The estimation of all the parameters in an un-known quantum state or measurement device, known as quantum state tomography (QST) and quantum detector tomography (QDT), is fundamental to the characterization and control of quantum systems. This paper presents a unified state space framework for QST and QDT based on observable measurement results at different evolution moments, applicable to both closed and Markovian open quantum systems. We derive lower bounds on the number of sampling points required for a unique estimation and analyze the computational complexity and mean squared error (MSE) scaling. As the estimator may yield unphysical results, we introduce new correction techniques to enforce physicality, which may also improve the infidelity scaling. The effectiveness of the proposed methods is validated through numerical simulations.
The estimation of all the parameters in an un-known quantum state or measurement device, known as quantum state tomography (QST) and quantum detector tomography (QDT), is fundamental to the characterization and control of quantum systems. This paper presents a unified state space framework for QST and QDT based on observable measurement results at different evolution moments, applicable to both closed and Markovian open quantum systems. We derive lower bounds on the number of sampling points required for a unique estimation and analyze the computational complexity and mean squared error (MSE) scaling. As the estimator may yield unphysical results, we introduce new correction techniques to enforce physicality, which may also improve the infidelity scaling. The effectiveness of the proposed methods is validated through numerical simulations.
The estimation of all the parameters in an unknown quantum state or measurement device, commonly known as quantum state tomography (QST) and quantum detector tomography (QDT), is crucial for comprehensively characterizing and controlling quantum systems. In this paper, we introduce a framework, in two different bases, that utilizes multiple quantum processes to simultaneously identify a quantum state and a detector. We develop a closed-form algorithm for this purpose and prove that the mean squared error (MSE) scales asO(1/N)for both QST and QDT, whereNdenotes the total number of state copies. This scaling aligns with established patterns observed in previous works that addressed QST and QDT as independent tasks. Furthermore, we formulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, where the physical constraints of the state and detector are characterized by polynomial equalities and inequalities. The effectiveness of our proposed methods is validated through numerical examples.
The estimation of all the parameters in an unknown quantum state or measurement device, commonly known as quantum state tomography (QST) and quantum detector tomography (QDT), is crucial for comprehensively characterizing and controlling quantum systems. In this paper, we introduce a framework, in two different bases, that utilizes multiple quantum processes to simultaneously identify a quantum state and a detector. We develop a closed-form algorithm for this purpose and prove that the mean squared error (MSE) scales as $O(1/N) $ for both QST and QDT, where $N $ denotes the total number of state copies. This scaling aligns with established patterns observed in previous works that addressed QST and QDT as independent tasks. Furthermore, we formulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, where the physical constraints of the state and detector are characterized by polynomial equalities and inequalities. The effectiveness of our proposed methods is validated through numerical examples.
Quantum state tomography (QST) has attracted considerable attention due to its fundamental role in quantum information processing. In this paper, we develop a unified theoretical framework for analyzing the sample complexity of structured QST under noisy observations arising from state preparation noise, measurement noise, and finite-shot statistical noise. The proposed framework applies to a broad family of structured quantum-state classes, including general mixed states, sparse states, low-rank states, matrix product states (MPSs), matrix product operators (MPOs), projected entangled-pair states (PEPSs), and projected entangled-pair operators (PEPOs), while further introducing physically consistent structured models---including low-rank and sparse states, low-rank MPOs (LR-MPOs), and low-rank PEPOs (LR-PEPOs)---that simultaneously exploit low-dimensional structures and preserve the physical constraint. Within this framework, we derive unified non-asymptotic sample complexity guarantees for two constrained least-squares estimators under noisy observations: a noise-aware estimator that incorporates the calibrated noise model and a noise-unaware estimator based on the ideal Born measurement model. For the noise-aware estimator, we derive unified trace-norm recovery guarantees that explicitly characterize the dependence of the sample complexity on three fundamental quantities: the complexity of the underlying structured state class, the complexity of the measurement ensemble, and the state preparation and measurement noise levels. For the noise-unaware estimator, we establish a unified non-asymptotic recovery guarantee consisting of a statistical error term and an additional deterministic bias term arising from the mismatch between the assumed reconstruction model and the noisy observation process.
The estimation of all the parameters in an unknown quantum state/measurement device, commonly referred to as quantum state/detector tomography (QST/QDT), plays a pivotal role in characterization and control of quantum systems. In this paper, we introduce a novel method to concurrently identify a quantum state and a detector using multiple Hamiltonians. We then develop two solutions, i.e., a closed-form algorithm and the sum of squares (SOS) optimization. The numerical examples demonstrate that the closed-form solution offers computational efficiency, while the SOS optimization method ensures accuracy.
Quantum state tomography (QST) is a technique for characterizing, benchmarking, and verifying quantum systems and devices. In this paper, we focus on collective QST using data from collective measurements performed on multiple copies of the state. We propose a closed-form solution and provide an analytical characterization of its computational complexity and mean squared error (MSE) scaling. Additionally, we reformulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, enabling the application of SOS tools for its solution. The effectiveness of the proposed methods is demonstrated through numerical simulations. Furthermore, we validate the algorithms using two-copy collective experimental data, where the entangled measurement provides information about the state purity. Compared to previous methods, our algorithms achieve lower MSEs and approach the collective MSE bound by leveraging this purity information more efficiently.
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance estimation precision, we also propose a two-step adaptive detector tomography algorithm and investigate how this adaptive strategy can be used to achieve efficient estimation of quantum detectors. Moreover, the superposition of coherent states are used as probe states for quantum detector tomography and the estimation error is analyzed. Numerical results demonstrate the effectiveness of both the proposed optimal and adaptive quantum detector tomography methods.