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.
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.
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.
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.
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) 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.
Recently, there is growing interest in studying quantum mechanics from the information geometry perspective, where a quantum state is depicted by a point in the projective Hilbert space (PHS). However, the absence of high-dimensional measures limits information geometry in the study of multiparameter systems. In this paper, we propose a measure of the intrinsic density of quantum states (IDQS) in the PHS with the volume element of quantum Fisher information (QFI). Theoretically, the IDQS is a measure to define the (over)completeness relation of a class of quantum states. As an application, the IDQS is used to study quantum measurement and multiparameter estimation. We find that the density of distinguishable states (DDS) for a set of efficient estimators is measured by the invariant volume element of the classical Fisher information, which is the classical counterpart of the QFI and serves as the metric of statistical manifolds. The ability to infer the IDQS via quantum measurement is studied with a determinant-form quantum Cram\'er-Rao inequality. As a result, we find a gap between the IDQS and the maximal DDS over the measurements. The gap has tight connections with the uncertainty relationship. Exemplified by the three-level system with two parameters, we find that the Berry curvature characterizes the square gap between the IDQS and the maximal attainable DDS. Specific to vertex measurements, the square gap is proportional to the square of the Berry curvature.
I analyse the geometry of the Uhlmann connection for a smooth family of finitedimensional density matrices in a neighbourhood of a pairwise eigenvalue-degeneracy locus,which by the von Neumann–Wigner theorem has codimension three in parameter space.First, I derive from the parallel-transport condition that the holonomy-generating weightattached to an eigenvalue pair (pa,pb) is wab = (√pa − √pb)2/(pa + pb), and verify thedefining operator identity numerically to 10−10; the frequently quoted heuristic weight√papb/(pa + pb) fails the same identity at order unity. Second, I establish an exact blockdecomposition of the Uhlmann curvature in the moving eigenframe: restricted to the twonearly degenerate bands, the curvature equals the Yang–Mills field strength of the reducedu(2)-valued connection plus a spectator residual ∆⊥ mediated by the remaining bands. Fortwo-level systems ∆⊥ vanishes identically; for three-level systems it is generically nonzeroand, together with every other Uhlmann-layer quantity, remains finite on the locus, because wab vanishes quadratically in the gap and cancels the eigenvector divergence—thepair connection itself vanishes linearly with the distance ϵ to the locus. The quantisedmonopole content (integer eigenbundle Chern numbers, verified by the Fukui–HatsugaiSuzuki method for spin-1/2 and spin-1 models) is carried by the eigenframe, not by thefinite Uhlmann curvature. Third, for the canonical depolarised-monopole family I obtainthe closed form ΦU = argcos(πwab) for the Uhlmann holonomy phase around a great circle,exhibiting a Z2 transition exactly at wab = 12, i.e. at Bloch radius rc = √3/2, depolarisationλc = 1−√3/2≈0.1340, purity Trρ2 = 7/8; the heuristic weight would instead forbid anytransition at finite mixedness. The results give a self-contained local field-theoretic characterisation of mixed-state gauge structure at level crossings and correct several statementsin the literature that rest on the heuristic weight, including the “half-Berry” limit of theUhlmann connection near degeneracy.
This post is the author's doctoral dissertation back in 1997. The dissertation covers following four kinds of problems: First, it studies achievable Cramer-Rao type bounds of various multi-parameter pure state models. Second, it relates CR-bouds, both of mixed state and pure states, with Berry-Uhlmann curvature. Roughly, it characterize how the statistical model differs from classical probability distribution family. Though the relation is rather qualitative for mixed states, the quantitative relation is obtained for pure state models. Third topic is relation between Berry-Uhlmann geometry and Amari-Nagaoka's quantum information geometry. Forth, various problems in quantum physics, uncertainty relations, measurement of temperature, time reversal symmetry, etc., are discussed using estimation theory.