Local and quasilocal conservation laws of three-state IRF cellular automata and their quantum deformations

Abstract

Using patch-matrix-product methods, we study two reversible three-state interaction-round-a-face cellular automata introduced by Klobas and Prosen [J. Phys. A 55, 094003 (2022)] - the species-preserving and species-flipping rules - and a coherent quantum deformation interpolating between them. Within an explicit translationally invariant ansatz, the two classical rules share a one-parameter family of quasi-local conservation laws with an auxiliary-dimension-three realization. Every regular member is also the first centered logarithmic derivative of an analytic auxiliary-dimension-two family passing through the identity observable. On the closed span of this family and the three elementary local charges, the only nonzero Euler velocities are $\pm\sqrt{3/23}$. Consequently, $\sqrt{3/23}$ is a rigorous lower bound on the maximal Euler speed in any larger conserved sector. For the species-flipping rule, this value agrees with the reported extrapolated value $0.361$ of Klobas and Prosen, strongly supporting completeness of the known sound-active charges. The species-preserving rule admits, in addition, a staggered generating family whose logarithmic derivatives form an infinite tower of strictly local charges. In the quantum deformation, the same algebraic structures yield exact low-bond-dimension invariant states (scar candidates), exponentially long-lived quasi-local quasimodes, and diagonal quasi-local charges that produce a nonzero Mazur bound after projection away from the three elementary local charges.