基于消息压缩的通信复杂性的直和定理
A direct sum theorem in communication complexity via message compression
摘要：
我们证明了两方有界误差随机多轮通信协议的直和问题的下界。我们的证明使用的概念的信息成本的协议，定义的Chakrabarti，施，Wirth和姚和进一步完善的酒吧，Yossef，Jayram，Kumar和Sivakumar。我们的主要技术结果是一个“压缩”定理，即对于输入上的任何概率分布$μ$，具有信息成本$c$的函数$f$的$k$轮私有硬币有界错误协议可以转换为具有有界分布错误和通信成本$O(kc)$的$k$轮确定性协议。我们证明了这一结果，使用相对熵和拒绝抽样参数的子状态定理。我们的直接和结果如下，从这个'压缩'的结果，通过基本的信息理论参数。我们还考虑了量子通信中的直和问题。使用概率参数，我们表明，消息不能以这种方式压缩，即使它们携带的信息很小。因此，可能需要新技术来解决量子通信中的直和问题。
英文摘要：
We prove lower bounds for the direct sum problem for two-party bounded error randomised multiple-round communication protocols. Our proofs use the notion of information cost of a protocol, as defined by Chakrabarti, Shi, Wirth and Yao and refined further by Bar-Yossef, Jayram, Kumar and Sivakumar. Our main technical result is a `compression' theorem saying that, for any probability distribution $μ$ over the inputs, a $k$-round private coin bounded error protocol for a function $f$ with information cost $c$ can be converted into a $k$-round deterministic protocol for $f$ with bounded distributional error and communication cost $O(kc)$. We prove this result using a substate theorem about relative entropy and a rejection sampling argument. Our direct sum result follows from this `compression' result via elementary information theoretic arguments. We also consider the direct sum problem in quantum communication. Using a probabilistic argument, we show that messages cannot be compressed in this manner even if they carry small information. Hence, new techniques may be necessary to tackle the direct sum problem in quantum communication.
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