Theories of dynamical quantum resources

对于任何给定的资源类型，都需要一个理论来回答基本问题，例如：哪些资源集合可以转换为其他资源集合，以及可以通过哪些方式完成给定的转换？化学符合这种模式，因为它回答了如何将大量可用化学物质转化为其他更有用的产品的问题。热力学是另一个例子，因为它回答了各种不平衡状态--热、机械、化学等--如何转化为其他状态的问题，例如，如何从不同温度下的一对热浴中提取有用的功。如今，资源理论方法的典范是纠缠理论。虽然纠缠的概念是在20世纪30年代引入的，但直到90年代中期，我们才开始对纠缠有一个正确的理解，因为它被认为对某些量子信息协议有用。换句话说，当研究人员开始将纠缠作为一种资源来研究时，转折点就出现了。从那时起，纠缠理论不仅在量子计算、密码学和通信中得到了应用，而且在物理的许多其他领域也得到了应用，包括相变的研究、多体系统基态的表征、量子场论中的全息术以及黑洞信息损失悖论。受这一成功的启发，除了作为资源被研究的纠缠之外，量子系统现在还有许多其他属性：例如，量子通道破坏对称性的程度，系统偏离热平衡的程度，以及量子态与上下文相关的程度。除了上述基本问题外，本提案旨在为以下问题提供答案：如何衡量不同动力资源的质量？如果一个特定的资源通道不能确定地转换到另一个，它能不确定地完成吗？如果是的话，概率是多少？如果有人能接触到催化剂呢？一个特别基本的问题是识别量子信道的等价类，这些等价类可以用来在资源的渐近多个副本的限制内相互模拟，并确定相互转换的速率。回答这些具体问题通常会导致对所讨论的物理或信息理论现象的性质--纠缠、不对称、耐热性等--的重大新见解，并提供一个框架，以组织有关这些现象的理论结果。正如纠缠理论所证明的那样，资源理论方法有能力彻底改变我们思考熟悉话题的方式。特别是，这项研究计划将开发量子信息的新方法，提高我们对量子物理的总体理解，并帮助加拿大建立其在量子科学方面的领导地位。

For any given kind of resource, one needs a theory to answer basic questions such as: which collections of resources can be converted into which others, and what are the ways in which a given conversion can be accomplished? Chemistry fits this mold, as it answers the question of how collections of chemicals available in abundance can be converted to other, more useful, products. Thermodynamics is another example, as it answers questions about how various sorts of nonequilibrium states-thermal, mechanical, chemical, etc-can be converted into others, for instance, how useful work can be extracted from a pair of heat baths at different temperatures. The poster child for the resource-theoretic approach today is entanglement theory. Although the notion of entanglement was introduced in the 1930s, it was not until the mid-nineties that we began to have a proper understanding of entanglement, after it was recognized to be useful for certain quantum information protocols. In other words, the turning point occurred when researchers began to study entanglement as a resource. Since then, entanglement theory has found applications not only in quantum computation, cryptography and communication, but in many other areas of physics, including the study of phase transitions, characterizing the ground states of many-body systems, holography in quantum field theories, and the black hole information-loss paradox. Inspired by this success, there are now many other properties of quantum systems, besides entanglement that are being studied as resources: for instance, the extent to which a quantum channel breaks symmetries, the extent to which a system deviates from thermal equilibrium, and the extent to which a quantum state is contextual. In addition to the basic questions above, this proposal aims to provide answers to questions such as: How does one measure the quality of different dynamical resources? If one particular resource channel cannot be converted to another deterministically, can it be done non-deterministically, and if so with what probability? What if one has access to a catalyst? A particularly fundamental problem is to identify the equivalence classes of quantum channels that can be used to simulate each other in the limit of asymptotically-many copies of the resource and to determine the rates of inter-conversion. Answering these concrete questions typically leads to significant new insights into the nature of the physical or information-theoretic phenomenon in question--entanglement, asymmetry, athermality, etc-- and provides a framework to organize theoretical results concerning those phenomena. As entanglement theory has demonstrated, the resource-theoretic approach has the capacity to revolutionize the way we think about familiar topics. Particularly, this research program will develop new methods in quantum information, improve our understanding of quantum physics in general, and help Canada establish its leadership in quantum science.