Local asymptotic normality for qubit states

We consider n identically prepared qubits and study the asymptotic properties of the joint state rho(circle times n). We show that for all individual states, rho situated in a local neighborhood of size 1 root n of a fixed state rho(0), the joint state converges to a displaced thermal equilibrium state of a quantum harmonic oscillator. The precise meaning of the convergence is that there exists physical transformations T-n (trace preserving quantum channels) which map the qubits states asymptotically close to their corresponding oscillator state, uniformly over all states in the local neighborhood. A few consequences of the main result are derived. We show that the optimal joint measurement in the Bayesian setup is also optimal within the point-wise approach. Moreover, this measurement converges to the heterodyne measurement which is the optimal joint measurement of position and momentum for the quantum oscillator. A problem of local state discrimination is solved using local asymptotic normality.