离散变量量子态和操作的相空间复杂度
Phase-space complexity of discrete-variable quantum states and operations
AI总结：
本文提出了一种用于离散变量量子系统的相空间复杂度度量，结合Wehrl熵和Fisher信息，分析了复杂度与纯度的关系，并扩展至量子通道，揭示了维度对复杂度生成的限制。
摘要：
我们引入了一个用于离散变量（DV）量子系统的相空间复杂度量化指标。受最近为连续变量系统开发的框架启发，我们基于旋相干态上的Husimi Q函数构建了一个量子态的复杂度度量。该量化指标将两个互补的信息论量结合为一个标量量：Wehrl熵，捕捉相空间的扩展，以及Fisher信息，捕捉局部化。我们推导了该度量的基本性质，包括其在SU(2)位移下的不变性。该复杂度被归一化，使得相干态具有单位复杂度，而完全混合态具有零复杂度，这一特性不同于连续变量的情况。我们为几种相关的状态家族提供了解析表达式，包括Gibbs和Dicke态，并对自旋压缩态、NOON态和随机生成的态进行了数值分析。数值结果揭示了复杂度和纯度之间单调但非确定性的关系，使我们推测最大复杂度由纯态获得，从而将问题与通过Majorana星座优化Wehrl熵联系起来。最后，我们将框架扩展到量子通道，定义了生成和破坏复杂度的度量。我们分析了常见单体门和幅度阻尼通道的性能，显示尽管低维系统可通过自旋压缩或NOON态实现最大复杂度，但在更高维中则不可能。这些结果突显了在生成相空间复杂度方面的维度依赖性限制，并在连续和离散变量领域内建立了统一的相空间复杂度方法。
英文摘要：
We introduce a quantifier of phase-space complexity for discrete-variable (DV) quantum systems. Motivated by a recent framework developed for continuous-variable systems, we construct a complexity measure of quantum states based on the Husimi Q-function defined over spin coherent states. The quantifier combines into a single scalar quantity two complementary information-theoretic quantities, the Wehrl entropy, which captures phase-space spread, and the Fisher information, which captures localization. We derive fundamental properties of this measure, including its invariance under SU(2) displacements. The complexity is normalized such that coherent states have unit complexity, while the completely mixed state has zero complexity, a feature distinct from the continuous-variable case. We provide analytic expressions for several relevant families of states, including Gibbs and Dicke states, and perform a numerical analysis of spin-squeezed states, NOON states, and randomly generated states. Numerical results reveal a monotonic, but not deterministic, relationship between complexity and purity, leading us to conjecture that maximal complexity is attained by pure states, thereby connecting the problem to the optimization of Wehrl entropy via Majorana constellations. Finally, we extend the framework to quantum channels, defining measures for both the generation and breaking of complexity. We analyze the performance of common unitary gates and the amplitude damping channel, showing that while low-dimensional systems can achieve maximal complexity via spin squeezing or NOON states, this becomes impossible in higher dimensions. These results highlight dimension-dependent limitations in the generation of phase-space complexity and establish a unified phase-space approach to complexity across both continuous and discrete variable regimes.
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