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arXiv:2410.11664 · Fundamental Research

Mathematical foundation of the UN(1) quantum geometric tensor

Xin Wang, Xu-Yang Hou, Jia-Chen Tang, Hao Guo

Published May 2, 2025• 0 citations
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Abstract

In this paper, we systematically establish the mathematical foundation for the U N ( 1 ) quantum geometric tensor (QGT) of mixed states Explicitly, we present a description based on the U N ( 1 ) principal bundle and derive a Pythagorean-like distance decomposition equation. Additionally, we offer a comprehensive comparison of its properties with those of the U(1) principal bundle description of the pure-state QGT. Finally, we prove a fundamental inequality for the U N ( 1 ) QGT and discuss its physical implication.

DOI
10.1016/j.fmre.2025.04.011

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In this paper, we systematically establish the mathematical foundation for the U N ( 1 ) quantum geometric tensor (QGT) of mixed states Explicitly, we present a description based on the U N ( 1 ) principal bundle and derive a Pythagorean-like distance decomposition equation. Additionally, we offer a comprehensive comparison of its properties with those of the U(1) principal bundle description of the pure-state QGT. Finally, we prove a fundamental inequality for the U N ( 1 ) QGT and discuss its physical implication.
Abstract The quantum geometric tensor (QGT) is a fundamental quantity for characterizing the geometric properties of quantum states and plays an essential role in elucidating various physical phenomena. The traditional QGT, defined only for pure states, has limited applicability in realistic scenarios where mixed states are common. To address this limitation, we generalize the definition of the QGT to mixed states using the purification bundle and the covariant derivative. Notably, our proposed definition reduces to the traditional QGT when mixed states approach pure states. In our framework, the real and imaginary parts of this generalized QGT correspond to the Bures metric and the mean gauge curvature, respectively, endowing it with a broad range of potential applications. Additionally, using our proposed mixed-state QGT, we derive the geodesic equation applicable to mixed states. This work establishes a unified framework for the geometric analysis of both pure and mixed states, thereby deepening our understanding of the geometric properties of quantum states.
Quantifying properties of quantum states through the limits of their manipulation is a central goal of quantum resource theories. For symmetry breaking, the quantum geometric tensor (QGT) governs asymptotic pure-state conversion, but a complete characterization for general mixed states has remained elusive. Here we fully resolve this problem for finite-dimensional systems under compact Lie group symmetries in the i.i.d. asymptotic regime. Specifically, we establish a single-letter formula for the optimal conversion rate between arbitrary states, with vanishing trace-distance error, in the resource theory of asymmetry. The rate is determined by a one-parameter family of metric adjusted QGTs, obtained by rescaling quantum Fisher information (QFI) matrices that interpolate between the symmetric logarithmic derivative and right logarithmic derivative (RLD) QFIs. For pure states, the RLD endpoint recovers the standard QGT. No state-independent finite subset of this family suffices in general, even for U(1)U(1)U(1) symmetry, revealing a qualitative distinction from pure-state conversion. Our formula further yields an exact pure-state distillation-rate formula in terms of a single asymmetry measure, characterizes asymptotically reversible interconversion, and identifies bound asymmetry for quantum clocks. Complementarity among members of the metric adjusted QGT family also reveals an activation mechanism: a state with zero conversion rate to a mixed clock state can still enhance another input's yield under joint processing. Our proof relies on two developments of independent interest. First, we extend quantum local asymptotic normality to unitary models with arbitrary rank and spectral degeneracy. Second, we characterize convertibility between quantum Gaussian shift models in terms of the same one-parameter family of QFIs.
The quantum geometric tensor (QGT) reveals local geometric properties and associated topological information of quantum states. Here a generalization of the QGT to mixed quantum states at finite temperatures based on the Sj\"oqvist distance is developed. The resulting Sj\"oqvist QGT is invariant under gauge transformations of individual spectrum levels of the density matrix. A Pythagorean-like relation connects the distances and gauge transformations, which clarifies the role of the parallel-transport condition. The real part of the QGT naturally decomposes into a sum of the Fisher-Rao metric and Fubini-Study metric, allowing a distinction between different contributions to the quantum distance. The imaginary part of the QGT is proportional to a weighted summation of the Berry curvatures, which leads to a geometric phase for mixed states under certain conditions. We present three examples of different dimensions to illustrate the temperature dependence of the QGT and a discussion on possible implications.
Geometrical concepts in quantum mechanics, such as the Berry phase or quantum metric, have been studied extensively in the case of pure quantum states, but their generalisation to density operators is still undergoing active research. It has become clear that more than one valid approach exists and different definitions of the quantum geometric tensor (QGT) may give inequivalent results. In this report, we cover some of the foundations of quantum geometry for pure states, motivate the need for mixed-state generalisations, and examine some approaches in that direction. The final section analyses more closely a recent proposal, deemed the “Sjöqvist QGT” and argues that its imaginary part does not produce a generalisation of the Berry curvature.
We examine the geometric interpretation of the quantum geometric tensor proposed in [Phys. Rev. B {\bf 110}, 035404 (2024)] for mixed quantum states, focusing on its imaginary part, which is proportional to a weighted sum of the Berry curvatures of the eigenstates of the density operator. While the real part naturally decomposes into Fisher--Rao and weighted Fubini--Study contributions, we show that the imaginary part does not, in general, coincide with the curvature of a connection on a globally defined U(1) line bundle whose holonomy yields a mixed-state geometric phase. Using a two-level system as an explicit example, we demonstrate that its surface integral depends on the choice of surface bounded by the same closed path, with ambiguities that are not integer multiples of 2π2π2π.