Complex numbers are widely used in both classical and quantum physics and are indispensable components for describing quantum systems and their dynamical behavior. Recently, the resource theory of imaginarity has been introduced, allowing for a systematic study of complex numbers in quantum mechanics and quantum information theory. In this work we develop theoretical methods for the resource theory of imaginarity, motivated by recent progress within theories of entanglement and coherence. We investigate imaginarity quantification, focusing on the geometric imaginarity and the robustness of imaginarity, and apply these tools to the state conversion problem in imaginarity theory. Moreover, we analyze the complexity of real and general operations in optical experiments, focusing on the number of unfixed wave plates for their implementation. We also discuss the role of imaginarity for local state discrimination, proving that any pair of real orthogonal pure states can be discriminated via local real operations and classical communication. Our study reveals the significance of complex numbers in quantum physics and proves that imaginarity is a resource in optical experiments.

中文翻译：

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虚构性资源理论：量化和状态转换

复数在古典物理学和量子物理学中均被广泛使用，并且是描述量子系统及其动力学行为必不可少的组成部分。最近，引入了虚构性资源理论，从而可以对量子力学和量子信息论中的复数进行系统的研究。在这项工作中，我们根据缠结和连贯性理论的最新进展，开发了虚构性资源理论的理论方法。我们研究了虚构量化，着重于几何虚构和虚构的鲁棒性，并将这些工具应用于虚构理论中的状态转换问题。此外，我们分析了光学实验中实际操作和一般操作的复杂性，重点是未固定波片的实现数量。我们还讨论了虚构性对局部状态歧视的作用，证明可以通过局部实际操作和经典通信来区分任意一对真实的正交纯态。我们的研究揭示了复数在量子物理学中的重要性，并证明了虚度是光学实验中的一种资源。