Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multi-mode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though non-universal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is saturated with random linear optics configurations, such as when the corresponding unitary transformation is Haar-random. Furthermore, we show that for N photons, any unitary 2N-design can be used to derive an analytical, though non-optimal, state reconstruction protocol.
The generation of quantum entangled states of many particles is a central goal of quantum information science. Characterizing such states is a complex task that demands exponentially large resources as particles are being added. Previously, we demonstrated a resource efficient source that can generate, in principal, entanglement between any number of photons. This source recursively fuse photon pairs generated by a pulsed laser into a multiphoton entangled state. In the current work, we perform quantum state tomography on the photon pair source and quantum process tomography on the fusion oper…
A scheme is discussed for measuring Nth-order coherences of two orthogonally polarized light fields in a single spatial mode at very limited experimental cost. To implement the scheme, the only measurements needed are the Nth-order intensity moments after the light beam has passed through two quarter-wave plates, one half-wave plate, and a polarizing beam splitter for specific settings of the wave plates. It is shown that this method can be applied for arbitrarily large N. A set of explicit values is given for the settings of the wave plates, constituting an optimal measurement of the Nth-orde…
In this work we develop an experimental procedure to interrogate the single- and multiphoton scattering matrices of an unknown quantum system interacting with propagating photons. Our proposal requires coherent state laser or microwave inputs and homodyne detection at the scatterer's output, and provides simultaneous information about multiple ---elastic and inelastic--- segments of the scattering matrix. The method is resilient to detector noise and its errors can be made arbitrarily small by combining experiments at various laser powers. Finally, we show that the tomography of scattering has…
We demonstrate fast counting and multiphoton detection abilities of a Silicon Photo Multiplier (SiPM). In fast counting mode we are able to detect two consecutive photons separated by only 2.3 ns corresponding to 430 MHz. The counting efficiency for small optical intensities at a wavelength of 532 nm was found to be around 8.3% with a dark count rate of 50 kHz at T=-7 degrees Celsius. Using the SiPM in multiphoton detection mode, we find a good signal discrimination for different numbers of simultaneous detected photons.
We present a method for spatially resolved multiphoton counting based on an intensified camera with the retrieval of multimode photon statistics fully accounting for non-linearities in the detection process. The scheme relies on one-time quantum tomographic calibration of the detector. Faithful, high-fidelity reconstruction of single- and two-mode statistics of multiphoton states is demonstrated for coherent states and their statistical mixtures. The results consistently exhibit classical values of Mandel and Fano parameters in contrast to raw statistics of camera photo-events. Detector operat…
Multiphoton entanglement plays a critical role in quantum information processing, and greatly improves our fundamental understanding of the quantum world. Despite tremendous efforts in either bulk media or fiber-based devices, nonlinear interactions in integrated circuits show great promise as an excellent platform for photon pair generation with its high brightness, stability and scalability \cite{Caspani2017}. Here, we demonstrate the generation of bi- and multiphoton polarization entangled qubits in a single silicon nanowire waveguide, and these qubits directly compatible with the dense wav…
We propose a systematic procedure to optimize quantum state tomography protocols for continuous variable systems based on excitation counting preceded by a displacement operation. Compared with conventional tomography based on Husimi or Wigner function measurement, the excitation counting approach can significantly reduce the number of measurement settings. We investigate both informational completeness and robustness, and provide a bound of reconstruction error involving the condition number of the sensing map. We also identify the measurement settings that optimize this error bound, and demo…
In terms of the characteristic functions of the quantum states, we present a complete operator description of a lossy photon-subtraction scheme. Feeding a single-mode squeezed vacuum into a variable beam splitter and counting the photons in one of the output channels, a broad class of multiphoton-subtracted squeezed vacuum states (MSSVSs) can be generated in other channel. Here the losses are considered in the beginning and the end channels in the circuit. Indeed, this scheme has been discussed in Ref. [Phys. Rev. A 100, 022341 (2019)]. However, different from the above work, we give all the d…
The relaxation of a quantum field stored in a high-$Q$ superconducting cavity is monitored by non-resonant Rydberg atoms. The field, subjected to repetitive quantum non-demolition (QND) photon counting, undergoes jumps between photon number states. We select ensembles of field realizations evolving from a given Fock state and reconstruct the subsequent evolution of their photon number distributions. We realize in this way a tomography of the photon number relaxation process yielding all the jump rates between Fock states. The damping rates of the $n$ photon states ($0\leq n \leq 7$) are found …
