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arXiv:1304.1866 · Physical Review A

Coarse-grained quantum state estimation for noisy measurements

Yong Siah Teo, Jaroslav Řeháček, Z. Hradil

Published Aug 14, 2013• 4 citations
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Abstract

We introduce a numerical coarse-graining scheme to reliably estimate the quantum state of a source given a set of noisy measurement outcomes that are not well calibrated. This scheme requires no additional knowledge or assumptions about the structures of the noisy outcomes and is, thus, applicable for very general tomography experiments with arbitrary noise perturbations. Further, this scheme involves an optimization procedure that is straightforward to implement and can readily be extended to any number of ill-calibrated noisy outcomes in a measurement setup. Compared with the standard estimation strategy where noise is neglected, coarse graining performed on simulated and experimental data for two-qubit quantum states and on simulated data from continuous-variable tomography shows that this scheme can significantly improve the accuracies of state estimators for noise levels ranging from low to moderately high values.

DOI
10.1103/physreva.88.022111

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We introduce a numerical coarse-graining scheme to reliably estimate the quantum state of a source given a set of noisy measurement outcomes that are not well calibrated. This scheme requires no additional knowledge or assumptions about the structures of the noisy outcomes and is, thus, applicable for very general tomography experiments with arbitrary noise perturbations. Further, this scheme involves an optimization procedure that is straightforward to implement and can readily be extended to any number of ill-calibrated noisy outcomes in a measurement setup. Compared with the standard estimation strategy where noise is neglected, coarse graining performed on simulated and experimental data for two-qubit quantum states and on simulated data from continuous-variable tomography shows that this scheme can significantly improve the accuracies of state estimators for noise levels ranging from low to moderately high values.
While quantum metrology enables measurement precision beyond classical limits, its performance is often susceptible to experimental imperfections. Most prior studies have focused on imperfections in quantum states and operations. Here, we investigate the effect of coarse graining in quantum measurement through both theoretical analysis and experimental demonstration. Using an interferometer with a squeezed vacuum and a laser input, we analyze how coarse graining in homodyne detection affects the precision of phase estimation. We evaluate the Fisher information under various coarse-graining conditions and determine, in each case, an optimal estimation strategy that saturates the Cramér-Rao bound. Remarkably, even extremely coarse-grained measurement -- with only two bins -- enables phase estimation beyond the standard quantum limit and even achieves a precision that follows the Heisenberg scaling. We experimentally demonstrate quantum-enhanced phase estimation under coarse-grained homodyne detection. To determine an optimal estimation strategy, we employ the method of moments and present calibration procedures that enable its application to general experimental settings. Using only two bins, we observe a quantum enhancement of 1.2 dB compared to the classical method using the ideal measurement, improving towards 3.8 dB as the bin number increases. These results highlight a practical pathway to achieving quantum enhancement under the presence of severe experimental imperfections.
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.
The reliable characterization of quantum states is a fundamental task in quantum information science. For this purpose, quantum state tomography provides a standard framework for reconstructing quantum states from measurement data, yet it is often degraded by experimental noise. Mitigating such noise is therefore essential for the accurate estimation of the states in realistic settings. In this work, we propose a scalable tomography method based on multilayer perceptron networks that mitigate unknown noise through supervised learning. This approach is data-driven and thus does not rely on explicit assumptions about the noise model or measurement, making it readily extendable to general quantum systems. Numerical simulations, ranging from special pure states to random mixed states, demonstrate that the proposed method effectively mitigates noise across a broad range of scenarios, compared with the case without mitigation.
In a multiqubit system, when a detector's resolution is insufficient to distinguish between qubits, individual qubit measurement becomes impractical. This limitation results in coarse-grained (CG) measurements that dilute system information and, more critically, hinder accurate assessment of qubit correlations, making quantum state tomography (QST) infeasible. To address this issue, we propose a scheme to utilize CG measurements for QST by integrating two-qubit gates and optimizing their corresponding positive operator-valued measures (POVMs) using parametrized quantum circuits. Specifically, we focus on constructing generalized symmetric informationally complete POVMs by maximizing the von Neumann entropy of the Gram matrix of these CG POVMs. Our numerical simulations demonstrate the effectiveness of this approach for QST using a two-qubit model and further verify the scalability of our scheme toward N -qubit systems.
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state and, therefore, have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions-a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.