Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators

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. First, 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. Second, 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. We give a new theoretical analysis of compressed tomography, based on the restricted isometry property (RIP) for low-rank matrices. Using these tools, we obtain near-optimal error bounds, for the realistic situation where the data contains 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 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 we describe a method for compressed quantum process tomography that works for processes with small Kraus rank.Comment: 16 pages, 3 figures. Matlab code included with the source file

Rank-based model selection for multiple ions quantum tomography

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) -- to models consising 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 4 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 4 ions experiment aimed at creating a Smolin state of rank 4. The two methods indicate that the optimal model for describing the data lies between ranks 6 and 9, and the Pearson $\chi^{2}$ test is applied to validate this conclusion. 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.Comment: 24 pages, 6 figures, 3 table