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The quest for precision in parameter estimation is a fundamental task in different scientific areas. The relevance of this problem thus provided the motivation to develop methods for the application of quantum resources to estimation protocols. Within this context, Bayesian estimation offers a complete framework for optimal quantum metrology techniques, such as adaptive protocols. However, the use of the Bayesian approach requires extensive computational resources, especially in the multiparameter estimations that represent the typical operational scenario for quantum sensors. Hence, the requirement to characterize protocols implementing Bayesian estimations can become a significant challenge. This work focuses on the crucial task of robustly benchmarking the performances of these protocols in both single and multiple-parameter scenarios. By comparing different figures of merits, evidence is provided in favor of using the median of the quadratic error in the estimations in order to mitigate spurious effects due to the numerical discretization of the parameter space, the presence of limited data, and numerical instabilities. These results, providing a robust and reliable characterization of Bayesian protocols, find natural applications to practical problems within the quantum estimation framework.

Recent developments have led to the possibility of embedding machine learning tools into experimental platforms to address key problems, including the characterization of the properties of quantum states. Leveraging on this, we implement a quantum extreme learning machine in a photonic platform to achieve resource-efficient and accurate characterization of the polarization state of a photon. The underlying reservoir dynamics through which such input state evolves is implemented using the coined quantum walk of high-dimensional photonic orbital angular momentum, and performing projective measurements over a fixed basis. We demonstrate how the reconstruction of an unknown polarization state does not need a careful characterization of the measurement apparatus and is robust to experimental imperfections, thus representing a promising route for resource-economic state characterisation.

Variational quantum metrology represents a powerful tool for optimizing generic estimation strategies, combining the principles of variational optimization with the techniques of quantum metrology. Such optimization procedures result particularly effective for multiparameter estimation problems, where traditional approaches, requiring prior knowledge of the system behavior, often suffer from limitations due to the curse of dimensionality and computational complexity. To overcome these challenges, we develop a variational approach able to efficiently optimize a multiparameter quantum phase sensor operating in a noisy environment. By exploiting the high reconfigurability of an integrated photonic device, we implement a hybrid quantum-classical feedback loop able to enhance the estimation performances, combining classical optimization techniques with quantum circuit evaluations. The latter allows us to compute the system partial derivatives with respect to the variational parameters by applying the parameter-shift rule, and thus reconstruct experimentally the Fisher information matrix. This in turn is adopted as the cost function of a derivative-free classical learning algorithm run to optimize the measurement settings. Our experimental results reveal significant improvements in terms of estimation accuracy and noise robustness, highlighting the potential of the implementation of variational techniques for practical applications in quantum sensing and more generally for quantum information processing with photonic circuits.

Quantum parameter estimation offers solid conceptual grounds for the design of sensors enjoying quantum advantage. This is realised not only by means of hardware supporting and exploiting quantum properties, but data analysis has its impact and relevance, too. In this respect, Bayesian methods have emerged as an effective and elegant solution, with the perk of incorporating naturally the availability of a priori information. In this article we present an evaluation of Bayesian methods for multiple phase estimation, assessed based on bounds that work beyond the usual limit of large samples assumed in parameter estimation. Importantly, such methods are applied to experimental data generated from the output statistics of a three-arm interferometer seeded by single photons. Our studies provide a blueprint for a more comprehensive data analysis in quantum metrology., Comment: 8 pages, 6 figures

Achieving quantum-enhanced performances when measuring unknown quantities requires developing suitable methodologies for practical scenarios, that include noise and the availability of a limited amount of resources. Here, we report on the optimization of sub-standard quantum limit Bayesian multiparameter estimation in a scenario where a subset of the parameters describes unavoidable noise processes in an experimental photonic sensor. We explore how the optimization of the estimation changes depending on which parameters are either of interest or are treated as nuisance ones. Our results show that optimizing the multiparameter approach in noisy apparata represents a significant tool to fully exploit the potential of practical sensors operating beyond the standard quantum limit for broad resources range.

Estimation of physical quantities is at the core of most scientific research and the use of quantum devices promises to enhance its performances. In real scenarios, it is fundamental to consider that the resources are limited and Bayesian adaptive estimation represents a powerful approach to efficiently allocate, during the estimation process, all the available resources. However, this framework relies on the precise knowledge of the system model, retrieved with a fine calibration that often results computationally and experimentally demanding. Here, we introduce a model-free and deep learning-based approach to efficiently implement realistic Bayesian quantum metrology tasks accomplishing all the relevant challenges, without relying on any a-priori knowledge on the system. To overcome this need, a neural network is trained directly on experimental data to learn the multiparameter Bayesian update. Then, the system is set at its optimal working point through feedbacks provided by a reinforcement learning algorithm trained to reconstruct and enhance experiment heuristics of the investigated quantum sensor. Notably, we prove experimentally the achievement of higher estimation performances than standard methods, demonstrating the strength of the combination of these two black-box algorithms on an integrated photonic circuit. This work represents an important step towards fully artificial intelligence-based quantum metrology.

