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We give a simplified version of the proofs that, outside of their isolated vertices, the complement of the enhanced power graph and of the power graph are connected of diameter at most $3$., Comment: 3 pages

Quantum Sensing is a rapidly growing branch of research within the area of quantum science and technology offering key resources, beyond classical ones, with potential for commercialisation of novel (quantum) sensors. The exploitation of quantum resources offered by photons can boost the performance of quantum sensors for innovative and challenging applications. In this paper we build on the idea of the Quantum Ghost Spectroscopy (QGS), i.e. the counterpart in the frequency domain of Quantum Ghost Imaging (QGI), targeting specific applications in the detection of possible threats. This is implemented by exploiting the opportunities offered by Quantum Optics, i.e. the generation of photon pairs characterized by spectral correlations. We will discuss our main results obtained with pure QGS experiments showing that it is possible to assess the presence of a target dealing with a low resources measurement. The time-frequency domain reveals a huge potential for several applications and frequency correlations represent a versatile tool that can be exploited to enable the spectral analysis of objects where a direct measurement would not be feasible (e.g. security). The use of nondegenerate sources of correlated photons allowed to reveal spectral features in the near infrared wavelengths employing the usual detectors for the visible region.

Reliably characterised pulses are the starting point of any application of ultrafast techniques. Unfortunately, experimental constraints do not always allow optimising the characterisation conditions. This dictates the need for refined analysis methods. Here we show that neutral networks can provide a viable characterisation when applied to data from SPIDER. We have adopted a cascade of convolutional networks, addressing the multiparameter structure of the interferogram with a reasonable computing power. In particular, the necessity of precalibration is reduced, thus pointing towards the introduction of neural networks in more generic arrangements.

We show how quantum coherence governs the quasiprobability statistics of outcome pairs, consecutively recorded at two distinct times, using weak measurements. In doing this, we have realised weak-sequential measurement with photonic qubits, where the first measurement is carried out by a positive operator-valued measure, whereas the second one is a projective operation. We determine the quasiprobability distributions associated to this procedure, based on both the commensurate and the Margenau-Hill approach, by establishing a link between these descriptions. Our results find application to quantum monitoring aimed at implementing or stabilising task without completely loosing the initial quantum coherence., Comment: 6 pages, 3 figures. Comments and feedback are welcome

Given a finite connected graph ${\Gamma}$ and a group $G$ acting transitively on the vertices of ${\Gamma}$, we prove that the number of vertices of ${\Gamma}$ and the cardinality of $G$ are bounded above by a function depending only on the cardinality of ${\Gamma}$ and on the exponent of $G$. We also prove that the number of generators of a group $G$ acting transitively on the arcs of a finite graph ${\Gamma}$ cannot be bounded by a function of the valency alone.

Fix $\varepsilon>0$. We say that a finite group $G$ is $\varepsilon$-quasirandom if every nontrivial irreducible complex representation of $G$ has degree at least $|G|^\varepsilon$. In this paper, we give a structure theorem for large $\varepsilon$-quasirandom groups, and we completely classify the $\frac{1}{5}$-quasirandom groups., Comment: 22 pages

Accurate phase estimation in the presence of unknown phase diffusive noise is a crucial yet challenging task in noisy quantum metrology. This problem is particularly interesting due to the detrimental impact of the associated noise. Here, we investigate the joint estimation of phase and phase diffusion using generalized Holland-Burnett states, known for their experimental accessibility. These states provide performance close to the optimal state in single-parameter phase estimation, even in the presence of photon losses. We adopt a twofold approach by analyzing the joint information extraction through the double homodyne measurement and the joint information availability across all probe states. Through our analysis, we find that the highest sensitivities are obtained by using states created by directing all input photons into one port of a balanced beam splitter. Furthermore, we infer that good levels of sensitivity persist even in the presence of moderate photon losses, illustrating the remarkable resilience of our probe states under lossy conditions., Comment: 11 pages, 5 figures

We find a lower bound on the proportion of derangements in a finite transitive group that depends on the minimal nontrivial subdegree. As a consequence, we prove that, if $\Gamma$ is a $G$-vertex-transitive digraph of valency $d\ge 1$, then the proportion of derangements in $G$ is greater than $1/2d$., Comment: 7 pages

Quantum steering captures the ability of one party, Alice, to control through quantum correlations the state at a distant location, Bob, with superior ability than allowed by a local hidden state model. Verifying the presence of quantum steering has implications for the certification of quantum channels, and its connection to the metrological power of the quantum state has been recently proved. This link is established by means of the violation of a Cram\'er-Rao bound holding for non-steerable states: its direct assessment would then require operation in the asymptotic regime of a large number of repetitions. Here, we extend previous work to account explicitly for the use of a limited number of resources, and put this modified approach to test in a quantum optics experiment. The imperfections in the apparatus demand an adaptation of the original test in the multiparameter setting. Our results provide guidelines to apply such a metrological approach to the validation of quantum channels.

