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Modern imaging technologies are widely based on classical principles of light or electromagnetic wave propagation. They can be remarkably sophisticated, with recent successes ranging from single molecule microscopy to imaging far-distant galaxies. However, new imaging technologies based on quantum principles are gradually emerging. They can either surpass classical approaches or provide novel imaging capabilities that would not otherwise be possible. {Here }we provide an overview {of the most recently developed quantum imaging systems, highlighting the non-classical properties of sources such as bright squeezed light, entangled photons, and single-photon emitters that enable their functionality.} We outline potential upcoming trends and the associated challenges, all driven by a central inquiry, which is to understand whether quantum light can make visible the invisible.

In continuous-variable quantum computation, identifying key elements that enable a quantum computational advantage is a long-standing issue. Starting from the standard results on the necessity of Wigner negativity, we develop a comprehensive and versatile framework that not only enables the identification of a potential quantum computational advantage, but also allows to pinpoint the contribution of each quantum gate in achieving this objective. As such, it can be straightforwardly applied to current continuous-variables quantum circuits, while also constraining the tolerable amount of losses above which any potential quantum advantage can be ruled out. We use $(s)$-ordered quasiprobability distributions on phase-space to capture the non-classical features in the protocol, and focus our model entirely on the ordering parameter $s$. This allows us to highlight the resourcefulness and robustness to loss of a universal set of unitary gates comprising three distinct Gaussian gates, and a fourth one, the cubic gate, providing important insight on the role of non-Gaussianity., Comment: 16 pages, 5 figures

Photon subtraction can enhance entanglement, which for pure states induces a decrease in the purity of reduced states. In contrast, by analyzing the purities of Gaussian states before and after subtracting a single photon, we prove that the purity of a Gaussian state can also be increased by less than 20%. On the one hand, it reveals that photon subtraction can reduce entanglement, and on the other hand, it reveals that it can achieve a limited amount of Gaussian state purification. Through the analysis of some examples, we demonstrate the inherent mechanism and applicable scope of photon-subtraction-based purification. In a multimode system, we find that photon subtraction can increase entanglement and purify some of the reduced states simultaneously. We thus present purification through the suppression of Gaussian noise as a new application for photon subtraction in continuous-variable quantum information processing., Comment: 7 pages, 3 figures, comments welcome!

Quantum correlations are at the core of the power of quantum information and are necessary to reach a quantum computational advantage. In the context of continuous-variable quantum systems, another necessary ressource for quantum advantages is non-Gaussianity. In this work, we propose a witness, based on previously known relations between metrological power and quantum correlations, to detect a strong form of entanglement that only non-Gaussian states possess and that cannot be undone by passive optical operations, i.e., entanglement in all mode bases. The strength of our witness is two-fold: it only requires measurements in one basis to check entanglement in any arbitrary mode basis; it can be made applicable experimentally using homodyne measurements and without requiring a full tomography of the state., Comment: 16 pages, 12 figures, comments are welcome!

Frequency-degenerate pulsed type-I parametric downconversion is a widely used source of squeezed light for numerous quantum optical applications. However, this source is typically spectrally multimode and the generated squeezing is distributed between many spectral modes with a limited degree of squeezing per mode. We show that in a nonlinear crystal, where the condition of complete group velocity matching (GVM) for the pump and the signal is satisfied, the number of generated modes may be as low as two or three modes. We illustrate the general theory with the example of the MgO-doped lithium niobate crystal pumped at 775 nm and generating squeezed light at 1.55 $\mu$m. Our model includes the derivation of the degree of squeezing from the properties of the pump and the crystal and shows that 12 dB of squeezing can be obtained in a periodically poled crystal of length 80 mm., Comment: 10 pages, 5 figures, final accepted version

The exploitation of the full structure of multimode light fields enables compelling capabilities in many fields including classical and quantum information science. We exploit data-encoding on the optical phase of the pulses of a femtosecond laser source for a photonic implementation of a reservoir computing protocol. Rather than intensity detection, data-reading is done via homodyne detection that accesses combinations of amplitude and phase of the field. Numerical and experimental results on NARMA tasks and laser dynamic predictions are shown. We discuss perspectives for quantum enhanced protocols.

