The Morse Gradient Test (EP4) demonstrated that qubit dissipation on IBM Eagle hardware follows the entropy gradient of the Bures density matrix manifold with a Morse residual of 0.0171 — meaning 98.3% of observed dissipative motion is accounted for by the manifold's geometric structure alone. This document states the physical consequences of that result. Quantum decay is not a random walk. It is a geometrically governed descent along a universal Morse foliation that exists independently of the physical hardware. The Second Law of Thermodynamics is encoded in the topology of the space that quantum information inhabits.
The Morse Gradient Test (EP4) demonstrated that qubit dissipation on IBM Eagle hardware follows the entropy gradient of the Bures density matrix manifold with a Morse residual of 0.0171 — meaning 98.3% of observed dissipative motion is accounted for by the manifold's geometric structure alone. This document states the physical consequences of that result. Quantum decay is not a random walk. It is a geometrically governed descent along a universal Morse foliation that exists independently of the physical hardware. The Second Law of Thermodynamics is encoded in the topology of the space that quantum information inhabits.
EP4 is the fourth experimental entry point of the EQ(vT) theory (Time as a Vector Field, DOI: 10.5281/zenodo.18917317). EQ(vT) predicts that the density matrix of any decohering quantum system evolves along a Bures–Morse gradient flow with attractor ρ → I/d (the maximum-entropy state). This deposit formalizes the EP4 test against real IBM quantum hardware data. Data source: IBM Open Science Prize 2021 (qiskit-community/open-science-prize-2021). System: 3-qubit XXX Heisenberg spin chain on ibmq_jakarta (qubits 1, 3, 5), initial state |110⟩, ISL-optimised Trotterisation (35 steps, t = π). Full quantum state tomography: 27 Pauli bases, 16192 shots/circuit, 8 repetitions. Methodology: Density matrices reconstructed from raw hardware data using an ignis-free pipeline (calibration matrix inversion + linear Pauli inversion + PSD projection). The test operates in the rotating frame, isolating the decoherent component by removing accumulated unitary evolution at each Trotter step. This is the correct and necessary frame for testing a decoherence model. Data provenance: Step 35 (t = π) uses the density matrix reconstructed directly from real ibmq_jakarta hardware. Steps 1–34 use an analytic T₁/T₂ decoherence model calibrated to ibmq_jakarta hardware parameters (T₁ = 120 μs, T₂ = 90 μs, gate time 15 ns/step). Model purity at step 35 agrees with hardware to <1%. Full per-step hardware tomography is a limitation noted in §5.2 of the formalization document. Results: Three test statistics all pass. T1 (best-fit Morse residual): median 0.0037 vs noise floor 0.011, ratio 0.33× — well below the 5× conventional significance threshold. T3 (monotone approach to I/8): Bures distance decreasing. S4 (scalar Morse ratio dS/|∇S|²_Bures): constant at 0.005 ± CV~0.005 across all 34 usable Trotter steps, confirming the gradient flow structure independently of any fitting procedure. Contents: EP4_MorseTest_Formalization.docx (formal write-up), MorseTest_IBM_v4.py (full implementation), MorseTest_IBM_v4.png (results figure). Related: EQ(vT) theory (DOI: 10.5281/zenodo.18917317) · EFT Standalone (DOI: 10.5281/zenodo.18963986) Michael Adis · ETN Invest AB · Södertälje · 2026
EP4-C documents the results of EP4A — a Lindblad Morse Gradient Test on the 3-qubit XXX Heisenberg spin chain using published ibmq_jakarta hardware parameters (T1=119 μs, T2=113 μs) — and identifies a structural consequence extending to the BCS superconductivity experiment (EP5-BCS). Core findings: (1) The Morse correlation dS = λ · D²_B is confirmed at 14.7× signal-to-noise on the Lindblad analytic trajectory. All thresholds were pre-declared before simulation. Decoherence parameters were taken from published hardware specs, not calibrated to match output. The Heisenberg revival F(|110⟩, U(π)|110⟩) = 1.000000 is verified. (2) The observed drift in λ is explained analytically with 85% accuracy and zero free parameters as a geometric consequence of the trajectory approaching the pure-state boundary ∂D(H) of the density matrix manifold near the revival at t=π. The drift is carried entirely by D_B shrinking as rotating-frame steps become geometrically short near the turning point; dS remains flat (CV=0.175). This is not a post-hoc explanation — the prediction follows from local eigenvalue structure of ρ at each step with no fitting. (3) Two independent aspects of the computation both indicate that the effective gravitational degree of freedom in EFT — f = |∇S|_B — is locally determined on D(H). First: λ(t) is predicted from local quantities only, with no non-local terms entering the drift explanation. Second: the Bures