Local Asymptotic Normality in Quantum Statistics

Guţǎ, Mădălin; Jenčová, Anna

doi:10.1007/s00220-007-0340-1

Cited by 55 publications

(80 citation statements)

References 55 publications

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“…For mixed states, the situation is more complicated. However, recent research by Guţȃ and Kahn indicates that the QCRB is asymptotically attainable if and only if [58][59][60] Tr…”

Section: Technical Preliminaries Of Qfimmentioning

confidence: 99%

Multiple phase estimation in quantum cloning machines

Yao

Xing

et al. 2014

Phys. Rev. A

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Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized d-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cramér-Rao bound is also explicitly discussed.

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“…For mixed states, the situation is more complicated. However, recent research by Guţȃ and Kahn indicates that the QCRB is asymptotically attainable if and only if [58][59][60] Tr…”

Section: Technical Preliminaries Of Qfimmentioning

confidence: 99%

Multiple phase estimation in quantum cloning machines

Yao

Xing

et al. 2014

Phys. Rev. A

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“…In this paper we give an asymptotically optimal measurement strategy for qubit states that is based on the technique of local asymptotic normality introduced in [22,23]. The technique is a quantum generalisation of Le Cam's classical statistical result [40], and builds on previous work of Hayashi and Matsumoto [25,29].…”

Section: Introductionmentioning

confidence: 99%

“…We then exploit this result to prove optimality of the second stage. A different approach to local asymptotic normality has been developed in [23] to which we refer for a more general exposition on the theory of quantum statistical models. A short discussion on the relation between the two approaches is given in the remark following Theorem 3.1.…”

Section: Introductionmentioning

confidence: 99%

Optimal Estimation of Qubit States with Continuous Time Measurements

Guţǎ

Janssens

Kahn

2007

Commun. Math. Phys.

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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 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 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 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 ρ 0 . 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.

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“…The existing results on quantum sufficiency [34] and quantum LAN [21,22] indicate the existence of a theory of quantum decision and convergence of models.…”

Section: Strong Convergence To Gaussian Shift Modelmentioning

confidence: 99%

“…The key tool in deriving our results is the theory of local asymptotic normality (LAN) for quantum states [19][20][21][22] which is the quantum extension of a fundamental concept in mathematical statistics introduced by Le Cam [23]. In the classical context this roughly means that a large i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Quantum-teleportation benchmarks for independent and identically distributed spin states and displaced thermal states

2010

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A successful state transfer (or teleportation) experiment must perform better than the benchmark set by the 'best' measure and prepare procedure. We consider the benchmark problem for the following families of states: (i) displaced thermal equilibrium states of given temperature; (ii) independent identically prepared qubits with completely unknown state. For the first family we show that the optimal procedure is heterodyne measurement followed by the preparation of a coherent state. This procedure was known to be optimal for coherent states and for squeezed states with the 'overlap fidelity' as figure of merit. Here we prove its optimality with respect to the trace norm distance and supremum risk. For the second problem we consider n i.i.d. spin-1 2 systems in an arbitrary unknown state ρ and look for the measurement-preparation pair (Mn, Pn) for which the reconstructed state ωn := Pn • Mn(ρ ⊗n ) is as close as possible to the input state, i.e. ωn − ρ ⊗n 1 is small. The figure of merit is based on the trace norm distance between input and output states. We show that asymptotically with n the this problem is equivalent to the first one. The proof and construction of (Mn, Pn) uses the theory of local asymptotic normality developed for state estimation which shows that i.i.d. quantum models can be approximated in a strong sense by quantum Gaussian models. The measurement part is identical with 'optimal estimation', showing that 'benchmarking' and estimation are closely related problems in the asymptotic set-up.

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Local Asymptotic Normality in Quantum Statistics

Cited by 55 publications

References 55 publications

Multiple phase estimation in quantum cloning machines

Multiple phase estimation in quantum cloning machines

Optimal Estimation of Qubit States with Continuous Time Measurements

Quantum-teleportation benchmarks for independent and identically distributed spin states and displaced thermal states

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