The problem of resolving point-like light sources not only serves as a benchmark for optical resolution but also holds various practical applications ranging from microscopy to astronomy. In this research, we aim to resolve two thermal sources sharing arbitrary mutual coherence using the spatial mode demultiplexing technique. Our analytical study includes scenarios where the coherence and the emission rate depend on the separation between the sources, and is not limited to the faint sources limit. We consider the fluorescence of two interacting dipoles to demonstrate that the dependence of emission characteristics on the parameter of interest can boost the sensitivity of the estimation and noticeably prolong the duration of information decay.

Historically, the resolution of optical imaging systems was dictated by diffraction, and the Rayleigh criterion was long considered an unsurpassable limit. In superresolution microscopy, this limit is overcome by manipulating the emission properties of the object. However, in passive imaging, when sources are uncontrolled, reaching sub-Rayleigh resolution remains a challenge. Here, we implement a quantum-metrolgy-inspired approach for estimating the separation between two incoherent sources, achieving a sensitivity five orders of magnitude beyond the Rayleigh limit. Using a spatial mode demultiplexer, we examine scenes with bright and faint sources, through intensity measurements in the Hermite-Gauss basis. Analysing sensitivity and accuracy over an extensive range of separations, we demonstrate the remarkable effectiveness of demultiplexing for sub-Rayleigh separation estimation. These results effectively render the Rayleigh limit obsolete for passive imaging.

Continuous variable encoding of quantum information requires the deterministic generation of highly correlated quantum states of light in the form of quantum networks, which, in turn, necessitates the controlled generation of a large number of squeezed modes. In this work, we present an experimental source of multimode squeezed states of light at telecommunication wavelengths. Generation at such wavelengths is especially important as it can enable quantum information processing, communication, and sensing beyond the laboratory scale. We use a single-pass spontaneous parametric down-conversion process in a non-linear waveguide pumped with the second harmonic of a femtosecond laser. Our measurements reveal significant squeezing in more than 21 frequency modes, with a maximum squeezing value exceeding 2.5 dB. We demonstrate multiparty entanglement by measuring the state's covariance matrix. Finally, we show the source reconfigurability by preparing few-node cluster states and measure their nullifier squeezing level. These results pave the way for a scalable implementation of continuous variable quantum information protocols at telecommunication wavelengths, particularly for multiparty, entanglement-based quantum communications. Moreover, the source is compatible with additional pulse-by-pulse multiplexing, which can be utilized to construct the necessary three-dimensional entangled structures for quantum computing protocols.

Quantum-optimal discrimination between one and two closely separated light sources can be achieved by ideal spatial-mode demultiplexing, simply monitoring whether a photon is detected in a single antisymmetric mode. However, we show that for any, no matter how small, imperfections of the demultiplexer, this simple statistical test becomes practically useless, i.e. as good as flipping a coin. While we identify a class of separation-independent tests with vanishing error probabilities in the limit of large numbers of detected photons, they are generally unreliable beyond that very limit. As a practical alternative, we propose a simple semi-separation-independent test, which provides a method for designing reliable experiments, through arbitrary control over the maximal probability of error., Comment: 5 pages + Supplementary Material

No quantum system can be considered totally isolated from its environment. In most cases the interaction between the system of interest and the external degrees of freedom deeply changes its dynamics, as described by open quantum system theory. Nevertheless, engineered environment can be turned into beneficial effects for some quantum information tasks. Here we demonstrate an optical simulator of a quantum system coupled to an arbitrary and reconfigurable environment built as a complex network of quantum interacting systems. We experimentally retrieve typical features of open quantum system dynamics like the spectral density and quantum non-Markovianity, by exploiting squeezing and entanglement correlation of a continuous variables optical platform. This opens the way to the experimental tests of open quantum systems in reconfigurable environments that are relevant in, among others, quantum information, quantum thermodynamics, quantum transport and quantum synchronization., Comment: 15 pages, 7 figures

Entanglement and non-Gaussianity are physical resources that are essential for a large number of quantum-optics protocols. Non-Gaussian entanglement is indispensable for quantum-computing advantage and outperforms its Gaussian counterparts in a number of quantum-information protocols. The characterization of non-Gaussian entanglement is a critical matter as it is in general highly demanding in terms of resources. We propose a simple protocol based on the Fisher information for witnessing entanglement in an important class of non-Gaussian entangled states: photon-subtracted states. We demonstrate that our protocol is relevant for the detection of non-Gaussian entanglement generated by multiple photon-subtraction and that it is experimentally feasible through homodyne detection., Comment: We have corrected errors found in Figures 8 and 9 of the previous version and improved the manuscript with an analysis of the entanglement produced by multi-photon subtraction. We have adapted the main text, abstract and title correspondingly. 16 pages, 14 figures

