Local asymptotic normality for qubit states

Abstract

We consider $n$ identically prepared qubits and study the asymptotic properties of the joint state ${\rho }^{\otimes n}$. We show that for all individual states, $\rho$ situated in a local neighborhood of size $1∕\sqrt{n}$ of a fixed state ${\rho }^{\mathbf{0}}$, 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.