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The growth of quantum technologies is attracting the interest of many students eager to learn concepts such as quantum entanglement or quantum superposition. However, the non-intuitive nature of these concepts poses a challenge to understanding them. Here, we present an entangled photon system which can perform a Bell test, i.e. the CHSH inequality, and can obtain the complete tomography of the two-photon state. The proposed setup is versatile, cost-effective and allows for multiple classroom operating modes. We present two variants, both facilitating the measurement of Bell inequalities and quantum state tomography. Experimental results showcase successful manipulation of the quantum state of the photons, achieving high-fidelity entangled states and significant violations of Bell's inequalities. Our setup's simplicity and affordability enhances accessibility for less specialized laboratories, allowing students to familiarize themselves with quantum physics concepts. [ABSTRACT FROM AUTHOR]

A nonlocal quantum approach is presented to polarimetry, leveraging the phenomenon of entanglement in photon pairs to enhance the precision in sample property determination. By employing two distinct channels, one containing the sample of interest and the other serving as a reference, the conditions are explored under which the inherent correlation between entangled photons can increase measurement sensitivity. Specifically, the quantum Fisher information (QFI) is calculated and compare the accuracy and sensitivity for the cases of single sample channel versus two channel quantum state tomography measurements. The theoretical results are verified by experimental analysis. The theoretical and experimental framework demonstrates that the nonlocal strategy enables enhanced precision and accuracy in extracting information about sample characteristics more than the local measurements. Depending on the chosen estimators and noise channels present, theoretical and experimental results show that noise‐induced bias decreases the precision for the estimated parameter. Such a quantum‐enhanced nonlocal polarimetry holds promise for advancing diverse fields including material science, biomedical imaging, and remote sensing, via high‐precision measurements through quantum entanglement. [ABSTRACT FROM AUTHOR]

Quantum entanglement detection is one of the fundamental tasks in quantum information science. Conventional methods for quantum state tomography exhibit limitations in scalability as the number of qubits increases, leading to exponential growth in the number of unknown parameters and required measurements. Consequently, the accuracy enhancement achieved by these methods is constrained. In response to this challenge, we developed a tailored convolutional neural network (CNN) model capable of effectively detecting entanglement in two-qubit quantum states, achieving an accuracy exceeding 97.5%. Notably, even in the presence of noise, this model retains its robust performance, displaying resilience against a tolerable level of noise contamination. Furthermore, the inherent generalization power of CNNs allows our model, which was initially trained on a specific spectrum of quantum states, to extend its applicability to wider states, positioning it as an outstanding tool for the further application of machine learning in the field of quantum computing, opening up new pathways for solving entanglement detection problems in quantum information. [ABSTRACT FROM AUTHOR]

Quantum state tomography (QST) aiming at reconstructing the density matrix of a quantum state plays an important role in various emerging quantum technologies. Recognizing the challenges posed by imperfect measurement data, we develop a unified neural network (NN)-based approach for QST under constrained measurement scenarios, including limited measurement copies, incomplete measurements, and noisy measurements. Through comprehensive comparison with other estimation methods, we demonstrate that our method improves the estimation accuracy in scenarios with limited measurement resources, showcasing notable robustness in noisy measurement settings. These findings highlight the capability of NNs to enhance QST with constrained measurements. [ABSTRACT FROM AUTHOR]

Quantum state tomography (QST) is one of the key steps in determining the state of the quantum system, which is essential for understanding and controlling it. With statistical data from measurements and Positive Operator-Valued Measures (POVMs), the goal of QST is to find a density operator that best fits the measurement data. Several optimization-based methods have been proposed for QST, and one of the most successful approaches is based on Accelerated Gradient Descent (AGD) with fixed step length. While AGD with fixed step size is easy to implement, it is computationally inefficient when the computational time required to calculate the gradient is high. In this paper, we propose a new optimal method for step-length adaptation, which results in a much faster version of AGD for QST. Numerical results confirm that the proposed method is much more time-efficient than other similar methods due to the optimized step size. [ABSTRACT FROM AUTHOR]

