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 O r d log () 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.

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 [36] 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.

We consider the statistical problem of 'compressive' estimation of low rank states with random basis measurements, where the estimation error is expressed terms of two metrics -the Frobenius norm and quantum infidelity. It is known that unlike the case of general full state tomography, low rank states can be identified from a reduced number of observables' expectations. Here we investigate whether for a fixed sample size N , the estimation error associated to a 'compressive' measurement setup is 'close' to that of the setting where a large number of bases are measured.In terms of the Frobenius norm, we demonstrate that for all states the error attains the optimal rate rd/N with only O(r log d) random basis measurements. We provide an illustrative example of a single qubit and demonstrate a concentration in the Frobenius error about its optimal for all qubit states. In terms of the quantum infidelity, we show that such a concentration does not exist uniformly over all states. Specifically, we show that for states that are nearly pure and close to the surface of the Bloch sphere, the mean infidelity 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. arXiv:1609.03758v1 [quant-ph]

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 and adaptive 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 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 . arXiv:1708.04941v2 [quant-ph]