Direct measurement of quantum non-Gaussianity requires some variant of a discrete photon-resolving detection, which is feasible only for low mean photon numbers. For a large mean photon number, intensity detection by linear photodiodes provides a continuous signal; therefore, the Fock probabilities of the unknown input state are not directly available. On the other hand, intensity moments can be measured directly, and photon-number moments can be estimated. Therefore, we derive and analyze a quantum non-Gaussianity witness based solely on the photon number mean and variance (or alternatively, the second-order correlation g^(2)) of an unknown state. Due to the simplicity of the used photon-number moments, the measurement results are easy to correct for losses and additive noise. We provide examples of simple amplification-based measurement schemes where our witness can be applied directly, thereby opening pathways to proof-of-principle tests and applications.

Non-Gaussian states of light are essential for numerous quantum information protocols; thus, certifying non-Gaussianity is crucial. Full quantum state tomography achieves this, but it implies observing the Wigner function negativity, which requires efficient detection. Certifying non-Gaussianity through directly measurable parameters is a simpler alternative, typically achieved by measuring photon-number probabilities - either directly using photon-number resolving detectors or through Hanbury Brown-Twiss type measurements with single-photon detectors. Here, we demonstrate theoretically and experimentally that optical parametric amplification combined with conventional intensity detectors can effectively replace this approach without the need for photon-number resolution. In our method, we measure the mean photon number and the second-order correlation function for the amplified state. Using it, we successfully certify the non-Gaussianity of a heralded quasi-single-photon state. Since optical parametric amplification is a multimode process, our method provides a foundation for developing high-dimensional quantum technologies utilizing multimode non-Gaussian states.

Current advances in nonlinear optics have made it possible to perform a homodyne-like tomography of an unknown state without highly efficient detectors or a strong local oscillator. Thereby, a new experimental direction has been opened into multimode and large-bandwidth quantum optics. An optical parametric amplifier (OPA) allows us to reconstruct the quadrature distribution of an unknown state directly from the measured intensity distribution with high precision. We propose adding a controllable displacement to the standard scheme, thus, obtaining a method applicable even to asymmetric and non-Gaussian states while significantly increasing estimation accuracy and lowering the OPA amplification requirement. To demonstrate the power of our method, we accurately detect the sub-Planck phase-space structure by obtaining distillable squeezing from the OPA estimates of various non-Gaussian states. With the improvements, OPA tomography became a generally applicable loss-tolerant and efficient alternative to homodyne detection.

The field of optical rogue waves is a rapidly expanding topic with a focus on explaining their emergence. To complement this research, instead of providing a microscopic model that generates extreme events, we concentrate on a general quantitative description of the observed behavior. We explore two complementary top-down approaches to estimating the exponent describing the power-law decaying distribution of optical rogue waves observed in supercontinuum generated in a single-mode fiber in the normal-dispersion regime by applying a highly fluctuating pump. The two distinct approaches provide consistent results, outperforming the standard Hill estimator. Further analysis of the distributions reveals the breakdown of power-law behavior due to pump depletion and detector saturation. Either of our methods is adaptable to analyze extreme-intensity events from arbitrary experimental data. While a lot of research efforts have been directed towards determining the emergence mechanism of optical rogue waves, less attention has been given to characterizing the level of "rogueness" in optical systems where rogue waves manifest. The authors provide such quantitative description for rogue waves resulting from supercontinuum generation

This work presents two distinct approaches to estimating the exponent related to the distribution of optical rogue waves observed in supercontinuum generated in a single-mode fiber, that is, to quantifying the rogueness. The first is a generalization of the well-known Hill estimator, and the second relies on estimating all parameters of a multi-parameter model. We show that the model shows a good correspondence with experimental data, and that the two estimating approaches provide consistent results, which are significantly more accurate than those obtained with earlier methods of estimation. Furthermore, alternative visualization through the tail function revealed the presence of pump depletion as well as detector saturation leading to the breakdown of power-law behavior for the largest observations. We characterized this breakdown via a combination of an exponential and a generalized Pareto distribution. Additionally, we have uncovered a weak memory effect in the data, which can be attributed to changes of the refractive index in the single-mode fiber.