Loophole-free Bell tests are essential if one wishes to perform device-independent quantum key distribution (QKD), since any loophole could be used by a potential adversary to undermine the security of the protocol. Crucial work by Eberhard demonstrated that weakly entangled two-qubit states have a far greater resistance to the detection loophole than maximally entangled states, allowing one to close the loophole with detection efficiency greater than 2/3. Here we demonstrate that this same limit holds for photon-counting CHSH Bell tests which can demonstrate non-locality for higher dimensiona…
We propose a design for a photon counting detector capable of resolving multiphoton events. The basic element of the setup is a fiber loop, which traps the input field with the help of a fast electrooptic switch. A single weakly coupled avalanche photodiode is used to detect small portions of the signal field extracted from the loop. We analyze the response of the loop detector to an arbitrary input field, and discuss both the reconstruction of the photon number distribution of an unknown field from the count statistics measured in the setup, and the application of the detector in conditional …
Unwanted multiphoton emission commonly reduces the degree of entanglement of photons generated by non-classical light sources and, in turn, hampers their exploitation in quantum information science and technology. Quantum emitters have the potential to overcome this hurdle but, so far, the effect of multiphoton emission on the quality of entanglement has never been addressed in detail. Here, we tackle this challenge using photon pairs from a resonantly-driven quantum dot and comparing quantum state tomography and second-order coherence measurements as a function of the excitation power. We obs…
Characterizing Photonic Partial Distinguishability Through Gram Matrix Tomography
Center for Quantum Enabled-Computing, Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心量子使能计算中心）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）；Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles（布鲁塞尔自由大学布鲁塞尔理工学院量子信息与通信中心）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）
In this article, we introduce a framework for quantum state tomography of qutrits by projective measurements. The framework is based on photon-counting with measurement results distorted due to the Poisson noise and dark counts. Two different measurement schemes are investigated numerically and compared in terms of their efficiency for distinct numbers of photons per measurement. The accuracy of state reconstruction is quantified by figures of merit which are presented on graphs versus the amount of noise.
We perform quantum state reconstruction of coherent and thermal states with a detector which has an enhanced multiphoton response. The detector is based on superconducting nanowires, where the bias current sets the dependence of the click probability on the photon number; this bias current is used as tuning parameter in the state reconstruction. The nonlinear response makes our nanowire-based detector superior to the linear detectors that are conventionally used for quantum state reconstruction.
In this article, we introduce a framework for entanglement characterization by time-resolved single-photon counting with measurement operators defined in the time domain. For a quantum system with unitary dynamics, we generate time-continuous measurements by shifting from the Schrodinger picture to the Heisenberg representation. In particular, we discuss this approach in reference to photonic tomography. To make the measurement scheme realistic, we impose timing uncertainty on photon counts along with the Poisson noise. Then, the framework is tested numerically on quantum tomography of qubits.…
All the $n(2n+3)$ mean and covariance parameters of an $n$-mode Gaussian states are expressed in terms of the expectation values of the same number of conjugates of the total number observable. This permits a complete tomography of the state. The same is applied to outputs of a Gaussian channel corresponding to selected coherent states to perform the complete tomography of the channel. This leads to some interesting problems concerning the distribution of the number operator and also tomographic complexity.
We present a comprehensive framework for quantum state tomography (QST) of time-bin qudits sent through a fiber. Starting from basic assumptions, we define a positive-operator valued measure (POVM) which is then applied to the quantum state reconstruction problem. A realistic scenario is considered where the time uncertainty of the detector is treated as a source of experimental noise. The performance of the quantum tomography framework is examined through a series of numerical simulations conducted for different parameters describing the apparatus. The quality of state recovery, quantified by…
Silicon single-photon detectors (SPDs) are the key devices for detecting single photons in the visible wavelength range. Here we present high detection efficiency silicon SPDs dedicated to the generation of multiphoton entanglement based on the technique of high-frequency sine wave gating. The silicon single-photon avalanche diodes (SPADs) components are acquired by disassembling 6 commercial single-photon counting modules (SPCMs). Using the new quenching electronics, the average detection efficiency of SPDs is increased from 68.6% to 73.1% at a wavelength of 785 nm. These sine wave gating SPD…
We propose a general methodology for efficient statistical reconstruction of a quantum state through collection and analysis of photon counting statistics. Our approach includes both strict quantitative criteria for adequacy and completeness of the statistical inverse problem, as well as a simple and reliable method for evaluating errors in the reconstructed state and approximation of a quantum state by means of the reduced finite-dimensional model.
