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Universal Information Hydrodynamics

Dunkley, J R

Published Nov 19, 2025• 0 citations
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Abstract

We work in the universal information hydrodynamics (UIH) framework developed in two companion papers, where reversible quantum dynamics and irreversible response are organised in a local metriplectic geometry generated by a convex free energy, a Fisher metric, and an antisymmetric no work channel. In this setting the irreversible drift is always a Fisher gradient flow in a weighted $H^{-1}$ geometry, equipped with a cost entropy inequality and a curvature coercivity bound that are invariant under the addition of any compatible reversible dynamics. In the present paper we show that this structure has a canonical realisation for Fokker Planck equations, finite Markov chains and GKLS semigroups, and that it can be reconstructed operationally from process and state tomography on real quantum hardware. First, starting from overdamped Langevin dynamics and its Fokker Planck generator, we identify a canonical Fisher Dirichlet operator $G = Q \operatorname{diag}(\pi)$ that plays the role of the symmetric mobility on a periodic lattice and saturates the cost entropy inequality mode by mode. We then demonstrate that in a wide class of thermal and driven GKLS models the symmetric part of the real generator in the stationary BKM metric has a density block that coincides exactly with this classical Fisher operator, including nonreversible chains and random qubit and qutrit ensembles, and that the global decay of entropy and Fisher energy is governed by the Markov spectral gap while the Fisher gap controls only local curvature. Finally, we implement a suite of IBM Quantum experiments on a superconducting qubit that realises the full UIH picture on hardware. Process tomography of idle circuits yields an effective real generator $K$ in the Pauli basis whose metric adjoint split in the BKM metric at the device stationary state produces a clean metriplectic decomposition $K = G + J$, with $G$ metric symmetric and $J$ metric skew. Independent idle depths are consistent with a single time homogeneous generator, the dissipative spectrum of $G$ sets an information theoretic speed limit for BKM Fisher decay, and a curvature test shows that the same BKM metric is the local Hessian of quantum relative entropy at the stationary state. All claims are backed by a reproducible Python archive that includes constructive Fokker Planck to Markov to GKLS lifts, hypocoercive decay diagnostics, and hardware K tomography. Taken together, the results in this paper show that this programme is not only mathematically coherent, but experimentally visible, showing strong signatures of universality.

