  We propose an adaptive, two steps strategy, for the estimation of mixed qubit
states. We show that the strategy is optimal in a local minimax sense for the
trace norm distance as well as other locally quadratic figures of merit. Local
minimax optimality means that given $n$ identical qubits, there exists no
estimator which can perform better than the proposed estimator on a
neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is
asymptotically Bayesian optimal for a large class of prior distributions.
  We present a physical implementation of the optimal estimation strategy based
on continuous time measurements in a field that couples with the qubits.
  The crucial ingredient of the result is the concept of local asymptotic
normality (or LAN) for qubits. This means that, for large $n$, the statistical
model described by $n$ identically prepared qubits is locally equivalent to a
model with only a classical Gaussian distribution and a Gaussian state of a
quantum harmonic oscillator.
  The term `local' refers to a shrinking neighborhood around a fixed state
$\rho_{0}$. An essential result is that the neighborhood radius can be chosen
arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by
which we first localize the state within a smaller neighborhood of radius
$n^{-1/2+\epsilon}$, and then use LAN to perform optimal estimation.
