We construct a composite variational functional K on the space of quantum density operators, built from three information-theoretic components: quantum relative entropy (measuring deviation from global consistency), record-consistency variance penalties (enforcing definiteness of macroscopic records), and boundary-compression divergence (penalizing deviations in boundary encoding). We prove that this single functional recovers three distinct pillars of physics in three limiting regimes: Helmholtz free-energy minimization in thermodynamic equilibrium (exactly, via the Jaynes identity), environment-induced superselection in the quantum measurement limit (for commuting record operators, reproducing Zurek's einselection), and the Einstein field equations in the semiclassical gravitational limit (via the Jacobson thermodynamic route, with the Clausius relation derived as a fixed-point condition rather than assumed). The gradient flow of K with respect to the Bures-Riemannian metric on density-operator space is proven to be well-posed for finite-dimensional systems, with exponential convergence following from the lambda-convexity of quantum relative entropy. We establish a uniqueness characterization: K is the unique functional (up to monotone reparameterization) satisfying scale universality, boundary sensitivity, record consistency, CPTP monotonicity, and reduction to known physics. The Lagrange multipliers in the record-consistency term are not free parameters but are determined by physically measurable decoherence timescales, yielding a framework with no adjustable constants beyond those already present in quantum mechanics and general relativity. This work provides a unified variational language for thermodynamics, decoherence, and gravitation, grounded in established mathematics: quantum information geometry, gradient flows in metric spaces, and thermodynamic derivations of gravitational dynamics.
We construct a composite variational functional K on the space of quantum density operators, built from three information-theoretic components: quantum relative entropy (measuring deviation from global consistency), record-consistency variance penalties (enforcing definiteness of macroscopic records), and boundary-compression divergence (penalizing deviations in boundary encoding). We prove that this single functional recovers three distinct pillars of physics in three limiting regimes: Helmholtz free-energy minimization in thermodynamic equilibrium (exactly, via the Jaynes identity), environment-induced superselection in the quantum measurement limit (for commuting record operators, reproducing Zurek's einselection), and the Einstein field equations in the semiclassical gravitational limit (via the Jacobson thermodynamic route, with the Clausius relation derived as a fixed-point condition rather than assumed). The gradient flow of K with respect to the Bures-Riemannian metric on density-operator space is proven to be well-posed for finite-dimensional systems, with exponential convergence following from the lambda-convexity of quantum relative entropy. We establish a uniqueness characterization: K is the unique functional (up to monotone reparameterization) satisfying scale universality, boundary sensitivity, record consistency, CPTP monotonicity, and reduction to known physics. The Lagrange multipliers in the record-consistency term are not free parameters but are determined by physically measurable decoherence timescales, yielding a framework with no adjustable constants beyond those already present in quantum mechanics and general relativity. This work provides a unified variational language for thermodynamics, decoherence, and gravitation, grounded in established mathematics: quantum information geometry, gradient flows in metric spaces, and thermodynamic derivations of gravitational dynamics.
We construct a composite variational functional K = K_ent + K_rec + K_bdry on the space of density operators, built from quantum relative entropy, record-consistency penalties, and boundary-compression divergence. We prove that K recovers three pillars of physics in three limits: free-energy minimization (exactly), environment-induced superselection (for commuting record operators), and the Einstein field equations (via the Jacobson route, with the Clausius relation and Bianchi identity derived rather than assumed). Beyond recovering known results, we show that the three sectors of K cycle through dominance as physical conditions change, and that the crossover regimes, where two sectors are simultaneously relevant, yield new, quantitative predictions inaccessible to any single-sector theory. We derive the crossover scales separating the thermodynamic, measurement, and gravitational regimes and identify a concrete laboratory prediction: population-dependent decoherence rates for quantum systems, governed by the competition between K_ent and K_rec. The gradient flow of K with respect to the Bures metric is proven well-posed with exponential convergence. The Lagrange multipliers in the record term are determined by physically measurable decoherence timescales, introducing no constants beyond those already present in quantum mechanics and general relativity.
We propose a thermodynamic variational framework in which quantum mechanics, classical dynamics, and gravitation emerge as equilibrium regimes of a single free-energy functional defined on probability distributions rather than on trajectories, wavefunctions, or spacetime metrics. The functional balances Fisher information, potential energy, and Shannon entropy, encoding an exploration–exploitation trade-off uniquely fixed by information-theoretic considerations. Matching the Fisher term to the quantum kinetic energy fixes its coefficient without free parameters. Extremization of the functional yields the continuity equation and the quantum Hamilton–Jacobi equation, and thus reproduces the Schrödinger equation as a thermodynamic equilibrium condition. At mesoscopic scales, competition between Fisher information and entropy introduces a characteristic quantum–classical crossover length that provides a thermodynamic perspective on decoherence. Measurement is interpreted as an irreversible thermodynamic transition, with energetic costs bounded by Landauer's principle. In the macroscopic regime, we show that requiring thermodynamic stability and local boundary response selects area-law entropy scaling as the leading contribution under stated assumptions. Given an area-law entropy, standard local arguments recover Einstein's field equations. The framework yields falsifiable predictions across quantum, mesoscopic, and gravitational regimes.
