Local asymptotic normality for quantum states.

Local asymptotic normality is an important result in mathematical statistics
which roughy states
the following: given N independent, identically distributed random variables
X(1), ..., X(N) whose
individual distributions depend on an unknown multidimensional parameter h
like P(h_0 + h /
sqrt{N}), then the
statistical information contained in the given sample is asymptotically equal
to that contained in
a
single normal variable centered at h and having a fixed variance: N(h, I^-1).

In quantum mechanics a similar result holds, with the random variables
replaced by independent
identically prepared quantum systems and the normal limit replaced by a so
called
Gaussian state of a quantum oscillator. Just like in the classical case,
results of this type are
very useful, for example by providing asymptotic lower bounds for the risk of
estimators of the
unknown parameters.

In a larger context, this is a first interesting example of convergence of
quantum statistical
experiments, an extension of the the classical theory to the world of quantum
mechanics.