Relevant metrological scenarios involve the simultaneous estimation of multiple parameters. The fundamental ingredient to achieve quantum-enhanced performances is based on the use of appropriately tailored quantum probes. However, reaching the ultimate resolution allowed by physical laws requires non trivial estimation strategies both from a theoretical and a practical point of view. A crucial tool for this purpose is the application of adaptive learning techniques. Indeed, adaptive strategies provide a flexible approach to obtain optimal parameter-independent performances, and optimize convergence to the fundamental bounds with limited amount of resources. Here, we combine on the same platform quantum-enhanced multiparameter estimation attaining the corresponding quantum limit and adaptive techniques. We demonstrate the simultaneous estimation of three optical phases in a programmable integrated photonic circuit, in the limited resource regime. The obtained results show the possibility of successfully combining different fundamental methodologies towards transition to quantum sensors applications.

Adopting quantum resources for parameter estimation discloses the possibility to realize quantum sensors operating at a sensitivity beyond the standard quantum limit. Such approach promises to reach the fundamental Heisenberg scaling as a function of the employed resources $N$ in the estimation process. Although previous experiments demonstrated precision scaling approaching Heisenberg-limited performances, reaching such regime for a wide range of $N$ remains hard to accomplish. Here, we show a method which suitably allocates the available resources reaching Heisenberg scaling without any prior information on the parameter. We demonstrate experimentally such an advantage in measuring a rotation angle. We quantitatively verify Heisenberg scaling for a considerable range of $N$ by using single-photon states with high-order orbital angular momentum, achieving an error reduction greater than $10$ dB below the standard quantum limit. Such results can be applied to different scenarios, opening the way to the optimization of resources in quantum sensing.

We apply the theory of semiparametric estimation to a Hong-Ou-Mandel interference experiment with a spectrally entangled two-photon state generated by spontaneous parametric downconversion. Thanks to the semiparametric approach we can evaluate the Cram\'er-Rao bound and find an optimal estimator for a particular parameter of interest without assuming perfect knowledge of the two-photon wave function, formally treated as an infinity of nuisance parameters. In particular, we focus on the estimation of the Hermite-Gauss components of the marginal symmetrised wavefunction, whose Fourier transform governs the shape of the temporal coincidence profile. We show that negativity of these components is an entanglement witness of the two-photon state.

The ultimate precision in any measurement is dictated by the physical process implementing the observation. The methods of quantum metrology have now succeeded in establishing bounds on the achievable precision for phase measurements over noisy channels. In particular, they demonstrate how the Heisenberg scaling of the precision can not be attained in these conditions. Here we discuss how the ultimate bound in presence of loss has a physical motivation in the Kramers-Kronig relations and we show how they link the precision on the phase estimation to that on the loss parameter.

The classification of big data usually requires a mapping onto new data clusters which can then be processed by machine learning algorithms by means of more efficient and feasible linear separators. Recently, Lloyd et al. have advanced the proposal to embed classical data into quantum ones: these live in the more complex Hilbert space where they can get split into linearly separable clusters. Here, we implement these ideas by engineering two different experimental platforms, based on quantum optics and ultra-cold atoms respectively, where we adapt and numerically optimize the quantum embedding protocol by deep learning methods, and test it for some trial classical data. We perform also a similar analysis on the Rigetti superconducting quantum computer. Therefore, we find that the quantum embedding approach successfully works also at the experimental level and, in particular, we show how different platforms could work in a complementary fashion to achieve this task. These studies might pave the way for future investigations on quantum machine learning techniques especially based on hybrid quantum technologies., Comment: 9 pages, 7 figures

Frequency correlations are a versatile and powerful tool which can be exploited to perform spectral analysis of objects whose direct measurement might be unfeasible. This is achieved through a so-called ghost spectrometer, that can be implemented with quantum and classical resources alike. While there are some known advantages associated to either choice, an analysis of their metrological capabilities has not yet been performed. Here we report on the metrological comparison between a quantum and a classical ghost spectrometer. We perform the estimation of the transmittivity of a bandpass filter using frequency-entangled photon pairs. Our results show that a quantum advantage is achievable, depending on the values of the transmittivity and on the number of frequency modes analyzed.