Quantum control techniques applied at macroscopic scales provide us with opportunities in fundamental physics and practical applications. Among them, measurement-based feedback allows efficient control of optomechanical systems and quantum-enhanced sensing. In this paper, we propose a near-resonant narrow-band force sensor with extremely low optically added noise in an optomechanical system subject to a feedback-controlled in-loop light. The membrane's intrinsic motion consisting of zero-point motion and thermal motion is affected by the added noise of measurement due to the backaction noise and imprecision noise. We show that, in the optimal low-noise regime, the system is analogous to an optomechanical system containing a near quantum-limited optical parametric amplifier coupled to an engineered reservoir interacting with the cavity. Therefore, the feedback loop enhances the mechanical response of the system to the input while keeping the optically added noise of measurement below the standard quantum limit. Moreover, the system based on feedback offers a much larger amplification bandwidth than the same system with no feedback.

A nanomechanical oscillator can be used as a sensitive probe of a small linearized mechanical force. We propose a simple quantum optomechanical scheme using a coherent light mode in the cavity and weak short-pulsed light-matter interactions. Our main result is that if we transfer some displacement to the mechanical mode in an initialization phase, then a much weaker optomechanical interaction is enough to obtain a high-precision multiparameter estimation of the unknown force. This approach includes not only estimating the displacement caused by the force but also simultaneously observing the phase shift and squeezing of the mechanical mode. We show that the proposed scheme is robust against typical experimental imperfections and demonstrate the feasibility of our scheme using orders of magnitude weaker optomechanical interactions than in previous related works. Thus, we present a simple, robust estimation scheme requiring only very weak light-matter interactions, which could open the way to new nanomechanical sensors.

In optical non-linear processes, rogue waves can be observed, which can be mathematically described by heavy-tailed distributions. These distributions are special since the probability of registering extremely high intensities is significantly higher than for the exponential distribution, which is most commonly observed in statistical and quantum optics. Our work provides a practical overview of the generic statistics toolkit concerning heavy-tailed distributions and proposes methods to deal with issues specific to non-linear optics. We take a closer look at supercontinuum generation, where rogue waves were already observed. We propose modifications to the Hill estimator to deal with detector saturation as well as corrections introduced by pumping the process by bright squeezed vacuum. The suggested methodology facilitates statistically reliable observation of heavy-tailed distribution in non-linear optics, nano-optics, atomic, solid-state processes, and optomechanics.

We investigate estimation of fluctuating channels and its effect on security of continuous-variable quantum key distribution. We propose a novel estimation scheme which is based on the clusterization of the estimated transmittance data. We show that uncertainty about whether the transmittance is fixed or not results in a lower key rate. However, if the total number of measurements is large, one can obtain using our method a key rate similar to the non-fluctuating channel even for highly fluctuating channels. We also verify our theoretical assumptions using experimental data from an atmospheric quantum channel. Our method is therefore promising for secure quantum communication over strongly fluctuating turbulent atmospheric channels.

We propose classical interferometry with low-intensity thermal radiation for the estimation of nonclassical independent Gaussian processes in material samples. We generally determine the mean square error of the phase-independent parameters of an unknown Gaussian process, considering a noisy source of radiation the phase of which is not locked to the pump of the process. We verify the sufficiency of passive optical elements in the interferometer, active optical elements do not improve the quality of the estimation. We also prove the robustness of the method against the noise and loss in both interferometric channels and the sample. The proposed method is suitable even for the case when a source of radiation sufficient for homodyne detection is not available.

We investigated the estimation of an unknown Gaussian process (containing displacement, squeezing and phase-shift) applied to a matter system. The state of the matter system is not directly measured; instead, we measure an optical mode which interacts with the system. We propose an interferometric setup exploiting a beam-splitter-type of light-matter interaction with homodyne detectors and two methods of estimation. We demonstrate the superiority of the interferometric setup over alternative non-interferometric schemes. Importantly, we show that even limited coupling strength and a noisy matter system are sufficient for very good estimation. Our work opens the way to many future investigations of light-matter interferometry for experimental platforms in quantum metrology of matter systems.