Superresolution is one of the key issues at the crossroads of contemporary quantum optics and metrology. Recently, it was shown that for an idealized case of two balanced sources, spatial mode demultiplexing (SPADE) achieves resolution better than direct imaging even in the presence of measurement crosstalk [Phys. Rev. Lett. 125, 100501 (2020)]. In this work, we consider arbitrarily unbalanced sources and provide a systematic analysis of the impact of crosstalk on the resolution obtained from SPADE. As we dissect, in this generalized scenario, SPADE's effectiveness depends non-trivially on the strength of crosstalk, relative brightness and the separation between the sources. In particular, for any source imbalance, SPADE performs worse than ideal direct imaging in the asymptotic limit of vanishing source separations. Nonetheless, for realistic values of crosstalk strength, SPADE is still the superior method for several orders of magnitude of source separations., Comment: 10 pages, 6 figures

Spectral- and time- multiplexing are currently explored to generate large multipartite quantum states of light for quantum technologies. In the continuous variable approach, the deterministic generation of large entangled states demands the generation of a large number of squeezed modes. Here, we demonstrate the simultaneous generation of 21 squeezed spectral modes at 156 MHz. We exploit the full repetition rate and the ultrafast shaping of a femtosecond light source to combine, for the first time, frequency- and time- multiplexing in multimode squeezing. This paves the way to the implementation of multipartite entangled states that are both scalable and fully reconfigurable., Comment: 9 pages, 6 figures

The power spectrum of an optical field can be acquired without a spectrally resolving detector by means of Fourier-transform spectrometry, based on measuring the temporal autocorrelation of the optical field. Analogously, we here perform temporal envelope measurements of ultrashort optical pulses without time resolved detection. We introduce the technique of Fourier transform chronometry, where the temporal envelope is acquired by measuring the frequency autocorrelation of the optical field in a linear interferometer. We apply our technique, which is the time-frequency conjugate measurement to Fourier-transform spectrometry, to experimentally measure the pulse envelope of classical and single photon light pulses., Comment: 4 figures

We determine analytically the quantum Cram\'er-Rao bound for the estimation of the separation between two point sources in arbitrary Gaussian states. Our analytical expression is valid for arbitrary sources brightness, and it allows to determine how different resources, such as mutual coherence (induced by thermal correlations or displacement) or squeezing affect the scaling of the ultimate resolution limit with the mean number of emitted photons. In practical scenarios, we find coherent states of the sources to achieve quantum optimal resolution., Comment: 5 pages, 3 figures, comments are welcome!

We show that instantaneous spatial singular modes of light in a dynamically evolving, turbulent atmosphere offer significantly improved high-fidelity signal transmission as compared to standard encoding bases corrected by adaptive optics. Their enhanced stability in stronger turbulence is associated with a subdiffusive algebraic decay of the transmitted power with evolution time., Comment: 6 pages, 4 figures (article) and 7 pages, 6 figures (supplemental material)

Quantum optical metrology aims to identify ultimate sensitivity bounds for the estimation of parameters encoded into quantum states of the electromagnetic field. In many practical applications, including imaging, microscopy, and remote sensing, the parameter of interest is not only encoded in the quantum state of the field, but also in its spatio-temporal distribution, i.e. in its mode structure. In this mode-encoded parameter estimation setting, we derive an analytical expression for the quantum Fisher information valid for arbitrary multimode Gaussian fields. To illustrate the power of our approach, we apply our results to the estimation of the transverse displacement of a beam and to the temporal separation between two pulses. For these examples, we show how the estimation sensitivity can be enhanced by adding squeezing into specific modes., Comment: 20 pages (14 main + 6 appendices), 3 figures. Sec. 4 rewritten to include a completely new set of examples that were not in v1. Comments are very welcome!

Quantum correlations are at the core of current developments in quantum technologies. Certification protocols of entanglement and steering, suitable for continuous-variable non-Gaussian states are scarce and generally highly demanding from an experimental point of view. We propose a protocol based on Fisher information for witnessing steering in general continuous-variable bipartite states, through homodyne detection. It proves to be relevant for the detection of non-Gaussian steering in scenarios where witnesses based on Gaussian features like the covariance matrix are shown to fail., Comment: 14 pages, 14 figures, comments are welcome!

