Reconstruction Of Density Matrix Research Articles

N-representable one-electron reduced density matrix reconstruction with frozen core electrons.

Recent advances in quantum crystallography have shown that, beyond conventional charge density refinement, a one-electron reduced density matrix (1-RDM) satisfying N-representability conditions can be reconstructed using jointly experimental X-ray structure factors and directional Compton profiles (DCP) through semidefinite programming. So far, such reconstruction methods for 1-RDM, not constrained to idempotency, have been tested only on a toy model system (CO2). In this work, a new method is assessed on crystalline urea [CO(NH2)2] using static (0 K) and dynamic (50 K) artificial experimental data. An improved model, including symmetry constraints and frozen core-electron contribution, is introduced to better handle the increasing system complexity. Reconstructed 1-RDMs, deformation densities and DCP anisotropy are analysed, and it is demonstrated that the changes in the model significantly improve the reconstruction quality, even when there is insufficient information and data corruption. The robustness of the model and the strategy are thus shown to be well adapted to address the reconstruction problem from actual experimental scattering data.

Electron density-based protocol to recover the interacting quantum atoms components of intermolecular binding energy.

A new approach for obtaining interacting quantum atoms-defined components of binding energy of intermolecular interactions, which bypasses the use of standard six-dimensional integrals and two-particle reduced density matrix (2-RDM) reconstruction, is proposed. To examine this approach, three datasets calculated within the density functional theory framework using the def2-TZVP basis have been explored. The first two, containing 53 weakly bound bimolecular associates and 13 molecular clusters taken from the crystal, were used in protocol refinement, and the third one containing other 20 bimolecular and three cluster systems served as a validation reference. In addition, to verify the performance of the proposed approach on an exact 2-RDM, calculations within the coupled cluster formalism were performed for part of the first set systems using the cc-pVTZ basis set. The process of optimization of the proposed parametric model is considered, and the role of various energy contributions in the formation of non-covalent interactions is discussed with regard to the obtained trends.

Intelligent optimization based density matrix reconstruction method with semi-positive constraint

Quantum state tomography (QST) is a technique used to reconstruct the density matrix of unknown quantum states based on experimentally obtained measurements. QST is a fundamental tool in the field of quantum information and quantum technology. It is commonly employed to assess the quality and limitations of experimental platforms. However, the density matrix reconstructed using the standard QST method often fails to guarantee semi-positive definiteness, which is physically unacceptable, due to limitations imposed by the randomness of quantum state measurements, noise in practical applications, and the number of measurements. To address this issue, a method is proposed that combines maximum likelihood (ML) estimation with population intelligence optimization algorithms. First, the issue of guaranteeing the semi-positive definiteness of the density matrix reconstructed using standard QST methods is analyzed. Subsequently, the ML method is introduced, and four commonly used population intelligence optimization algorithms are applied to find the density matrix that maximizes the likelihood of reproducing the experimental measurements. Finally, the superiority of the proposed method is demonstrated using an IBM Quantum (IBMQ) processor in scenarios involving separable and entangled states of multiple qubits, and compared with the standard QST method.

Fully on-chip photonic turnkey quantum source for entangled qubit/qudit state generation

Integrated photonics has recently become a leading platform for the realization and processing of optical entangled quantum states in compact, robust and scalable chip formats, with applications in long-distance quantum-secured communication, quantum-accelerated information processing and nonclassical metrology. However, the quantum light sources developed so far have relied on external bulky excitation lasers, making them impractical prototype devices that are not reproducible, hindering their scalability and transfer out of the laboratory into real-world applications. Here we demonstrate a fully integrated quantum light source that overcomes these challenges through the integration of a laser cavity, a highly efficient tunable noise suppression filter (>55 dB) exploiting the Vernier effect, and a nonlinear microring for entangled photon-pair generation through spontaneous four-wave mixing. The hybrid quantum source employs an electrically pumped InP gain section and a Si3N4 low-loss microring filter system, and demonstrates high performance parameters, that is, pair emission over four resonant modes in the telecom band (bandwidth of ~1 THz) and a remarkable pair detection rate of ~620 Hz at a high coincidence-to-accidental ratio of ~80. The source directly creates high-dimensional frequency-bin entangled quantum states (qubits/qudits), as verified by quantum interference measurements with visibilities up to 96% (violating Bell’s inequality) and by density matrix reconstruction through state tomography, showing fidelities of up to 99%. Our approach, leveraging a hybrid photonic platform, enables scalable, commercially viable, low-cost, compact, lightweight and field-deployable entangled quantum sources, quintessential for practical, out-of-laboratory applications such as in quantum processors and quantum satellite communications systems.

