nLab quantum metric

Idea

Given:

1. a complex Hilbert space $\mathcal{H}$, with projective space $P(\mathcal{H})$ of pure quantum states,

2. a smooth family $\Psi$ of pure quantum states, hence a smooth map $X \overset{\Psi}{\longrightarrow} P(\mathcal{H})$ from a smooth manifold $X$,

then the quantum metric on this family is the pullback of the Fubini-Study metric to $X$.

In quantum information theory this is also called the quantum Fisher information metric or quantum information metric.

Related concepts

• Berry curvature

References

General

Original discussion:

• Jean-Pierre Provost, Gérard Vallée: Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76 (1980) 289–301 [doi:10.1007/BF02193559, euclid:cmp/1103908308]

• William K. Wootters: Statistical distance and Hilbert space, Phys. Rev. D 23 (1981) 357 [doi:10.1103/PhysRevD.23.357]

Review:

• Michael Kolodrubetz, Dries Sels, Pankaj Mehta, Anatoli Polkovnikov: Geometry and non-adiabatic response in quantum and classical systems, Physics Reports 697 (2017) 1–88 [doi:10.1016/j.physrep.2017.07.001, arXiv:1602.01062]

Review in the context of condensed matter theory:

• Tianyu Liu, Xiao-Bin Qiang, Hai-Zhou Lu, X. C. Xie: Quantum geometry in condensed matter, National Science Review 12 3 (2025) nwae334 [doi:10.1093/nsr/nwae334, arXiv:2409.13408]

• Jiabin Yu, B. Andrei Bernevig, Raquel Queiroz, Enrico Rossi, Päivi Törmä, Bohm-Jung Yang: Quantum Geometry in Quantum Materials, npj Quantum Materials 10 (2025) 101 [doi:10.1038/s41535-025-00801-3, arXiv:2501.00098]

• Arpit Arora, Joel Î.-J. Wang, Tse-Ming Chen: Quantum geometry in condensed matter: Fundamentals and applications, Appl. Phys. Lett. 129 (2026) 130401 [doi:10.1063/5.0354797]

On experimental measurement:

• Tomoki Ozawa, Nathan Goldman: Extracting the quantum metric tensor through periodic driving, Phys. Rev. B 97 (2018) 201117 [arXiv:10.1103/PhysRevB.97.201117, arXiv:1803.05818]

Further discussion:

• Frédéric Piéchon, Arnaud Raoux, Jean-Noël Fuchs, Gilles Montambaux: Geometric orbital susceptibility: quantum metric without Berry curvature, Phys. Rev. B 94 (2016) 134423 [doi:10.1103/PhysRevB.94.134423, arXiv:1605.01258]

  (for 2-band systems: (8) on p. 2)

• Ansgar Graf, Frédéric Piéchon: Berry Curvature and Quantum Metric in $N$-band systems – an Eigenprojector Approach, Phys. Rev. B 104 (2021) 085114 [arXiv:10.1103/PhysRevB.104.085114, arXiv:2102.09899]

For Grassmannian targets

Generalization from target complex projective spaces to complex Grassmannians:

• Bruno Mera, Tomoki Ozawa: Kähler geometry and Chern insulators: Relations between topology and the quantum metric, Phys. Rev. B 104 (2021) 045104 [doi:10.1103/PhysRevB.104.045104, arXiv:2103.11583]

• Johannes Mitscherling, Alexander Avdoshkin, Joel E. Moore: Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals, Phys. Rev. B 112 (2025) 085104 [doi:10.1103/qscv-qxqt]