Abstract
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.
Highlights
As expected from the theoretical results, we find that all estimators perform significantly better than the LSE on low rank states; the physical estimator has slightly smaller estimation error than the others, including the oracle estimator
Since full quantum tomography becomes unfeasible for large dimensional systems, it is useful to identify lower dimensional models with good approximation properties for physically relevant states, and to develop estimation methods tailored for such models
In this work we analysed several estimation algorithms targeted at estimating low rank states in multiple ions tomography (MIT)
Summary
In this corrigendum to the paper Butucea et al Theoretical results describing the upper bound to the operator norm error of the least squares estimator. Proposition 2, theorem 1, corollary 1, and theorem 2 establish error rates for estimators obtained by normalising, penalising or thresholding the least square estimator. The proofs of these results use the operator norm error rate n ( ) as a generic expression, and are not affected by its concrete dependence on the number of atoms k. The upper bounds on the Frobenius square norm error in corollary 1 and theorem 2, will scale as r · nc (e)2 = r3k N rather than rd N = r2k N.
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