Rank-based model selection for multiple ions quantum tomography

Abstract

The statistical analysis of measurement data has become a key component of many quantum engineering experiments. As standard full state tomography becomes unfeasible for large dimensional quantum systems, one needs to exploit prior information and the 'sparsity' properties of the experimental state in order to reduce the dimensionality of the estimation problem. In this paper we propose model selection as a general principle for finding the simplest, or most parsimonious explanation of the data, by fitting different models and choosing the estimator with the best trade-off between likelihood fit and model complexity. We apply two well established model selection methods-the Akaike information criterion (AIC) and the Bayesian information criterion (BIC)-two models consisting of states of fixed rank and datasets such as are currently produced in multiple ions experiments. We test the performance of AIC and BIC on randomly chosen low rank states of four ions, and study the dependence of the selected rank with the number of measurement repetitions for one ion states. We then apply the methods to real data from a four ions experiment aimed at creating a Smolin state of rank 4. By applying the two methods together with the Pearson χ 2 test we conclude that the data can be 2 suitably described with a model whose rank is between 7 and 9. Additionally we find that the mean square error of the maximum likelihood estimator for pure states is close to that of the optimal over all possible measurements. Contents

Crystallizing highly-likely subspaces that contain an unknown quantum state of light

“…Recently, methods employed in classical statistical-model selection were used to localize the signal (see, for example, refs 23, 24, 25, 26, 27, 28). These methods involve the consideration of the popular Akaike criterion and the Bayesian information criterion to penalize the likelihood function for the problem and restrict models for up to a certain number of parameters.…”

Practical Bayesian tomography

“…In SMC, we need to be able to draw samples from a prior, see (11). In this Section we briefly review how to draw samples from several well-established priors [16].…”

“…Quantum tomography has seen many improvements since its inception [6]. In particular, tomography has enjoyed advances in providing maximum-likelihood estimators [7], region estimators [8][9][10], model selection [11,12], hedging [13], and compressed sensing [14,15].…”

Pseudo-Bayesian quantum tomography with rank-adaptation

“…The data was kindly provided by M. Guţȃ and T. Monz. It had been used in [1,26]. We apply two proposed estimators to the real data set of a system of 4 ions which is Smolin state further manipulated.…”

“…Rank-penalized maximum likelihood (BIC) was introduced in [26] while a rankpenalized least-square estimatorρ rank−pen was proposed in [1], together with a proof of its consistency. More specifically, when the density matrix of the system is ρ 0 with r = rank(ρ 0 ), the authors of [1] proved that the Frobenius norm of the estimation error satisfies ρ rank−pen − ρ 0 2 F = O(r4 n /N ) where N is the number of quantum measurements.…”