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arXiv:2604.15752

Quantifying Uhlmann curvature from Yang-Mills action and its implications in quantum multiparameter estimation

Yi-Lin Ge, Bing-Shu Hu, Ling-Yun Deng, Xiao-Ming Lu

quant-phSubmitted Apr 17, 2026• Announced Apr 20, 2026• 0 citations• Revised Sep 13, 2026 (v2)
v1 · Apr 17, 2026v2 · Sep 13, 2026
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Abstract

The geometry of quantum states has profound implications in quantum multiparameter estimation. While the Riemannian structure of quantum state space is well understood, the full understanding of the curvature structure of mixed quantum states is still an open problem. Inspired by the Yang-Mills action in non-Abelian gauge theory, we propose a scalar quantifying the Uhlmann curvature and establish its connection to the measurement incompatibility in quantum multiparameter estimation problems. We show that this curvature measure is gauge invariant, reparametrization invariant, and vanishes if and only if the Uhlmann curvature vanishes. We also explicitly calculate the Uhlmann curvature for the joint estimation of phase and phase diffusion as an example.

Comments
5 pages, 1 figure, accepted for publication in Communications in Theoretical Physics

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•Aug 17, 2026•0 citations
The geometry of quantum states has profound implications in quantum multiparameter estimation. While the Riemannian structure of quantum state space is well understood, the full understanding of the curvature structure of mixed quantum states is still an open problem. Inspired by the Yang-Mills action in non-Abelian gauge theory, we propose a scalar quantifying the Uhlmann curvature and establish its connection to the measurement incompatibility in quantum multiparameter estimation problems. We show that this curvature measure is gauge invariant, reparametrization invariant, and vanishes if and only if the Uhlmann curvature vanishes. We also explicitly calculate the Uhlmann curvature for the joint estimation of phase and phase diffusion as an example.
Abstract The geometry of quantum states has profound implications in quantum multiparameter estima- tion. While the Riemannian structure of quantum state space is well understood, the full under- standing of the curvature structure of mixed quantum states is still an open problem. Inspired by the Yang-Mills-type action density in non-Abelian gauge theory, we propose a scalar quantifying the Uhlmann curvature and establish its connection to the measurement incompatibility in quan- tum multiparameter estimation problems. We show that this curvature measure is gauge invariant, reparametrization invariant, and vanishes if and only if the Uhlmann curvature vanishes. We also explicitly calculate the Uhlmann curvature for the joint estimation of phase and phase diffusion as an example.
Achieving joint quantum-enhanced precision in multiparameter sensing requires both high sensitivity and measurement compatibility. These two aspects are characterized by the quantum Fisher information matrix (QFIM) and the Uhlmann curvature matrix (UCM), respectively, with weak compatibility corresponding to the vanishing of the relevant UCM elements. Here, we develop a symmetry-projection framework that classifies phase generators into subspace-preserving and subspace-changing sectors. For probe states confined to a symmetry subspace, symmetry projection imposes a common block-diagonal structure on the QFIM and UCM, rendering cross-sector parameters simultaneously free from information cross-talk and measurement incompatibility. When the subspace-changing generators act as scalars within the occupied subspace, the corresponding QFIM block reduces to four times the symmetrized covariance matrix, even for mixed probe states. For parity-protected collective SU(2)\mathrm{SU}(2)SU(2) systems, this structure singles out the transverse anti-squeezed quadrature and the longitudinal mean-spin direction as natural optimal sensing axes. Applied to a dissipative one-axis twisting model, the dynamically generated probe state exhibits identically vanishing UCM elements for transverse--longitudinal parameter pairs, while maintaining nearly balanced, Heisenberg-scaled QFIM components over a broad transient window. Our work opens a route to symmetry-protected, weakly compatible multiparameter sensing in interacting quantum many-body systems.
For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.
Geometric phases play a fundamental role in understanding the geometric structure of quantum states, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.
The geometry and topology of quantum systems have deep connections to quantum dynamics. In this Letter, I show how to measure the non-Abelian Berry curvature and its related topological invariant, the second Chern number, using dynamical techniques. The second Chern number is the defining topological characteristic of the four-dimensional generalization of the quantum Hall effect and has relevance in systems from three-dimensional topological insulators to Yang-Mills field theory. I illustrate its measurement using the simple example of a spin-3/2 particle in an electric quadrupole field. I show how one can dynamically measure diagonal components of the Berry curvature in an overcomplete basis of the degenerate ground state space and use this to extract the full non-Abelian Berry curvature. I also show that one can accomplish the same ideas by stochastically averaging over random initial states in the degenerate ground state manifold. Finally, I show how this system can be manufactured and the topological invariant measured in a variety of realistic systems, from superconducting qubits to trapped ions and cold atoms.
Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.