Calibration of sensors is a fundamental step to validate their operation. This can be a demanding task, as it relies on acquiring a detailed modelling of the device, aggravated by its possible dependence upon multiple parameters. Machine learning provides a handy solution to this issue, operating a mapping between the parameters and the device response, without needing additional specific information on its functioning. Here we demonstrate the application of a Neural Network based algorithm for the calibration of integrated photonic devices depending on two parameters. We show that a reliable characterization is achievable by carefully selecting an appropriate network training strategy. These results show the viability of this approach as an effective tool for the multiparameter calibration of sensors characterized by complex transduction functions.

Characterizing a system often demands learning its response function to an applied field. Such knowledge is rooted on the experimental evaluation of punctual fiducial response and interpolation to access prediction at arbitrary values. Quantum metrological resources are known to provide enhancement in assessing these fiducial points, but the implications for improved function estimation have only recently been explored, and have not been yet demonstrated. Here we show an experimental realization of function estimation based on a photonic achitecture. The phase response of a liquid-crystal to a voltage has been reconstructed by means of quantum and classical phase estimation, providing evidence of the superiority of the former and highlighting the interplay between punctual statistical error and interpolation error. Our results show how quantum resources should successfully be employed to access the rich information contained in continuous signals.

We present an experimental method to detect lower bounds to the quantum capacity of two-qubit communication channels. We consider an implementation with polarisation degrees of freedom of two photons and report on the efficiency of such a method in the presence of correlated noise for varying values of the correlation strength. The procedure is based on the generation of separable states of two qubits and local measurements at the output. We also compare the performance of the correlated two-qubit channel with the single-qubit channels corresponding to the partial trace on each of the subsystems, thus showing the beneficial effect of properly taking into account correlations to achieve a larger quantum capacity.

Non-equilibrium states of quantum systems in contact with thermal baths help telling environments with different temperatures or different statistics apart. We extend these studies to a more generic problem that consists in discriminating between two baths with disparate constituents at unequal temperatures. Notably there exist temperature regimes in which the presence of coherence in the initial state preparation is beneficial for the discrimination capability. We also find that non-equilibrium states are not universally optimal, and detail the conditions in which it becomes convenient to wait for complete thermalisation of the probe. These concepts are illustrated in a linear optical simulation., Comment: Few typos corrected, bibliography expanded

The characterization of quantum features in large Hilbert spaces is a crucial requirement for testing quantum protocols. In the continuous variables encoding, quantum homodyne tomography requires an amount of measurements that increases exponentially with the number of involved modes, which practically makes the protocol intractable even with few modes. Here we introduce a new technique, based on a machine learning protocol with artificial Neural Networks, that allows to directly detect negativity of the Wigner function for multimode quantum states. We test the procedure on a whole class of numerically simulated multimode quantum states for which the Wigner function is known analytically. We demonstrate that the method is fast, accurate and more robust than conventional methods when limited amounts of data are available. Moreover the method is applied to an experimental multimode quantum state, for which an additional test of resilience to losses is carried out.

We characterize the energetic footprint of a two-qubit quantum gate from the perspective of non-equilibrium quantum thermodynamics. We experimentally reconstruct the statistics of energy and entropy fluctuations following the implementation of a controlled-unitary gate, linking them to the performance of the gate itself and the phenomenology of Landauer principle at the single-quantum level. Our work thus addresses the energetic cost of operating quantum circuits, a problem that is crucial for the grounding of the upcoming quantum technologies., Comment: This version matches the published article

Quantum metrology derives its capabilities from the careful employ of quantum resources for carrying out measurements. This advantage, however, relies on refined data postprocessing, assessed based on the variance of the estimated parameter. When Bayesian techniques are adopted, more elements become available for assessing the quality of the estimation. Here we adopt generalized classical Cram\'er-Rao bounds for looking in detail into a phase-estimation experiment performed with quantum light. In particular, we show that the third-order absolute moment can give a superior capability in revealing biases in the estimation, compared to standard approaches. Our studies point to the identification of an alternative strategy that brings a possible advantage in monitoring the correct operation of high-precision sensors., Comment: 6 pages, 5 figures

Postselection following weak measurements has long been investigated for its peculiar manifestation of quantum signatures. In particular, the postselected events can give rise to anomalous values lying outside the spectrum of the measured quantity, and may provide enhanced Fisher information. Furthermore, the Pusey inequality highlights that, for extremely weak measurements, non-contextual models can account for the outcome probabilities. It is then interesting to investigate whether these are linked in a unified framework. Here we discuss on the existence of a possible connection in the case of qubits. We show that when performing generic postselected measurements there exist no one-to-one mapping between them, an instance that leads to drawing more involved considerations.

