[论文解读] Distinguishability and Accessible Information in Quantum Theory

Christopher A. Fuchs|ArXiv.org|Jan 23, 1996

Quantum Information and Cryptography参考文献 456被引用 204

一句话总结

本论文建立了一个严格的信息论框架，用于使用三种关键度量——保真度、Kullback-Leibler散度和互信息——量化非正交量子态的可区分性。该研究推导出这些量的精确表达式及最优测量方法，主要贡献包括对Holevo关于量子互信息上界证明的简化版本，以及一种通过非线性矩阵方程生成越来越紧致下界的新方法。

ABSTRACT

This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. These measures should have important appli- cations in quantum cryptography and quantum computation theory. The results reported include the following. An exact expression for the quantum fidelity between two mixed states is derived. The optimal measurement that gives rise to it is studied in detail. Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. Several upper and lower bounds to the quan- tum Kullback relative information are derived. The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. This is tackled in two ways. The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. This includes the possibility of correlation or quantum entanglement between the systems. This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states.

研究动机与目标

• 以经典信息论度量作为基础，形式化非正交量子态的操作可区分性。
• 解决非正交量子态无法完美区分的根本问题，从而需要采用统计度量来衡量可区分性。
• 通过在量子测量上进行优化，推导出经典可区分度量——保真度、Kullback-Leibler散度和互信息——的量子版本。
• 建立一个将量子测量中的推理与扰动联系起来的形式化体系，最终证明未知量子态的不可广播定理。
• 提供量子可区分度量的精确表达式与紧致界，应用于量子密码学、计算与通信领域。

提出的方法

• 以经典信息论度量（保真度、Kullback-Leibler散度、互信息）为起点，定义量子可区分性。
• 通过对正算子值测度（POVMs）进行优化，将‘量子可区分性’定义为每种度量的最优值。
• 采用先进的矩阵分析技术，包括求解非线性矩阵方程，推导出量子Kullback信息的逐步紧致下界。
• 利用迹不等式与谱论推导量子互信息的界，包括Holevo于1973年提出的上界的简化推导。
• 应用纯化映射与共轭线性型技术，分析量子态可区分性与测量扰动。
• 发展一种代数形式体系，表达量子测量中推理（信息获取）与扰动（态扰动）之间的权衡。

实验结果

研究问题

• RQ1 经典可区分度量如何被调整以量化非正交量子态的可区分性？
• RQ2 在三种可区分度量——保真度、Kullback-Leibler散度和互信息——中，使它们最大化或最小化的最优量子测量是什么？
• RQ3 量子互信息的最紧致上界与下界是什么？如何通过矩阵分析推导出这些界？
• RQ4 能否开发一种系统性方法，以生成量子Kullback-Leibler信息的越来越紧致的下界？
• RQ5 在不扰动量子态的前提下，能在多大程度上推断其身份？其基本限制是什么？

主要发现

• 推导出量子保真度的精确表达式，且保真度最大化的最优测量被完全表征。
• 提供了Holevo于1973年提出的量子互信息上界的新简化推导，阐明了其结构与假设。
• 识别出一个显式且更紧致的量子互信息上界，该界在简化Holevo界推导过程中隐含出现。
• 开发了一种系统性方法，通过仅求解高阶非线性矩阵方程，生成量子Kullback-Leibler信息的逐级更紧致下界。
• 证明了不可广播定理：即使复制态并非单独纯态，也无法在无关联的两个系统上创建未知量子态的两个副本。
• 建立了推理与扰动之间权衡的形式体系，表明该问题可代数化表达，类似于可区分度量优化问题。