Neural networks are becoming ubiquitous in various areas of physics as a successful machine learning (ML) technique for addressing different tasks. Based on ML technique, we propose and experimentally demonstrate an efficient method for state reconstruction of the widely used Sagnac polarization-entangled photon source. By properly modeling the target states, a multi-output fully connected neural network is well trained using only six of the sixteen measurement bases in standard tomography technique, and hence our method reduces the resource consumption without loss of accuracy. We demonstrate the ability of the neural network to predict state parameters with a high precision by using both simulated and experimental data. Explicitly, the mean absolute error for all the parameters is below 0.05 for the simulated data and a mean fidelity of 0.99 is achieved for experimentally generated states. Our method could be generalized to estimate other kinds of states, as well as other quantum information tasks. [ABSTRACT FROM AUTHOR]

This paper introduces a novel and efficient technique for quantum state estimation, coined as low-rank matrix-completion quantum state tomography for characterizing pure quantum states, as it requires only non-entangling bases and 2 n + 1 local Pauli operators. This significantly reduces the complexity of the process and increases the accuracy of the state estimation, as it eliminates the need for the entangling bases, which are experimentally difficult to implement on quantum devices. The required minimal post-processing, improved accuracy and efficacy of this matrix-completion-based method make it an ideal benchmarking tool for investigating the properties of quantum systems, enabling researchers to verify the accuracy of quantum devices, characterize their performance, and explore the underlying physics of quantum phenomena. Our numerical results demonstrate that this method outperforms contemporary techniques in its ability to accurately reconstruct multi-qubit quantum states on real quantum devices, making it an invaluable contribution to the field of quantum state characterization and an essential step toward the reliable deployment of intermediate- and large-scale quantum devices. [ABSTRACT FROM AUTHOR]

This paper proposed a quantum state tomography approach based on the extreme learning machine (ELM), which is available in the reconstruction of quantum states via a lightweight neural network. The key step of the proposed tomography approach is to employ the ELM to approximate the complex mapping between the measurement values sequence and the real density matrix. After obtaining the output of the ELM-based estimator, a matrix transformation technique is used to make the network outputs satisfy quantum state constraints. Compared with deep learning-based tomography approaches, our proposed ELM-based approach enables both high-fidelity and high-efficiency quantum state tomography with only one training process under the condition of very few numbers of training samples, network layers and hidden layer nodes. In addition, the proposed tomography approach is robust to noisy measurement values, since the ELM-based estimator is quite lightweight. Simulations on the tomography of eigenstates, superposition states and mixed states are presented to verify our theoretical findings. Also, the superiority of the ELM-based tomography approach is demonstrated in comparison with that based on the radial basis function network, convolutional neural network and maximum likelihood estimation approach. [ABSTRACT FROM AUTHOR]

The potential of achieving computational hardware with quantum advantage depends heavily on the quality of quantum gate operations. However, the presence of imperfect two-qubit gates poses a significant challenge and acts as a major obstacle in developing scalable quantum information processors. Google's Quantum AI and collaborators claimed to have conducted a supremacy regime experiment. In this experiment, a new two-qubit universal gate called the Sycamore gate is constructed and employed to generate random quantum circuits (RQCs), using a programmable quantum processor with 53 qubits. These computations were carried out in a computational state space of size 9 × 10 15 . Nevertheless, even in strictly-controlled laboratory settings, quantum information on quantum processors is susceptible to various disturbances, including undesired interaction with the surroundings and imperfections in the quantum state. To address this issue, we conduct both quantum state tomography (QST) and quantum process tomography (QPT) experiments on Google's Sycamore gate using different artificial architectural superconducting quantum computer. Furthermore, to demonstrate how errors affect gate fidelity at the level of quantum circuits, we design and conduct full QST experiments for the five-qubit eight-cycle circuit, which was introduced as an example of the programability of Google's Sycamore quantum processor. These quantum tomography experiments are conducted in three distinct environments: noise-free, noisy simulation, and on IBM Quantum's genuine quantum computer. Our results offer valuable insights into the performance of IBM Quantum's hardware and the robustness of Sycamore gates within this experimental setup. These findings contribute to our understanding of quantum hardware performance and provide valuable information for optimizing quantum algorithms for practical applications. [ABSTRACT FROM AUTHOR]

