Recent experimental progress in the preparation and control of quantum systems has brought to light the importance of Quantum State Tomography (QST) in validating the results. In this thesis we investigate several aspects of QST, whose central problem is to devise estimation schemes for the recovery of an unknown state, given an ensemble of n independent identically prepared systems. The key issues in tackling QST for large dimensional systems is the construction of physically relevant low dimensional state models, and the design of appropriate measurements. Inspired by compressed sensing tomography, in chapters 4, 5 we consider the statistical problem of estimating low rank states (r ≪ d) in the set-up of Multiple Ions Tomography (MIT), where r and d are the rank and the dimension of the state respectively. We investigate how the estimation error behaves with a reduction in the number of measurement settings, compared to ‘full’ QST in two setups - Pauli and random bases measurement designs. We study the estimation errors in this ‘incomplete’ measurement setup in terms of a concentration of the Fisher information matrix. For the random bases design we demonstrate that O(r logd) settings suffice for the mean square error w.r.t the Frobenius norm to achieve the optimal O(1/n) rate of estimation. When the error functions are locally quadratic, like the Frobenius norm, then the expected error (or risk) of standard procedures achieves this optimal rate. However, for fidelity based errors such as the Bures distance we show that no ‘compressive’ recovery exists for states close to the boundary, and it is known that even with conventional ‘full’ tomography schemes the risk scales as O(1/√n) for such states and error functions. For qubit states this boundary is the surface of the Bloch sphere. Several estimators have been proposed to improve this scaling with ‘adaptive’ tomography. In chapter 6 we analyse this problem from the perspective of the maximum Bures risk over all qubit states. We propose two adaptive estimation strategies, one based on local measurements and another based on collective measurements utilising the results of quantum local asymptotic normality. We demonstrate a scaling of O(1/n) for the maximum Bures risk with both estimation strategies, and also discuss the construction of a minimax optimal estimator. In chapter 7 we return to the MIT setup and systematically compare several tomographic estimators in an extensive simulation study. We present and analyse results from this study, investigating the performance of the estimators across various states, measurement designs and error functions. Along with commonly used estimators like maximum likelihood, we propose and evaluate a few new ones. We finally introduce two web-based applications designed as tools for performing QST simulations online.
Recent experimental progress in the preparation and control of quantum systems has brought to light the importance of Quantum State Tomography (QST) in validating the results. In this thesis we investigate several aspects of QST, whose central problem is to devise estimation schemes for the recovery of an unknown state, given an ensemble of n independent identically prepared systems. The key issues in tackling QST for large dimensional systems is the construction of physically relevant low dimensional state models, and the design of appropriate measurements. Inspired by compressed sensing tomography, in chapters 4, 5 we consider the statistical problem of estimating low rank states (r ≪ d) in the set-up of Multiple Ions Tomography (MIT), where r and d are the rank and the dimension of the state respectively. We investigate how the estimation error behaves with a reduction in the number of measurement settings, compared to ‘full’ QST in two setups - Pauli and random bases measurement designs. We study the estimation errors in this ‘incomplete’ measurement setup in terms of a concentration of the Fisher information matrix. For the random bases design we demonstrate that O(r logd) settings suffice for the mean square error w.r.t the Frobenius norm to achieve the optimal O(1/n) rate of estimation. When the error functions are locally quadratic, like the Frobenius norm, then the expected error (or risk) of standard procedures achieves this optimal rate. However, for fidelity based errors such as the Bures distance we show that no ‘compressive’ recovery exists for states close to the boundary, and it is known that even with conventional ‘full’ tomography schemes the risk scales as O(1/√n) for such states and error functions. For qubit states this boundary is the surface of the Bloch sphere. Several estimators have been proposed to improve this scaling with ‘adaptive’ tomography. In chapter 6 we analyse this problem from the perspective of the maximum Bures risk over all qubit states. We propose two adaptive estimation strategies, one based on local measurements and another based on collective measurements utilising the results of quantum local asymptotic normality. We demonstrate a scaling of O(1/n) for the maximum Bures risk with both estimation strategies, and also discuss the construction of a minimax optimal estimator. In chapter 7 we return to the MIT setup and systematically compare several tomographic estimators in an extensive simulation study. We present and analyse results from this study, investigating the performance of the estimators across various states, measurement designs and error functions. Along with commonly used estimators like maximum likelihood, we propose and evaluate a few new ones. We finally introduce two web-based applications designed as tools for performing QST simulations online.
