Non-perturbative constraints on stability and renormalization group flows in nonequilibrium matter

We derive constraints on renormalization group (RG) flows and stability of phases in nonequilibrium systems using quantum information inequalities. These constraints involve conditional mutual information (CMI), which quantifies correlations between spatially separated regions not mediated by their surroundings. First, assuming CMI is UV finite, we derive a monotonicity constraint on its crossover scaling function. Under certain assumptions, this implies that the CMI scaling exponent cannot increase along the RG flow. Second, we bound the CMI of a convex mixture of states in terms of the CMI of individual components. We use this inequality to infer perturbative stability of spontaneous symmetry breaking states against quantum channels that explicitly break symmetry. We illustrate these constraints through several examples, including decoherence-driven transitions in classical symmetry-broken states, area-law CMI in anisotropic conserved dynamics, and even transitions in pure quantum states. We also discuss implications for classical nonequilibrium steady states.