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Quantum Sphere-Packing Bounds With Polynomial Prefactors

机译：具有多项式预因子的量子球堆积边界

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摘要

We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(log n), indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.
机译：我们以低于容量的速率（通常称为量子球堆积界限）研究经典量子信道上经典编码中最佳错误概率的下限。温特（Winter）和达赖（Dalai）推导了经典量子信道的界线。但是，仅当通道为经典通道时，范围内的指数才重合。在本文中，我们证明了这两个指数均接受变分表示，并且与Golden-Thompson不等式相关，从而重申了达赖的表达在一般的经典量子通道中更强。其次，我们为经典量子通道建立了一个有限的块长球体填充边界，这极大地提高了达赖因数从次幂到多项式的阶次。此外，获得的恒定成分码误差指数与最著名的经典随机编码指数之间的差距以o（log n / n）的量级消失，这表明我们的球体堆积界限在高速率下几乎是精确的。最后，对于一类特殊的对称经典量子信道，我们无需完全恒定的组成码假设就可以完全表征其最佳误差概率。主要技术贡献是用于量子假设测试的两个逆霍夫定界和误差指数函数的鞍点性质。

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  • 来源
    《IEEE Transactions on Information Theory》 |2019年第5期|2872-2898|共27页
  • 作者

    Cheng Hao-Chung; Hsieh Min-Hsiu; Tomamichel Marco;

  • 作者单位

    Univ Technol Sydney, Fac Engn & Informat Technol, Ctr Quantum Software & Informat, Ultimo, NSW 2007, Australia|Natl Taiwan Univ, Grad Inst Commun Engn, Taipei 10617, Taiwan|Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge CB3 0WA, England;

    Univ Technol Sydney, Ctr Quantum Software & Informat, Ultimo, NSW 2007, Australia|Univ Technol Sydney, Fac Engn & Informat Technol, Ultimo, NSW 2007, Australia;

    Univ Technol Sydney, Ctr Quantum Software & Informat, Ultimo, NSW 2007, Australia|Univ Technol Sydney, Fac Engn & Informat Technol, Ultimo, NSW 2007, Australia;

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  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Error analysis; error probability; channel coding; information entropy; quantum mechanics;

    机译：误差分析;误差概率;信道编码;信息熵;量子力学;

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