Enzymes are essential to maintain organisms alive. Some of the reactions they catalyze are associated with a change in reagents chirality, hence their activity can be tracked by using optical means. However, illumination affects enzyme activity: the challenge is to operate at low-intensity regime avoiding loss in sensitivity. Here we apply quantum phase estimation to real-time measurement of invertase enzymatic activity. Control of the probe at the quantum level demonstrates the potential for reducing invasiveness with optimized sensitivity at once. This preliminary effort, bringing together methods of quantum physics and biology, constitutes an important step towards full development of quantum sensors for biological systems.

Introducing quantum sensors as solution to real-world problem demands reliability and controllability outside laboratory conditions. Producers and operators ought to be assumed to have limited resources ready available for calibration, and yet, they should be able to trust the devices. Neural networks are almost ubiquitous for similar tasks for classical sensors: here we show the applications of this technique to calibrating a quantum photonic sensor. This is based on a set of training data, collected only relying on the available probe states, hence reducing overheads. We found that covering finely the parameter space is key to achieve uncertainties close to their ultimate level. This technique has potential to become the standard approach to calibrate quantum sensors.

The superposition of quantum states lies at the heart of physics and has been recently found to serve as a versatile resource for quantum information protocols, defining the notion of quantum coherence. In this contribution, we report on the implementation of its complementary concept, coherence from quantum measurements. By devising an accessible criterion which holds true in any classical statistical theory, we demonstrate that noncommutative quantum measurements violate this constraint, rendering it possible to perform an operational assessment of the measurement-based quantum coherence. In particular, we verify that polarization measurements of a single photonic qubit, an essential carrier of one unit of quantum information, are already incompatible with classical, i.e., incoherent, models of a measurement apparatus. Thus, we realize a method that enables us to quantitatively certify which quantum measurements follow fundamentally different statistical laws than expected from classical theories and, at the same time, quantify their usefulness within the modern framework of resources for quantum information technology., Comment: close to published version

Quantum photonics has demonstrated its potential for enhanced sensing. Current sources of quantum light states tailored to measuring, allow to monitor phenomena evolving on time scales of the order of the second. These are characteristic of product accumulation in chemical reactions of technologically interest, in particular those involving chiral compounds. Here we adopt a quantum multiparameter approach to investigate the dynamic process of sucrose acid hydrolysis as a test bed for such applications. The estimation is made robust by monitoring different parameters at once.

In this work we demonstrate the use of stimulated emission tomography to characterize a hyper-entangled state generated by spontaneous parametric down-conversion in a CW-pumped source. In particular, we consider the generation of hyper-entangled states consisting of photon pairs entangled in polarisation and path. These results extend the capability of stimulated emission tomography beyond the polarisation degree of freedom, and demonstrate the use of this technique to study states in higher dimension Hilbert spaces.

The simplicity of a question such as wondering if correlations characterize or not a certain system collides with the experimental difficulty of accessing such information. Here we present a low demanding experimental approach which refers to the use of a metrology scheme to obtain a conservative estimate of the strength of frequency correlations. Our testbed is the widespread case of a photon pair produced per downconversion. The theoretical architecture used to put the correlation degree on a quantitative ground is also described.

Phase estimation is the most investigated protocol in quantum metrology, but its performance is affected by the presence of noise, also in the form of imperfect state preparation. Here we discuss how to address this scenario by using a multiparameter approach, in which noise is associated to a parameter to be measured at the same time as the phase. We present an experiment using two-photon states, and apply our setup to investigating optical activity of fructose solutions. Finally, we illustrate the scaling laws of the attainable precisions with the number of photons in the probe state.

The emergence of realistic properties is a key problem in understanding the quantum-to-classical transition. In this respect, measurements represent a way to interface quantum systems with the macroscopic world: these can be driven in the weak regime, where a reduced back-action can be imparted by choosing meter states able to extract different amounts of information. Here we explore the implications of such weak measurement for the variation of realistic properties of two-level quantum systems pre- and post-measurement, and extend our investigations to the case of open systems implementing the measurements.

Quantum networks of growing complexity play a key role as resources for quantum computation; the ability to identify the quality of their internal correlations will play a crucial role in addressing the buiding stage of such states. We introduce a novel diagnostic scheme for multipartite networks of entangled particles, aimed at assessing the quality of the gates used for the engineering of their state. Using the information gathered from a set of suitably chosen multiparticle Bell tests, we identify conditions bounding the quality of the entangled bonds among the elements of a register. We demonstrate the effectiveness, flexibility, and diagnostic power of the proposed methodology by characterizing a quantum resource engineered combining two-photon hyperentanglement and photonic-chip technology. Our approach is feasible for medium-sized networks due to the intrinsically modular nature of cluster states, and paves the way to section-by-section analysis of large photonics resources., Comment: 5 pages, 3 figures, RevTex4-1