Assuming Gaussian dynamics and arbitrary, possibly non-Markovian noise, we derive necessary and sufficient conditions for entanglement. For a vacuum input, the entangling-disentangling transition is set by the environmental noise alone and is independent of the optomechanical coupling strength. Furthermore, frequency-dependent squeezing cannot entangle a system that is separable when driven by vacuum. In the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO), we find that the present level of classical noise, dominated by low-frequency seismic noise, makes significant stationary entanglement unlikely. We then turn to quantum frequency estimation and present two schemes. The first scheme harnesses entanglement as a metrological resource, approaching Heisenberg-limited precision up to a constant factor using blocks of Greenberger-Horne-Zeilinger (GHZ) states prepared with logarithmic-depth circuits and read out with local, adaptive measurements. The second scheme addresses the precision-bandwidth trade-off through a weak-measurement Ramsey protocol that recovers the classical precision limit over an extended dynamic range. Finally, we harness dissipation by introducing the 0-n Fock qubit, a noise-biased bosonic encoding stabilized by frequency-selective dissipation, with bit-flip errors exponentially suppressed in n (the highest Fock level). Used as a bias-preserving ancilla for cat-qubit architectures, it enables noise-biased controlled-X gates and low logical error rates with experimentally realistic parameters. Together, these results explore the regimes where entanglement and quantum advantage can survive noise, and show how measurement and dissipation can be engineered to protect them.
This thesis explores various protocols which can be applied to quantum metrology, such that precision of the measurement can beat the best classical approaches. We first demonstrate that ancilla-assisted schemes can improve phase estimation in the presence of noise, and we propose an experiment to demonstrate these effects. Given that the use of ancillae helps, it is no surprise that there are noisy channels for which the optimal strategy may depend on the noise level. We show that there is a non-trivial crossover between the ancilla-assisted strategy and the parallel-entangled strategy for Pauli noise channels and the amplitude damping channel. We see that under certain circumstances, the environment cannot access the parameter of interest. We then integrated techniques from quantum communication, to derive a general framework that renders quantum metrology protocols into cryptographic ones. Here the protocol is unconditionally secure if the trusted parties wish only to achieve the standard quantum limit. If Heisenberg-limited precision is to be achieved, they sacrifice some security, where adversaries can access information, but only at the risk of getting caught at it (cheat-sensitivity). In many scenarios, the quantum Fisher information is achieved for only one value of the phase. In a particular phase estimation scheme that measures the parity at the output of a Mach-Zehnder interferometer where the input is a two-mode squeezed vacuum, this problem is particularly imminent. We apply an adaptive technique to resolve this issue, showing that one can achieve precisions substantially below the shot noise limit. The key to a quadratically enhanced improvement in quantum metrology is entanglement. Finally, we demonstrate in a photonic system how high-dimensional entanglement can be certified, and provide an alternate interpretation of entanglement from an information-theoretic perspective.
Quantum optimal control (QOC) schemes can be employed to enhance the sensitivity of quantum metrology (QM) protocols undergoing Markovian noise, which can limit their precision to a standard quantum limit (SQL)-like scaling. In this paper, we propose a QOC scheme for QM that leverages entanglement and optimized coupling interactions with an ancillary system to provide enhanced metrological performance under general Markovian dynamics. We perform a comparative analysis of our entanglement-enhanced scheme against the unentangled scheme conventionally employed in QOC-enabled QM for varying evolution times and decoherence levels, revealing that the entanglement-enhanced scheme enables significantly better noise performance, even when a noisy ancilla is employed. We further extend our investigation to time-inhomogeneous noise models, specifically focusing on a noisy frequency estimation scenario within a spin-boson bath, and evaluate the protocol's performance under completely dissipative and dephasing dynamics. Our findings indicate that, in certain situations, schemes employing coherent control of a single particle are severely limited. In such cases, employing the entanglement-enhanced scheme can provide improved performance.
Bosonic systems, particularly in quantum optics and atomic physics, are leading platforms for achieving quantum enhanced precision in parameter estimation. By exploiting properties such as mode and particle entanglement, it is possible to attain precisions that surpass the shot noise limit with respect to key resources like probe number or energy. Yet the mechanisms by which these bosonic resources enable quantum enhancement remain unclear. Consequently, the design of optimal probes and evolutions often relies on case by case analyses, where continuous and discrete variable regimes are treated separately and their connection is still unclear. We develop a comprehensive framework for quantum metrology that unifies all known precision enhancement mechanisms based on bosonic systems. Our approach employs a superselection rule compliant representation of the electromagnetic field that explicitly includes the phase reference, thereby enforcing total particle number conservation and bridging the discrete and continuous limits of quantum optics and symmetric massive systems. Within this unified formalism, of which established results emerge as special cases, we identify the distinct roles of mode and particle entanglement for quantum enhanced precision. The framework further provides general measurement optimization strategies for arbitrary multimode entangled probe states and naturally incorporates noise and non-unitary dynamics, ensuring applicability to realistic experimental conditions.
Quantum metrology aims to exploit quantum phenomena to overcome classical limitations in the estimation of relevant parameters. We consider a probe undergoing a phase shift $\ensuremath{\varphi}$ whose generator is randomly sampled according to a distribution with unknown concentration $\ensuremath{\kappa}$, which introduces a physical source of noise. We then investigate strategies for the joint estimation of the two parameters $\ensuremath{\varphi}$ and $\ensuremath{\kappa}$ given a finite number N of interactions with the phase imprinting channel. We consider both single qubit and multipartite entangled probes, and identify regions of the parameters where simultaneous estimation is advantageous, resulting in up to a twofold reduction in resources. Quantum enhanced precision is achievable at moderate N, while for sufficiently large N classical strategies take over and the precision follows the standard quantum limit. We show that full-scale entanglement is not needed to reach such an enhancement, as efficient strategies using significantly fewer qubits in a scheme interpolating between the conventional sequential and parallel metrological schemes yield the same effective performance. These results may have relevant applications in optimization of sensing technologies.
The traditional framework of quantum metrology commonly assumes unlimited access to resources, overlooking resource constraints in realistic scenarios. As such, the optimal strategies therein can be infeasible in practice. Here, we investigate quantum metrology where the total energy consumption of the probe state preparation, intermediate control operations, and the final measurement is subject to a constraint. We establish a comprehensive theoretical framework for characterizing energy-constrained multistep quantum processes, based on which we develop a general optimization method for energy-constrained quantum metrology that determines both the optimal precision and the corresponding strategy. Using the method, we determine the ultimate precision limit of energy-constrained phase estimation and identify a novel advantage of quantum superpositions of causal orders in enhancing the energy efficiency of adaptive quantum estimation.