In this paper, we establish a dynamical quantum state tomography framework. Under this framework, it is feasible to obtain complete knowledge of any unknown state of a d -level system via only one operator of a special type of positive operator-valued measure (POVM) in dimension d. We define a new channel, referred to as the time-dependent average channel. Utilizing this channel, we show that we can acquire a collection of projective operators that is informationally complete (IC) and thus obtain the corresponding informationally complete POVMs (IC-POVMs). Zauner conjectured that for any dimension d there exists a fiducial vector, such that all remaining d 2 − 1 elements of the desired symmetric informationally complete POVM (SIC-POVM) can be obtained by acting on said vector with unitary matrices representing elements of the Weyl-Heisenberg group. We show that under certain condition, it is possible to obtain infinite families of projective operators that are IC, and obtain infinite families of corresponding IC-POVMs; otherwise, Zauner's conjecture is incorrect. We also show how to simulate a SIC-POVM on any unknown quantum state by using the time-dependent average channel. [ABSTRACT FROM AUTHOR]

Quantum entanglement detection is one of the fundamental tasks in quantum information science. Conventional methods for quantum state tomography exhibit limitations in scalability as the number of qubits increases, leading to exponential growth in the number of unknown parameters and required measurements. Consequently, the accuracy enhancement achieved by these methods is constrained. In response to this challenge, we developed a tailored convolutional neural network (CNN) model capable of effectively detecting entanglement in two-qubit quantum states, achieving an accuracy exceeding 97.5%. Notably, even in the presence of noise, this model retains its robust performance, displaying resilience against a tolerable level of noise contamination. Furthermore, the inherent generalization power of CNNs allows our model, which was initially trained on a specific spectrum of quantum states, to extend its applicability to wider states, positioning it as an outstanding tool for the further application of machine learning in the field of quantum computing, opening up new pathways for solving entanglement detection problems in quantum information.

Quantum state tomography (QST) is one of the key steps in determining the state of the quantum system, which is essential for understanding and controlling it. With statistical data from measurements and Positive Operator-Valued Measures (POVMs), the goal of QST is to find a density operator that best fits the measurement data. Several optimization-based methods have been proposed for QST, and one of the most successful approaches is based on Accelerated Gradient Descent (AGD) with fixed step length. While AGD with fixed step size is easy to implement, it is computationally inefficient when the computational time required to calculate the gradient is high. In this paper, we propose a new optimal method for step-length adaptation, which results in a much faster version of AGD for QST. Numerical results confirm that the proposed method is much more time-efficient than other similar methods due to the optimized step size.

Currently, quantum communication is gradually moving from theory to reality. Considering the practical application of quantum communication, more effective and feasible quantum communication schemes are needed. The feasibility of quantum communication schemes can be effectively verified through the three-dimensional reconstruction technology in quantum computers, namely, quantum state tomography. Therefore, this paper creatively proposes a scheme for separating arbitrary two-qubit quantum information using three-qubit entangled states and Bell states as quantum channels and uses IBM Experience platform and quantum state tomography technology for experimental verification. A corresponding quantum circuit is designed on the IBM Quantum Experience cloud platform for experimental simulation. Without loss of generality, we use the RawFeatureVector circuit and a random function of Qiskit to generate the two-qubit state in our scheme. And on this basis, entirely and comprehensively extracted massive data of quantum state information to calculate fidelity, proving that the scheme is safe and effective. Finally, to ensure the communication security, we have analyzed the security of the scheme. [ABSTRACT FROM AUTHOR]

Matrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits. [ABSTRACT FROM AUTHOR]

The physical problem behind informationally complete (IC) measurements is determining an unknown state statistically by measurement outcomes, known as state tomography. It is of central importance in quantum information processing such as channel estimating, device testing, quantum key distribution, etc. However, constructing such measurements with good properties is a long‐standing problem. In this work, projective realizations of IC measurements are investigated. Conditions of informational completeness are presented with proofs first. Then the projective realizations of IC measurements, including proposing the first general construction of minimal projective IC measurements (MPICM) in no prime power dimensional systems, as well as determining an unknown state in Cn$C^{n}$ via a single projective measurement with some kinds of optimalities in a larger system, are investigated. Finally, The results can be extended to local state tomography. Some discussions on employing several kinds of optimalities are also provided. [ABSTRACT FROM AUTHOR]