We propose a self-consistent axiomatic framework that unifies quantum unitary evolution, thermal dissipation, and spacetime geometry based on three fundamental principles: the \emph{complex-time principle}, the \emph{information-geometry equivalence principle}, and the \emph{maximal entropy gradient principle}. Starting from these axioms, we rigorously derive three core equations: a complex-time quantum-thermal master equation, an information-induced gravitational field equation, and a unified gradient-flow dynamics that couples quantum states, thermodynamics, and curvature. From these, we further derive four key derived equations: a generalized Einstein field equation in the low-energy limit, a quantum-thermodynamic uncertainty relation, a relation between entropy production and curvature scalar, and a conservation law for quantum information current. The framework naturally recovers standard quantum mechanics, thermodynamics, and general relativity in appropriate limits, while offering new predictions about the relation between quantum entanglement and spacetime structure. All derivations are carried out in full mathematical detail, and the resulting equations are original, closed-form, and internally consistent.
Existing information-theoretic derivations of quantum mechanics from Fisher informationlack a thermodynamic embedding: they contain no temperature, no entropy, and no free-energy balance, and therefore cannot address the quantum–classical boundary within thesame variational structure. We show that this limitation is removed by a Helmholtz-typefree-energy functional on probability distributions in which Fisher information provides theenergy and relative Shannon entropy, weighted by an effective temperature ϑ, provides theentropic cost. Chentsov’s theorem and the Shore–Johnson axioms constrain this functionalup to two coefficients, one of which is fixed by consistency with the quantum kinetic energy.At ϑ= 0 the functional reproduces the Fisher-information variational principle of Reginattoand Hall–Reginatto; extremisation of the associated action recovers the Schr¨odinger equationvia the Madelung transformation. Quantum mechanics is thereby identified as the zero-temperature fixed point of the Helmholtz structure. The finite-temperature sector ϑ > 0,which has no counterpart in previous approaches, yields three results: a gauge-invariantbalance scale Lc = ℏ/2√mϑ at which both the free-energy descent and the gradient-flowspreading law change character — the spreading exponent crossing over from the quantum-pressure value 1/4 to the diffusive value 1/2 — together with, under harmonic confinement,a genuine variationally stable minimum interpolating between the exact quantum ground-state and classical equipartition widths; a dissipation bound for localisation transitions,derived from a Wasserstein gradient flow with Lyapunov structure, from which Landauer’sprinciple follows as a corollary in the entropy-dominated regime; and a thermodynamicstability property that singles out the Born weights as the unique stable basin fractionsunder the dissipative dynamics, presented as a consistency check.
We formulate a geometric extension of open quantum dynamics in which the density matrix evolves as a Riemannian gradient flow on the Bures manifold. The evolution equation \frac{d\rho}{dt} = -i[H,\rho] - \lambda \nabla_B \Phi[\rho] drives the state toward regions of minimal information degradation, defined by the functional \Phi[\rho], the instantaneous rate of relative-entropy change under the bare Lindbladian. This deterministic flow preserves trace and positivity, and introduces a single new universal constant \lambda>0 while recovering standard Lindblad dynamics as \lambda\to0. Under a Unitary Orthogonality Condition – exactly satisfied for pure dephasing and approximately in many-body localized phases – \Phi becomes a Lyapunov functional, and its critical points (decoherence-free subspaces) are asymptotically stable attractors. Applied to disordered spin chains, the framework reproduces the known entropy‑suppression scaling \Phi \lesssim \gamma N (J/W)^2. The theory offers a falsifiable, information‑geometric principle where quantum states intrinsically minimise their own information loss. Keywords: quantum information geometry, Bures metric, gradient flow, Lindblad dynamics, decoherence-free subspaces, many-body localization, Lyapunov functional
We present a fundamental unified framework based on information geometry, in which quantum mechanics, statistical thermodynamics, and general relativity emerge from a single information-theoretic ontology. Starting from three rigorous axioms that establish quantum information as the primary physical substrate, we construct the universal information action functional using the Bures–Fisher–Rao metric on the manifold of quantum density matrices. Full variational derivation yields a unified field equation that governs the evolution of quantum information and emergent spacetime geometry. In appropriate limiting regimes, we recover the Schrödinger equation, equilibrium statistical mechanics, and the Einstein field equations. This work provides the foundational layer of a complete theory of everything based on the principle: geometry is information.