Concentration of measure on quantum states

量子信息处理是一门探索信息在物理系统中的行为方式的科学，这些系统可以进行足够精确的控制，从而产生量子力学效应。当这些效应开始出现时，我们关于信息行为的基本直觉的很大一部分不再成立；例如，如果不干扰未知量子系统，就无法确定它的状态。这对密码学和信息论等领域产生了影响，密码学处理的是在恶意对手存在的情况下进行的信息处理，信息论研究的是数据压缩和纠错等任务的基本限制。在这两个领域出现的主要困难之一是，许多数学工具不能直接从经典世界转移到量子世界，例如基于测量集中现象的结果收集。这是大数定律的广义版本，它告诉我们，几次实验结果的平均值几乎总是接近其期望值。这些技术在计算机科学、物理学等领域有着非常广泛的应用，它们本身已经成为数学的一个子领域。然而，只有当实验结果是经典值时，它们才适用；当它们本身是量子系统时，它们通常不再起作用，需要新的方法。这项研究的目标是系统地发展量子态的测量集中理论，以及它在量子密码学和量子信息论中的应用。从基本问题到我计划研究的更实用的主题，有大量的公开问题。在基本问题中，甚至某些涉及量子态的基本采样问题也是悬而未决的。在密码学方面，我计划为“设备无关”协议和连续变量协议开发新的工具。在这种协议中，设备可以仅根据它们的输入输出行为进行认证，因此可以由不可信的制造商提供。连续变量协议在实验上更容易实现，但在数学上更难分析。在信息论中，我计划探索量子纠错的基本极限，并更好地理解量子计算机中可能出现的各种噪声模型。这一研究计划既解决了关于物理世界中信息的性质的核心问题，也解决了当前的技术挑战，并具有近期和长期影响的潜力。

Quantum information processing is the science that explores the way information behaves in physical systems that can be controlled precisely enough for quantum mechanical effects to emerge. When these effects begin to appear, a large part of our basic intuition about how information behaves no longer holds; for instance, one cannot determine the state of an unknown quantum system without disturbing it. This has consequences on areas such as cryptography, which deals with information processing in the presence of malicious adversaries, and information theory, which studies the fundamental limits of tasks such as data compression and error correction. One of the main difficulties that arises in both of these fields is that many mathematical tools cannot be transposed directly from the classical world to the quantum world, such as the collection of results based on the concentration of measure phenomenon. This is a generalized version of the law of large numbers, which tells us that the average of the results of several experiments is almost always close to its expected value. These techniques have a very broad range of applications in computer science, physics, and far beyond, and have become a subfield of mathematics in their own right. However, they apply only when the experiment results are classical values; when they are themselves quantum systems, they often no longer work, and new methods are needed. The goal of this research program is to systematically develop a theory of concentration of measure for quantum states, together with its applications in quantum cryptography and quantum information theory. There is a vast array of open problems ranging from fundamental questions to more applied topics that I plan to investigate. Among the fundamental questions, even certain basic sampling problems involving quantum states remain open. On the cryptography side, I plan to develop new tools for "device-independent" protocols, in which devices can be certified from their input-output behaviour alone and can therefore be supplied by an untrusted manufacturer, and for continuous-variable protocols, which are far easier to implement experimentally but much more difficult to analyze mathematically. In information theory, I plan to explore the fundamental limits of quantum error correction and to gain a better understanding of the kinds of noise models that can arise in a quantum computer. This research program addresses both core questions about the nature of information in the physical world as well as current technological challenges, and has the potential for both near-future and long-term impact.