Noisy intermediate-scale quantum computers (NISQ) have demonstrated superior performance compared to conventional computing schemes. Nonetheless, errors associated with two-qubit entangling gates remain a substantial obstacle in fully harnessing the potential of current superconducting quantum computers. The achievement of quantum computation and quantum information processing critically depends on the presence of entangled states. These states are indispensable for quantum computers to surpass classical computers in terms of computational power. This paper presents a comprehensive characterization of a two-qubit entangling gate, known as Dagwood Bumstead (DB) gate, to investigate the challenges posed by errors in the DB gate on NISQ devices. Firstly, we propose a scheme for the realization of the DB gate using two capacitively coupled Josephson Superconducting qubits, based on the experimental setup carried out in Bialczak et al. (2010). Additionally, we introduce and characterize a novel two-qubit quantum gate. Our proposed gate showcases an impressive average gate fidelity of 0.85450. The DB gate’s performance is fully described through quantum process tomography (QPT) and quantum state tomography (QST) experiments conducted in three different environments: noisy, noiseless, and on a real IBM quantum computer. The process fidelity of the DB gate is measured to be 0.8069705, and the state fidelity is determined to be 0.8858750. This study offers valuable insights into the performance of the DB gate on a NISQ device, showcasing its potential for executing intricate algorithms and operations on an authentic quantum computer.

Practical entanglement distillation is a critical component in quantum information theory. Entanglement distillation is often utilized for designing quantum computer networks and quantum repeaters. The practical entanglement distillation problem is formulated as a bilevel optimization problem. A fuzzy formulation is introduced to estimate the quantum state (density matrix) from pseudo-likelihood functions (i.e., quantum state tomography). A scale-independent relationship between fuzzy relations in terms of the pseudo-likelihood functions is obtained. The entanglement distillation optimization problem is solved using the combined coupled map lattice and dual annealing approach. Comparative analysis of the results is then conducted against a standard dual annealing algorithmic implementation. [ABSTRACT FROM AUTHOR]

The purpose of this research work is to learn the quantum states in an ideal environments analytically, computationally, and graphically. The analysis starts with the learning of quantum states in identity channels with the help of the Regularized Follow the Leader (RFTL) algorithm. Our machine will try to learn the states based on the previous information, which is called the online learning model. The objective of this problem is to minimize regret by utilizing a learning algorithm that successively anticipates quantum states through observed measurements and losses. We have to produce many copies of quantum state ρ to perform analysis on them, which indicates the use of the shadow tomography approach in an ideal situation. Our goal is to learn the shadow of the state ρ by using a series of measurement operators that have two outcomes in nature. Aaronson et al. [1] developed an online setting for a non-realizable case, where the maximum possible loss is O( √ T n) for the best possible state up to T−measurements . It is noteworthy that this outcome is an extension of the Aaronson PAC-like findings [2]. [ABSTRACT FROM AUTHOR]

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.

Entanglement, that is, quantum correlations that do not have a classical counterpart, is a precondition to establishing communication protocols beyond the existing classical protocols, such as quantum key distribution, that achieves a higher level of security without computational assumptions. In this work, we present a proof of demonstration of detecting various entangled states, prepared by time-bin encoding with photons that are natural resources for long-distance quantum communication. We generate a maximally entangled state in time-bin qubits and verify the state in two ways. We first consider measurements that realize entanglement witnesses for the verification of entanglement. We then perform a quantum state tomography for the full characterization. Experimental resources are also discussed.

Quantum state tomography (QST) refers to any method that allows one to reconstruct the accurate representation of a quantum system based on data obtainable from an experiment. In this paper, we concentrate on theoretical methods of quantum tomography, but some significant experimental results are also presented. Due to a considerable body of literature and intensive ongoing research activity in the field of QST, this overview is restricted to presenting selected ideas, methods, and results. First, we discuss tomography of pure states by distinguishing two aspects—complex vector reconstruction and wavefunction measurement. Then, we move on to the Wigner function reconstruction. Finally, the core section of the article is devoted to the stroboscopic tomography, which provides the optimal criteria for state recovery by including the dynamics in the scheme. Throughout the paper, we pay particular attention to photonic tomography, since multiple protocols in quantum optics require well-defined states of light.

Quantum state tomography (QST) is a central technique to fully characterize an unknown quantum state. However, standard QST requires an exponentially growing number of quantum measurements against the system size, which limits its application to smaller systems. Here, we explore the sparsity of underlying quantum state and propose a QST scheme that combines the matrix product states' representation of the quantum state with a supervised machine learning algorithm. Our method could reconstruct the unknown sparse quantum states with very high precision using only a portion of the measurement data in a randomly selected basis set. In particular, we demonstrate that the Wolfgang states could be faithfully reconstructed using around 25 % of the whole basis, and that the randomly generated quantum states, which could be efficiently represented as matrix product states, could be faithfully reconstructed using a number of bases that scales sub-exponentially against the system size. [ABSTRACT FROM AUTHOR]

