Secure quantum network coding for controlled repeater networks

Shang, Tao; Li, Jiao; Liu, Jianwei

doi:10.1007/s11128-016-1323-y

Cited by 9 publications

(1 citation statement)

References 15 publications

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“…It used two pairs of Bell states shared in advance by the two senders to achieve perfect transmission of two qubits in the butterfly network. Since then, more and more scholars have started to focus on this area, and various QNC protocols [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] have been proposed. In 2009, Kobayashi et al [10] introduced classical communication into quantum network coding, and achieved lossless quantum state transmission with fidelity of 1 by simulating a classical linear network coding scheme.…”

Section: Introductionmentioning

confidence: 99%

School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|•| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|•| 2 ) < ∞ for some ε > 0. RésuméDes conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de super-Poincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|•| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et étend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilité à support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|•| 2 ) < ∞ pour quelque ε > 0.

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