Wednesday, September 26, 2012

1209.5737 (Cyril Stark)

Simultaneous Estimation of Dimension, States and Measurements: Gram
estimations
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Cyril Stark
We study a systematic procedure to find effective quantum models describing measurement data. Experiments (e.g., involving superconducting qubits) have shown that we do not always have a good understanding of how to model the measurements with positive operator valued measures (POVMs). It turns out that the ad hoc postulation of POVMs can lead to inconsistencies. For example when doing asymptotic state tomography via linear inversion, one sometime ends up with density matrices that are not positive semidefinite. We propose an alternative procedure where we do not make any a priori assumptions on the quantum model, i.e., on the Hilbert space dimension, the states or the POVMs. In this paper, we take the first steps along this program by estimating the Gram matrix associated with the states and the measurements. The Gram matrix specifies the Hilbert space dimension and determines all the states and the POVM elements, up to simultaneous rotations in the space of Hermitian matrices. We are guided by Occam's razor, i.e., we search for the minimal quantum model consistent with the data. In an upcoming paper we will show how the explicit valid density matrices and POVM elements can be found, using a heuristic algorithm that takes the state-measurement Gram matrix as input.
View original: http://arxiv.org/abs/1209.5737

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