[论文解读] Lower bound on expected communication cost of quantum Huffman coding

Anurag Anshu, Ankit Garg+2 more|arXiv (Cornell University)|May 15, 2016

Quantum Computing Algorithms and Architecture被引用 1

一句话总结

本文建立了量子霍夫曼编码预期通信成本的下界，证明即使在交互式通信下，任何单次方案也无法实现由冯·诺伊曼熵给出的渐近量子压缩速率。该结果揭示了经典与量子单次压缩之间存在根本性差距，对量子通信复杂性和单次量子逆向香农定理具有影响。

ABSTRACT

Data compression is a fundamental problem in quantum and classical information theory. A typical version of the problem is that the sender Alice receives a (classical or quantum) state from some known ensemble and needs to transmit them to the receiver Bob with average error below some specified bound. We consider the case in which the message can have a variable length and the goal is to minimize its expected length. For classical messages this problem has a well-known solution given by Huffman coding. In this scheme, the expected length of the message is equal to the Shannon entropy of the source (with a constant additive factor) and the scheme succeeds with zero error. This is a single-shot result which implies the asymptotic result, viz. Shannon's source coding theorem, by encoding each state sequentially. For the quantum case, the asymptotic compression rate is given by the von-Neumann entropy. However, we show that there is no one-shot scheme which is able to match this rate, even if interactive communication is allowed. This is a relatively rare case in quantum information theory when the cost of a quantum task is significantly different than the classical analogue. Our result has implications for direct sum theorems in quantum communication complexity and one-shot formulations of Quantum Reverse Shannon theorem.

研究动机与目标

• 研究单次量子压缩方案是否能够实现由冯·诺伊曼熵给出的渐近量子压缩速率。
• 确定交互式通信是否能够弥合经典与量子单次压缩性能之间的差距。
• 建立变量长度编码方案中量子通信成本的根本限制。
• 探讨其对量子通信复杂性中直接和定理的影响。
• 在压缩背景下，检验单次形式的量子逆向香农定理的有效性。

提出的方法

• 分析在具有有界平均误差的变长编码下，量子霍夫曼编码的预期通信成本。
• 将单次量子压缩成本与由冯·诺伊曼熵给出的渐近速率进行比较。
• 使用信息论技术，推导出单次情形下预期通信成本的下界。
• 考虑交互式通信协议对压缩成本的影响，表明其无法弥合差距。
• 应用量子信息理论结果，证明在单次设置下无法匹配渐近速率的不可能性。
• 依赖于量子态和系综的结构性质，证明该下界。

实验结果

研究问题

• RQ1 单次量子霍夫曼编码能否实现由冯·诺伊曼熵定义的渐近压缩速率？
• RQ2 交互式通信是否能使量子单次方案匹配渐近压缩速率？
• RQ3 量子变长编码的预期通信成本的根本下界是什么？
• RQ4 在熵匹配方面，单次量子压缩成本与经典霍夫曼编码相比如何？
• RQ5 该下界对量子通信复杂性中直接和定理有何影响？

主要发现

• 不存在任何单次量子编码方案能够实现由冯·诺伊曼熵给出的渐近压缩速率。
• 即使在交互式通信下，量子单次编码的预期通信成本也无法匹配渐近速率。
• 预期通信成本的下界严格大于冯·诺伊曼熵，表明经典与量子单次压缩之间存在根本性差距。
• 该结果意味着经典单次霍夫曼编码的优势无法延伸至量子领域。
• 该发现挑战了能够匹配渐近速率的单次形式量子逆向香农定理的可能性。
• 该结果凸显了量子信息理论中一个罕见案例：量子任务成本显著超过其经典对应物。