Density matrix reconstruction using non-negative matrix product states

Quantum state tomography is a key technique for quantum information processing but is challenging due to the exponential growth of its complexity with the system size. In this work we propose an algorithm which iteratively finds the best non-negative matrix product state approximation based on a set of measurement outcomes whose size does not necessarily grow exponentially. Compared to the tomography method based on neural network states, our scheme utilizes a so-called tensor train representation that allows straightforward recovery of the unknown density matrix in the matrix product operator form. As applications, the effectiveness of our algorithm is numerically demonstrated to reconstruct the ground state of the XXZ spin chain under depolarizing noise.

N-representable one-electron reduced density matrix reconstruction at non-zero temperatures

<i>N</i>-representable one-electron reduced density matrix reconstruction at non-zero temperatures

Optimal sparse eigenspace and low-rank density matrix estimation for quantum systems

Quantum state tomography, which aims to estimate quantum states that are described by density matrices, plays an important role in quantum science and quantum technology. This paper examines the eigenspace estimation and the reconstruction of large low-rank density matrix based on Pauli measurements. Both ordinary principal component analysis (PCA) and iterative thresholding sparse PCA (ITSPCA) estimators of the eigenspace are studied, and their respective convergence rates are established. In particular, we show that the ITSPCA estimator is rate-optimal. We present the reconstruction of the large low-rank density matrix and obtain its optimal convergence rate by using the ITSPCA estimator. A numerical study is carried out to investigate the finite sample performance of the proposed estimators.

Reconstructing quantum states with generative models

A major bottleneck in the quest for scalable many-body quantum technologies is the difficulty in benchmarking their preparations, which suffer from an exponential `curse of dimensionality' inherent to their quantum states. We present an experimentally friendly method for density matrix reconstruction based on deep neural-network generative models. The learning procedure comes with a built-in approximate certificate of the reconstruction and makes no assumptions on the state under scrutiny, making it both reliable and unconditional. It can efficiently handle a broad class of complex systems including prototypical states in quantum information, as well as ground states of local spin models common to condensed matter physics. The key insight is to reduce the state tomography task to an unsupervised learning problem of the statistics of an informationally complete set of quantum measurements. This constitutes a modern machine learning approach to the validation of large quantum devices, which may prove relevant as a neural-network ansatz over mixed states suitable for variational optimization.

Ultrafast Mapping of Coherent Dynamics and Density Matrix Reconstruction in a Terahertz-Assisted Laser Field.

A time-resolved spectroscopic protocol exploiting terahertz-assisted photoionization is proposed to reconstruct transient density matrix. Population and coherence elements are effectively mapped onto spectrally separated peaks in photoionization spectra. The beatings of coherence dynamics can be temporally resolved beyond the pulse duration, and the relative phase between involved states is directly readable from the oscillatory spectral distribution. As demonstrated by a photoexcited multilevel open quantum system, the method shows potential applications for subfemtosecond time-resolved measurements of coherent dynamics with free electron lasers and tabletop laser fields.

Invited Article: Time-bin entangled photon pairs from Bragg-reflection waveguides

Semiconductor Bragg-reflection waveguides are well-established sources of correlated photon pairs as well as promising candidates for building up integrated quantum optics devices. Here, we use such a source with optimized non-linearity for preparing time-bin entangled photons in the telecommunication wavelength range. By taking advantage of pulsed state preparation and efficient free-running single-photon detection, we drive our source at low pump powers, which results in a strong photon-pair correlation. The tomographic reconstruction of the state's density matrix reveals that our source exhibits a high degree of entanglement. We extract a concurrence of $88.9\pm 1.8\%$ and a fidelity of $94.2 \pm 0.9\%$ with respect to a Bell state.

Generation and Coherent Control of Pulsed Quantum Frequency Combs.

We present a method for the generation and coherent manipulation of pulsed quantum frequency combs. Until now, methods of preparing high-dimensional states on-chip in a practical way have remained elusive due to the increasing complexity of the quantum circuitry needed to prepare and process such states. Here, we outline how high-dimensional, frequency-bin entangled, two-photon states can be generated at a stable, high generation rate by using a nested-cavity, actively mode-locked excitation of a nonlinear micro-cavity. This technique is used to produce pulsed quantum frequency combs. Moreover, we present how the quantum states can be coherently manipulated using standard telecommunications components such as programmable filters and electro-optic modulators. In particular, we show in detail how to accomplish state characterization measurements such as density matrix reconstruction, coincidence detection, and single photon spectrum determination. The presented methods form an accessible, reconfigurable, and scalable foundation for complex high-dimensional state preparation and manipulation protocols in the frequency domain.

Hybrid reconstruction of quantum density matrix: when low-rank meets sparsity

Both the mathematical theory and experiments have verified that the quantum state tomography based on compressive sensing is an efficient framework for the reconstruction of quantum density states. In recent physical experiments, we found that many unknown density matrices in which people are interested in are low-rank as well as sparse. Bearing this information in mind, in this paper we propose a reconstruction algorithm that combines the low-rank and the sparsity property of density matrices and further theoretically prove that the solution of the optimization function can be, and only be, the true density matrix satisfying the model with overwhelming probability, as long as a necessary number of measurements are allowed. The solver leverages the fixed-point equation technique in which a step-by-step strategy is developed by utilizing an extended soft threshold operator that copes with complex values. Numerical experiments of the density matrix estimation for real nuclear magnetic resonance devices reveal that the proposed method achieves a better accuracy compared to some existing methods. We believe that the proposed method could be leveraged as a generalized approach and widely implemented in the quantum state estimation.