Quantum light is described not only by a quantum state but also by the shape of the electromagnetic modes on which the state is defined. Optical precision measurements often estimate a ``mode parameter'' that determines properties such as frequency, temporal shape and the spatial distribution of the light field. By deriving quantum precision limits, we establish the fundamental bounds for mode parameter estimation. Our results reveal explicit mode-design recipes that enable the estimation of any mode parameter with quantum enhanced precision. Our approach provides practical methods for optimizing mode parameter estimation with relevant applications, including spatial and temporal positioning, spectroscopy, phase estimation, and superresolution imaging., Comment: 5 + 11 pages

We analyze the problem of resolving two point-sources in the case of mutually coherent sources with arbitrary quantum statistics, mutual phase, relative and absolute intensity. We use a sensitivity measure based on the method of moments and compare direct imaging with spatial mode demultiplexing, analytically proving the advantage of the latter. We show that the sensitivity of spatial mode demultiplexing saturates the quantum Fisher information, for all known cases, even for non-Gaussian states of the sources.

Quantum key distribution with continuous variables already uses advantageous high-speed single-mode homodyne detection with low electronic noise at room temperature. Together with continuous-variable information encoding to nonclassical states, the distance for secure key transmission through lossy channels can approach 300 km in current optical fibers. Such protocols tolerate higher channel noise and also limited data processing efficiency compared to coherent-state protocols. The secret key rate can be further increased by increasing the system clock rates, and, further, by a suitable frequency-mode-multiplexing of optical transmission channels. However, the multiplexed modes couple together in the source or any other part of the protocol. Therefore, multiplexed communication will experience crosstalk and the gain can be minuscule. Advantageously, homodyne detectors allow solving this crosstalk problem by proper data processing. It is a potential advantage over protocols with single-photon detectors, which do not enable similar data processing techniques. We demonstrate the positive outcome of this methodology on the experimentally characterized frequency-multiplexed entangled source of femtosecond optical pulses with natural crosstalk between eight entangled pairs of modes. As the main result, we predict almost 15-fold higher secret key rate. This experimental test and analysis of frequency-multiplexed entanglement source opens the way for the field implementation of high-capacity quantum key distribution with continuous variables., Comment: 14 pages, 5 figures; to be published in Photonics Research

Spectro-temporal modes of light can be exploited for the generation of high-dimensional Gaussian quantum states. Such states are at the basis of continuous variable quantum information protocols where they have to support mode-selective non-Gaussian operations. We develop a general framework for single-photon addition on multimode states of light via parametric down conversion processes. We identify the analytical conditions for single-mode and mode-selective photon addition. We show that spectral mode selectivity can be achieved in the Type-II collinear down conversion, while single-mode condition are retrieved for noncollinear Type-I and Type-II processes. Numerical results are shown for photon addition in parametric down conversion process at near-infrared and telecommunications wavelengths.

Sensitivity limits are usually determined using the Cram\'er-Rao bound. Recently this approach has been used to obtain the ultimate resolution limit for the estimation of the separation between two incoherent point sources. However, methods that saturate these resolution limits, usually require the full measurement statistics, which can be challenging to access. In this work, we introduce a simple superresolution protocol to estimate the separation between two thermal sources which relies only on the average value of a single accessible observable. We show how optimal observables for this technique may be constructed for arbitrary thermal sources, and we study their sensitivities when one has access to spatially resolved intensity measurements (direct imaging) and photon counting after spatial mode demultiplexing. For demultiplexing, our method is optimal, i.e. it saturates the quantum Cram\'er-Rao bound. We also investigate the impact of noise on the optimal observables, their measurement sensitivity and on the scaling with the number of detected photons of the smallest resolvable separation. For low signals in the image plane, we demonstrate that our method saturates the Cram\'er-Rao bound even in the presence of noise., Comment: 18 pages, 12 figures, 1 table. Comments are very welcome!

