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.
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.
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.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small ε, estimating an unknown state on Cd of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(ε2drmax{1,tr}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor t. Further, measuring order r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate.
For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
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.
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.
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 Compressed sensing (CS) has been verified that it offers a significant performance improvement for large quantum systems comparing with the conventional quantum tomography approaches, because it reduces the number of measurements from O ( d 2 ) to O ( rd log( d )) in particular for quantum states that are fairly pure. Yet few algorithms have been proposed for quantum state tomography using CS specifically, let alone basis analysis for various measurement sets in quantum CS. To fill this gap, in this paper an efficient and robust state reconstruction algorithm based on compressive sensing is developed. By leveraging the fixed point equation approach to avoid the matrix inverse operation, we propose a fixed-point alternating direction method algorithm for compressive quantum state estimation that can handle both normal errors and large outliers in the optimization process. In addition, properties of five practical measurement bases (including the Pauli basis) are analyzed in terms of their coherences and reconstruction performances, which provides theoretical instructions for the selection of measurement settings in the quantum state estimation. The numerical experiments show that the proposed algorithm has much less calculating time, higher reconstruction accuracy and is more robust to outlier noises than many existing state reconstruction algorithms.