Noise affects the performance of quantum technologies, hence the importance of elaborating operative figures of merit that can capture its impact in exact terms. In quantum metrology, the introduction of the Fisher information measurement noise susceptibility now allows to quantify the robustness of measurement for single-parameter estimation. Here we extend this notion to the multiparameter quantum estimation scenario. We provide its mathematical definition in the form of a semidefinite program. Although a closed formula could not be found, we further derive an upper and a lower bound to the susceptibility. We then apply these techniques to two paradigmatic examples of multiparameter estimation: the joint estimation of phase and phase-diffusion and the estimation of the different parameters describing the incoherent mixture of optical point sources. Our figure of merit provides clear indications on conditions allowing or hampering robustness of multiparameter measurements.

The relative fixity of a digraph $\Gamma$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $\Gamma$ and the number of vertices of $\Gamma$. We characterize the vertex-primitive digraphs whose relative fixity is at least $1/3$, and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant., Comment: 16 pages

The introduction of quantum methods in spectroscopy can provide enhanced performance and technical advantages in the management of noise. We investigate the application of quantum illumination in a pump and probe experiment. Thermal lensing in a suspension of gold nanorods is explored using a classical beam as the pump and the emission from parametric downconversion as the probe. We obtain an insightful description of the behaviour of the suspension under pumping with a method known to provide good noise rejection. Our findings are a further step towards investigating effects of quantum light in complex plasmonic media.

We present a device that exploits spatial and spectral correlations in parametric downconversion at once. By using a ghost imaging arrangement, we have been able to reconstruct remotely the frequency profile of a composite system. The presence of distinct spectral regions is corroborated by a model-independent statistical analysis that constitutes an intriguing possibility also in the low count regime.

In the past years, optical fluorescence microscopy (OFM) made steady progress towards increasing the localisation precision of fluorescent emitters in biological samples. The high precision achieved by these techniques has prompted new claims, whose rigorous validation is an outstanding problem. For this purpose, local estimation theory (LET) has emerged as the most used mathematical tool. We establish a novel multi-parameter estimation framework that captures the full complexity of single-emitter localisation in an OFM experiment. Our framework relies on the fact that there are other unknown parameters alongside the emitter's coordinates, such as the average number of photons emitted (brightness), that are correlated to the emitter position, and affect the localisation precision. The increasing complexity of a multi-parameter approach allows for a more accountable assessment of the precision. We showcase our method with MINFLUX microscopy, the OFM approach that nowadays generates images with the best resolution. Introducing the brightness as an unknown parameter, we shed light on features that remain obscure in the conventional approach: the precision can be increased only by increasing the brightness, (i.e., illumination power or exposition time), whereas decreasing the beam separation offers limited advantages. We demonstrate that the proposed framework is a solid and general method for the quantification of single-emitter localisation precision for any OFM approach on equal footing, evaluating the localization precision of stimulated emission depletion (STED) microscopy and making a comparison with MINFLUX microscopy., Comment: 20 pages, 4 figures

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

Assessing the presence of chemical, biological, radiological and nuclear threats is a crucial task which is usually dealt with by analyzing the presence of spectral features in a measured absorption profile. The use of quantum light allows to perform these measurements remotely without compromising the measurement accuracy through ghost spectrometry. However, in order to have sufficient signal-to-noise ratio, it is typically required to wait long acquisition times, hence subtracting to the benefits provided by remote sensing. In many instances, though, reconstructing the full spectral lineshape of an object is not needed and the interest lies in discriminating whether a spectrally absorbing object may be present or not. Here we show that this task can be performed fast and accurately through ghost spectrometry by comparing the low resources measurement with a reference. We discuss the experimental results obtained with different samples and complement them with simulations to explore the most common scenarios.

We prove that, if $\Gamma$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\Gamma$ of order at least $6$, or the number of vertices of $\Gamma$ is bounded above by an absolute constant., Comment: 20 pages

Recovering the original spectral lineshapes from data obtained by instruments with extended transmission profiles is a basic tenet in spectroscopy. By using the moments of the measured lines as basic variables, we turn the problem into a linear inversion. However, when only a finite number of these moments are relevant, the rest of them act as nuisance parameters. These can be taken into account with a semiparametric model, which allows us to establish the ultimate bounds on the precision attainable in the estimation of the moments of interest. We experimentally confirm these limits with a simple ghost spectroscopy demonstration.

Two-photon interference underlies the functioning of many quantum photonics devices. It also serves as the prominent tool for testing the indistinguishability of distinct photons. However, as their time-spectral profile becomes more involved, extracting relevant parameters, foremost the central frequency difference, may start suffering difficulties. In a parametric approach, these arise from the need for an exhaustive model combined with limited count statistics. Here we discuss a solution to curtail these effects on the evaluation of frequency separation relying on a semiparametric method. The time trace of the quantum interference pattern of two photons from two independent solid-state emitters is preprocessed by means of singular spectral analysis before inspecting its spectral content. This approach allows to single out the relevant oscillations from both the envelope and the noise, without resorting to fitting. This opens the way for robust and efficient on-line monitoring of quantum emitters.