Our ability to generate new distributions of light has been remarkably enhanced in recent years. At the most fundamental level, these light patterns are obtained by ingeniously combining different electromagnetic modes. Interestingly, the modal superposition occurs in the spatial, temporal as well as spatio-temporal domain. This generalized concept of structured light is being applied across the entire spectrum of optics: generating classical and quantum states of light, harnessing linear and nonlinear light-matter interactions, and advancing applications in microscopy, spectroscopy, holography, communication, and synchronization. This Roadmap highlights the common roots of these different techniques and thus establishes links between research areas that complement each other seamlessly. We provide an overview of all these areas, their backgrounds, current research, and future developments. We highlight the power of multimodal light manipulation and want to inspire new eclectic approaches in this vibrant research community., Comment: Under review in J. Opt

Wigner negativity, as a well-known indicator of nonclassicality, plays an essential role in quantum computing and simulation using continuous-variable systems. Recently, it has been proven that Einstein-Podolsky-Rosen steering is a prerequisite to generate Wigner negativity between two remote modes. Motivated by the demand of real-world quantum network, here we investigate the shareability of generated Wigner negativity in the multipartite scenario from a quantitative perspective. By establishing a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality, we show that the amount of Wigner negativity cannot be freely distributed among different modes. Moreover, for photon subtraction -- one of the main experimentally realized non-Gaussian operations -- we provide a general method to quantify the remotely generated Wigner negativity. With this method, we find that there is no direct quantitative relation between the Gaussian steerability and the amount of generated Wigner negativity. Our results pave the way for exploiting Wigner negativity as a valuable resource for numerous quantum information protocols based on non-Gaussian scenario.

Non-Gaussian states of an optical field are important as a proposed resource in quantum information applications. While conditional preparation is a highly successful approach to preparing such states, their quality is limited by detector non-idealities such as dead time, narrow dynamic range, limited quantum efficiency and dark noise. Mesoscopic photon counters, with peak performance at higher photon number, offer many practical advantages over single-photon level conditioning detectors. Here we propose a novel approach involving displacement of the ancilla field into the regime where mesoscopic detectors can be used. We explore this strategy theoretically and present simulations accounting for experimental non-idealities such as loss and amplification noise, showing that precise photon-number resolution is not necessary to herald highly nonclassical states. We conclude that states with strong Wigner negativity can be prepared at high rates by this technique under experimentally attainable conditions.

Entanglement is an indispensable quantum resource for quantum information technology. In continuous-variable quantum optics, photon subtraction can increase the entanglement between Gaussian states of light, but for mixed states the extent of this entanglement increase is poorly understood. In this work, we use an entanglement measure based the R\'enyi-2 entropy to prove that single-photon subtraction increases bipartite entanglement by no more than log 2. This value coincides with the maximal amount of bipartite entanglement that can be achieved with one photon. The upper bound is valid for all Gaussian input states, regardless of the number of modes and the purity.

Recent works identified resolution limits for the distance between incoherent point sources. However, it remains unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cram\'er-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, crosstalk and detector noise., Comment: 6 pages, 4 figures. Comments are welcome

We use complex network theory to study a class of continuous-variable quantum states that present both multipartite entanglement and non-Gaussian statistics. We consider the intermediate scale of several dozens of components at which such systems are already hard to characterize. In particular, the states are built from an initial imprinted cluster state created via Gaussian entangling operations according to a complex network structure. We then engender non-Gaussian statistics via multiple photon subtraction operations acting on a single node. We replicate in the quantum regime some of the models that mimic real-world complex networks in order to test their structural properties under local operations. We then go beyond the already known single-mode effects, by studying the emergent network of photon-number correlations via complex networks measures. We analytically prove that the imprinted network structure defines a vicinity of nodes, at a distance of four steps from the photon-subtracted node, in which the emergent network changes due to photon subtraction. Moreover, our numerical analysis shows that the emergent structure is greatly influenced by the structure of the imprinted network. Indeed, while the mean and the variance of the degree and clustering distribution of the emergent network always increase, the higher moments of the distributions are governed by the specific structure of the imprinted network. Finally, we show that the behaviour of nearest neighbours of the subtraction node depends on how they are connected to each other in the imprinted structure., Comment: 26 pages (incl. appendix), 20 figures

We derive a theoretical framework for the experimental certification of non-Gaussian features of quantum states using double homodyne detection. We rank experimental non-Gaussian states according to the recently defined stellar hierarchy and we propose practical Wigner negativity witnesses. We simulate various use-cases ranging from fidelity estimation to witnessing Wigner negativity. Moreover, we extend results on the robustness of the stellar hierarchy of non-Gaussian states. Our results illustrate the usefulness of double homodyne detection as a practical measurement scheme for retrieving information about continuous variable quantum states., Comment: 9+9 pages, comments welcome