We generalize an approach to studying the quantum state tomography (QST) of open systems in terms of the dynamical map in Kraus representation within the framework of dynamic generation of informationally complete positive operator-valued measures. As applications, we use the generalized approach to theoretically study the QST of qubit systems in the presence of nonequilibrium environments which exhibit nonstationary and non-Markovian random telegraph noise statistical properties. We derive the time-dependent measurement operators for the quantum state reconstruction of the single qubit and two-qubit systems in terms of the polarization operator basis. It is shown that the behavior of the time-dependent measurement operators is closely associated with the dynamical map of the qubit systems. [ABSTRACT FROM AUTHOR]

We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent (MiFGD), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, MiFGD converges provably close to the true density matrix at an accelerated linear rate asymptotically in the absence of experimental and statistical noise, under common assumptions. With this manuscript, we present the method, prove its convergence property and provide the Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real (noisy) experiments, performed on the IBM's quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than the state-of-the-art approaches, with similar or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite the presence of experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems. [ABSTRACT FROM AUTHOR]

Multi-qubit state tomography is a key problem in quantum information technology, which has been studied extensively. In this work, we focus on multi-qubit state tomography based on neural network estimators and typical conventional estimation approaches. For multi-qubit pure states, fully connected neural networks and restricted Boltzmann machine networks are designed, respectively, to carry out state tomography of 2-qubit (low-dimensional) systems and 5-qubit (high-dimensional) systems. Comparisons are made with maximum likelihood estimation and least squares estimation, where performance indicators are selected as reconstruction accuracy, time cost and the number of parameters. Simulation results indicate that intelligent approaches have significant advantages over conventional approaches for state tomography of low-dimensional systems; for high-dimensional systems, however, the conventional approach is still dominant when the measurement is complete, while the restricted Boltzmann machine network can achieve higher reconstruction accuracy when the measurement is incomplete. [ABSTRACT FROM AUTHOR]

We introduce in this paper a tomographic analysis of the properties of a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe with a perfect fluid. We first review previous works where the Schutz's parametrization in terms of Clebsch potentials was adopted to describe the perfect fluid. This approach allows to introduce a fiducial time in the Wheeler–De Witt equation. We revisit the properties of the perfect fluid and the introduced Clebsch potentials. In particular, we see that thermasy plays an extremely important role in the definition of fiducial time. The definition of a time and the condition a ≥ 0 for the expansion factor imply that the Hamiltonian operator must be self-adjoint which implies a restriction on the initial conditions for the wave packet. We show that these allow to obtain well-defined tomograms. Tomograms are marginal functions which incorporate all the information contained in the wave function of the universe, but have the properties of classical probability functions. They can be defined for classical distributions on the phase space as well, enabling us to describe quantum and classical states with the same family of functions. The aim of this paper is to compare the difference between classical tomograms where the Hawking and Penrose theorems imply the inevitability of an initial singular state and the well-defined initial quantum states found in literature. Finally the introduction of a time in the Wheeler–DeWitt allows us to consider the evolution of the classical and quantum initial states of the universe which can be accomplished by introducing a transition probability function for tomograms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method of dark count correction in quantum state tomography of entangled photon pairs. The framework is based on a linear model of dark counts, which is imposed on the measurement formalism. The method is tested on empirical data derived from an experiment on polarization-entangled photons. We demonstrate that the numerical approach to dark count correction guarantees more reliable state reconstruction compared with standard estimation. Most importantly, however, the conceptually simple theoretical approach proves to be more efficient at entanglement quantification than experimental techniques. [ABSTRACT FROM AUTHOR]

Entanglement states serve as the central resource for a number of important applications in quantum information science, including quantum key distribution, quantum precision measurement, and quantum computing. In pursuit of more promising applications, efforts have been made to generate entangled states with more qubits. However, the efficient creation of a high-fidelity multiparticle entanglement remains an outstanding challenge due to the difficulty that increases exponentially with the number of particles. We design an interferometer that is capable of coupling the polarization and spatial paths of photons and prepare 2-D four-qubit GHZ entanglement states. Using quantum state tomography, entanglement witness, and the violation of Ardehali inequality against local realism, the properties of the prepared 2-D four-qubit entangled state are analyzed. The experimental results show that the prepared four-photon system is an entangled state with high fidelity. [ABSTRACT FROM AUTHOR]