DOI
10.5281/zenodo.17651781

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We work in the universal information hydrodynamics (UIH) framework developed in two companion papers, where reversible quantum dynamics and irreversible response are organised in a local metriplectic geometry generated by a convex free energy, a Fisher metric, and an antisymmetric no work channel. In this setting the irreversible drift is always a Fisher gradient flow in a weighted H−1H^{-1}H−1 geometry, equipped with a cost entropy inequality and a curvature coercivity bound that are invariant under the addition of any compatible reversible dynamics. In the present paper we show that this structure has a canonical realisation for Fokker Planck equations, finite Markov chains and GKLS semigroups, and that it can be reconstructed operationally from process and state tomography on real quantum hardware. First, starting from overdamped Langevin dynamics and its Fokker Planck generator, we identify a canonical Fisher Dirichlet operator G=Qdiag⁡(π)G = Q \operatorname{diag}(\pi)G=Qdiag(π) that plays the role of the symmetric mobility on a periodic lattice and saturates the cost entropy inequality mode by mode. We then demonstrate that in a wide class of thermal and driven GKLS models the symmetric part of the real generator in the stationary BKM metric has a density block that coincides exactly with this classical Fisher operator, including nonreversible chains and random qubit and qutrit ensembles, and that the global decay of entropy and Fisher energy is governed by the Markov spectral gap while the Fisher gap controls only local curvature. Finally, we implement a suite of IBM Quantum experiments on a superconducting qubit that realises the full UIH picture on hardware. Process tomography of idle circuits yields an effective real generator KKK in the Pauli basis whose metric adjoint split in the BKM metric at the device stationary state produces a clean metriplectic decomposition K=G+JK = G + JK=G+J, with GGG metric symmetric and JJJ metric skew. Independent idle depths are consistent with a single time homogeneous generator, the dissipative spectrum of GGG sets an information theoretic speed limit for BKM Fisher decay, and a curvature test shows that the same BKM metric is the local Hessian of quantum relative entropy at the stationary state. All claims are backed by a reproducible Python archive that includes constructive Fokker Planck to Markov to GKLS lifts, hypocoercive decay diagnostics, and hardware K tomography. Taken together, the results in this paper show that this programme is not only mathematically coherent, but experimentally visible, showing strong signatures of universality.
We study irreversible response for coarse grained densities in Fisher-regularised quantum hydrodynamics, working within a local metriplectic framework. The state space, boundary class and a uniformly elliptic symmetric mobility G are fixed once and for all, and all constructions take place in the weighted H−1 ρ (G) geometry. Three instantaneous objects are singled out: the realised irreversible drift generated by G, a cost-entropy inequality that links control cost to entropy production, and a curvature coercivity bound on the Fisher functional. All three are invariant under the addition of any reversible drift generated by an antisymmetric operator J satisfying a weighted Liouville constraint. Equality in the cost-entropy bound picks out a one dimensional irreversible ray, and a simple "equality dial" quantifies the reversible content of a given evolution. Assumptions are minimal, convex free energy, strictly positive densities, symmetric uniformly elliptic G, and the H−1 ρ tangent model. All identities are supported by operational diagnostics and reproducible code. Read together with the companion paper, the present results identify the dissipative metriplectic channel compatible with the same Fisher geometry and Wasserstein-Otto tangent. The combined picture gives a minimal reversible-irreversible split for Fisher-regularised quantum hydrodynamics: the reversible current is the Fisher selected Schrödinger flow, while the present work fixes the local irreversible geometry and its equality and curvature certificates. A final scalar Fisher sector shows that the same weighted operator Lρ,G and Fisher quadratic form support a log density potential with a controlled Newtonian limit and a simple coupling to the Madelung Hamilton-Jacobi equation, kept deliberately within a scalar, weak field regime. All claims are necessity statements inside the stated axioms; no uniqueness of G or of the scalar dynamics is asserted beyond this local setting. The intended scope is quantum hydrodynamics and quantum information geometry for Fisher regularised Schrödinger dynamics with metriplectic dissipation.
In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.
We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation, Euclidean invariance with a global phase symmetry, reversibility, and convex regularity. Within the resulting class of first order local Hamiltonian field theories, these axioms single out the canonical Poisson bracket on density and phase under the Dubrovin and Novikov assumptions for local hydrodynamic brackets. Using a pointwise, gauge covariant complex change of variables that maps density and phase to a single complex field, we show that the only convex, rotationally invariant, first derivative local functional of the density whose Euler Lagrange term yields a reversible completion that is exactly projectively linear is the Fisher functional. When its coefficient equals Planck constant squared divided by twice the mass, the dynamics reduce to the linear Schrodinger equation. For many body systems, a single local complex structure across sectors enforces the same relation species by species, fixing a single Planck constant. Galilean covariance appears through the Bargmann central extension, with the usual superselection consequences. Comparison with the Doebner and Goldin family identifies the reversible zero diffusion corner with linear Schrodinger dynamics. We provide operational falsifiers via residual diagnostics for the continuity and Hamilton Jacobi equations and report numerical minima at the Fisher scale that are invariant under Galilean boosts. In this setting, quantum mechanics emerges as a reversible fixed point of Fisher regularised information hydrodynamics. A code archive enables direct numerical checks, including a superposition stress test that preserves exact projective linearity within our axioms.
