Local Asymptotic Normality for Finite Dimensional Quantum Systems

Kahn, Jonas; Guţǎ, Mădălin

doi:10.1007/s00220-009-0787-3

Cited by 106 publications

(177 citation statements)

References 41 publications

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“…Guţȃ and Kahn [11] introduced a different tool based on (strong) quantum local asymptotic normality to prove the qubit case. This was further generalized to full models on any finite dimensional Hilbert space [12]. However, all these proofs depend on a specific parametrization of quantum states.…”

Section: B the Holevo Boundmentioning

confidence: 99%

“…Since one cannot do better than the best collective measurement, the ultimate precision bound is the one that is asymptotically achieved by a sequence of the best collective measurements as the number of copies tends to infinity. This fundamental question has been addressed by several authors before [2][3][4][5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem

Suzuki

2016

Journal of Mathematical Physics

101

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The main contribution of this paper is to derive an explicit expression for the fundamental precision bound, the Holevo bound, for estimating any two-parameter family of qubit mixed-states in terms of quantum versions of Fisher information. The obtained formula depends solely on the symmetric logarithmic derivative (SLD), the right logarithmic derivative (RLD) Fisher information, and a given weight matrix. This result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models; the Holevo bound coincides with the SLD Cramér-Rao bound and it does with the RLD Cramér-Rao bound. One of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property: The structure of the Holevo bound changes smoothly when the weight matrix varies. In particular, it always coincides with the RLD Cramér-Rao bound for a certain choice of the weight matrix. Several examples illustrate these findings.

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Section: B the Holevo Boundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem

Suzuki

2016

Journal of Mathematical Physics

101

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show abstract

“…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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“…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%

“…The qubit case is described in detail in Section III and the precise result is formulated in Theorem III.4. The LAN theory has been used to find asymptotically optimal estimation procedures for qubits [20] and qudits [21] and to show that the Holevo bound for state estimation is achievable [24]. Here we use it to solve the benchmark problem for qubits by casting it into the corresponding one for displaced thermal states.…”

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 for Finite Dimensional Quantum Systems

Cited by 106 publications

References 41 publications

Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem

Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem

Multiple phase estimation in quantum cloning machines

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

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