Local geometry and quantum geometric tensor of mixed states,Physical Review B

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Local geometry and quantum geometric tensor of mixed states
Physical Review B ( IF 3.2 ) Pub Date : 2024-07-17 , DOI: 10.1103/physrevb.110.035144
Xu-Yang Hou ₁ , Zheng Zhou ₁ , Xin Wang ₁ , Hao Guo _{1,

2} , Chih-Chun Chien ₃

Affiliation

The quantum geometric tensor (QGT) is a fundamental concept for characterizing the local geometry of quantum states. After casting the geometry of pure quantum states and extracting the QGT, we generalize the geometry to mixed quantum states via the density matrix and its purification. The gauge-invariant QGT of mixed states is derived, whose real and imaginary parts are the Bures metric and the Uhlmann form, respectively. In contrast to the imaginary part of the pure-state QGT that is proportional to the Berry curvature, the Uhlmann form vanishes identically for ordinary physical processes. Moreover, there exists a Pythagorean-like equation that links different local distances and reflect the underlying fibration. The Bures metric of mixed states is shown to reduce to the corresponding Fubini-Study metric of the ground states as temperature approaches zero, establishing a correspondence despite the different underlying fibrations. We also present two examples with contrasting local geometries and discuss experimental implications.

中文翻译：

混合态的局部几何和量子几何张量

量子几何张量（QGT）是表征量子态局部几何的基本概念。在铸造纯量子态的几何结构并提取 QGT 后，我们通过密度矩阵及其纯化将几何结构推广到混合量子态。推导了混合态规范不变QGT，其实部和虚部分别为Bures度量和Uhlmann形式。与与贝里曲率成正比的纯态 QGT 虚部相比，乌尔曼形式对于普通物理过程同样消失。此外，存在一个类似毕达哥拉斯方程，它连接不同的局部距离并反映潜在的纤维化。当温度接近零时，混合态的 Bures 度量会减少到相应的基态 Fubini-Study 度量，尽管底层纤维不同，但仍建立了对应关系。我们还提出了两个具有对比局部几何形状的例子，并讨论了实验意义。

更新日期：2024-07-19

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