This work develops a quantum–statistical framework for analyzing non-local linear kernels by embedding them as Gibbs states of self-adjoint Hamiltonians on a tensor-product Hilbert space. In the symmetric setting, the construction yields a well-defined equilibrium ensemble and separates exact operator-level identities from conditional diagnostics that depend on explicit spectral assumptions. In the classical limit, the conditional Gibbs marginals recover row-wise softmax weights by construction, while a finite kinetic scale introduces a controlled within-block deformation that preserves block structure. Independent of equilibrium considerations, any kernel defines an operator-induced bipartite state whose reduced spectra are determined by the singular values of the kernel. The paper formulates a quantitative purity flatness condition under which the associated operator entanglement approaches the maximal logarithmic scaling allowed by the natural bipartition. Thermodynamic observables are linked to information geometry by identifying the Fisher–Rao metric along the inverse-temperature direction with the Hamiltonian variance, motivating energetic susceptibility and finite-size scaling diagnostics for large operator families. The framework distinguishes operator-state diagnostics, which are well defined for arbitrary kernels, from equilibrium and spectral diagnostics, which require a Hermitian representative, and provides a set of mathematically explicit tools for studying non-local kernels within quantum statistical mechanics. This is a preprint version.
This work develops a quantum–statistical framework for analyzing non-local linear kernels by embedding them as Gibbs states of self-adjoint Hamiltonians on a tensor-product Hilbert space. In the symmetric setting, the construction yields a well-defined equilibrium ensemble and separates exact operator-level identities from conditional diagnostics that depend on explicit spectral assumptions. In the classical limit, the conditional Gibbs marginals recover row-wise softmax weights by construction, while a finite kinetic scale introduces a controlled within-block deformation that preserves block structure. Independent of equilibrium considerations, any kernel defines an operator-induced bipartite state whose reduced spectra are determined by the singular values of the kernel. The paper formulates a quantitative purity flatness condition under which the associated operator entanglement approaches the maximal logarithmic scaling allowed by the natural bipartition. Thermodynamic observables are linked to information geometry by identifying the Fisher–Rao metric along the inverse-temperature direction with the Hamiltonian variance, motivating energetic susceptibility and finite-size scaling diagnostics for large operator families. The framework distinguishes operator-state diagnostics, which are well defined for arbitrary kernels, from equilibrium and spectral diagnostics, which require a Hermitian representative, and provides a set of mathematically explicit tools for studying non-local kernels within quantum statistical mechanics. This is a preprint version.
Fisher information appears in quantum theory in two mathematically distinct settings: as a classical probability functional in variational reconstructions of Schrodinger dynamics and as the quantum Fisher metric of parameterized density operators. Here we examine how far these roles can be connected without identifying the two objects. Starting from a Hamilton-Jacobi ensemble action, we recover the Fisher term that generates the quantum potential and isolate its contribution to the kinetic-energy expectation. We then analyze the quantum Fisher information matrix (QFIM) of bipartite states. For any pure two-qubit state, the optimized magnitude of the off-diagonal QFIM element for normalized local generators equals the concurrence, while a collective phase generator yields FQ = 4 C^2. Extending the optimized pure-state quantity by a convex roof gives an exact identity with Wootters concurrence for arbitrary mixed two-qubit states. This differs from evaluating the QFIM directly on a mixed density operator: for Werner states the raw optimized cross-QFIM is 2 p^2/(1 + p) and remains nonzero inside the separable region. The gap between the raw and convex-roof quantities therefore isolates, for this family, correlation geometry not captured by entanglement. Finally, in a two-path interferometer with a which-way marker, the optimal phase Fisher information available from local output statistics is I_phi^opt = V^2. These results place probability-gradient energy, bipartite correlation geometry, entanglement, and decoherence-induced loss of local phase sensitivity in a common mathematical setting while preserving the distinctions between classical Fisher information, QFIM geometry, Bell nonlocality, and global phase quantization.