In this article, we introduce a framework for two-qutrit Werner states tomography with Gaussian noise. The measurement scheme is based on the symmetric, informationally complete positive operator-valued measure. To make the framework realistic, we impose the Gaussian noise on the measured states numbers. Through numerical simulation, we successfully reconstructed the two-qutrit Werner states in various experimental scenarios and analyzed the optimal scenario from four aspects: fidelity, purity, entanglement, and coherence. [ABSTRACT FROM AUTHOR]

In recent years, neural network estimator-based quantum state tomography has gained its popularity. Inspired by advances in the field of state-of-the-art deep learning techniques, we apply the U-net model and its variants with an encoder-decoder architecture to the field of quantum state tomography as well as taking full account of measurement imperfections. A major advantage of the proposed scheme is that it is capable of learning multi-scale features and global information, which contributes to the success of overcoming the effects of noisy and missing measurement values. Since the fully convolutional architecture allows the U-net model and its variants to adapt to various scales of input and output and avoid the problem of exponential growth of network parameters caused by fully connected networks, the proposed scheme can be extended to higher-dimensional quantum state tomography directly with considerable scalability. Theoretically, arbitrary convolutional neural networks with an encoder-decoder architecture can be used for quantum state tomography under the guidance of the proposed framework. Finally, comparisons with back-propagation network-based quantum state tomography and maximum likelihood quantum state tomography are carried out. Simulation results demonstrate the powerful capability of the proposed scheme to achieve high-fidelity quantum state tomography under imperfect measurement. [ABSTRACT FROM AUTHOR]

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 Schrödinger 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. Next, we investigate the accuracy of the model for polarization-entangled photon pairs. Entanglement detection and precision of state reconstruction are quantified by figures of merit and presented on graphs versus the amount of time uncertainty. [ABSTRACT FROM AUTHOR]

Extracting information from quantum devices has long been a crucial problem in the field of quantum mechanics. By performing elaborate measurements, quantum state tomography, an important and fundamental tool in quantum science and technology, can be used to determine unknown quantum states completely. In this study, methods to determine multi‐qubit pure quantum states uniquely and directly are explored. Two adaptive protocols are proposed, with their respective quantum circuits. Herein, two or three observables are sufficient, while the number of measurement outcomes are either the same as or fewer than those in existing methods. Additionally, experiments on the IBM 5‐qubit quantum computer, as well as numerical investigations, demonstrate the feasibility of the proposed protocols. [ABSTRACT FROM AUTHOR]

Practical entanglement distillation is a critical component in quantum information theory. Entanglement distillation is often utilized for designing quantum computer networks and quantum repeaters. The practical entanglement distillation problem is formulated as a bilevel optimization problem. A fuzzy formulation is introduced to estimate the quantum state (density matrix) from pseudo-likelihood functions (i.e., quantum state tomography). A scale-independent relationship between fuzzy relations in terms of the pseudo-likelihood functions is obtained. The entanglement distillation optimization problem is solved using the combined coupled map lattice and dual annealing approach. Comparative analysis of the results is then conducted against a standard dual annealing algorithmic implementation.

In this paper, we introduce a numerical framework for quantum tomography and entanglement quantification of three-qubit generalized Werner states. The scheme involves the single-qubit symmetric, informationally complete, positive operator-valued measure (SIC-POVM), which is then generalized to perform three-qubit measurements. We introduce random errors into the scheme by imposing the Poisson noise on the measured photon counts. The precision of state estimation is quantified and presented on graphs. As a special case, we compare the efficiency of the framework for the pure three-qubit generalized Werner state and the W state. The reconstructed states are presented graphically and discussed. [ABSTRACT FROM AUTHOR]

With the power to find the best fit to arbitrarily complicated symmetry, machine-learning (ML)-enhanced quantum state tomography (QST) has demonstrated its advantages in extracting complete information about the quantum states. Instead of using the reconstruction model in training a truncated density matrix, we develop a high-performance, lightweight, and easy-to-install supervised characteristic model by generating the target parameters directly. Such a characteristic model-based ML-QST can avoid the problem of dealing with a large Hilbert space, but cab keep feature extractions with high precision, capturing the underlying symmetry in data. With the experimentally measured data generated from the balanced homodyne detectors, we compare the degradation information about quantum noise squeezed states predicted by the reconstruction and characteristic models; both are in agreement with the empirically fitting curves obtained from the covariance method. Such a ML-QST with direct parameter estimations illustrates a crucial diagnostic toolbox for applications with squeezed states, from quantum information process, quantum metrology, advanced gravitational wave detectors, to macroscopic quantum state generation. [ABSTRACT FROM AUTHOR]