We develop a general formalism, based on the Wigner function representation of continuous-variable quantum states, to describe the action of an arbitrary conditional operation on a multimode Gaussian state. We apply this formalism to several examples, thus showing its potential as an elegant analytical tool for simulating quantum optics experiments. Furthermore, we also use it to prove that EPR steering is a necessary requirement to remotely prepare a Wigner-negative state., Comment: 12 pages, 3 figures

We present the experimental implementation of simultaneous spatial multimode demultiplexing as a distance measurement tool. We first show a simple and intuitive derivation of the Fisher information in the presence of Poissonian noise. We then estimate the distance between two incoherent beams in both directions of the transverse plane, and find a perfect accordance with theoretical prediction, given a proper calibration of the demultiplexer. We find that, even though sensitivity is limited by the cross-talks between channels, we can perform measurements in 2 dimensions much beyond Rayleigh limit with a large dynamic.

Multi-plane light conversion allows to perform arbitrary transformations on a finite set of spatial modes with no theoretical restriction to the quality of the transformation. Even though the number of shaped modes is in general small, the number of modes transmitted by a multi-plane light converter (MPLC) is extremely large. In this work, we aim to characterize the transmission properties of a multi-plane light converter inside and, for the first time, outside the design-modes subspace. By numerically reconstructing the transmission matrix of such systems, we individuate new ways to evaluate their efficiency in performing the design transformation. Moreover, we develop an analytical random matrix model that suggests that in the regime of a large number of shaped modes an MPLC behaves like a random scattering medium with limited number of controlled channels., Comment: New version of Fig. 1 and significant rewriting

In the current quest for efficient and experimentally feasible platforms for implementation of multimode squeezing and entanglement in the continuous variable regime, we underpin and complement our results on the generation of versatile multimode entanglement and cluster states in nonlinear waveguide arrays presented by Barral et al., Phys. Rev. Appl. $\bf{14}$, 044025 (2020). We present detailed derivations of the equations that describe the propagation of light through this system, and then we focus on parameter regimes where these equations can be solved analytically. These analytical solutions build an intuition for the wide landscape of quantum states that are accessible through the activation of pumping, coupling and measurement schemes. Furthermore, we showcase the acquired insights by using one of the identified analytical solutions to exhibit the generation, optimization and scalability of spatial linear cluster states., Comment: 20 pages, 9 figures, v2 closer to the published version. arXiv admin note: substantial text overlap with arXiv:1912.11154

In the last decade a lot of research activity focused on the use of quantum entanglement as a resource for remote target detection, i.e. on the design of a quantum radar. The literature on this subject uses tools of quantum optics and quantum information theory, and therefore it often results obscure to radar engineers. This review has been written with the purpose of removing this obscurity.As such, it contains a review of the main advances in the quantum radar literature accompanied by a thorough introduction of the quantum optics background necessary for its understanding., Comment: 24 pages, 11 figures, 1 table, review on quantum radars intended for readers with no background in quantum optics: figures and text have been updated to improve the clarity of the manuscript for radar engineers

Superresolution techniques based on intensity measurements after a spatial mode decomposition can overcome the precision of diffraction-limited direct imaging. However, realistic measurement devices always introduce finite crosstalk in any such mode decomposition. Here, we show that any nonzero crosstalk leads to a breakdown of superresolution when the number $N$ of detected photons is large. Combining statistical and analytical tools, we obtain the scaling of the precision limits for weak, generic crosstalk from a device-independent model as a function of the crosstalk probability and $N$. The scaling of the smallest distance that can be distinguished from noise changes from $N^{-1/2}$ for an ideal measurement to $N^{-1/4}$ in the presence of crosstalk., Comment: 5 + 6 pages, 5 + 3 figures

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 present a single-pass source of broadband multimode squeezed light with potential application in quantum information and quantum metrology. The source is based on a type I parametric down-conversion (PDC) process inside a bulk nonlinear crystal in a non-collinear configuration. The generated squeezed light exhibits a spatiotemporal multimode behavior that is probed using a homodyne measurement with a local oscillator shaped both spatially and temporally. Finally we follow a covariance matrix based approach to reveal the distribution of the squeezing among several independent temporal and spatial modes. This unambiguously validates the multimode feature of our source.

