This manuscript presents a methodological synthesis of short-time survival probabilities, purity loss, and subsystem-resolved transition rates in finite-dimensional quantum dynamics. Starting from a pure initial state, the analysis relates the initial survival-loss rate to the trace of the established transition-rate operator and to the initial rate of linear-entropy growth. These relations are formulated for general right-differentiable physical state trajectories, without requiring a Lindblad master equation or Markovian dynamics. The GKSL setting supplies an explicit representation in terms of jump operators. For pure product preparations, compatible local projectors resolve the orthogonal state block into nonnegative departure-sector weights. A standard coverage transform connects these weights to subsystem survival slopes and an initial linear-entropy balance. The manuscript also presents a direct measurement procedure, with finite-time bias bounds and statistical uncertainty estimates. Worked examples clarify the limitations of survival-based diagnostics: overlap revival need not indicate non-Markovianity, simultaneous subsystem departures need not generate entanglement, and an entangled conditionally projected state need not imply a nonclassical noise source. The discussion also connects the transition-operator spectrum to boundary entropy asymptotics and distinguishes initial purity loss from the established coherent entanglement-timescale result. Version 9 substantially revises the mathematical assumptions, attribution, and interpretation of earlier LCQI formulations. Established results are explicitly credited. The contribution is their organization, subsystem application, and operational analysis; no new fundamental dynamical law or thermodynamic arrow of time is claimed.
This manuscript presents a methodological synthesis of short-time survival probabilities, purity loss, and subsystem-resolved transition rates in finite-dimensional quantum dynamics. Starting from a pure initial state, the analysis relates the initial survival-loss rate to the trace of the established transition-rate operator and to the initial rate of linear-entropy growth. These relations are formulated for general right-differentiable physical state trajectories, without requiring a Lindblad master equation or Markovian dynamics. The GKSL setting supplies an explicit representation in terms of jump operators. For pure product preparations, compatible local projectors resolve the orthogonal state block into nonnegative departure-sector weights. A standard coverage transform connects these weights to subsystem survival slopes and an initial linear-entropy balance. The manuscript also presents a direct measurement procedure, with finite-time bias bounds and statistical uncertainty estimates. Worked examples clarify the limitations of survival-based diagnostics: overlap revival need not indicate non-Markovianity, simultaneous subsystem departures need not generate entanglement, and an entangled conditionally projected state need not imply a nonclassical noise source. The discussion also connects the transition-operator spectrum to boundary entropy asymptotics and distinguishes initial purity loss from the established coherent entanglement-timescale result. Version 9 substantially revises the mathematical assumptions, attribution, and interpretation of earlier LCQI formulations. Established results are explicitly credited. The contribution is their organization, subsystem application, and operational analysis; no new fundamental dynamical law or thermodynamic arrow of time is claimed.
This work introduces a minimal and operational framework for describing decoherence in open quantum systems as a finite-time process. Rather than proposing a new dynamical theory or modifying the axioms of quantum mechanics, the article introduces a bounded, causal, and dimensionless scalar quantity — a local temporal coherence field C(x,t), designed to diagnose the persistence of temporal correlations over a finite causal window. The coherence field is constructed from experimentally accessible observables or reduced-state quantities via a kernel-weighted causal average. It does not represent a new physical degree of freedom and does not introduce additional equations of motion. Instead, it provides a compact diagnostic layer that makes the finite-time structure of coherence explicit, complementing standard descriptions based on memory kernels, time-dependent generators, and Lindblad master equations. The framework is shown to interface naturally with non-Markovian dynamics, coherence-weighted decoherence processes, and effective decay rates, while remaining consistent with standard GKSL structures in the appropriate limits. Its scope, limitations, and relation to existing non-Markovianity measures are explicitly discussed. The coherence-field perspective is intended as an operational and experimentally falsifiable diagnostic tool, applicable across a range of contemporary platforms where finite-time coherence effects are relevant (e.g. superconducting circuits, hybrid light–matter systems, and driven non-equilibrium regimes).This article forms part of the broader Ranesis conceptual framework, which investigates the role of finite-time persistence, coherence, and temporal organization across physical and informational systems. Within that context, the present work constitutes a self-contained and domain-specific application to open quantum systems. No assumptions, postulates, or formal structures from the broader framework are required to read or validate the results presented here.
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.
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.
