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CENTAUR: The system
 In Software Development Environments (SDE
, 1988
"... asymptotic normality for finite dimensional quantum ..."
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asymptotic normality for finite dimensional quantum
Local asymptotic normality in quantum statistics
, 2006
"... The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕn θ0+u / √ n consisting of joint states of n identically pr ..."
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The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕn θ0+u / √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all “local parameters ” u ∈ Rm such that θ = θ0 + u / √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ Rm. In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and show that weak and strong convergence are equivalent in the case of finite number of parameters for experiments based on type I algebras with discrete center. For reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory.
Optimal estimation of qubit states with continuous time measurements
, 2006
"... 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 norm one distance as well as other locally quadratic figures of merit. Local minimax optimality means that given n identical qubits, there exist ..."
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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 norm one 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 measurement 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 ρ0. 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+ǫ, and then use LAN to perform optimal estimation. 1 1
A Note on FisherHelstrom Information Inequality in Pure State Models
"... This paper concerns the design problem of choosing the measurement that provides the maximum Fisher information for the unknown parameter of a quantum system. We show that when the system under investigation is described by a oneparameter ndimensional pure state model an optimal measurement exists ..."
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This paper concerns the design problem of choosing the measurement that provides the maximum Fisher information for the unknown parameter of a quantum system. We show that when the system under investigation is described by a oneparameter ndimensional pure state model an optimal measurement exists, such that Fisher information attains the upper bound constituted by Helstrom information. A characterisation theorem and two strategies of implementations are derived and discussed. These results generalise to ndimensional spaces those obtained for n = 2 by BarndorffNielsen and Gill (2000). AMS (2000) subject classification. Primary 62B05; secondary 62F10.
Local asymptotic normality for finite . . .
, 2008
"... The previous results on local asymptotic normality (LAN) for qubits [20, 17] are extended to quantum systems of arbitrary finite dimension d. LAN means that the quantum statistical model consisting of n identically prepared ddimensional systems with joint state ρ⊗n converges as n → ∞ to a statisti ..."
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The previous results on local asymptotic normality (LAN) for qubits [20, 17] are extended to quantum systems of arbitrary finite dimension d. LAN means that the quantum statistical model consisting of n identically prepared ddimensional systems with joint state ρ⊗n converges as n → ∞ to a statistical model consisting of classical and quantum Gaussian variables with fixed and known covariance matrix, and unknown means related to the parameters of the density matrix ρ. Remarkably, the limit model splits into a product of a classical Gaussian with mean equal to the diagonal parameters, and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the offdiagonal elements. As in the qubits case [17], LAN is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown ddimensional quantum states. This measurement strategy shall be described in a forthcoming paper [19].