Multimode entanglement is an essential resource for quantum information in continuous-variable systems. Light-based quantum technologies will arguably not be built upon table-top bulk setups, but will presumably rather resort to integrated optics. Sequential bulk optics-like proposals based on cascaded integrated interferometers are not scalable with the current state-of-the-art low-loss materials used for continuous variables. We analyze the multimode continuous-variable entanglement capabilities of a compact currently-available integrated device without bulk-optics analog: the array of nonlinear waveguides. We theoretically demonstrate that this simple and compact structure, together with a reconfigurable input pump distribution and multimode coherent detection of the output modes, is a versatile entanglement synthesizer in the spatial domain. We exhibit this versatility through analytical and numerically optimized multimode squeezing, entanglement, and cluster state generation in different encodings. Our results re-establish spatial encoding as a contender in the game of continuous-variable quantum information processing., Comment: 17 pages, 5 figures, v4 closer to the published version

A few decades ago, quantum optics stood out as a new domain of physics by exhibiting states of light with no classical equivalent. The first investigations concerned single photons, squeezed states, twin beams and EPR states, that involve only one or two modes of the electromagnetic field. The study of the properties of quantum light then evolved in the direction of more and more complex and rich situations, involving many modes, either spatial, temporal, frequency, or polarization modes. Actually, each mode of the electromagnetic field can be considered as an individual quantum degree of freedom. It is then possible, using the techniques of nonlinear optics, to couple different modes, and thus to build in a controlled way a quantum network in which the nodes are optical modes and that is endowed with a strong multipartite entanglement. In addition, such networks are easily reconfigurable and subject only to weak decoherence. They open indeed many promising perspectives for optical communications and computation. Because of the linearity of Maxwell equations a linear superposition of two modes is another mode. This means that a "modal superposition principle" exists hand in hand with the regular quantum state superposition principle. The purpose of the present review is to show the interest of considering these two aspects of multimode quantum light in a global way. Indeed using different sets of modes allows to consider the same quantum state under different perspectives: a given state can be entangled in one basis, factorized in another. We will show how to produce, characterize and tailor multimode quantum light. Finally we will describe some applications to quantum technologies, such as parameter estimation and measurement-based quantum computation., Comment: Review paper

Negativity of the Wigner function is seen as a crucial resource for reaching a quantum computational advantage with continuous variable systems. However, these systems, while they allow for the deterministic generation of large entangle states, require an extra element such as photon subtraction to obtain such negativity. Photon subtraction is known to affect modes beyond the one where the photon is subtracted, an effect which is governed by the correlations of the state. In this manuscript, we build upon this effect to remotely prepare states with Wigner-negativity. More specifically, we show that photon subtraction can induce Wigner-negativity in a correlated mode if and only if that correlated mode can perform Einstein-Podolsky-Rosen steering in the mode of subtraction., Comment: 5 pages and 2 figures

Multipartite Einstein-Podolsky-Rosen steering is an essential resource for quantum communication networks where the reliability of equipment at all of the nodes cannot be fully trusted. Here, we present experimental generation of a highly versatile and flexible repository of multipartite steering using an optical frequency comb and ultrafast pulse shaping. Simply modulating the optical spectral resolution of the detection system using the pulse shaper, this scheme is able to produce on-demand 4, 8 and 16-mode Gaussian steering without changing the photonics architecture. We find that the steerability increases with higher spectral resolution. For 16-mode state, we identify as many as 65534 possible bipartition steering existing in this intrinsic multimode quantum resource, and demonstrate that the prepared state steerability is robust to mode losses. Moreover, we verify four types of monogamy relations of Gaussian steering and demonstrate strong violation for one of them. Our method offers a powerful foundation for constructing quantum networks in real-world scenario., Comment: 8 pages, 7 figures

Multimode photon-subtraction provides an experimentally feasible option to construct large non-Gaussian quantum states in continuous-variable quantum optics. The non-Gaussian features of the state can lead towards the more exotic aspects of quantum theory, such as negativity of the Wigner function. However, the pay-off for states with such delicate quantum properties is their sensitivity to decoherence. In this paper, we present a general model that treats the most important source of decoherence in a purely optical setting: losses. We use the framework of open quantum systems and master equations to describe losses in n-photon-subtracted multimode states, where each photon can be subtracted in an arbitrary mode. As a main result, we find that mode-dependent losses and photon-subtraction generally do not commute. In particular, the losses do not only reduce the purity of the state, they also change the modal structure of its non-Gaussian features. We then conduct a detailed study of single-photon subtraction from a multimode Gaussian state, which is a setting that lies within the reach of present-day experiments., Comment: 14 pages, 8 figures

