Abstract Precise device characterisation is a fundamental requirement for a large range of applications using photonic hardware, and constitutes a multi-parameter estimation problem. Estimates based on measurements using single photons or classical light have precision which is limited by shot noise, while quantum resources can be used to achieve sub-shot-noise precision. However, there are many open questions with regard to the best quantum protocols for multi-parameter estimation, including the ultimate limits to achievable precision, as well as optimal choices for probe states and measurements. In this paper, we develop a formalism based on Fisher information to tackle these questions for setups based on linear-optical components and photon-counting measurements. A key ingredient of our analysis is a mapping for equivalent protocols defined for photonic and spin systems, which allows us to draw upon results in the literature for general finite-dimensional systems. Motivated by the protocol in Zhou, et al Optica 2, 510 (2015), we present new results for quantum-enhanced tomography of unitary processes, including a comparison of Holland-Burnett and NOON probe states.
Abstract Precise device characterisation is a fundamental requirement for a large range of applications using photonic hardware, and constitutes a multi-parameter estimation problem. Estimates based on measurements using single photons or classical light have precision which is limited by shot noise, while quantum resources can be used to achieve sub-shot-noise precision. However, there are many open questions with regard to the best quantum protocols for multi-parameter estimation, including the ultimate limits to achievable precision, as well as optimal choices for probe states and measurements. In this paper, we develop a formalism based on Fisher information to tackle these questions for setups based on linear-optical components and photon-counting measurements. A key ingredient of our analysis is a mapping for equivalent protocols defined for photonic and spin systems, which allows us to draw upon results in the literature for general finite-dimensional systems. Motivated by the protocol in Zhou, et al Optica 2, 510 (2015), we present new results for quantum-enhanced tomography of unitary processes, including a comparison of Holland-Burnett and NOON probe states.
A fundamental task in photonics is to characterise an unknown optical process, defined by properties such as birefringence, spectral response, thickness and flatness. Amongst many ways to achieve this, single-photon probes can be used in a method called quantum process tomography (QPT). Furthermore, QPT is an essential method in determining how a process acts on quantum mechanical states. For example for quantum technology, QPT is used to characterise multi-qubit processors and quantum communication channels; across quantum physics QPT of some form is often the first experimental investigation of a new physical process, as shown in the recent research into coherent transport in biological mechanisms. However, the precision of QPT is limited by the fact that measurements with single-particle probes are subject to unavoidable shot noise---this holds for both single photon and laser probes. In situations where measurement resources are limited, for example, where the process is rapidly changing or the time bandwidth is constrained, it becomes essential to overcome this precision limit. Here we devise and demonstrate a scheme for tomography which exploits non-classical input states and quantum interferences; unlike previous QPT methods our scheme capitalises upon the possibility to use simultaneously multiple photons per mode. The efficiency---quantified by precision per photon used---scales with larger photon number input states. Our demonstration uses four-photon states and our results show a substantial reduction of statistical fluctuations compared to traditional QPT methods---in the ideal case one four-photon probe state yields the same amount of statistical information as twelve single probe photons.
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 measurements, alongside quantum states and processes, form a cornerstone of quantum information processing. However, unlike states and processes, their efficient characterisation remains relatively unexplored. We resolve this asymmetry by introducing a comprehensive framework for efficient detector estimation that reveals the fundamental limits to extractable parameter information and errors arising in detector analysis - the detector quantum Fisher information. Our development eliminates the need to optimise for the best probe state, while highlighting aspects of detector analysis that fundamentally differ from quantum state estimation. Through proofs, examples and experimental validation, we demonstrate the relevance and robustness of our proposal for current quantum detector technologies. By formalising a dual perspective to state estimation, our framework completes and connects the triad of efficient state, process, and detector tomography, advancing quantum information theory with broader implications for emerging technologies reliant on precisely calibrated measurements.
Accurately estimating the overlap between quantum states is a fundamental task in quantum information processing. While various strategies using distinct quantum measurements have been proposed for overlap estimation, the lack of experimental benchmarks on estimation precision limits strategy selection in different situations. Here we compare the performance of four practical strategies for overlap estimation, including tomography-tomography, tomography-projection, Schur collective measurement and optical swap test using photonic quantum systems. We encode the quantum states on the polarization and path degrees of freedom of single photons. The corresponding measurements are performed by photon detection on certain modes following single-photon mode transformation or two-photon interference. We further propose an adaptive strategy with optimized precision in full-range overlap estimation. Our results shed new light on extracting the parameter of interest from quantum systems, prompting the design of efficient quantum protocols.
With the rapid development of quantum technologies in recent years, the need for high-sensitivity measuring techniques has become a key issue. In particular, optical sensors based on quantum states of light have proven to be optimal resources for high-precision interferometry. Nevertheless, their performance may be severely affected by the presence of noise or imperfections. In this work, we derive the quantum Fisher information matrix associated with the simultaneous estimation of an interferometric phase and the indistinguishability characterizing the probe state consisting of an even number of photons. We find the optimal measurement attaining the ultimate precision for both parameters in a single setup and perform an experiment based on a pair of photons with an unknown degree of indistinguishability entering a two-port interferometer.
Quantum metrology promises phase sensitivity surpassing the shot-noise limit by exploiting entanglement and photon-number correlations. NOON states-maximally path-entangled N-photon superpositions (∣N,0⟩+∣0,N⟩)/2 -achieve the Heisenberg limit 1/N for single-parameter estimation, as demonstrated experimentally by Afek et al. (2010) using hybrid coherent-plus-squeezed light up to N=5. We present an end-to-end differentiable quantum-optical framework-implemented in Strawberry Fields (Killoran et al., 2019) with a TensorFlow backend -that learns optimal circuit parameters by maximising the classical Fisher information (CFI) across all coincidence channels for N=2,3,4,5. Starting from proper numerical reproductions of the Afek et al. coincidence fringes, verified by FFT analysis and parity measurements, we apply gradient descent (Adam) to the eight trainable circuit parameters. Raw CFI improvements grow dramatically with photon number: +153% (N=2), +834% to +956% (N=3), +829% to +1598% (N=4), and +1775% (N=5), alongside post-selection rate improvements of +153% to +3269%, and an 8× to 133× improvement in useful measurement events per pulse across N=2-5. A fundamental inter-channel trade-off is identified at N=2 but weakens at higher N where the Afek initialisation is further from optimal. These results provide numerically rigorous benchmarks for adaptive single-parameter quantum sensing and demonstrate that the Afek working point is significantly suboptimal at N≥3. QFI calculations confirm that the optimised probe reaches 82% of the Heisenberg limit at N=2 and improves from 36% to 58% at N=5, while useful measurement events per pulse improve by 8× to 133× across all N, making quantum-enhanced sensing at N≥3 experimentally practical.