Quantum state tomography (QST) is the task of statistically constructing the density matrix of an unknown quantum state by measuring its several copies. The presence of noise in the QST setup can considerably degrade the fidelity between the constructed density matrix and the actual state. We consider a noisy QST setup with depolarizing noise and attempt to mitigate the effects of noise by quantum error mitigation (QEM). We compare the performance of different QEM methods with the same resources and find that the measurement error mitigation and zero noise extrapolation provide the best performance in terms of maximizing the fidelity between the state and its density matrix.

Reliable and efficient reconstruction of pure quantum states under the processing of noisy measurement data is a vital tool in fundamental and applied quantum information sciences owing to communication, sensing, and computing. Specifically, the purity of such reconstructed quantum systems is crucial in surpassing the classical shot-noise limit and achieving the Heisenberg limit, regarding the achievable precision in quantum sensing. However, the noisy reconstruction of such resourceful sensing probes limits the quantum advantage in precise quantum sensing. For this, we formulate a pure quantum state reconstruction method through eigenvalue decomposition. We show that the proposed method is robust against the depolarizing noise; it remains unaffected under high strength white noise and achieves quantum state reconstruction accuracy similar to the noiseless case. [ABSTRACT FROM AUTHOR]

We develop a method to quantify the superposition state of two different Laguerre–Gaussian modes. By analyzing the characteristics of the intensity distribution obtained in a single measurement, including the petal number, the position and value of the extremum intensity, one can quantify the angular momentum index, the radial node index and the superposition coefficient simultaneously. Experimentally, we measure a series of superposition states, whose angular momentum index ranges from −47 to 53, radial node index from 0 to 3 and superposition weight from 0.1 to 0.9. The average trace distance and the mean fidelity of these states are lower than 0.053 ± 0.001 and higher than 0.982 ± 0.002, respectively. Our method can further obtain the superposition coefficient compared with previous mode verification ones and can reduce the number of measurement settings compared with the traditional quantum state tomography, thus more applicable in practice. [ABSTRACT FROM AUTHOR]

We compare reconstructed quantum state images of a birefringent sample using direct quantum state tomography and inverse numerical optimization technique. Qubits are used to characterize birefringence in a flat transparent plastic sample by means of polarization sensitive measurement using density matrices of two-level quantum entangled photons. Pairs of entangled photons are generated in a type-II nonlinear crystal. About half of the generated photons interact with a birefringent sample, and coincidence counts are recorded. Coincidence rates of entangled photons are measured for a set of sixteen polarization states. Tomographic and inverse numerical techniques are used to reconstruct the density matrix, the degree of entanglement, and concurrence for each pixel of the investigated sample. An inverse numerical optimization technique is used to obtain a density matrix from measured coincidence counts with the maximum probability. Presented results highlight the experimental noise reduction, greater density matrix estimation, and overall image enhancement. The outcome of the entanglement distillation through projective measurements is a superposition of Bell states with different amplitudes. These changes are used to characterize the birefringence of a 3M tape. Well-defined concurrence and entanglement images of the birefringence are presented. Our results show that inverse numerical techniques improve overall image quality and detail resolution. The technique described in this work has many potential applications. [ABSTRACT FROM AUTHOR]

We propose an efficient scheme for qutrit state tomography based on optimal unambiguous discrimination of three nonorthogonal pure qutrit states. In our scheme, we first introduce an ancillary qubit initially prepared in a known state and apply a joint unitary transformation on the whole system composed of a qutrit state and an ancillary qubit state. Afterwards, we make a local von Neumann measurement (VNM) on the ancillary qubit. If it is projected onto the state |0〉, we finally perform mutually unbiased measurements (MUMs) on the collapsed qutrit state arbitrarily chosen from a complete set of four mutually unbiased bases (MUBs). With only six properly-chosen projectors, the real and imaginary parts of non-diagonal elements can be entirely obtained. Together with the diagonal elements obtained directly from one VNM in the computational bases, the density matrix of an arbitrary unknown qutrit state can be fully determined. Compared with the previous work (R. Salazar and A. Delgado 2012), our scheme consumes much less resources. In comparison with MUBs-based tomography scheme (H. Yuan, et al., 2016), our scheme requires more measurements and less projectors, but pays the price of a loss of certain fraction identically prepared states. Our proposal can be extended to the case of other d-dimensional quantum states if their d MUBs are exactly known.

