[论文解读] Activated zero-error classical capacity of quantum channels in the presence of quantum no-signalling correlations.

Runyao Duan, Xin Wang|arXiv (Cornell University)|Oct 19, 2015

Quantum Computing Algorithms and Architecture被引用 4

一句话总结

本文引入了量子无信道相关辅助下的激活量子无信道性零误差经典容量，将其形式化为一个半定规划（SDP），并考虑了无噪声正向通信。研究表明，仅需1比特无噪声经典通信即可完全激活任意经典-量子信道以实现其渐近容量，并建立了与黄金分割比例及超密集编码极限相关的可加性与边界。

ABSTRACT

Recently the one-shot quantum no-signalling assisted zero-error classical capacity of a quantum channel has been formulated as a semidefinite programming (SDP) depending only on the Choi-Kraus operator space of the channel. In this paper, we study the extit{activated quantum no-signalling assisted zero-error classical capacity} by first allowing the assistance from some noiseless forward communication channel and later paying back the cost of the helper. We show that the one-shot activated capacity can also be formulated as a SDP and derive a number of striking properties of this number. In particular, this number is additive under direct sum, and is always greater than or equal to the super-dense coding bound. An a remarkable consequence, we find that one bit noiseless classical communication is able to fully activate any classical-quantum channel to achieve its asymptotic capacity, or the semidefinite fractional packing number. We also discuss the condition under which a noisy channel can activate an activatable channel. Interestingly, a channel is able to activate itself if its one-shot capacity is greater than or equal to the Golden Ratio - $(1+\sqrt{5})/2$. We also show that the asymptotic activated capacity is still equal to the usual no-signalling assisted capacity. Finally, we show that general the asymptotic no-signalling assisted zero-error capacity does not equal to the semidefinite (fractional) packing number by an explicit construction.

研究动机与目标

• 定义并形式化量子信道的激活量子无信道相关辅助零误差经典容量。
• 确定实现信道完全渐近容量所需的最小经典通信成本。
• 研究噪声信道激活可激活信道的条件。
• 将渐近激活容量与标准无信道相关辅助容量及半定分数打包数进行比较。
• 分析黄金分割比例作为信道自激活阈值的作用。

提出的方法

• 基于信道的Choi-Kraus算子空间，将单次激活零误差容量形式化为半定规划（SDP）问题。
• 引入两阶段协议：首先通过无噪声经典正向信道激活信道；其次通过反向通信偿还辅助者的成本。
• 利用SDP的对偶性与结构特性，证明在信道直和下激活容量的可加性。
• 通过将激活容量与超密集编码边界及半定分数打包数比较，推导出边界。
• 构造显式反例，表明渐近无信道相关辅助容量并不总等于半定分数打包数。
• 应用黄金分割比例阈值条件，基于单次容量确定信道的自激活能力。

实验结果

研究问题

• RQ1 单次激活零误差容量能否形式化为半定规划？
• RQ2 实现经典-量子信道完全渐近容量所需的最小无噪声经典通信量是多少？
• RQ3 在何种条件下，噪声信道可激活可激活信道？
• RQ4 渐近激活容量是否等于标准无信道相关辅助容量？
• RQ5 渐近无信道相关辅助零误差容量是否总等于半定分数打包数？

主要发现

• 单次激活零误差容量可形式化为仅依赖于信道Choi-Kraus算子空间的半定规划。
• 仅需1比特无噪声经典通信即可完全激活任意经典-量子信道，以实现其渐近容量，或等价于半定分数打包数。
• 在信道直和下，激活容量具有可加性。
• 激活容量始终大于或等于超密集编码边界。
• 若其单次激活容量至少为黄金分割比例 $(1+\bar{5})/2$，则信道可自激活。
• 渐近激活容量等于标准无信道相关辅助容量，但渐近无信道相关辅助零误差容量一般不等于半定分数打包数，如显式反例所示。