61 research outputs found

    Optimal estimation of qubit states with continuous time measurements

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    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 nn identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size n1/2n^{-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 nn, the statistical model described by nn 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 ρ0\rho_{0}. An essential result is that the neighborhood radius can be chosen arbitrarily close to n1/4n^{-1/4}. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius n1/2+ϵn^{-1/2+\epsilon}, and then use LAN to perform optimal estimation.Comment: 32 pages, 3 figures, to appear in Commun. Math. Phy
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