Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation
Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation
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量子率失真、逆香农定理和源通道分离
DOI：
10.1109/tit.2012.2215575
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发表时间：
2011
影响因子：
2.5
通讯作者：
M. Wilde
中科院分区：
文献类型：
--
作者：
N. Datta;Min;M. Wilde
我们推导了经典信息论中两个关键定理的量子对应定理，即率失真定理和信源信道分离定理。率失真定理给出了有损数据压缩的最终限制，并且源-信道分离定理意味着由压缩和信道编码组成的两阶段协议对于在无记忆信道上传输无记忆源是最佳的。尽管它们在经典领域很重要，但在量子信息理论中，这些领域的工作却少得令人惊讶。在本文中，我们证明了量子率失真函数是根据净化的正则化纠缠给出的。我们还确定了一个单字母表示的纠缠辅助量子率失真函数，我们证明了它作为一个下界的无辅助量子率失真函数。这意味着无辅助的量子率失真函数是非负的，并且通常不等于源和失真输出之间的相干信息（尽管巴纳姆猜想相干信息在这里是相关的）。此外，我们证明了几个量子源-通道分离定理。其中最强的是在纠缠辅助设置，在其中我们建立了一个必要和充分条件，通过一个无记忆的量子信道传输一个无记忆的源到给定的失真。
We derive quantum counterparts of two key theorems of classical information theory, namely, the rate-distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In this paper, we prove that the quantum rate-distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate-distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate-distortion function. This implies that the unassisted quantum rate-distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). Moreover, we prove several quantum source-channel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.