经典BSC与量子PSC对偶性的半经典证明
A Semiclassical Proof of Duality Between the Classical BSC and the Quantum PSC
摘要：
在2018年，Renes [IEEE Trans. Inf. Theory，第64卷，第1期，第111页]。577-592（2018）]（arXiv：1701.05583）发展了经典输入量子输出（CQ）通道的通道对偶性的一般理论。该结果表明，一些著名的二元擦除信道上的线性码的对偶结果可以扩展到一般的经典信道，但代价是使用本质上是量子力学的对偶问题。这种二元性的一个特殊情况是纠错编码（分别为窃听保密性）和用于窃听保密性的编码（分别纠错）。虽然这一结果对经典编码有重要意义，但对于没有强大量子信息论背景的研究人员来说，一般对偶结果背后的机制是相当具有挑战性的。在这项工作中，我们利用先前的结果对PSC的线性码给出了一个替代推导上述特殊情况下，通过计算性能指标的封闭形式的表达式。指出以前的结果包括最优性的平方根测量（SRM）的线性码上的PSC和傅立叶对偶的线性码。我们还表明，SRM形成了一个次优的测量信道编码上的BSC（当解释为一个CQ问题）和秘密通信上的PSC。我们的证明只需要线性代数和基本群论，虽然为了方便我们使用量子狄拉克符号。
英文摘要：
In 2018, Renes [IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 577-592 (2018)] (arXiv:1701.05583) developed a general theory of channel duality for classical-input quantum-output (CQ) channels. That result showed that a number of well-known duality results for linear codes on the binary erasure channel could be extended to general classical channels at the expense of using dual problems which are intrinsically quantum mechanical. One special case of this duality is a connection between coding for error correction (resp. wire-tap secrecy) on the quantum pure-state channel (PSC) and coding for wire-tap secrecy (resp. error correction) on the classical binary symmetric channel (BSC). While this result has important implications for classical coding, the machinery behind the general duality result is rather challenging for researchers without a strong background in quantum information theory. In this work, we leverage prior results for linear codes on PSCs to give an alternate derivation of the aforementioned special case by computing closed-form expressions for the performance metrics. The noted prior results include optimality of the square-root measurement (SRM) for linear codes on the PSC and the Fourier duality of linear codes. We also show that the SRM forms a suboptimal measurement for channel coding on the BSC (when interpreted as a CQ problem) and secret communications on the PSC. Our proofs only require linear algebra and basic group theory, though we use the quantum Dirac notation for convenience.
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