  Quantum state tomography on a d-dimensional system demands resources that
grow rapidly with d. They may be reduced by using model selection to tailor the
number of parameters in the model (i.e., the size of the density matrix). Most
model selection methods typically rely on a test statistic and a null theory
that describes its behavior when two models are equally good. Here, we consider
the loglikelihood ratio. Because of the positivity constraint $\rho \geq 0$,
quantum state space does not generally satisfy local asymptotic normality,
meaning the classical null theory for the loglikelihood ratio (the Wilks
theorem) should not be used. Thus, understanding and quantifying how positivity
affects the null behavior of this test statistic is necessary for its use in
model selection for state tomography. We define a new generalization of local
asymptotic normality, metric-projected local asymptotic normality, show that
quantum state space satisfies it, and derive a replacement for the Wilks
theorem. In addition to enabling reliable model selection, our results shed
more light on the qualitative effects of the positivity constraint on state
tomography.