Abstract The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/ n . However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as 1 / n for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states. We propose qubit estimation strategies based on separate adaptive measurements, and collective measurements, that achieve 1/ n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us to derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate O ( n − 1 log n ) under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the ‘standard’ rate n −1 .
Abstract We consider the statistical problem of ‘compressive’ estimation of low rank states ( r ≪ d ) with random basis measurements, where r , d are the rank and dimension of the state respectively. We investigate whether for a fixed sample size N , the estimation error associated with a ‘compressive’ measurement setup is ‘close’ to that of the setting where a large number of bases are measured. We generalise and extend previous results, and show that the mean square error (MSE) associated with the Frobenius norm attains the optimal rate rd / N with only O ( r log d ) random basis measurements for all states. An important tool in the analysis is the concentration of the Fisher information matrix (FIM). We demonstrate that although a concentration of the MSE follows from a concentration of the FIM for most states, the FIM fails to concentrate for states with eigenvalues close to zero. We analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the MSE about its optimal despite a lack of concentration of the FIM for states close to the boundary of the Bloch sphere. We also consider the estimation error in terms of a different metric–the quantum infidelity. We show that a concentration in the mean infidelity (MINF) does not exist uniformly over all states, highlighting the importance of loss function choice. Specifically, we show that for states that are nearly pure, the MINF scales as 1 / N but the constant converges to zero as the number of settings is increased. This demonstrates a lack of ‘compressive’ recovery for nearly pure states in this metric.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random selection of settings, and that the number of settings can be significantly reduced with only a negligible increase in error. We present an argument to explain these findings based on a concentration inequality for the Fisher information matrix. In the more general setup of random basis measurements we use this argument to show that for certain rank r states it suffices to measure in bases to achieve the average Fisher information over all bases. We present numerical evidence for random states of up to eight atoms, which suggests that a similar behaviour holds in the case of Pauli bases measurements, for randomly chosen states. The relation to similar problems in compressed sensing is also discussed.
The estimation of high dimensional quantum states is an important statistical problem arising in current quantum technology applications. A key example is the tomography of multiple ions states, employed in the validation of state preparation in ion trap experiments (Häffner et al 2005 Nature 438 643 ). Since full tomography becomes unfeasible even for a small number of ions, there is a need to investigate lower dimensional statistical models which capture prior information about the state, and to devise estimation methods tailored to such models. In this paper we propose several new methods aimed at the efficient estimation of low rank states and analyse their performance for multiple ions tomography. All methods consist in first computing the least squares estimator, followed by its truncation to an appropriately chosen smaller rank. The latter is done by setting eigenvalues below a certain 'noise level' to zero, while keeping the rest unchanged, or normalizing them appropriately. We show that (up to logarithmic factors in the space dimension) the mean square error of the resulting estimators scales as where r is the rank, is the dimension of the Hilbert space, and N is the number of quantum samples. Furthermore we establish a lower bound for the asymptotic minimax risk which shows that the above scaling is optimal. The performance of the estimators is analysed in an extensive simulations study, with emphasis on the dependence on the state rank, and the number of measurement repetitions. We find that all estimators perform significantly better than the least squares, with the 'physical estimator' (which is a bona fide density matrix) slightly outperforming the other estimators.
Abstract As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators. The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of Guta et al (2018 (arXiv:1809.11162)) which point to strong optimality properties, and high computational efficiency of the projected linear estimators. To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online.