Can spatial distance and proper time be defined operationally — by how a system leaks information to its environment — rather than postulated as primitive geometric quantities? This paper answers affirmatively for both, within Markovian open quantum dynamics, building on the survival-expansion framework of the companion paper "Local Curvature of Quantum Information" (doi:10.5281/zenodo.18792424). Spatial distance is defined as the excess irreversibility generated by superposing two states: when a system is superposed over two locations, the environment learns which at a rate growing with their separation. This distance is proven to be a genuine metric — it satisfies the triangle inequality via an isometric embedding into environmental "shadow space", recovers the Euclidean metric of R³ for independent position channels, and becomes undefined before decoherence even for orthogonal states, so that geometry and classicality emerge together. Proper time is defined as elapsed background time minus the minimal dissipation needed to traverse a worldline. The reversed triangle inequality of relativity (the twin paradox) then follows automatically from the subadditivity of optimal costs, and the exact Minkowski interval corresponds to a dissipation rate that saturates at high transport velocity: the light cone as a bound on irreversibility production. Both Lorentzian signatures are traced to a single convex-analytic root: spatial distance is directly a subadditive object (a norm), while proper time is an additive duration minus a subadditive optimal cost, hence superadditive. In an exactly solvable driven damped-oscillator model, the information deficit coincides identically with Clausius entropy production, the minimal transport deficit obeys Λ∗=γm2kBTbΔx2T \Lambda^* = \frac{\gamma m}{2 k_B T_b} \frac{\Delta x^2}{T} Λ∗=2kB​Tb​γm​TΔx2​, and the emergent light speed is c2=κkBTbγm=κD c^2 = \frac{\kappa k_B T_b}{\gamma m} = \kappa D c2=γmκkB​Tb​​=κD, with DDD the Einstein diffusion coefficient. The universality of the emergent light cone is shown to be equivalent to a Compton-clock scaling of the dissipation–time conversion rate (κ=2mc2ℏ \kappa = \frac{2 m c^2}{\hbar} κ=ℏ2mc2​), with the quantum–classical crossover condition kBTb=ℏγ2 k_B T_b = \frac{\hbar \gamma}{2} kB​Tb​=2ℏγ​ arising as an internal consistency requirement rather than a postulate. A junction-cost theorem establishes that velocity discontinuities incur strictly positive excess dissipation, so the turning twin ages strictly less even at equal speeds. All results are verified numerically on lattice decoherence models, exact linear–quadratic optimal control, and Fock-space computations; the verification scripts accompany the manuscript. Locality of the environmental couplings is assumed, not derived, and the construction has analog-gravity status: a Lorentzian interval in a preferred frame defined by the bath.
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.
A mathematically explicit extension of quantum hydrodynamics is developed in which a scalar coherence field ΦG modifies phase transport and is driven by the information-geometric structure of the probability density together with probability transport itself. The construction starts from the Madelung representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S) sector together with a dissipative Onsager-type relaxation law for ΦG. This yields a closed PDE system consisting of a continuity equation, a modified Hamilton–Jacobi equation, and a coherence-field equation that reduces in the overdamped limit to an elliptic closure. The source of ΦG is neither bare mass density nor bare probability density: it is a local Fisher-information density together with probability transport, so that ΦG measures coherence demand. The field equations are written explicitly; the coupling sign is fixed by an energy-stability argument, the transport-forcing sign by a dissipation argument, and the induced coherence-pressure term is computed exactly. A structural no-signalling condition is obtained by restricting the nonlinearity to Fisher-marginal source functionals that are blind to the entanglement Fisher excess under marginalisation. A coarse-grained Yukawa sector is shown to admit a conditional Newtonian limit under explicitly stated screening and additivity assumptions; this remains a conjectural macroscopic regime rather than a completed derivation of gravity. The paper closes with a falsifiable numerical programme—ablation studies, dispersion diagnostics, Fisher-information monitoring, and an explicit pilot solver scaffold—together with a claim-status ledger, a testability map, and a development roadmap that separate what is defined, what is derived, what is heuristic, and what remains unproven. Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung dynamics; coherence field; no-signalling; information geometry.