Affine rigidity of Lindblad generators and stationary-state entanglement

Complete positivity can determine information apparently missing from a measured decay operator. We study all Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generators with a prescribed restriction A to traceless operators. Conditional complete positivity gives an exact compact convex set of compatible stationary states. Our main theorem establishes a sharp universal rank threshold: in Hilbert-space dimension d≥3, any generator whose compressed Choi matrix has rank at most d−1 is uniquely determined by A, even among completions with unrestricted Hamiltonians and jump operators. Common-target pumping is a special case. Pure-target reset generators have rank d and already exhibit affine ambiguity, establishing optimality of the threshold. A spectral compression inequality also gives the exact maximal affine-source displacement under depolarizing contamination, (d−1)/dη, and explicit stationary-state bounds under errors in A. A general necessary- and-sufficient criterion resolves cases beyond the rank guarantee through support–kernel coupling. Calibrated paired preparations isolate the homogeneous dynamics, yielding a two- step stationary-state reconstruction without direct source measurement whenever rigidity and uniqueness hold. Two-qubit examples demonstrate both robust entanglement certification and complete loss of entanglement information in non-rigid families, despite matching dynamical invariants and local responses. Complete proofs, explicit benchmark matrices and reproducible finite-dimensional checks are provided.