Local asymptotic normality for qubit states

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We consider n identically prepared qubits and study the asymptotic properties of the joint state ρ⊗ ⁿ. We show that for all individual states ρsituated in a local neighborhood of size 1/√n of a fixed state ρ⁰, the joint state converges to a displaced thermal equilibrium state of a quantum harmonic oscillator. The precise meaning of the convergence is that there exist physical transformations Tₙ (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 set-up is also optimal within the pointwise 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.