A single photon, delocalized over two optical modes, is characterized by means of quantum homodyne tomography. The reconstructed four-dimensional density matrix extends over the entire Hilbert space and thus reveals, for the first time, complete information about the dual-rail optical quantum bit as a state of the electromagnetic field. The experimental data violate the Bell inequality albeit with a loophole similar to the detection loophole in photon counting experiments.
Subnatural-linewidth single-photon source is a potential candidate for exploring the time degree of freedom in photonic quantum information science. This type of single-photon source has been demonstrated to be generated and reshaped in atomic ensembles without any external cavity or filter, and is typically characterized through photon-counting technology. However, the full complex temporal mode function(TMF) of the photon source is not able to be revealed from direct photon counting measurement. Here, for the first time, we demonstrate the complete temporal mode of the subnatural-linewidth s…
We discuss methods of quantum state tomography for solid-state systems with a large nuclear spin $I=3/2$ in nanometer-scale semiconductors devices based on a quantum well. Due to quadrupolar interactions, the Zeeman levels of these nuclear-spin devices become nonequidistant, forming a controllable four-level quantum system (known as quartit or ququart). The occupation of these levels can be selectively and coherently manipulated by multiphoton transitions using the techniques of nuclear magnetic resonance (NMR) [Yusa et al., Nature (London) 434, 101 (2005)]. These methods are based on an uncon…
We propose a technique for performing quantum state tomography of photonic polarization-encoded multi-qubit states. Our method uses a single rotating wave plate, a polarizing beam splitter and two photon-counting detectors per photon mode. As the wave plate rotates, the photon counters measure a pseudo-continuous signal which is then Fourier transformed. The density matrix of the state is reconstructed using the relationship between the Fourier coefficients of the signal and the Stokes' parameters that represent the state. The experimental complexity, i.e. different wave plate rotation frequen…
We train convolutional neural networks to predict whether or not a set of measurements is informationally complete to uniquely reconstruct any given quantum state with no prior information. In addition, we perform fidelity benchmarking based on this measurement set without explicitly carrying out state tomography. The networks are trained to recognize the fidelity and a reliable measure for informational completeness. By gradually accumulating measurements and data, these trained convolutional networks can efficiently establish a compressive quantum-state characterization scheme by acceleratin…
Quantum networks illustrate the use of connected nodes of quantum systems as the backbone of distributed quantum information processing. When the network nodes are entangled in graph states, such a quantum platform is indispensable to almost all the existing distributed quantum tasks. Unfortunately, real networks unavoidably suffer from noise and technical restrictions, making nodes transit from quantum to classical at worst. Here, we introduce a figure of merit in terms of the number of classical nodes for quantum networks in arbitrary graph states. Such a network property is revealed by expl…
We present a continuous-variable experimental analysis of a two-photon Fock state of free-propagating light. This state is obtained from a pulsed non-degenerate parametric amplifier, which produces two intensity-correlated twin beams. Counting two photons in one beam projects the other beam in the desired two-photon Fock state, which is analyzed by using a pulsed homodyne detection. The Wigner function of the measured state is clearly negative. We developed a detailed analytic model which allows a fast and efficient analysis of the experimental results.
