When working with quantum states, analysis of the final quantum state generated through probabilistic measurements is essential. This analysis is typically conducted by constructing the density matrix from either partial or full tomography measurements of the quantum state. While full tomography measurement offers the most accurate reconstruction of the density matrix, limited measurements pose challenges for reconstruction algorithms, often resulting in nonphysical density matrices with negative eigenvalues. This is often remedied using maximum likelihood estimators, which have a high computing time or by other estimation methods that decrease the reconstructed fidelity. In this paper, we show that when restricting the measurement sample size, improvement over existing algorithms can be achieved. Our findings underline the multiplicity of solutions in the reconstruction problem, depending upon the generated state and measurement model utilized, thus motivating further research towards identifying optimal algorithms tailored to specific experimental contexts.
When working with quantum states, analysis of the final quantum state generated through probabilistic measurements is essential. This analysis is typically conducted by constructing the density matrix from either partial or full tomography measurements of the quantum state. While full tomography measurement offers the most accurate reconstruction of the density matrix, limited measurements pose challenges for reconstruction algorithms, often resulting in nonphysical density matrices with negative eigenvalues. This is often remedied using maximum likelihood estimators, which have a high computing time or by other estimation methods that decrease the reconstructed fidelity. In this paper, we show that when restricting the measurement sample size, improvement over existing algorithms can be achieved. Our findings underline the multiplicity of solutions in the reconstruction problem, depending upon the generated state and measurement model utilized, thus motivating further research towards identifying optimal algorithms tailored to specific experimental contexts.
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.
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.
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.
We present a numerical algorithm for the maximum-likelihood estimation of the density matrix, and apply it to the homodyne tomography of a single-mode radiation field. The algorithm is based on a specific form of the Gauss decomposition for positive definite Hermitian matrices. Results from Monte Carlo simulated experiments are presented. PACS: 42.50.Dv, 03.65.Bz Over past several years a great deal of interest has been paid to the reconstruction of the complete quantum state from experimental data [1]. Theoretical research in this field, apart from designing new measurement schemes, explores also the problem of extracting the optimal amount of information from data available in a realistic experimental setup. In this communication, we present a numerical algorithm for reconstructing the density matrix using the maximum-likelihood method [2, 3]. We discuss this algorithm in the context of quantum homodyne tomography [4, 5], which is actually the most successful method for quantum state reconstruction, and, in fact, the unique technique that has been so far experimentally implemented to measure nonclassical states of light [4, 6]. The realistic homodyne measurement of the quadrature is described by the positive operator-valued measure
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.