Cafaro Carlo, Ray Shannon, Alsing Paul M
SUNY Polytechnic Institute, Albany, New York 12203, USA.
Air Force Research Laboratory, Information Directorate, Rome, New York 13441, USA.
Phys Rev E. 2022 Mar;105(3-1):034143. doi: 10.1103/PhysRevE.105.034143.
We present an information geometric characterization of quantum driving schemes specified by su(2;C) time-dependent Hamiltonians in terms of both complexity and efficiency concepts. Specifically, starting from pure output quantum states describing the evolution of a spin-1/2 particle in an external time-dependent magnetic field, we consider the probability paths emerging from the parametrized squared probability amplitudes of quantum origin. The information manifold of such paths is equipped with a Riemannian metrization specified by the Fisher information evaluated along the parametrized squared probability amplitudes. By employing a minimum action principle, the optimum path connecting initial and final states on the manifold in finite time is the geodesic path between the two states. In particular, the total entropy production that occurs during the transfer is minimized along these optimum paths. For each optimum path that emerges from the given quantum driving scheme, we evaluate the so-called information geometric complexity (IGC) and our newly proposed measure of entropic efficiency constructed in terms of the constant entropy production rates that specify the entropy minimizing paths being compared. From our analytical estimates of complexity and efficiency, we provide a relative ranking among the driving schemes being investigated. Moreover, we determine that the efficiency and the temporal rate of change of the IGC are monotonic decreasing and increasing functions, respectively, of the constant entropic speed along these optimum paths. Then, after discussing the connection between thermodynamic length and IGC in the physical scenarios being analyzed, we briefly examine the link between IGC and entropy production rate. Finally, we conclude by commenting on the fact that an higher entropic speed in quantum transfer processes seems to necessarily go along with a lower entropic efficiency together with a higher information geometric complexity.
我们从复杂性和效率概念的角度，给出了由su(2;C)含时哈密顿量指定的量子驱动方案的信息几何特征。具体而言，从描述自旋1/2粒子在外部含时磁场中演化的纯输出量子态出发，我们考虑由量子起源的参数化平方概率幅产生的概率路径。此类路径的信息流形配备了一种黎曼度规，该度规由沿参数化平方概率幅评估的费希尔信息指定。通过采用最小作用量原理，在有限时间内连接流形上初始态和终态的最优路径是这两个态之间的测地线。特别地，沿这些最优路径，转移过程中发生的总熵产生最小化。对于从给定量子驱动方案中出现的每条最优路径，我们评估所谓的信息几何复杂性（IGC）以及我们新提出的熵效率度量，该度量是根据指定被比较的熵最小化路径的恒定熵产生率构建的。根据我们对复杂性和效率的解析估计，我们对所研究的驱动方案进行了相对排序。此外，我们确定，沿着这些最优路径，效率和IGC的时间变化率分别是恒定熵速度的单调递减和递增函数。然后，在讨论所分析物理场景中热力学长度与IGC之间的联系之后，我们简要考察IGC与熵产生率之间的联系。最后，我们评论了这样一个事实，即在量子转移过程中，较高的熵速度似乎必然伴随着较低的熵效率以及较高的信息几何复杂性，以此作为总结。