Overview: This paper develops the state-to-geometry interface of the OT–GKSL architecture for open quantum sources. Its starting point is a familiar low-energy tension: classical general relativity couples spacetime geometry to matter through the Einstein equations, whereas quantum matter may be open, coherent, correlated, preparation dependent, or subject to measurement and environmental records. Semiclassical gravity and stochastic-gravity approaches have long made clear that the relation between quantum sources and classical geometry is not conceptually trivial (Page & Geilker, 1981; Kuo & Ford, 1993; Hu & Verdaguer, 2008). The OT–GKSL architecture addresses this problem by not assuming classical spacetime as part of the native quantum state description. Instead, it separates a persistent quantum/open-state domain from later classical descendants. At the native level, states belong to a faithful finite-dimensional working stratum, ρ ∈ D°(H_eff), and evolve under a declared GKSL dynamics, dρξ/dξ = L*(ρξ). The completely positive Markovian structure follows the standard theory of Gorini, Kossakowski & Sudarshan (1976) and Lindblad (1976). On a faithful reversible detailed-balance component, the dissipative generator admits the noncommutative optimal-transport gradient representation developed in the quantum Markov setting by Carlen & Maas (2017): L*_db(ρ) = - grad_gOT D(ρ || π). Here g_OT is a positive state-space metric and D(ρ || π) = Tr[ρ(log ρ - log π)] is the relative entropy to a faithful stationary state π. The detailed-balance entropy production is σ_db(ρ) = ||grad_gOT D(ρ || π)||²_gOT ≥ 0. This identity concerns the detailed-balance dissipative component, not automatically the complete GKSL trajectory. If L* = L_coh + L_db + L*_rem, then the full relative-entropy balance retains the coherent and remainder contributions separately. The native evolution parameter ξ, the accumulated entropic ordering variable t_ent, and physical readout time t_ro are therefore distinct: ξ ≠ t_ent ≠ t_ro. A physical clock requires a stabilized and calibrated clock record. Its identification with the temporal sector of the Lorentzian spacetime readout requires the additional clock–coframe and proper-time compatibility conditions of the certified spacetime branch. From records to geometric readout: Classical geometry is not identified directly with the native OT geometry. The architecture first constructs physical records. Protocol-complete certified records determine event identity, while a separate geometric coarse-graining determines which event distinctions are used for localization. A 2 rank-D family then supplies local geometric coordinates. On the observed spacetime branch, D = 4. native states → certified physical records → event classes → geometric coarse-graining → rank-D coordinate records → horizontal readout sector → section and soldering →Lorentzian coframe → classical metricThe important distinction is complete event-record content ≠ rank-D coordinate content.Additional records may encode matter properties, preparation data, measurement outcomes, orientation, gauge information, clock information, or internal structure without becoming additional spacetime dimensions. On a regular geometric patch, the coordinate map defines a vertical sector and its OT-orthogonalhorizontal complement, V = ker DΠ_coord, H = V^(⊥ gOT). A local section selects native representatives and a calibrated soldering map transfers the readablehorizontal data to the physical frame bundle. The physical coframe e^a then defines g_ro = η_ab e^a ⊗ e^b, with η_ab = diag(-1,+1,+1,+1)on the observed Lorentzian branch. The positive metric g_OT and the Lorentzian metric g_ro are therefore different mathematical objects with different domains and physical roles. Lorentzian signature is not obtained by changing the signature of g_OT; it enters through the calibrated readout branch.This distinction is compatible with the broader operational idea that physical reference frames and records may themselves be quantum systems rather than external classical structures (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019), while the specific record-to-geometry ordering is a commitment of the OT-GKSL architecture Horizontal connection and holonomy bridge: A central result of the paper is the typed separation between native state-space geometry and readout geometry. Let ∇^OT denote the Levi-Civita connection of the applicable OT metric. The connection relevant for the geometric bridge is the projected connection on the horizontal bundle, ∇^H_X Y = P_H ∇^OT_X Y.Its curvature R^H is not generally the restriction of the ambient OT curvature R^OT. In particular, R^OT ≠ 0 does not by itself imply R^H ≠ 0.The readout bridge therefore transports the horizontal connection, rather than silently identifying ambient OT curvature with spacetime curvature. With a local section s and soldering map Ω, the transferred connection has the standard inhomogeneous transformation structure, Γ_B,s = Ω (s*Γ_H) Ω^(-1) - (dΩ) Ω^(-1).It may then be compared with the independently constructed readout connection. The corresponding bridge defect is controlled on the declared certified domain. For a fixed loop C, a connection-level defect gives a length-type transport estimate of Duhamel form. An area-type small-loop estimate requires the stronger input of curvature control, a suitable small contractible loop, and compatible gauge/trivialization data. The paper therefore distinguishes four statements: ambient OT curvature; horizontal curvature and holonomy; transferred/readout holonomy; detector-visible holonomic response. None is identified with the next without an explicit bridge or visibility condition.3. This separation is related conceptually to earlier geometric treatments of mixed-state parallel transport and quantum holonomy (Uhlmann, 1986; Andersson & Heydari, 2014), although the OT-GKSL horizontal/readout bridge is an architecture-specific construction. Finite holonomy and preparation sensitivity. The consolidated formulation distinguishes a nontrivial finite loop signal from its sensitivity to preparation. A finite holonomy may satisfy H(ρ*) ≠ I while nevertheless Dρ H(ρ*) = 0. Thus anontrivial background holonomy does not prove that the holonomy varies under the preparation tested by an experiment.Two operational observables are accordingly kept distinct.Y_H^fin = L_H^fin(H - I) is