Quantum state tomography (QST) is an important and challenging task in the field of quantum information, which has attracted a lot of attentions in recent years. Machine learning models can provide a classical representation of the quantum state after trained on the measurement outcomes, which are part of effective techniques to solve QST problem. In this work, we use a variational autoencoder (VAE) to learn the measurement distribution of two quantum states generated by MPS circuits. We first consider the Greenberger–Horne–Zeilinger (GHZ) state which can be generated by a simple MPS circuit. Simulation results show that a VAE can reconstruct 3- to 8-qubit GHZ states with a high fidelity, i.e., 0.99, and is robust to depolarizing noise. The minimum number ( N s ∗ ) of training samples required to reconstruct the GHZ state up to 0.99 fidelity scales approximately linearly with the number of qubits (N). However, for the quantum state generated by a complex MPS circuit, N s ∗ increases exponentially with N , especially for the quantum state with high entanglement entropy. [ABSTRACT FROM AUTHOR]

In this paper, we propose an online state tomography method for N -qubit quantum system based on the continuous weak measurement and compressed sensing (CS). The quantum system is described by stochastic master equation. The continuous weak measurement operators for the N -qubit quantum system, which are indirectly acted on the quantum system, are derived according to the measurement operator results of two-level quantum system. The online time-varying measurement operators are obtained by means of the dynamic evolution equation of the system. The quantum state is online estimated by solving the optimization problem of minimizing the two-norm with the positive definite constraints of density matrix, and we use the nonnegative least squares algorithm to solve the optimization problem. CS theory is used to reduce the number of the measurements in the process of online state estimation. In the numerical experiments, we study the effects of external control field, measurement rate and the different numbers of qubits on the performance in the proposed method. The minimum required numbers of measurements for 2, 3, 4 and 5 qubits are found. The normalized distance and fidelity of our proposed method can achieve satisfying accuracy with small number of measurements. [ABSTRACT FROM AUTHOR]

Quantum state tomography strategy via weak value is valid for arbitrary dimensional systems, and can be easily realized. However, the efficiency of quantum state tomography strategy via weak value is lower than that of the conventional quantum state tomography (QST), especially for high-dimensional systems. To improve the efficiency of the scheme, we propose a two-step adaptive quantum state tomography via weak value for pure states (AQSTW). In AQSTW, by using appropriate measurement strength, the efficiency of AQSTW is significantly larger than that of the conventional QST, especially for high-dimensional systems. And AQSTW can be easily implemented. These results demonstrate AQSTW may be useful in the measurements of the unknown quantum states.

The quantum state associated to an unknown experimental preparation procedure can be determined by performing quantum state tomography. If the statistical uncertainty in the data dominates over other experimental errors, then a tomographic reconstruction procedure must express this uncertainty. A rigorous way to accomplish this is via statistical confidence regions (CRs) in state space. Naturally, the size of this region decreases when increasing the number of samples, but it also depends critically on the construction method of the region. We compare recent methods for constructing CRs as well as a reference method based on a Gaussian approximation. For the comparison, we propose an operational measure with the finding, that there is a significant difference between methods, but which method is preferable can depend on the details of the state preparation scenario.

We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent (MiFGD), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, MiFGD converges provably close to the true density matrix at an accelerated linear rate asymptotically in the absence of experimental and statistical noise, under common assumptions. With this manuscript, we present the method, prove its convergence property and provide the Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real (noisy) experiments, performed on the IBM’s quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than the state-of-the-art approaches, with similar or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite the presence of experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems.

We generalize an approach to studying the quantum state tomography (QST) of open systems in terms of the dynamical map in Kraus representation within the framework of dynamic generation of informationally complete positive operator-valued measures. As applications, we use the generalized approach to theoretically study the QST of qubit systems in the presence of nonequilibrium environments which exhibit nonstationary and non-Markovian random telegraph noise statistical properties. We derive the time-dependent measurement operators for the quantum state reconstruction of the single qubit and two-qubit systems in terms of the polarization operator basis. It is shown that the behavior of the time-dependent measurement operators is closely associated with the dynamical map of the qubit systems.