This QBN mathematical preprint proves exact Gromov–Hausdorff equivalence between Fubini–Study quantum metric and Fisher–Rao classical information geometry under deficit gradient flow coarse-graining, with fixed scaling d_FR=2 d_FS. It establishes the complete bidirectional Chentsov–Fubini–Study correspondence linking unitary invariance (quantum) and Markov invariance (classical), unifying quantum and classical information geometry within the universal QBN metric framework.
This QBN mathematical preprint proves exact Gromov–Hausdorff equivalence between Fubini–Study quantum metric and Fisher–Rao classical information geometry under deficit gradient flow coarse-graining, with fixed scaling d_FR=2 d_FS. It establishes the complete bidirectional Chentsov–Fubini–Study correspondence linking unitary invariance (quantum) and Markov invariance (classical), unifying quantum and classical information geometry within the universal QBN metric framework.
We provide an experimentally measurable local gauge U(1) invariant Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states, it also captures only the quantum part of the uncertainty in the evolution Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus arrive at a more general and measurable bound. Upon imposing the monotonicity condition, it reduces to the square-root derivative quantum Fisher Information. We show that on the Fisher information metric space dynamical phase is zero. A relation between square root derivative and logarithmic derivative is formulated such that both give the same Fisher information. We generalize the Fubini-Study metric for mixed states further and arrive at a set of Fubini-Study metric---called α metric. This newly defined α metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity condition on this metric, we derive the monotone α metric. It reduces to the Fisher information metric for α=1.
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.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but the geometric description of the latent world (modaldimension) still lacks a rigorous metric theory. This paper elevates the latent worldalgebraic framework to the level of information geometry, proving that the naturalmetric carried by the latent world subalgebra Alat is precisely the Bures-Wassersteinmetric, and that the measurement projection π : A → Aobs corresponds to a gradient flow from the Wasserstein-2 space to the Fisher-Rao space. The core discoveryis that the critical condition for the information potential difference ∆µ ≥ ln2 isequivalent to the curvature singularity of the Bures distance, and the projection dynamics is exactly the blow-up solution of the Wasserstein gradient flow at the divergence of the Ricci curvature. This paper establishes the exact relation Klat = −14FQ between the latent world curvature Klat and the quantum Fisher information FQ,proposes testable predictions of information-geometric phase transitions, and extends the framework to noncommutative geometry (spectral triples) and interfaceswith quantum gravity. This work provides a unified perspective of differential geometry and optimal transport for understanding the quantum-classical transition.
We present a complete, mathematically rigorous formulation of the Information Geometry Unification Principle (IGUP). Starting from three fundamental axioms establishing quantum information as the primary ontological substrate, we construct the Bures–Fisher–Rao metric on the manifold of density matrices, derive covariant derivatives, curvature tensors, and the universal information action. We perform full variational derivations for both the quantum state ρ and the information metric gμν, obtaining closed-form unified field equations that govern quantum dynamics, thermodynamics, and gravitational geometry. We show explicitly how the Schrödinger, Liouville, Boltzmann, and Einstein equations arise as limiting cases. This paper constitutes the foundational core of a complete theory of physics based on the identification of geometry with information.
This Volume II extends the group-invariance framework of Nielsen and Okamura (2026) into quantum information geometry. We provide rigorous definitions for quantum states, Umegaki relative entropy and Petz f-divergences, unitary conjugation actions, the Bures metric, and monotone metrics. We establish the quantum analogues of the classical invariance theorems and derive an explicit maximal-invariant reduction for qubit pairs under simultaneous unitary conjugation: the three scalars (r₁, r₂, cos θ) from Bloch vectors. These form the direct quantum counterpart of the classical relative location + relative scale reduction. A comprehensive QuTiP simulation verifies invariance to machine precision (~10⁻¹³) and demonstrates clean collapse of the quantum relative entropy onto these three invariants. The work unifies the symmetry-reduction principle across classical f-divergences, Fisher–Rao distance, quantum relative entropy, and Bures geometry. Also available on Academia.edu: https://www.academia.edu/attachments/133246712/download_file?s=portfolio
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.