CPTP-SSM: Expanding State-Tracking to Continuous Groups with Uncertainty

Recent selective state-space models (SSMs) have increased transition expressivity to support the state tracking of finite-state automata (FSA). Recently, SSMs are increasingly applied to long sequences from the real world beyond language modeling. We extend state tracking to continuous groups with uncertainty, requiring input-selective transitions that can be either conservative on the continuous state or dissipative under uncertainty. Rather than enlarging the transition, we constrain the state space and admit only transitions that map it into itself. We introduce CPTP-SSM, a selective SSM on the compact convex set of qudit density matrices. The transitions are completely positive trace-preserving (CPTP) maps, a single family that admits both conservative and dissipative transitions. Expressing the Kraus representation in the generalized Gell-Mann basis turns each transition into a real affine map, enabling the recurrence to run as a parallel associative scan. We establish expressivity and stability guarantees for CPTP-SSM and further characterize how the qudit dimension governs the representable groups, while the Kraus rank governs the capacity for dissipation. Experimentally, CPTP-SSM achieves strong length extrapolation on both finite and continuous state tracking, with competitive performance in long time-series classification and continuous sensor prediction.