Characterization of quantum objects, being them states, processes, or measurements, complemented by previous knowledge about them is a valuable approach, especially as it leads to routine procedures for real-life components. To this end, Machine Learning algorithms have demonstrated to successfully operate in presence of noise, especially for estimating specific physical parameters. Here we show that a neural network (NN) can improve the tomographic estimate of parameters by including a convolutional stage. We applied our technique to quantum process tomography for the characterization of several quantum channels. We demonstrate that a stable and reliable operation is achievable by training the network only with simulated data. The obtained results show the viability of this approach as an effective tool based on a completely new paradigm for the employment of NNs operating on classical data produced by quantum systems.

High-performing nanomaterials are paramount for advanced photonic technologies, like sensing, lasing, imaging, data storage, processing, and medical and biological applications. Metal nanoparticles play a key role, because the localized surface plasmonic resonances enhance the linear and nonlinear optical properties of hosting materials, thus increasing the range of applications. The improved optical response results in an enriched and largely unexplored phenomenology. In the present work, we show the formation and interaction of two dispersive shock waves emerging by illuminating an aqueous suspension of gold nanorods with a pulsed laser beam. By analyzing the characteristic undular bores that regularize a shock front in a dispersive material, we were able to distinguish dispersive shock waves of different origins, i.e., electronic and thermal, first coexisting and then interacting to form a double front of equal intensity. The observed scenario agrees with existing literature reporting an electronic and thermal contribution to the nonlinear refractive index of material containing gold nanoparticles. Although a full comprehension of the reported results deserves an analytical description, they already pave the way to using materials containing metal nanoparticles as a platform for studying fundamental extreme nonlinear phenomena and developing novel solutions for sensing and control.

Measurements in the quantum domain can exceed classical notions. This concerns fundamental questions about the nature of the measurement process itself, as well as applications, such as their function as building blocks of quantum information processing protocols. In this paper we explore the notion of entanglement for detection devices in theory and experiment. A method is devised that allows one to determine nonlocal quantum coherence of positive-operator-valued measures via negative contributions in a joint distribution that fully describes the measurement apparatus under study. This approach is then applied to experimental data for detectors that ideally project onto Bell states. In particular, we describe the reconstruction of the aforementioned entanglement quasidistributions from raw data and compare the resulting negativities with those expected from theory. Therefore, our method provides a versatile toolbox for analyzing measurements regarding their quantum-correlation features for quantum science and quantum technology., Comment: version accepted for publication

Quantum coherence is a central ingredient in quantum physics with several theoretical and technological ramifications. In this work we consider a figure of merit encoding the information on how the coherence generated on average by a quantum gate is affected by unitary errors (coherent noise sources). We provide numerical evidences that such information is well captured by the statistics of local energy measurements on the output states of the gate. These findings are then corroborated by experimental data taken in a quantum optics setting.

We prove that, if $\Gamma$ is a finite connected $3$-valent vertex-transitive, or $4$-valent vertex- and edge-transitive graph, then either $\Gamma$ is part of a well-understood family of graphs, or every non-identity automorphism of $\Gamma$ fixes at most $1/3$ of the edges. This answers a question proposed by Primo\v{z} Poto\v{c}nik and the third author., Comment: 14 pages

We give a sufficient and necessary condition for a Praeger-Xu graph to be a Cayley graph.

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.

Capturing specific kinds of quantum correlation is of paramount importance for quantum networking. Different routes can be taken to achieve this task, highlighting different novel aspects of such quantum correlations. Following the recent theoretical results by Yadin, Fadel and Gessner [Nat. Commun. 12, 2410 (2021)], we demonstrate experimentally how steering manifests in the metrological abilities of a bipartite state. Our results confirm the relevance of this novel approach, and compare the outcome with already employed alternatives., Comment: New version including the material in the Erratum

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.

Multiphase estimation is a paradigmatic example of a multiparameter problem. When measuring multiple phases embedded in interferometric networks, specially-tailored input quantum states achieve enhanced sensitivities compared with both single-parameter and classical estimation schemes. Significant attention has been devoted to defining the optimal strategies for the scenario in which all of the phases are evaluated with respect to a common reference mode, in terms of optimal probe states and optimal measurement operators. As well, the strategies assume unlimited external resources, which is experimentally unrealistic. Here, we optimize a generalized scenario that treats all of the phases on an equal footing and takes into account the resources provided by external references. We show that the absence of an external reference mode reduces the number of simultaneously estimatable parameters, owing to the immeasurability of global phases, and that the symmetries of the parameters being estimated dictate the symmetries of the optimal probe states. Finally, we provide insight for constructing optimal measurements in this generalized scenario. The experimental viability of this work underlies its immediate practical importance beyond fundamental physics., Comment: 11 pages, 2 figures, 1 table; comments welcome!

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 dimensionality of a thermometer is key in the design of quantum thermometry schemes. In general, the phenomenology that is typical of finite-dimensional quantum thermometry does not apply to infinite dimensional ones. We analyse the dynamical and metrological features of non-equilibrium Gaussian Quantum Thermometers: on one hand, we highlight how quantum entanglement can enhance the readiness of composite Gaussian thermometers; on the other hand, we show that non-equilibrium conditions do not guarantee the best sensitivities in temperature estimation, thus suggesting the reassessment of the working principles of quantum thermometry.

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