Due to finite statistics, experimentally estimated density matrices are either not physical or biased, depending on the chosen reconstruction schemes; linear inversion of measurement results or Maximum-Likelihood estimations. We present in this work an alternative approach inspired by device-independent quantum information protocols. We embed a tomographic complete set of measurement operators within the framework of a Bell scenario such that statistical fluctuations are tamed by regularizing the corresponding input-output probability distributions. Numerical simulations based on entangled bipartite quantum states subjected to random noise show an improvement of the state estimation.
Due to finite statistics, experimentally estimated density matrices are either not physical or biased, depending on the chosen reconstruction schemes; linear inversion of measurement results or Maximum-Likelihood estimations. We present in this work an alternative approach inspired by device-independent quantum information protocols. We embed a tomographic complete set of measurement operators within the framework of a Bell scenario such that statistical fluctuations are tamed by regularizing the corresponding input-output probability distributions. Numerical simulations based on entangled bipartite quantum states subjected to random noise show an improvement of the state estimation.
In quantum state tomography, the estimated frequencies do not correspond directly to a physical quantum state, due to statistical fluctuations. Thus, one resorts to point estimators that return the state that matches observations the best, and a variety of estimators have been proposed -- linear inversion, least squares, maximum likelihood (ML) -- each making different trade-offs. In this short note, we investigate an alternative approach inspired by device-independent quantum information protocols. We embed a tomographic complete set of measurement operators within the framework of a Bell scenario, and first regularize the estimated frequencies using a device-independent maximum likelihood algorithm. We then run the standard maximum likelihood algorithm to estimate the underlying quantum state. In this preliminary investigation, we do not observe significant differences in the reconstructed state when using the new algorithm.
Tomography of a quantum state is usually based on a positive-operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows us to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside from the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multidimensional Laplace integrals appearing in Bayesian mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum nondemolition measurements relying on Rydberg atoms, and heterodyne fluorescence measurements of a superconducting qubit.
Quantum state tomography aims at reconstructing the state of a quantum system. The standard approach is based on a tomographically complete set of observables and a maximum likelihood estimation of the density matrix. However, the number of measurements scales exponentially with the number of qubits or qudits of the system. By leveraging some properties of any physical density matrix, we propose a method in which the number of necessary measurements depends on the state under investigation but without the need for any a priori information. In some cases, this can lead to a significant reduction of the resources necessary for state characterization.
In this paper, we propose a method for denoising experimental density matrices that combines standard quantum state tomography with an attention-based neural network architecture. The algorithm learns the noise from the data itself, without knowledge of its sources. Firstly, we show how the proposed protocol can improve the averaged fidelity of reconstruction over linear inversion and maximum likelihood estimation in the finite-statistics regime, reducing at least by an order of magnitude the amount of necessary training data. Next, we demonstrate its use for out-of-distribution data in realistic scenarios. In particular, we consider squeezed states of few spins in the presence of depolarizing noise and measurement/calibration errors and certify its metrologically useful entanglement content. The protocol introduced here targets experiments involving few degrees of freedom and afflicted by a significant amount of unspecified noise. These include NISQ devices and platforms such as trapped ions or photonic qudits. Published by the American Physical Society 2024
The device-independent approach to physics is one where conclusions are drawn directly and solely from the observed correlations between measurement outcomes. In quantum information, this approach allows one to make strong statements about the properties of the underlying devices via the observation of Bell-inequality-violating correlations. However, since one can only perform a finite number of experimental trials, an important gap remains between the many theoretical tools developed for such purposes and the experimentally obtained raw data. In particular, a sensible way to estimate the underlying quantum distribution has so far remained elusive. Here, we propose a few methods to bridge this gap. Under the assumption that the experimental trials are independent and identically distributed, our methods allow one to achieve the analog of quantum state estimation in the device-independent setting by generating unique point estimates of the true quantum distribution. In addition, we provide strong numerical evidence showing that these estimates converge toward the underlying distribution with a rate at least as good as the relative frequencies. We further demonstrate how these estimates of the true quantum distribution can be used to provide sensible estimates of certain desired properties of the system, such as the amount of entanglement present in the measured system.
Assumption-free reconstruction of quantum states from measurements is essential for benchmarking and certifying quantum devices, but it remains difficult due to the extensive measurement statistics and experimental resources it demands. An approach to alleviating these demands is provided by adaptive measurement strategies, which can yield up to a quadratic improvement in reconstruction accuracy for pure states by dynamically optimizing measurement settings during data acquisition. A key open question is whether these asymptotic advantages remain in realistic experiments, where readout is inevitably noisy. In this work, we analyze the impact of readout noise on adaptive quantum state tomography with readout-error mitigation, focusing on the challenging regime of reconstructing pure states using mixed-state estimators. Using analytical arguments based on Fisher information optimization and extensive numerical simulations using Bayesian inference, we show that any nonzero readout noise eliminates the asymptotic quadratic scaling advantage of adaptive strategies. We numerically investigate the behavior for finite measurement statistics for single- and two-qubit systems with exact readout-error mitigation and find a gradual transition from ideal to sub-optimal scaling. We furthermore investigate realistic scenarios where detector tomography is performed with a limited number of state copies for calibration, showing that insufficient detector characterization leads to estimator bias and limited reconstruction accuracy. Although our result imposes an upper bound on the reconstruction accuracy that can be achieved with adaptive strategies, we nevertheless observe numerically a constant-factor gain in reconstruction accuracy, which becomes larger as the readout noise decreases. This indicates potential practical benefits in using adaptive measurement strategies in well-calibrated experiments.