Geometric Structure of Hidden-State Trajectories

• Geometric structure of hidden-state trajectories is defined by the organization of latent states through evolution equations, revealing manifolds, phases, and statistical signatures in various dynamical systems.
• Methodologies span from quantum master equations and Markov models to graph-based techniques and neural network activations, providing diverse insights into the hidden geometry.
• Understanding these trajectories improves system control, inference methods, and algorithm design, with practical implications in quantum computation and advanced signal processing.

The geometric structure of hidden-state trajectories refers to the organization, evolution, and mathematical properties of latent states in dynamical and statistical systems, especially as these states evolve, interact, and manifest in observable or unobservable behaviors. Across disciplines—quantum physics, nonlinear dynamics, machine learning, statistical inference, and signal processing—this concept organizes both the dynamics of state evolution and the inference or interpretation of hidden structures underlying observable outputs. Geometric descriptors (such as phases, manifolds, tensors, or statistical signatures) reveal both the constraints and freedoms inherent in a system's latent space, expose ambiguities and invariances, and often serve as the foundation for learning, control, or verification procedures.

1. Mathematical Formulations of Hidden-State Trajectories

Latent (hidden) states in dynamical systems are typically encoded by vectors, matrices, or more general objects that are not directly observable but determine a system’s evolution. Their trajectories acquire a geometric structure through the governing equations, such as:

• Quantum Open Systems: The evolution of the density matrix $\rho(t)$ is governed by the Lindblad master equation,

$$\dot{\rho}(t) = -i[H(t), \rho(t)] + \lambda \sum_m (L_m \rho(t) L_m^\dagger - \tfrac{1}{2} \{ L_m^\dagger L_m, \rho(t) \}),$$

with unravelings into quantum trajectories that may depend on hidden parameters. The path or "trajectory" of %%%%2%%%% in projective Hilbert space can be associated with a geometric phase,

$$\gamma_{nj} = \arg\langle \psi(0) | \psi(T) \rangle + \int_0^T \langle \psi(t) | H(t) | \psi(t) \rangle dt,$$

which is itself a geometric quantity reflecting properties of the trajectory, not just the endpoint (Pawlus et al., 2010).

• Markovian Models: The state is represented as a (possibly signed) density $Q$ in the space of Hermitian matrices, evolving via trace-preserving operators. The hidden trajectory is then a sequence $\mu^t(Q)$, $t=0,1,\ldots$, forming a path in finite-dimensional affine space, with geometric and ergodic properties akin to classical Markov chains (Faigle et al., 2010).
• Quantum Information Geometry: On the quantum state space $Q_n$, trajectories governed by the extended averaged Hebbian learning equation (EAHLE) are shown to be e-geodesics—autoparallel curves with respect to an exponential-type connection:

$$\frac{d p}{dt} = p C + C p - 2 \operatorname{Tr}(C p) p,$$

yielding evolution along geodesics with explicit analytic representation (Uwano, 2016).

• Graph-based and Sobolev Structures: In learning frameworks, hidden states are modeled as fields or vector assignments on the nodes of a graph, equipped with Sobolev-type norms,

$$\|f\|_{H^1(V)}^2 = \|f\|^2_{L^2(V)} + \sum_{(u,v) \in E} |f(u) - f(v)|^2 \rho(u,v),$$

where the geometric structure of trajectories in function space is intimately tied to the topology of the underlying graph (Pasechnyuk-Vilensky et al., 27 Jul 2025).

• Latent Trajectories in Machine Learning: In deep networks, especially transformer-based LLMs, the trajectory is a sequence of hidden state activations through layers and time steps. The difference (delta) between the state at the beginning and end of a reasoning process $\Delta h(T) = h_\text{end}(T) - h_\text{start}(T)$ encodes a geometric signature of the computation, such that correct and incorrect outputs cluster separately in this hidden activation space (Liang et al., 2 Oct 2025).

