We propose probability-time (τ) as an information-geometric ordering parameter on the manifold of quantum states undergoing irreversible evolution under completely positive trace-preserving (CPTP) dynamics. Probability-time is defined as the accumulated information-geometric arc length of the entropy-producing component of a trajectory, distinguishing it from chronological time (t), which parameterizes duration rather than irreversible progression. (Revised Version)
We propose probability-time (τ) as an information-geometric ordering parameter on the manifold of quantum states undergoing irreversible evolution under completely positive trace-preserving (CPTP) dynamics. Probability-time is defined as the accumulated information-geometric arc length of the entropy-producing component of a trajectory, distinguishing it from chronological time (t), which parameterizes duration rather than irreversible progression. (Revised Version)
This paper develops a mathematical foundation for probability-time, an ordering parameter that measures irreversible progression along quantum state-space trajectories distinct from chronological time. In the setting of finite-dimensional CPTP dynamics with nonnegative entropy production, probability-time is defined as the accumulated information-geometric arc length of the entropy-producing component of the system’s progression. The paper describes an axiomatic characterization that isolates the invariant content of the construction: monotonicity on dissipative trajectories, stationarity under reversible evolution, and equivalence under strictly monotone reparameterizations. Within this framework, admissible contractive monotone metrics are shown to generate a common irreversible ordering while producing generally distinct numerical parameterizations. For commuting trajectories, the metrics yield only metric-dependent scalings, whereas the noncommuting trajectories exhibit genuinely nonlinear metric dependence. In detailed-balance regimes the relation between probability-time and Spohn entropy production is analyzed, demonstrating the failure of global strong canonicality but establishing a near-equilibrium quadratic relation under the Kubo–Mori metric. We thereby clarify in what sense probability-time provides a geometric measure of irreversible progression.
We extend the probability-time framework—defined as the information-geometric arc-length of irreversible trajectories in state space— to a two-qubit open-system setting and sharpen its interpretation using a cross-metric amplitude-damping simulation. Probability-time is defined as the accumulated information-geometric arc-length of the irreversible component of a trajectory in density-operator space. For a Bell-state initial condition evolved under local dephasing and local amplitude damping, the dephasing case shows approximate collapse of concurrence curves when expressed against probability-time, whereas amplitude damping does not. The latest simulation further shows that, for the same two-qubit amplitude-damping trajectory, both the Bures--Helstrom and Kubo--Mori constructions produce monotone probability-time while yielding different numerical parameterizations. This supports a more precise claim: the metric-independent content of probability-time is the induced irreversible ordering, while the numerical parameterization remains metric-dependent.
Abstract The ontological status of time remains unresolved in fundamental physics. In quantum mechanics, time appears as an external classical parameter, whereas in general relativity it is dynamical and intertwined with spacetime geometry. Canonical approaches to quantum gravity, most notably the Wheeler–DeWitt formulation, eliminate time altogether, leading to the well-known “problem of time.” These tensions motivate frameworks in which time is not fundamental but emergent. In this work, we propose a discrete, update-based framework in which the universe evolves through a sequence of global quantum states indexed by an abstract update parameter rather than by a physical time coordinate. Evolution is formulated as a completely positive, trace-preserving (CPTP) quantum channel acting on the universal density operator, ensuring exact conservation of probability and global quantum information at every update. The framework is deliberately conservative: it preserves the formal structure of quantum mechanics and quantum field theory, enforces vacuum stability, and supports global conservation laws without introducing new dynamical postulates. Within this setting, time emerges relationally at the level of subsystems. As subsystems become increasingly entangled with their environments, entropy gradients arise that define an effective arrow of time, closely aligned with the thermodynamic arrow. The resulting temporal flow is relative, statistical, and observer-dependent, while the global evolution remains information-preserving and, in the closed-system limit, reversible. We explore the implications of this framework for cosmology and black hole physics. The Big Bang is reinterpreted as an initial quantum state rather than a singular beginning of time, avoiding temporal singularities. Black hole evaporation is shown to be compatible with unitary evolution when the full quantum state is considered, naturally reproducing the expected Page curve and resolving the apparent information paradox. Overall, the framework provides a coherent, minimal setting in which time, irreversibility, and cosmological history emerge from quantum correlations rather than from a fundamental temporal background.
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.
We develop an event-based extension of quantum theory in which irreversible interaction events are taken as physically primary, and apply this framework to the emergence of time and effective gravitational behavior. An event is defined as a non-invertible completely positive, trace-preserving (CPTP) map acting on quantum states. Unitary dynamics are treated as an effective interpolation between events rather than as fundamental evolution. Time is defined as a partial ordering induced by irreversible events, while continuous time arises as a coarse-grained limit at high event density. We introduce event density as an operational quantity governing both decoherence and temporal accumulation, and show that spatial gradients of event density naturally give rise to effective gravitational time-dilation phenomena. No new force laws, spacetime curvature, or quantum gravity dynamics are postulated. Instead, gravity and time are shown to emerge as scale-dependent descriptions arising from irreversible event structure. The formulation is compatible with standard quantum mechanics, open-system dynamics, and general relativity in their respective domains, while clarifying the physical role of irreversibility in linking quantum theory and spacetime descriptions.
The foundations of modern theoretical physics rest on the interplay between quantum dynamics, thermodynamics, and spacetime geometry. While quantum mechanics describes microscopic dynamics through unitary time evolution, thermodynamic phenomena introduce irreversibility and entropy production. In this work we propose a unified framework in which quantum evolution is formulated on a complex-time manifold that naturally incorporates both dynamical and thermodynamic structures. We introduce a complex-time coordinate combining physical time and thermal flow and derive a generalized evolution equation from a variational principle defined on a complex-time geometric manifold. The resulting equation contains both Hamiltonian dynamics and geometric diffusion terms governed by the Laplace–Beltrami operator. This formulation provides a geometric interpretation of thermodynamic irreversibility within quantum theory and establishes connections with Euclidean quantum field theory, geometric quantum mechanics, and thermal time hypotheses. Possible implications for quantum statistical mechanics and information flow in gravitational systems are discussed.