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This research establishes a fundamental geometric structure linking distance measures and curvature for both pure and mixed quantum states, revealing a universal fluctuation-dissipation principle that applies across different state types. By connecting linear response functions directly to the mixed-state quantum geometric tensor, the study provides new tools for characterizing transport behaviors in condensed matter systems and quantum information algorithms.
Fluctuation-dissipation structure of quantum geometry
arXiv:2608.25006v1 Announce Type: new Abstract: The geometry of quantum states is a fundamental research area with applications ranging from band theory in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are usually studied, while mixed ones are needed in general, for instance to allow for finite temperatures. The geometry of mixed states, however, is much more challenging, with important aspects yet to be explored. Here, we provide one missing piece by identifying the structure that relates distance measure and curvature for general (mixed or pure) quantum states. This structure is shown to carry the physical meaning of fluctuation-dissipation---by directly relating distance measure with fluctuations and curvature with linear response---and it turns into the corresponding well-known structure for pure states, where it is K\"ahler, and only then. We thus obtain a general, simple fluctuation-dissipation picture, which, among its consequences, implies that response functions inherently probe the mixed-state quantum geometric tensor. We end by using this picture to give geometric characterizations of generic transport behaviors.
Abstract:The geometry of quantum states is a fundamental research area with applications ranging from band theory in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are usually studied, while mixed ones are needed in general, for instance to allow for finite temperatures. The geometry of mixed states, however, is much more challenging, with important aspects yet to be explored. Here, we provide one missing piece by identifying the structure that relates distance measure and curvature for general (mixed or pure) quantum states. This structure is shown to carry the physical meaning of fluctuation-dissipation---by directly relating distance measure with fluctuations and curvature with linear response---and it turns into the corresponding well-known structure for pure states, where it is Kähler, and only then. We thus obtain a general, simple fluctuation-dissipation picture, which, among its consequences, implies that response functions inherently probe the mixed-state quantum geometric tensor. We end by using this picture to give geometric characterizations of generic transport behaviors.
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
arXiv:2608.25006 [quant-ph]
(or arXiv:2608.25006v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2608.25006 arXiv-issued DOI via DataCite (pending registration)
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