Operator Formulation of Quantum Causal Entropy: Hilbert-Space Representation and Spectral Properties

The dynamical structure of quantum mechanics is conventionally formulated through Hamiltoniandriven unitary evolution, while entropy is typically treated as a statistical descriptor associatedwith measurement outcomes or environmental interactions. In this work, we propose the QuantumCausal Entropy (QCE) framework, an entropy-driven extension of quantum dynamics in whichentropy-weighted causal operators contribute directly to the evolution of quantum states. The proposed formulation introduces an entropy-dependent correction to the dynamical generator whilepreserving Hermiticity, probability conservation, and the Hilbert-space structure of conventionalquantum theory. Within this approach, standard Schr¨odinger dynamics emerge as a limiting casecorresponding to vanishing entropy-coupling strength, ensuring consistency with established experimental results. The framework predicts entropy-dependent modifications to transition amplitudes, decoherence rates, and state-selection probabilities in systems exhibiting significant entropygradients in state space. Possible experimental signatures may arise in high-coherence quantumplatforms, including superconducting qubits, atomic interferometers, and quantum optical systems.By incorporating entropy as an explicit causal element of quantum evolution, the QCE formulation provides a unified information-theoretic perspective linking quantum dynamics, decoherenceprocesses, and entropy-driven information flow, offering a foundation for future theoretical andexperimental investigations of entropy-mediated quantum phenomena.