On the central limit theorem for unsharp quantum random variables

Dimić, Aleksandra; Dakić, Borivoje

doi:10.1088/1367-2630/aacd68

Cited by 4 publications

(4 citation statements)

References 35 publications

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“…The reduction of resources can be further traced down in the case where 𝛼(n) grows in n. In this case, for a sufficiently large system (large n) this number is reduced to the logical minimum leading to the single-copy detection. [23,48] This possibility is presented in detail in the next section.…”

Section: A Certain Binary Cost Function Of Settings and Outcomes Smentioning

confidence: 99%

Quantum Verification and Estimation with Few Copies

et al. 2022

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As quantum technologies advance, the ability to generate increasingly large quantum states has experienced rapid development. In this context, the verification and estimation of large entangled systems represent one of the main challenges in the employment of such systems for reliable quantum information processing. Though the most complete technique is undoubtedly full tomography, the inherent exponential increase of experimental and post‐processing resources with system size makes this approach infeasible even at moderate scales. For this reason, there is currently an urgent need to develop novel methods that surpass these limitations. This review article presents novel techniques focusing on a fixed number of resources (sampling complexity), and thus prove suitable for systems of arbitrary dimension. Specifically, a probabilistic framework requiring at best only a single copy for entanglement detection is reviewed, together with the concept of selective quantum state tomography, which enables the estimation of arbitrary elements of an unknown state with a number of copies that is low and independent of the system's size. These hyper‐efficient techniques define a dimensional demarcation for partial tomography and open a path for novel applications.

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Section: A Certain Binary Cost Function Of Settings and Outcomes Smentioning

confidence: 99%

Quantum Verification and Estimation with Few Copies

et al. 2022

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“…Considering it in the opposite direction: the confidence for entanglement detection grows exponentially fast in the number of repetitions N which constitutes what we dub the few-copy detection regime [24] where we achieve the high confidence detection by measuring only (thus the name) a few copies of the system (see Section 2 2.3). The reduction of resources can be further traced down in the case where α(n) grows in n. In this case, for a sufficiently large system (large n) this number is reduced to the logical minimum leading to the single-copy detection [23,47]. This possibility is presented in detail in the next section.…”

Section: If Smentioning

confidence: 99%

Quantum verification and estimation with few copies

Morris,

Saggio,

Gočanin

et al. 2021

Preprint

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As quantum technologies advance, the ability to generate increasingly large quantum states has experienced rapid development. In this context, the verification of large entangled systems represents one of the main challenges in the employment of such systems for reliable quantum information processing. Though the most complete technique is undoubtedly full tomography, the inherent exponential increase of experimental and postprocessing resources with system size makes this approach unfeasible even at moderate scales. For this reason, there is currently an urgent need to develop novel methods that surpass these limitations. This review article presents novel techniques focusing on a fixed number of resources (sampling complexity), and thus prove suitable for systems of arbitrary dimension. Specifically, a probabilistic framework requiring at best only a single copy for entanglement detection is reviewed, together with the concept of shadow and selective quantum state tomography, which enables the estimation of arbitrary elements of an unknown state with a number of copies that is independent of the system's size. These hyper-efficient techniques define a dimensional demarcation for partial tomography and open a path for novel applications.

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“…The first identity (75) follows along the lines of the second one ( 76) and so we focus on verifying the latter. Using the quantum Plancherel identity (26) and the relation (47), we apply the triangle inequality and split the integration domain into two disjoint sets such that…”

Section: Local-tail Decompositionmentioning

confidence: 99%

“…For a more detailed list of papers on noncommutative or quantum central limit theorems (QCLT), see for example [19,25] and references therein. A partially quantitative central limit theorem for unsharp measurements has been obtained in [26].…”

Section: Introductionmentioning

confidence: 99%

Convergence rates for the quantum central limit theorem

Becker,

Datta,

Lami

et al. 2019

Preprint

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A. Mathematical notation B. Definitions 1. Quantum information with continuous variables 2. Phase space formalism C. Moments D. Quantum convolution III. Cushen and Hudson's quantum central limit theorem IV. Main results A. Quantitative bounds in the QCLT B. Optimality of convergence rates and necessity of finite second moments in the QCLT C. Applications to capacity of cascades of beam splitters with non-Gaussian environment17 D. New results on quantum characteristic functions V. New results on quantum characteristic functions: Proofs A. Quantum-classical correspondence B. Decay estimates on the quantum characteristic function VI. Quantitative bounds in the QCLT: Proofs A. Preliminary steps 1. Williamson form 2. Local-Tail decomposition B. Proofs of convergence rates in Hilbert-Schmidt distance C. Convergence in trace distance and relative entropy VII. Optimality of convergence rates and necessity of finite second moments in the QCLT: Proofs A. Failure of convergence for states with unbounded energy B. Optimality of the convergence rates VIII. Cascade of beam splitters: Proofs A. Generalities of the cascade channels B. On the effective environment state C. Approximating cascade channels Acknowledgements A. Standard moments vs phase space moments: the integer case B. Standard moments vs phase space moments: the fractional case C. Standard moments vs phase space moments: a partial converse References 1 Throughout this paper we set " 1. 2 The characteristic function of a complex-valued random variable X is defined by χX pzq . ." E " e zX ˚´z ˚X ‰ .

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On the central limit theorem for unsharp quantum random variables

Cited by 4 publications

References 35 publications

Quantum Verification and Estimation with Few Copies

Quantum Verification and Estimation with Few Copies

Quantum verification and estimation with few copies

Convergence rates for the quantum central limit theorem

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