Southwest Quantum Information and Technology

Towards a model selection rule for quantum state tomography

Presenting Author: Travis Scholten, Blume-Kohout group (Sandia)

Quantum tomography on continuous variable systems poses a challenge: the density matrix comprises infinitely many parameters, but only finite data is available. Allowing all those parameters to vary will incorporate excessive noise, producing a poor estimate. Fortunately, model selection techniques can be used to fix (or exclude) some parameters. Model selection has been used in tomography to determine the best rank for an estimate, characterize sources of entanglement, and detect drift in state preparation. But these methods rely implicitly or explicitly on the Wilks Theorem, which predicts the behavior of the loglikelihood ratio statistic (LLRS) used to choose between models. Until now, it was not known whether the Wilks Theorem is accurate for quantum state tomography. We investigated the behavior of the LLRS using Monte Carlo simulations, and found that Wilks' prediction fails dramatically. Instead, the distribution of the LLRS is heavily distorted by boundaries (in state space and between models). We construct a model for the behavior of the LLRS, derive an almost analytic prediction for its mean value, and compare it to numerical experiments. The new model improves on existing methods (e.g. the Wilks Theorem), but is still imperfect. We conclude that LLRS-based model selection techniques like Akaike’s AIC may not be reliable for quantum tomography.

(Session 9b : Friday from 6:00 pm - 6:30 pm)