具有引力和热类比的量子Groverian测地线
Quantum Groverian Geodesic Paths with Gravitational and Thermal Analogies
摘要：
我们给出了一个统一的变分法推导的格罗佛测地线的量子态矢量和量子概率振幅。在第一种情况下，我们表明，水平仿射参数化测地线路径的希尔伯特空间的归一化向量出现从指定的长度最小化的Hilbert空间射线流形上的Fubini-研究度量。在第二种情况下，我们表明，测地线路径的概率幅出现最小化的长度表示的Fisher信息。在这两个推导中，我们发现测地线方程是由简谐振子（SHO）描述的。然而，虽然在第一次求导中振荡频率与哈密顿系统的（恒定）能量色散成比例，但在第二次求导中振荡频率与（恒定）Fisher信息的平方根成比例。有趣的是，通过设置这两个频率彼此相等，我们恢复了著名的Anandan-Aharonov关系，该关系将哈密顿系统的平方演化速度与其能量色散联系起来。最后，在过渡远离量子设置，我们讨论了SHO型的测地线运动的出现在守恒量的存在下，通过分析两个具体的现象的引力和物理起源，分别的普遍性。
英文摘要：
We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini-Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations are described by simple harmonic oscillators (SHOs). However, while in the first derivation the frequency of oscillations is proportional to the (constant) energy dispersion of the Hamiltonian system, in the second derivation the frequency of oscillations is proportional to the square-root of the (constant) Fisher information. Interestingly, by setting these two frequencies equal to each other, we recover the well-known Anandan-Aharonov relation linking the squared speed of evolution of an Hamiltonian system with its energy dispersion. Finally, upon transitioning away from the quantum setting, we discuss the universality of the emergence of geodesic motion of SHO type in the presence of conserved quantities by analyzing two specific phenomena of gravitational and thermodynamical origin, respectively.
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