Continuous-variable quantum key distribution is a practical application of quantum information theory that is aimed at generation of secret cryptographic key between two remote trusted parties and that uses multi-photon quantum states as carriers of key bits. Remote parties share the secret key via a quantum channel, that presumably is under control of of an eavesdropper, and which properties must be taken into account in the security analysis. Well-studied fiber-optical quantum channels commonly possess stable transmittance and low noise levels, while free-space channels represent a simpler, less demanding and more flexible alternative, but suffer from atmospheric effects such as turbulence that in particular causes a non-uniform transmittance distribution referred to as fading. Nonetheless free-space channels, providing an unobstructed line-of-sight, are more apt for short, mid-range and potentially long-range (using satellites) communication and will play an important role in the future development and implementation of QKD networks. It was previously theoretically shown that coherent-state CV QKD should be in principle possible to implement over a free-space fading channel, but strong transmittance fluctuations result in the significant modulation-dependent channel excess noise. In this regime the post-selection of highly transmitting sub-channels may be needed, which can even restore the security of the protocol in the strongly turbulent channels. We now report the first proof-of-principle experimental test of coherent state CV QKD protocol using different levels Gaussian modulation over a mid-range (1.6-kilometer long) free-space atmospheric quantum channel. The transmittance of the link was characterized using intensity measurements for the reference but channel estimation using the modulated coherent states was also studied. We consider security against Gaussian collective attacks, that were shown to be optimal against CV QKD protocols . We assumed a general entangling cloner collective attack (modeled using data obtained from the state measurement results on both trusted sides of the protocol), that allows to purify the noise added in the quantum channel . Our security analysis of coherent-state protocol also took into account the effect of imperfect channel estimation, limited post-processing efficiency and finite data ensemble size on the performance of the protocol. In this regime we observe the positive key rate even without the need of applying post-selection. We show the positive improvement of the key rate with increase of the modulation variance, still remaining low enough to tolerate the transmittance fluctuations. The obtained results show that coherent-state CV QKD protocol that uses real free-space atmospheric channel can withstand negative influence of transmittance fluctuations, limited post-processing efficiency, imperfect channel estimation and other finite-size effects, and be successfully implemented. Our result paves the way to the full-scale implementation of the CV QKD in real free-space channels at mid-range distances.

Vladyslav C. Usenko,1, ∗ Kirill Yu. Spasibko,2, 3, 4 Laszlo Ruppert,1 Maria V. Chekhova,2, 3, 4 Radim Filip,1 and Gerd Leuchs2, 3 Department of Optics, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1/B24, 91058 Erlangen, Germany University of Erlangen-Nürnberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany Physics Department, Moscow State University, Leninskiye Gory 1-2, Moscow 119991, Russia

For systems analogous to a linear harmonic oscillator, the simplest way to characterize the state is by a covariance matrix containing the symmetrically-ordered moments of operators analogous to position and momentum. We show that using Stokes-like detectors without direct access to either position or momentum, the estimation of the covariance matrix of a macroscopic signal is still possible using interference with a classical noisy and low-intensity reference. Such a detection technique will allow to estimate macroscopic quantum states of electromagnetic radiation without a coherent high-intensity local oscillator. It can be directly applied to estimate the covariance matrix of macroscopically bright squeezed states of light.

We analyze homodyne detection of macroscopically bright multimode nonclassical states of light and propose their application in quantum communication. We observe that the homodyne detection is sensitive to a mode-matching of the bright light to the highly intense local oscillator. Unmatched bright modes of light result in additional noise which technically limits detection of Gaussian entanglement at macroscopic level. When the mode-matching is sufficient, we show that multimode quantum key distribution with bright beams is feasible. It finally merges the quantum communication with classical optical technology of visible beams of light.

In this paper, we examine a generalization of the symmetric informationally complete POVMs. SIC-POVMs are the optimal measurements for full quantum tomography, but if some parameters of the density matrix are known, then the optimal SIC POVM should be orthogonal to a subspace. This gives the concept of the conditional SIC-POVM. The existence is not known in general, but we give a result in the special case when the diagonal is known of the density matrix.

We investigate the main limitations which prevent the continuous-variable quantum key distribution protocols from achieving long distances in the finite-size setting. We propose a double-modulation protocol which allows using each state for both channel estimation and key distribution. As opposed to the standard method, we optimize the parameters of the protocol and consider squeezed as well as coherent states as a signal. By optimally combining the resources the key rate can approach the theoretical limit for long distances, and one can obtain about ten times higher key rate using ten times shorter block size than in the current state-of-the-art implementation.

In this paper we estimate the parameters of the qubit Pauli channel using the channel matrix formalism. The main novelty of this work is that we do not assume the directions of the Pauli channel to be known, but they are determined through the tomography process, too. The results show that for optimally estimating the contraction parameters and the channel matrix we should have input qubits and measurements in the channel directions. However, for optimally estimating the channel directions, we should use different tomography conditions.

In this paper we examine a generalization of the symmetric informationally complete POVMs. SIC-POVMs are the optimal measurements for full quantum tomography, but if some parameters of the density matrix are known, then the optimal SIC POVM should be orthogonal to a subspace. This gives the concept of the conditional SIC-POVM. The existence is not known in general, but we give a result in the special cases when the diagonal is known of the density matrix.