We study noise propagation dynamics in a femtosecond oscillator by injecting external noise on the pump intensity. We utilize a spectrally-resolved homodyne detection technique that enables simultaneous measurement of amplitude and phase quadrature noises of different spectral bands of the oscillator. To reveal the impact of added pump noise on the oscillator noise, we perform a modal analysis of the oscillator noise in which each mode corresponds to a particular temporal/spectral shape of the pulsed light. We compare this modal approach with the conventional noise detection methods and find the superiority of our method in particular unveiling a complete physical picture of noise distribution in femtosecond oscillator.

Even though Gaussian quantum states of multimode light are promising quantum resources due to their scalability, non-Gaussianity is indispensable for quantum technologies, in particular to reach quantum computational advantage. However, embodying non-Gaussianity in a multimode Gaussian state remains a challenge as controllable non-Gaussian operations are hard to implement in a multimode scenario. Here, we report the first generation of non-Gaussian quantum states of a multimode light field by subtracting a photon in a desired mode from multimode Gaussian states, and observe negativity of the Wigner function. For entangled Gaussian states, we observe that photon subtraction makes non-Gaussianity spread among various entangled modes. In addition to applications in quantum technologies, our results shed new light on non-Gaussian multimode entanglement with particular emphasis on quantum networks.

The noise dynamics of an Optical Frequency Comb (OFC) based on a mode-locked Ti-Sapphire laser is analyzed in terms of noise modes. A spectrally resolved multipixel homodyne detection enables the simultaneous measurement of the amplitude and phase noises of several optical frequency channels, from which the covariance matrices of the amplitude and phase quadratures of the laser field are calculated. The decomposition of these matrices into the four most significant time/frequency modes of the field enables the tracking of the origin of the noises and the correlations between the noise modes. In particular, the correlations between the amplitude and phase noises are measured. These measurements are well reproduced by a model taking into account the correlations between the CEO phase noise and the amplitude noise induced by the group velocity dispersion of the laser cavity.

We have theoretically investigated the possibility of using any of several continuous-variable Bell-type inequalities - for which the dichotomic measurements are achieved with coarse-grained quadrature (homodyne) measurements - in a multi-party configuration where each participant is given a section, in the frequency domain, of the output of an optical parametric oscillator which has been synchronously-pumped with a frequency comb. Such light sources are undergoing intense study due to their novel properties, including the potential for production of light entangled in many hundreds of physical modes - a critical component for many proposals in optical or hybrid-optical quantum computation proposals. The situation we study notably uses only highly-efficient optical homodyne detection, meaning that in such systems the fair-sampling loophole would be relatively easy to avoid., Comment: 11 Pages, 5 Figures, Link to computer code to be added later

Graph states are the backbone of measurement-based continuous-variable quantum computation. However, experimental realisations of these states induce Gaussian measurement statistics for the field quadratures, which poses a barrier to obtain a genuine quantum advantage. In this letter, we propose mode-selective photon addition and subtraction as viable and experimentally feasible pathways to introduce non-Gaussian features in such continuous-variable graph states. In particular, we investigate how the non-Gaussian properties spread among the vertices of the graph, which allows us to show the degree of control that is achievable in this approach., Comment: 5 pages, 3 figures

In this work, we investigate three different compact fibered systems generating vacuum squeezing that involve optical cavities limited by the end surface of a fiber and by a curved mirror and containing a thin parametric crystal. These systems have the advantage to couple squeezed states directly to a fiber, allowing the user to benefit from the flexibility of fibers in the use of squeezing. Three types of fibers are investigated: standard single-mode fibers, photonic-crystal large mode area single-mode fibers and short multi-mode fiber taped to a single-mode fiber. The observed squeezing is modest (-0.56 dB, -0.9 dB, -1 dB), but these experiments open the way to miniaturized squeezing devices that could be a very interesting advantage in scaling up quantum systems for quantum processing, opening new perspectives in the domain of integrated quantum optics.