Steven T. Flammia, David Gross, Yi-Kai Liu, Jens Eisert
Intuitively, if a density operator has only a few non-zero eigenvalues, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We exhibit two complementary ways of making this intuition precise. On the one hand, we show that the sample complexity decreases with the rank of the density operator. In other words, fewer copies of the state need to be prepared in order to estimate a low-rank density matrix. On the other hand---and maybe more surprisingly---we prove that unknown low-rank states may be reconstructed using an incomplete set of measurement settings. The method does not require any a priori assumptions about the unknown state, uses only simple Pauli measurements, and can be efficiently and unconditionally certified. Our results extend earlier work on compressed tomography, building on ideas from compressed sensing and matrix completion. Instrumental to the improved analysis are new error bounds for compressed tomography, based on the restricted isometry property (RIP) for low-rank matrices. These bounds are much stronger than those known previously, allowing us to show that compressed tomography essentially achieves the information-theoretic limit. Moreover, these bounds show that even for states which are only approximately low rank, we can still achieve the aforementioned improvements in the complexity. We then consider the numerical performance of various estimators and study their accuracy subject to a detailed accounting of the required experimental and computational resources. We exhibit methods which consistently outperform standard maximum likelihood estimation, achieving higher fidelity state reconstructions using faster processing and fewer measurement settings. Finally, we apply these methods to quantum process tomography, and use them to characterize quantum processes with small Kraus rank.
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http://arxiv.org/abs/1205.2300
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