[论文解读] Uncertainty relations for multiple measurements with applications

Omar Fawzi|arXiv (Cornell University)|Aug 29, 2012

Scientific Measurement and Uncertainty Evaluation参考文献 21被引用 5

一句话总结

本论文为多重量子测量提出了强有力的不确定性关系，并将其应用于密码学与通信任务。论文提出了度量不确定性关系与量子到经典提取器（QC-extractors）的显式构造，实现了首个显式的资讯锁存方案，并通过量子容量界限在噪声存储模型中证明了安全性。

ABSTRACT

Uncertainty relations express the fundamental incompatibility of certain observables in quantum mechanics. Far from just being puzzling constraints on our ability to know the state of a quantum system, uncertainty relations are at the heart of why some classically impossible cryptographic primitives become possible when quantum communication is allowed. This thesis is concerned with strong notions of uncertainty relations and their applications in quantum information theory. One operational manifestation of such uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be "locked". Furthermore, knowing the key, it is possible to recover, that is "unlock", the message. We give new efficient constructions of bases satisfying strong uncertainty relations leading to the first explicit construction of an information locking scheme. We also give several other applications of our uncertainty relations both to cryptographic and communication tasks. In addition, we define objects called QC-extractors, that can be seen as strong uncertainty relations that hold against quantum adversaries. We provide several constructions of QC-extractors, and use them to prove the security of cryptographic protocols for two-party computations based on the sole assumption that the parties' storage device is limited in transmitting quantum information. In doing so, we resolve a central question in the so-called noisy-storage model by relating security to the quantum capacity of storage devices.

研究动机与目标

• 为多重量子测量开发强有力的不确定性关系，以实现新型量子密码原原子。
• 构建显式、高效的基底，满足强有力的不确定性关系，以实现实际应用。
• 引入QC-extractors作为对抗量子对手的不确定性关系工具。
• 利用量子容量作为安全参数，证明两方量子协议在噪声存储模型下的安全性。
• 通过将协议安全性与存储设备的量子容量相联系，解决量子密码学中的基础性问题。

提出的方法

• 使用度量不确定性关系，通过算子范数量化多重量子测量之间的不相容性。
• 利用互为正交基的完整集合（MUBs）和酉2-设计，构造显式的实现不确定性关系的基底。
• 将QC-extractors定义为量子到经典提取器，可在存在量子侧信息时仍保持不确定性。
• 应用去耦合定理与最小熵界限，证明使用小尺寸经典密钥的资讯锁存。
• 利用交换技巧与算子切尔诺夫界限，分析不确定性关系中的保真度与测量结果。
• 通过QC-extractor构造，将噪声存储模型中的安全性与存储设备的量子容量相联系。

实验结果

研究问题

• RQ1 能否为多重量子测量开发显式且高效的不确定性关系构造？
• RQ2 不确定性关系如何用于构建使用小尺寸经典密钥的资讯锁存方案？
• RQ3 量子侧信息在多大程度上限制对手提取经典信息的能力？
• RQ4 能否构造QC-extractors以确保在两方协议中对抗量子对手的安全性？
• RQ5 存储设备的量子容量如何决定在噪声存储模型中协议的安全性？

主要发现

• 首次通过使用MUBs的度量不确定性关系，实现了资讯锁存方案的显式构造。
• 仅使用泡利算符无法实现资讯锁存，因其不确定性特性有限。
• QC-extractors通过MUBs的完整集合与按位构造方法实现，可确保对量子对手的不确定性。
• 通过将安全性与存储设备的量子容量相联系，证明了在噪声存储模型中的安全性，解决了核心开放问题。
• 基于保真度的去耦合论证表明，最小熵界限可推导出有效资讯锁存，且泄漏可忽略不计。
• 推导出界限 $1 - \epsilon - 2^{-n} \leq 2^n / t$，将密钥大小 $t$ 与消息长度 $n$ 在资讯锁存中联系起来。