Quantum metrology is an auspicious discipline of quantum information which is currently witnessing a surge of experimental breakthroughs and theoretical developments. The main goal of quantum metrology is to estimate unknown parameters as accurately as possible. By using quantum resources as probes, it is possible to attain a measurement precision that would be otherwise impossible using the best classical strategies. For example, with respect to the task of phase estimation, the maximum precision (the Heisenberg limit) is a quadratic gain in precision with respect to the best classical strategies. Of course, quantum metrology is not the sole quantum technology currently undergoing advances. The theme of this thesis is exploring how quantum metrology can be enhanced with other quantum techniques when appropriate, namely: graph states, error correction and cryptography. Graph states are an incredibly useful and versatile resource in quantum information. We aid in determining the full extent of the applicability of graph states by quantifying their practicality for the quantum metrology task of phase estimation. In particular, the utility of a graph state can be characterised in terms of the shape of the corresponding graph. From this, we devise a method to transform any graph state into a larger graph state (named a bundled graph state) which approximately saturates the Heisenberg limit. Additionally, we show that graph states are a robust resource against the effects of noise, namely dephasing and a small number of erasures, and that the quantum Cramér-Rao bound can be saturated with a simple measurement strategy. Noise is one of the biggest obstacles for quantum metrology that limits its achievable precision and sensitivity. It has been showed that if the environmental noise is distinguishable from the dynamics of the quantum metrology task, then frequent applications of error correction can be used to combat the effects of noise. In practise however, the required frequency of error correction to maintain Heisenberg-like precision is unobtainable for current quantum technologies. We explore the limitations of error correction enhanced quantum metrology by taking into consideration technological constraints and impediments, from which, we establish the regime in which the Heisenberg limit can be maintained in the presence of noise. Fully implementing a quantum metrology problem is technologically demanding: entangled quantum states must be generated and measured with high fidelity. One solution, in the instance where one lacks all of the necessary quantum hardware, is to delegate a task to a third party. In doing so, several security issues naturally arise because of the possibility of interference of a malicious adversary. We address these issues by developing the notion of a cryptographic framework for quantum metrology. We show that the precision of the quantum metrology problem can be directly related to the soundness of an employed cryptographic protocol. Additionally, we develop cryptographic protocols for a variety of cryptographically motivated settings, namely: quantum metrology over an unsecured quantum channel and quantum metrology with a task delegated to an untrusted party. Quantum sensing networks have been gaining interest in the quantum metrology community over the past few years. They are a natural choice for spatially distributed problems and multiparameter problems. The three proposed techniques, graph states, error correction and cryptography, are a natural fit to be immersed in quantum sensing network. Graph states are an well-known candidate for the description of a quantum network, error correction can be used to mitigate the effects of a noisy quantum channel, and the cryptographic framework of quantum metrology can be used to add a sense of security. Combining these works formally is a future perspective.
Quantum metrology is an auspicious discipline of quantum information which is currently witnessing a surge of experimental breakthroughs and theoretical developments. The main goal of quantum metrology is to estimate unknown parameters as accurately as possible. By using quantum resources as probes, it is possible to attain a measurement precision that would be otherwise impossible using the best classical strategies. For example, with respect to the task of phase estimation, the maximum precision (the Heisenberg limit) is a quadratic gain in precision with respect to the best classical strategies. Of course, quantum metrology is not the sole quantum technology currently undergoing advances. The theme of this thesis is exploring how quantum metrology can be enhanced with other quantum techniques when appropriate, namely: graph states, error correction and cryptography. Graph states are an incredibly useful and versatile resource in quantum information. We aid in determining the full extent of the applicability of graph states by quantifying their practicality for the quantum metrology task of phase estimation. In particular, the utility of a graph state can be characterised in terms of the shape of the corresponding graph. From this, we devise a method to transform any graph state into a larger graph state (named a bundled graph state) which approximately saturates the Heisenberg limit. Additionally, we show that graph states are a robust resource against the effects of noise, namely dephasing and a small number of erasures, and that the quantum Cramér-Rao bound can be saturated with a simple measurement strategy. Noise is one of the biggest obstacles for quantum metrology that limits its achievable precision and sensitivity. It has been showed that if the environmental noise is distinguishable from the dynamics of the quantum metrology task, then frequent applications of error correction can be used to combat the effects of noise. In practise however, the required frequency of error correction to maintain Heisenberg-like precision is unobtainable for current quantum technologies. We explore the limitations of error correction enhanced quantum metrology by taking into consideration technological constraints and impediments, from which, we establish the regime in which the Heisenberg limit can be maintained in the presence of noise. Fully implementing a quantum metrology problem is technologically demanding: entangled quantum states must be generated and measured with high fidelity. One solution, in the instance where one lacks all of the necessary quantum hardware, is to delegate a task to a third party. In doing so, several security issues naturally arise because of the possibility of interference of a malicious adversary. We address these issues by developing the notion of a cryptographic framework for quantum metrology. We show that the precision of the quantum metrology problem can be directly related to the soundness of an employed cryptographic protocol. Additionally, we develop cryptographic protocols for a variety of cryptographically motivated settings, namely: quantum metrology over an unsecured quantum channel and quantum metrology with a task delegated to an untrusted party. Quantum sensing networks have been gaining interest in the quantum metrology community over the past few years. They are a natural choice for spatially distributed problems and multiparameter problems. The three proposed techniques, graph states, error correction and cryptography, are a natural fit to be immersed in quantum sensing network. Graph states are an well-known candidate for the description of a quantum network, error correction can be used to mitigate the effects of a noisy quantum channel, and the cryptographic framework of quantum metrology can be used to add a sense of security. Combining these works formally is a future perspective.