球面上的Monge度规与量子态几何
Monge Metric on the Sphere and Geometry of Quantum States
摘要：
分析了N维希尔伯特空间中混合量子态集合的拓扑和几何性质。假设相应的经典动力学发生在球上，我们用矢量SU（2）相干态和广义Husimi分布来定义任意两个密度矩阵之间的Monge距离。蒙赫度规有一个简单的半经典解释，并导致一个非平凡的几何。在所有纯态中，相干态与最大混合态的距离ρ * 最大，与单位矩阵成正比，而离域混沌态与ρ * 接近。这与标准（trace，Hilbert-Schmidt或Bures）度规所诱导的几何结构形成了对比，在标准度规中，对于所有纯态，到ρ * 的距离都是相同的。我们讨论了可能的物理后果，包括酉时间演化和退相干过程。我们还引入了一种简化的Monge度规，它定义在纯量子态空间中，更适合于数值计算。
英文摘要：
Topological and geometrical properties of the set of mixed quantum states in the N-dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU(2) coherent states and the generalised Husimi distributions to define the Monge distance between arbitrary two density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ_*, proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized 'chaotic' states are close to ρ_*. This contrasts the geometries induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, where the distance from ρ_* is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states, and more suitable for numerical computation.
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