a finite-loop detector. A body-fixed or Maurer-Cartan response may instead use Ξ_H = L̂_H(H^(-1) ∇_prep H).These are distinct calibrated observables. They coincide only under an explicitly declared translation/calibration relation. Preparation differentiation must also be gauge controlled, by freezing a physical record-defined frame, differentiating a gauge-invariant functional, or using a covariant derivative.Source-side constitutive branch The source/matter descendant is constructed independently of the geometric descendant. Within the Dirac/HS-NJL working class considered in the paper, an internal operator Y defines the positive quadratic tracer κ_Y = Tr(Y†Y). A frozen calibration maps this quantity to an operational coordinate p_κ. The scalar source-side constitutive law is written μ = μ(p_κ), with local susceptibility β_μ(p_κ) = - ∂_(pκ) ln μ(p_κ). A certification gate may determine whether the working point remains admissible, but its derivative is not part of the physical constitutive susceptibility in the certified interior. The scalar constitutive channel and the holonomic channel are mathematically distinct and possess different experimental symmetry handles. Einstein lock: The observed gravitational branch is constrained by an Einstein lock. The two-derivative gravitational kinetic block is fixed to the Einstein-Hilbert form,S_EH[g_ro; G0] = (c³ / 16πG0) ∫ d⁴x sqrt(-g_ro) R[g_ro], with G0 = constant.This is a branch-selection and recovery condition of the OT-GKSL corpus. It is not claimed to be the unique consequence of GKSL dynamics alone. The architecture forbids a readable preparation-dependent factor multiplying the Einstein-Hilbert curvature term on the locked branch. State dependence is instead carried by independently specified source, response, gauge, interface, readout, or higher-order sectors. The gravitational comparison is performed only after geometry and source data have beenindependently fixed: 4 C_E = G_μν[g_ro] - κ0 T^eff_μν - Δ^if_μν. A gravitational claim requires the residual C_E to satisfy its declared norm and tolerance. TheBianchi identity acts as a compatibility condition on these already constructed sectors; it does not define the source after the fact. The branch-resolved Trilemma: The central result is a local and conditional separation theorem for the visible constitutive and holonomic branches. On a finite-dimensional admissible test sector, define the kinematic analysis using the unweighted source coordinate and the holonomic preparation response, h → (dp_κ[h], ζ_H[h]). The constitutive coefficient β_μ is applied only after this kinematic conditioning. Consequently, testing β_μ,* = 0 does not make the underlying analysis map singular by definition. For a normed visible output space, the declared first-order closure has the form DO_vis[h] = N_E(β_μ,* dp_κ[h]) + N_H(ζ_H[h]) + B_oth[h] + R_aud[h]. Here B_oth contains independently constructed visible branches not removed by the frozen protocol, and R_aud is the declared audit remainder. The factorization of the scalar constitutive term through β_μ is an explicit visible/source closure assumption; it is not derived solely from OT variational calculus. At a working point where β_μ,* = 0, if, for a tested variation h, ||N_H(ζ_H[h])|| > ||B_oth[h] + R_aud[h]||, then DO_vis[h] ≠ 0. The final implication follows from the reverse triangle inequality. The nontrivial content lies upstream: construction of the regular record/geometric quotien
This paper studies a minimal time-symmetric operational framework in which information from initial preparation and post-selection is used to define an operational description of quantum evolution. Quantum randomness is interpreted as epistemic incompleteness rather than ontological indeterminacy. The central dynamical object is the informational coupling map Λκ(ρ) = (1 − κ)ρ + κσZ, which mixes the actual state ρ with a fixed reference state σZ. The channel itself is mathematically standard; the contribution of the manuscript is the operational interpretation of the dimensionless coupling parameter κ as a measure of boundary alignment, its entropy-based estimation under explicitly stated assumptions, and its embedding in a delayed-choice quantum random number generator (QRNG) protocol. A Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) representation of the corresponding continuous-time dynamics is given using a separate relaxation rate γ, complete positivity is explicit, and no-signaling is preserved under the proposed local action. In the limit γ → 0, standard unitary quantum mechanics is recovered, whereas nonzero γ produces small deviations that can be interpreted as effective informational bias associated with post-selection. In the present framework, σZ is not treated as a universal microscopic final state, but as a context-dependent operational reference state determined by the post-selection structure used in the model.
This report synthesizes a definitive paradigm shift in quantum thermodynamics, formalizing the proposition that the thermodynamic arrow of time is not a global inevitability but a localized, structurally dependent, and mathematically malleable vector that can be actively engineered at the microscopic scale. This is achieved by analyzing the architecture of irreversibility through the rigorous mathematical lens of information geometry. Quantum states are represented as coordinates on a statistical manifold equipped with Petz monotone metrics (like the Bures-Helstrom metric). Within this framework, thermodynamic irreversibility and the arrow of time emerge as precisely defined geometric flows. The core thesis is that the arrow of time is the natural gradient flow of information-theoretic entropy (Rényi entropy) along the geodesics of the quantum statistical manifold. The report details three primary methods for active manipulation: (1) Active reversal using continuous measurement and Feedback Hamiltonians, which function as the exact score function of the quantum trajectory distribution to steer states against the natural entropy gradient. (2) Decentralizing the arrow of time by defining it as a local vector field subordinate to Quantum Causal Influence (QCI), which explains phenomena like coexisting and periodically inverting temporal directions in quantum many-body scars. Also available on Academia.edu: https://www.academia.edu/attachments/132724601/download_file?s=portfolio