2. Geometric Phases and Hidden Parameters

A significant class of phenomena arises when geometric phases—quantities accumulated along the path of evolution—depend on hidden structure. In open quantum systems, the geometric phase of a trajectory can depend on hidden parameters associated with the choice of jump operators in the Lindblad equation. Making a shift $L_m \rightarrow L_m - f_m(t) \hat{1}$ leaves the density matrix evolution invariant but changes the effective Hamiltonian for individual trajectories, yielding an $f$-dependent geometric phase,

$$\gamma_{nj} = -\pi + \frac{\omega}{4f\lambda} \ln\left(e^{\frac{4\pi}{\omega}f\lambda} \cos^2\frac{\theta_0}{2} + e^{-\frac{4\pi}{\omega}f\lambda} \sin^2\frac{\theta_0}{2}\right),$$

even though $\rho(t)$ is $f$-independent (Pawlus et al., 2010). This ambiguity calls into question the physical significance of trajectory-assigned geometric phases in the absence of a fixed measurement protocol.

In monitored quantum systems, the geometric phase becomes a stochastic variable distributed over quantum trajectories due to random quantum jumps. The distribution can display topological transitions dependent on dissipation and driving parameters, with critical behavior visible only at the trajectory (not density matrix) level (Viotti et al., 2023). For continuously measured qubits, the geometric phase along self-closing trajectories undergoes sharp transitions as a function of measurement strength, with transitions characterized by winding numbers and Berry curvature integrated over the trajectory manifold (Shea et al., 2023).

In addition, geometric phases with complex values—having both real and imaginary parts—arise when the trajectory moves through non-Hermitian or complexified state spaces, where the imaginary component reflects geometric quantum diffusion or decoherence (Yang et al., 2014).

3. Hidden Geometry in Markov and Stochastic Models

Markovian frameworks for quantum and classical stochastic processes naturally introduce a geometric structure in the space of states—often affine or projective spaces of matrices or probability distributions. Here, hidden-state trajectories are the orbits or paths traced by these elements under the action of Markov or quantum channels.

A salient insight is that violations of Bell-type inequalities within this framework do not indicate the absence of hidden states, but instead reflect a lack of joint observability. The state space is large enough to contain negative components—even as observable outcomes remain classically interpretable—and the geometry of trajectories under Markov operators can be analyzed for their stationary points, convergence, and long-term behavior (Faigle et al., 2010).

Histogrammatic techniques can further resolve hidden geometric structures: By collecting statistics of observed transitions conditioned on history, one constructs histograms whose multimodality and convergence rate reveal parallel hidden pathways, memory depth, or lack of the local Markov property in projected dynamics (Zhao et al., 14 Mar 2025).

4. Learning and Inferring Hidden Geometry

In data-driven settings, the geometry of hidden-state trajectories determines representational capacity, sample complexity, and inference limits.

For LTI state-space systems, the trajectory of hidden states is inferred via estimation of the system’s Markov parameters, optimally regularized by the Hankel nuclear norm to enforce low effective dimension. The subsequent minimal realization—via a Ho–Kalman-type reconstruction—embeds the recovered dynamics in a geometric manifold, and allows non-asymptotic bounds on estimation error and order recovery (Djehiche et al., 2022).

In transfer learning and domain adaptation, aligning not only distributions but also geometric structure (underlying data manifolds) becomes critical. Here, constraints such as low-rank and sparsity regularization align the “hidden” structure of source and target, enforcing similarity of local neighbor relations ("hidden trajectories") across domains (Luo et al., 2017).

For neural networks, especially LLMs, the internal trajectory of hidden states encodes correctness and solution quality as a geometrically separable signal. Non-parametric methods (such as CLUE) cluster the activation deltas of reasoning traces, using nearest-centroid classification in hidden space for verification without any gradient-based training; such methods robustly distinguish correct from incorrect outputs and can be leveraged for reranking and calibration (Liang et al., 2 Oct 2025).