metric singularity near pure states is controlled entirely by the local eigenvalue spectrum of ρ at that point, directly analogous to how spacetime curvature in GR is determined by the local stress-energy tensor. (4) The BCS experiment (EP5-BCS, pending CAN Superconductors disc samples) has the identical geometric structure: the BCS ground state at T=0 is a pure state on ∂D(H). The clean Morse window — where the Pearson correlation between dS and D²_B is predicted to be high (> 0.85) and λ to track the zero-parameter geometric prediction — is 0.2 < T/T_c < 0.8, derived from EFT geometry before any BCS measurement. Boundary drift zones near T=0 and T=T_c are predicted as positive confirmations of the same structure. The document includes: full pre-declared threshold tables, per-step trajectory data, the zero-parameter λ prediction, all figures, reproducible Python code (NumPy/SciPy/Matplotlib only, no Qiskit required, runs in under 30 seconds), and the complete reasoning chain from initial results through geometric diagnosis to the locality finding. Verification chain:- EP4A (this document): Lindblad analytic trajectory — COMPLETE, 14.7× signal- EP4B: IBM hardware tomography at steps 5, 10, 20, 30, 35 — PENDING (script ready)- EP5-BCS: Superconducting disc, CAN Superconductors — PENDING, clean window 0.2 < T/T_c < 0.8 Related work:- EFT standalone: DOI 10.5281/zenodo.18963986- EQ(vT): DOI 10.5281/zenodo.18917317- EP4A (Morse Gradient Test, Lindblad): DOI 10.5281/zenodo.18969071- BCS pre-registration: DOI 10.5281/zenodo.18964362
This work investigates the emergence of state-dependent dissipative structures in open quantum systems and develops a geometric framework for constructing physically admissible non-linear extensions of Markovian quantum dynamics. Building upon the coarse-grained completely positive generators developed in previous work, we consider the effects of finite and dynamically polarizable environments, leading naturally to state-dependent Kossakowski structures and non-linear dissipative flows. To constrain such non-linear dynamics and prevent the appearance of unphysical states, we formulate the evolution as a gradient flow on the manifold of quantum states endowed with the Bures metric. A locality-preserving Extension Consistency Condition is introduced to ensure the admissibility of non-linear lifts to composite systems and to eliminate non-local ancillary back-action. Within this framework, we establish a Boundary Repulsion Theorem showing that the rank-deficient boundary of the positive state manifold is located at an infinite geometric distance along the flow trajectories. As a consequence, positivity of the density operator is preserved for all finite evolution times. A worked example based on a three-level V-system demonstrates the regularizing effect of the Bures geometry in the presence of state-dependent dissipation. The results suggest that information-geometric structures may provide a natural route toward physically consistent non-linear quantum master equations while preserving the operational integrity of quantum state evolution.
The second law does more than set the direction of thermodynamic evolution: it endows nonequilibrium transformations with an underlying geometry. In this work, we provide a unified geometric description of entropy production in classical and quantum thermodynamics based on Wasserstein-2 structures arising from gradient flows of free energy. We review how relaxation to equilibrium, in overdamped diffusions, discrete detailed-balanced Markov chains, and dissipative Lindblad dynamics, can be formulated as a gradient flow on the space of states. The associated Wasserstein-2 distance bounds entropy production, yielding a finite-time refinement of the second law. We extend this framework beyond purely dissipative dynamics by introducing generalized Wasserstein-2 metrics that incorporate conservative (Hamiltonian) dynamics in both classical inertial systems and open quantum systems, yielding intrinsic distances that exactly characterize minimal entropy production under fixed dissipative mobilities. We establish equivalence bounds between purely dissipative and Hamiltonian-dissipative geometries, explicitly quantifying how inertial or coherent dynamics can reduce dissipation. Finally, when restricted to equilibrium distributions, we recover the thermodynamic length of linear response-including the quantum thermodynamic length-thereby linking optimal transport, thermodynamic length, and counterdiabatic protocols within a single geometric framework. All in all, our results extend the Riemannian program of thermodynamics further from equilibrium and provide a geometric foundation for optimal protocols beyond the overdamped setting.
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 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.