信息与量子边信息结合的界限
Bounds on Information Combining With Quantum Side Information
摘要：
“信息组合的界限”是熵不等式，它决定了一组随机变量的信息（熵）在以某些规定的方式组合时如何变化。这样的界限在经典信息论中起着重要的作用，特别是在编码和香农理论中;熵功率不等式是它们的特殊例子。可以说，最基本的信息组合是两个二进制随机变量（CNOT门）的相加，所得的量在置信传播和极化编码中起着重要作用。我们调查这个问题的设置中的量子边信息是可用的，这已被公认为一个硬设置的熵功率不等式。我们的主要技术结果是一个非平凡的，接近最佳的，下界的组合熵，这可以被看作是一个几乎最佳的“量子格伯夫人引理”。我们的证明使用了三个主要成分：（1）冯诺伊曼熵的一个新的界，它在低的两两态熵的情况下是紧的;（2）由于Fawzi-Renner的强次可加性的定量改进，其中我们设法处理恢复映射上的最小化;（3）Renes等人关于经典量子信道的最新对偶性结果，并进一步给出了在量子边信息下的最优上下界，得到了有趣的分析观察和强有力的数字证据的支持。最后，我们将我们的界应用于二进制输入经典量子信道的极化编码，并给出了以下三个结果：（A）偶非平稳信道在极化变换下的极化。(B)接近对称容量所需的块长度在容量间隙中最多以次指数方式缩放。(C)根据上述下限猜想，在间隙中的块长度多项式就足够了。
英文摘要：
"Bounds on information combining" are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in Belief propagation and Polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal "quantum Mrs. Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; (2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; (3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to Polar coding for binary-input classical-quantum channels, and show the following three results: (A) Even non-stationary channels polarize under the polar transform. (B) The blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity. (C) Under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