5. Topological and Modular Hidden Structures

In quantum and signal systems, hidden subsystems can be defined by tensor decompositions of the Hilbert space—sometimes non-canonically—where basis index arithmetic (e.g., $n = Nk + l$) maps a simple system to a tensor product of "external" and "internal" degrees of freedom. This gives rise to rich modular or multi-qudit structures even in ostensibly single-particle systems, which can underlie quantum computational universality, permit the formulation of modular operators, and explain the formal emulation of quantum computation in classical analog circuits by mapping signals onto subsystem coordinates in Hilbert space (Czachor, 2023).

Such hidden tensor decompositions equip the state space with a multi-layered geometric organization, where each layer corresponds to a modular or occupation number variable, enabling the construction of universal gates and resource states for computation and information protocols.

6. Geometry and Topology in Nonlinear and Complex Systems

Nonlinear dynamical systems possessing hidden attractors display intricate geometric organization in state space. Detailed visualization of attraction basins shows that hidden attractors’ basins are structurally separated from unstable equilibria by regions of the phase space that act as barriers, often formed as unbounded cylinders or higher-dimensional analogues. Advanced graphical tools enable interactive exploration of these geometric structures, revealing fractal boundaries, coexistence of multiple attractors, and confirming the “hidden” character by demonstrating that no trajectory started in the immediate neighborhood of unstable equilibria is attracted to the hidden attractor. This geometric insight is critical for understanding multistability and the control of nonlinear systems (Danca et al., 2018).

In semiclassical systems, the geometry of hidden-state trajectories is tied to the organization of saddle points in complexified phase space. Exposed saddles (accessed by Newton–Raphson searches from real classical trajectories) and hidden saddles (connected with tunneling or classically forbidden processes) define different regimes of quantum evolution; their location, multiplicity, and relative action values govern interference and global structure in propagated wavepackets and coherent states (Wang et al., 2021).

7. Ambiguities, Physical Interpretation, and Operational Consequences

An overarching theme is that the geometric structure of hidden-state trajectories often brings with it ambiguity, especially regarding quantities not uniquely fixed by observable dynamics. In quantum open systems, the geometric phase can depend on choices of unraveling or hidden parameterization; in stochastic and Markovian models, the structure of negative entries, memory, or "non-Markovianity" in projected observables complicate standard interpretations.

Resolution strategies include:

• Fixing operational procedures (e.g., measurement schemes) that select a preferred unraveling, rendering the hidden parameters physically meaningful (Pawlus et al., 2010).
• Seeking invariant formulations (e.g., mixed-state or interferometric geometric phases) that are robust to freedom in hidden parameterization.
• Employing data-driven clusterings or geometric summaries in neural networks that, while not unique, are empirically validated and robust across domains (Liang et al., 2 Oct 2025).

These considerations are critical for both foundational issues (such as the definition of physical phases, the reality of hidden variables, or the existence of memory effects) and applied design (as in tradeoffs of space-time complexity in master equation implementations (Wolpert et al., 2017), generalization error scaling with manifold topology (Pasechnyuk-Vilensky et al., 27 Jul 2025), or robustness of quantum gates to geometric noise).

For further in-depth paper, see:

• Open-system geometric phases and ambiguity in unravelings (Pawlus et al., 2010)
• Markovian models with hidden states and negative probabilities (Faigle et al., 2010)
• Quantum trajectories, imaginary geometric phases, and Berry curvature (Yang et al., 2014)
• EAHLE and e-geodesics in quantum information geometry (Uwano, 2016)
• Master equation space-time tradeoffs (Wolpert et al., 2017)
• CLUE hidden-state clustering in LLMs (Liang et al., 2 Oct 2025)
• Hidden tensor structures and quantum emulation (Czachor, 2023)
• Action formalisms and topological transitions for measurement-induced geometric phases (Shea et al., 2023)
• Feed-anywhere graph-based hidden state geometry (Pasechnyuk-Vilensky et al., 27 Jul 2025)
• Markov-state holography and histograms of hidden trajectories (Zhao et al., 14 Mar 2025)