We develop an information-geometric framework for describing quantum state-update processes associated with measurement and statistical distinguishability. The approach is based on the quantum relative entropy and the quantum Fisher information metric, which together induce a natural Riemannian geometry on the manifold of quantum states. Using the second-order expansion of relative entropy, we show how the Fisher metric governs the local structure of distinguishability between nearby states and defines a corresponding thermodynamic length. This geometric structure provides an effective description of finite quantum state transitions in terms of fluctuation geometry and information-space distance. The formalism is applied to thermal two-level systems and harmonic oscillator states, illustrating how the Fisher metric encodes susceptibilities, fluctuations, and geometric transition costs. We also discuss the relation between thermodynamic length, dissipation bounds, and optimal paths in state space. Within this framework, wave function collapse is interpreted not as a microscopic dynamical mechanism, but as an effective state-update process that admits a geometric characterization in the manifold of density operators. The resulting perspective unifies concepts from quantum information theory, thermodynamics, and differential geometry within a common operational framework based on statistical distinguishability. Possible connections with quantum speed limits, entanglement geometry, and holographic relations between relative entropy and gravitational dynamics are briefly discussed.
We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.
Non-Hermitian degeneracies are usually discussed through spectral coalescence, whereas entanglement is a property of eigenvectors and need not be fixed by the eigenvalues alone. We formulate a compact bipartite model that separates these two notions. A Hermitian operator with a degenerate eigenspace is transformed by an invertible non-unitary similarity map. The resulting Hamiltonian is non-Hermitian and retains a non-defective degeneracy at every finite value of the non-Hermiticity parameter. For an exactly solvable two-qubit realization, the right eigenstates evolve continuously from product states to maximally entangled states although the spectrum is unchanged. We distinguish the positive right-state reduced density matrix from the generally non-positive biorthogonal reduction, for which entropy may become complex. The same two-qubit solution gives a closed partial-transpose negativity and a Schmidt-gauged non-local magic. Entanglement grows monotonically with the non-Hermiticity parameter, whereas the non-local magic vanishes for both the product and maximally entangled limits and is largest at an intermediate coupling. In larger bipartite spaces, the Page entropy and Haar-averaged purity provide reference values for eigenstate typicality. These diagnostics separate non-defective degeneracy, exceptional-point sensitivity, Haar-typical entanglement, and non-stabilizer correlations without relying on a proliferation of basis-dependent spectral quantities.
Abstract One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher–Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective space of pure states, the Fubini–Study tensor, which is preserved under unitary evolution. In this work, we explore how the modification of the Hermitian tensor structure on the projective space may affect the geometry of pure quantum states, and whether such generalisations can be used to define dual connections with a direct correspondence to classical probability distribution functions, modified by the presence of a non-trivial phase. We show that it is indeed possible to construct a family of connections that are dual to each other in a generalised sense with respect to the real-valued sector of the Fubini–Study tensor. Using this biorthogonal formalism, we systematically classify the four types of tensors that can arise when the dynamics of a quantum system are governed by a non-Hermitian Hamiltonian, identifying both the complex-valued metric and the Berry curvature. Finally, we elucidate the role of the metric in a quantum natural gradient descent optimisation problem, generalised to the non-Hermitian case for a suitable choice of cost function.
We present a statistical framework for extracting spatially resolved entanglement directly from an ensemble of marginal (one-body) wavefunctions in Time-Dependent Quantum Monte Carlo (TDQMC). Treating the guide waves as a statistical mixture in Hilbert space, we show that the Gram matrix acts as a covariance operator whose spectrum coincides with the Schmidt spectrum. The associated functional standard deviation closely tracks the von Neumann entanglement entropy both globally and locally via walker partitioning, providing a physically transparent real-space diagnostic of quantum correlations without requiring construction of the full many-body wavefunction. Applications to one-dimensional two-electron bosonic and fermionic systems (helium atom and hydrogen-like molecule) demonstrate excellent agreement with strict conditional-wave results for opposite-spin electrons. For same-spin fermions, TDQMC's statistical treatment of exchange symmetry avoids the unphysical negative local entropies that arise from naive ln(2) subtraction, yielding spatial profiles that remain positive throughout. The method establishes a direct bridge between classical ensemble statistics and quantum entanglement measures, offering a computationally efficient real-space diagnostic tool for mapping the spatial distribution of correlations in many-body systems.