[论文解读] Strong converse bounds in quantum network information theory: distributed hypothesis testing and source coding

Hao–Chung Cheng, Nilanjana Datta+1 more|arXiv (Cornell University)|May 2, 2019

Wireless Communication Security Techniques参考文献 43被引用 10

一句话总结

该论文为量子网络信息任务建立了强有力的合流界限，特别是分布式量子假说检验和具有经典侧信息的源编码。证明了 Stein 指数由正则化的量子相对熵给出，在备择假设为乘积态且鲍勃无通信约束时，退化为单字母公式，利用了量子信息瓶颈特性及量子去极化半群的反向超收缩性。

ABSTRACT

We consider a distributed quantum hypothesis testing problem with communication constraints, in which the two hypotheses correspond to two different states of a bipartite quantum system, multiple identical copies of which are shared between Alice and Bob. They are allowed to perform local operations on their respective systems and send quantum information to Charlie at limited rates. By doing measurements on the systems that he receives, Charlie needs to infer which of the two different states the original bipartite state was in, that is, which of the two hypotheses is true. We prove that the Stein exponent for this problem is given by a regularized quantum relative entropy. The latter reduces to a single letter formula when the alternative hypothesis consists of the products of the marginals of the null hypothesis, and there is no rate constraint imposed on Bob. Our proof relies on certain properties of the so-called quantum information bottleneck function. The second part of this paper concerns the general problem of finding finite blocklength strong converse bounds in quantum network information theory. In the classical case, the analogue of this problem has been reformulated in terms of the so-called image size characterization problem. Here, we extend this problem to the classical-quantum setting and prove a second order strong converse bound for it. As a by-product, we obtain a similar bound for the Stein exponent for distributed hypothesis testing in the special case in which the bipartite system is a classical-quantum system, as well as for the task of quantum source coding with compressed classical side information. Our proofs use a recently developed tool from quantum functional inequalities, namely, the tensorization property of reverse hypercontractivity for the quantum depolarizing semigroup.

研究动机与目标

• 在通信约束下推导分布式量子假说检验的强合流界限。
• 将经典图像大小刻画推广至经典-量子设置，以获得二阶强合流界限。
• 为具有经典侧信息的量子源编码建立二阶强合流界限。
• 通过量子泛函不等式，将点对点量子信道结果推广至网络设置。
• 将多用户量子系统中的量子信息瓶颈函数与 Stein 指数联系起来。

提出的方法

• 利用量子去极化半群的反向超收缩性的张量化性质，推导有限块长界限。
• 应用量子相对熵的变分公式，将概率测度上的优化问题与算子表达式关联起来。
• 采用量子信息瓶颈函数，刻画分布式设置下信息压缩与保真度之间的权衡。
• 将强合流问题重新表述为经典-量子域中的图像大小刻画问题。
• 利用相对熵与迹不等式的对偶性，推导错误概率的界限。
• 通过输入分布与算子上的优化，将 Stein 指数简化为正则化的量子相对熵表达式。

实验结果

研究问题

• RQ1 在通信约束下，分布式量子假说检验中的 Stein 指数的确切形式是什么？
• RQ2 当备择假设在鲍勃无速率约束下为零假设各边缘的乘积时，强合流行为如何变化？
• RQ3 能否为具有经典侧信息的量子源编码建立二阶强合流界限？
• RQ4 量子信息瓶颈函数在多用户量子系统中，对分布式量子推断的基本极限刻画程度如何？
• RQ5 经典图像大小刻画在何种程度上可推广至经典-量子网络设置？

主要发现

• 分布式量子假说检验的 Stein 指数由正则化的量子相对熵给出，为错误衰减提供了紧致的渐近界。
• 当备择假设为零假设各边缘的乘积且鲍勃无通信速率时，Stein 指数退化为单字母公式。
• 为经典-量子图像大小问题建立了二阶强合流界限，意味着在可实现区域外错误概率以指数速度收敛至 1。
• 该方法为解码器处具有经典侧信息的量子源编码提供了二阶强合流界限。
• 证明依赖于量子去极化半群的反向超收缩性的张量化，从而实现有限块长分析。
• 使用了量子相对熵的变分公式以对偶化优化问题，将经典概率测度与量子算子联系起来。