Theories of dynamical quantum resources

对于任何一种给定的资源，人们都需要一个理论来回答一些基本问题，比如：哪些资源可以转换为哪些资源，以及给定的转换可以通过什么方式完成？化学符合这种模式，因为它回答了如何将大量可用的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，因为它回答了各种各样的非平衡状态——热的、机械的、化学的等等——如何转化为其他状态的问题，例如，如何从一对不同温度的热浴中提取有用的功。当今资源理论方法的典型代表是纠缠理论。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，在人们认识到纠缠对某些量子信息协议有用之后，我们才开始对纠缠有了正确的理解。换句话说，当研究人员开始将纠缠作为一种资源进行研究时，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，而且在物理学的许多其他领域也得到了应用，包括相变研究、表征多体系统的基态、量子场论中的全息学和黑洞信息损失悖论。受到这一成功的启发，除了纠缠之外，现在还有许多量子系统的其他特性正在作为资源进行研究：例如，量子通道打破对称性的程度，系统偏离热平衡的程度，以及量子状态上下文的程度。除了上述基本问题外，本提案还旨在回答以下问题：如何衡量不同动态资源的质量？如果一个特定的资源通道不能确定性地转换为另一个资源通道，那么它是否可以非确定性地转换？如果可以，转换的概率是多少？如果有人能接触到催化剂呢？一个特别基本的问题是确定量子信道的等价类，这些量子信道可以在资源的渐近多副本的限制下用于相互模拟，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论现象（纠缠、不对称、热性等）本质的重要新见解，并提供一个框架来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，资源理论的方法有能力彻底改变我们对熟悉话题的思考方式。特别是，该研究项目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.