Optimal interferometric certification of multi-photon indistinguishability
Université libre de Bruxelles（布鲁塞尔自由大学）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术实验室）；Center for Quantum Enabled-Computing, Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心量子使能计算中心）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）
We present methods and results of shot-by-shot correlation of noisy measurements to extract entangled state and process tomography in a superconducting qubit architecture. We show that averaging continuous values, rather than counting discrete thresholded values, is a valid tomographic strategy and is in fact the better choice in the low signal-to-noise regime. We show that the effort to measure $N$-body correlations from individual measurements scales exponentially with $N$, but with sufficient signal-to-noise the approach remains viable for few-body correlations. We provide a new protocol to…
The high-fidelity analysis of many-body quantum states of indistinguishable atoms requires the accurate counting of atoms. Here we report the tomographic reconstruction of an atom-number-resolving detector. The tomography is performed with an ultracold rubidium ensemble that is prepared in a coherent spin state by driving a Rabi coupling between the two hyperfine clock levels. The coupling is followed by counting the occupation number in one level. We characterize the fidelity of our detector and show that a negative-valued Wigner function is associated with it. Our results offer an exciting p…
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available wit…
Optical coherence tomography has become an important imaging technology in cardiology and ophthalmology, with other applications under investigations. Major advances in optical coherence tomography (OCT) imaging are likely to occur through a quantum field approach to the technology. In this paper, which is the first part in a series on the topic, the quantum basis of OCT first order correlations is expressed in terms of full field quantization. Specifically first order correlations are treated as the linear sum of single photon interferences along indistinguishable paths. Photons and the elect…
The coexistence of quantum and classical signals over the same optical fiber with minimal degradation of the transmitted quantum information is critical for operating large-scale quantum networks over the existing communications infrastructure. Here, we systematically characterize the quantum channel that results from simultaneously distributing approximate single-photon polarization-encoded qubits and classical light of varying intensities through fiber-optic channels of up to 15~km. Using spectrally resolved quantum process tomography with a Bayesian reconstruction method we developed, we es…
We employed an electrically-driven polarization controller to implement anisotropic depolarizing quantum channels for the polarization state of single photons. The channels were characterized by means of ancilla-assisted quantum process tomography using polarization-entangled photons generated in the process of spontaneous parametric down-conversion. The demonstrated depolarization method offers good repeatability, low cost, and compatibility with fiber-optic setups. It does not perturb the modal structure of single photons, and therefore can be used to verify experimentally protocols for mana…
Using detector tomography, we investigate the detection mechanism in NbN-based superconducting single photon detectors (SSPDs). We demonstrate that the detection probability uniquely depends on a particular linear combination of bias current and energy, for a large variation of bias currents, input energies and detection probabilities, producing a universal detection curve. We obtain this result by studying multiphoton excitations in a nanodetector with a sparsity-based tomographic method that allows factoring out of the optical absorption. We discuss the implication of our model system for th…
The article undertakes the problem of pure state estimation from projective measurements based on photon counting. Two generic frames for qubit tomography are considered -- one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases (MUBs). Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to hi…
Continuous-Variable Quantum State Tomography Enabled by Quantum Mirrors
Universidad de Concepción（康塞普西翁大学）；ANID - Millennium Science Iniciative Program - Millennium Institute for Research in Optics（智利国家研究与发展局-千年科学倡议计划-光学研究千年研究所）；Universidad de Chile（智利大学）
We experimentally demonstrate that a non-classical state prepared in an atomic memory can be efficiently transferred to a single mode of free-propagating light. By retrieving on demand a single excitation from a cold atomic gas, we realize an efficient source of single photons prepared in a pure, fully controlled quantum state. We characterize this source using two detection methods, one based on photon-counting analysis, and the second using homodyne tomography to reconstruct the density matrix and Wigner function of the state. The latter technique allows us to completely determine the mode o…
In recent years, a close connection between the description of open quantum systems, the input-output formalism of quantum optics, and continuous matrix product states in quantum field theory has been established. So far, however, this connection has not been extended to the condensed-matter context. In this work, we substantially develop further and apply a machinery of continuous matrix product states (cMPS) to perform tomography of transport experiments. We first present an extension of the tomographic possibilities of cMPS by showing that reconstruction schemes do not need to be based on l…
The distinguishing of the multiphoton quantum interference effect from the classical one forms one of the most important issues in modern quantum mechanics and experimental quantum optics. For a long time, the two-photon interference (TPI) of correlated photons has been recognized as a pure quantum effect that cannot be simulated with classical lights. In the meantime, experiments have been carried out to investigate the classical analogues of the TPI. In this study, we conduct TPI experiments with uncorrelated photons with different center frequencies from a luminescent light source, and we c…
A Concept of Two-Point Propagation Field of a Single Photon: A Way to X-ray Picometer Displacement Detection and Nanometer Resolution 3D X-ray Micro-Tomography
