Fluctuation-dissipation structure of quantum geometry
Abstract
Geometry of quantum states is a fundamental research area with applications ranging from topological materials in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are often studied, while mixed states are generally needed, for instance to have nonzero temperatures or other statistical mixtures. Mixed-state geometry, however, is much more challenging and has so far lacked any overall structure that relates distance and curvature, like the Kähler structure does for pure states. This work introduces that missing structure. We show that it has the physical meaning of fluctuation-dissipation---by identifying distance with fluctuations and curvature with linear response---and becomes the known structure for pure states, but is non-Kähler otherwise. This paints a simple picture that unifies fluctuation-dissipation and geometry of quantum states. It also applies on parameter spaces, as encountered in condensed matter or estimation theory. Our picture implies that the geometry contains all linear responses and fluctuations, proves the usual fluctuation-dissipation relation, explains that quantum geometry is present in any sum rule, and allows one to geometrically characterize generic transport behaviors.
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BibTeX
@misc{moosavi2026fluctuationdissipation,
title = {{Fluctuation-dissipation structure of quantum geometry}},
author = {Per Moosavi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25006},
eprint = {2608.25006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25006},
}
RIS
TY - UNPB
TI - Fluctuation-dissipation structure of quantum geometry
AU - Per Moosavi
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2608.25006
UR - https://arxiv.org/abs/2608.25006
AB - Geometry of quantum states is a fundamental research area with applications ranging from topological materials in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are often studied, while mixed states are generally needed, for instance to have nonzero temperatures or other statistical mixtures. Mixed-state geometry, however, is much more challenging and has so far lacked any overall structure that relates distance and curvature, like the Kähler structure does for pure states. This work introduces that missing structure. We show that it has the physical meaning of fluctuation-dissipation---by identifying distance with fluctuations and curvature with linear response---and becomes the known structure for pure states, but is non-Kähler otherwise. This paints a simple picture that unifies fluctuation-dissipation and geometry of quantum states. It also applies on parameter spaces, as encountered in condensed matter or estimation theory. Our picture implies that the geometry contains all linear responses and fluctuations, proves the usual fluctuation-dissipation relation, explains that quantum geometry is present in any sum rule, and allows one to geometrically characterize generic transport behaviors.
ER -