Optimal estimation of parameters of an entangled quantum state

Two-photon entangled quantum states are a fundamental tool for quantum information and quantum cryptography. A complete description of a generic quantum state is provided by its density matrix: the technique allowing experimental reconstruction of the density matrix is called quantum state tomography. Entangled states density matrix reconstruction requires a large number of measurements on many identical copies of the quantum state. An alternative way of certifying the amount of entanglement in two-photon states is represented by the estimation of specific parameters, e.g., negativity and concurrence. If we have a priori partial knowledge of our state, it’s possible to develop several estimators for these parameters that require lower amount of measurements with respect to full density matrix reconstruction. The aim of this work is to introduce and test different estimators for negativity and concurrence for a specific class of two-photon states.

Density matrix reconstruction of three-level atoms via Rydberg electromagnetically induced transparency

We present combined measurements of the spatially resolved optical spectrum and the total excited-atom number in an ultracold gas of three-level atoms under electromagnetically induced transparency conditions involving high-lying Rydberg states. The observed optical transmission of a weak probe laser at the center of the coupling region exhibits a double peaked spectrum as a function of detuning, while the Rydberg atom number shows a comparatively narrow single resonance. By imaging the transmitted light onto a charge-coupled-device camera, we record hundreds of spectra in parallel, which are used to map out the spatial profile of Rabi frequencies of the coupling laser. Using all the information available we can reconstruct the full one-body density matrix of the three-level system, which provides the optical susceptibility and the Rydberg density as a function of spatial position. These results help elucidate the connection between three-level interference phenomena, including the interplay of matter and light degrees of freedom and will facilitate new studies of many-body effects in optically driven Rydberg gases.

Quantum state tomography for strongly coupled nuclear spin systems

We perform quantum state tomography (QST) in a three-qubit system consisting of strongly coupled nuclear spins, known in the NMR literature as $ABX$ systems. We find that the number of experiments necessary to perform QST in such systems can be reduced with respect to those containing three qubits weakly coupled, which reduces the experimental effort required for the complete density-matrix reconstruction. To test the procedure we implement the full protocol for quantum teleportation. The tomographic results demonstrate that the density matrix can be reconstructed with fewer operations and high fidelity.

Density matrix reconstruction by off-axis electron holography

Extended abstract of a paper presented at Microscopy and Microanalysis 2013 in Indianapolis, Indiana, USA, August 4 – August 8, 2013.

Density matrix reconstruction from displaced photon number distributions

We consider state reconstruction from the measurement statistics of phase space observables generated by photon number states. The results are obtained by inverting certain infinite matrices. In particular, we obtain reconstruction formulae, each of which involves only a single phase space observable.

Quantum state tomography for quadrupolar nuclei using global rotations of the spin system

In this paper, we describe a quantum state tomography method based on global rotations of the spin system which, together with a coherence selection scheme, enables the complete density matrix reconstruction. The main advantage of this technique, in respect to previous proposals, is the use of much shorter rf pulses, which decreases significantly the time necessary for algorithm quantum state tomography. In this case, under adequate experimental conditions, the rf pulses correspond to simple spatial rotations of the spin states, and its analytical description is conveniently given in the irreducible tensor formalism. Simulated results show the feasibility of the method for a single spin 72 nucleus. As an experimental result, we exemplify the application of this method by tomographing the steps during the implementation of the Deutsch algorithm. The algorithm was implemented in a (23)Na quadrupole nucleus using the strongly modulated pulses technique. We also extended the tomography method for a 3-coupled homonuclear spin 12 system, where an additional evolution under the internal Hamiltonian is necessary for zero order coherences evaluation.

Two-body Reduced Density Matrix Reconstruction for Van der Waals Systems

A method for the calculation of the electronic energy of a correlated system is presented. This approach is based on the reconstruction of the total two-body reduced density matrix by doing separate configurations interaction calculations on fragments. The method has been tested on Van der Waals systems and has been implemented by considering restrictive N-representability conditions. It is shown that the computational strategy presented in this work can describe with good accuracy weak dispersion interactions, and considerably lowers the size-consistency error of a classical configuration interaction calculation.

Generalized density functional theories using the k-electron densities: Development of kinetic energy functionals

Several explicit formulas for the kinetic energy of a many-electron system as a functional of the k-electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using density-matrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of density-functional theory. Together with N-representability constraints on the k-density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional density-functional theory, notably regarding the development of more accurate density functionals for the kinetic energy.