Quantum critical points admit "universal" diagnostics insensitive to microscopic details.This talk unifies three: entanglement Rényi entropy (Shannon in the Schmidt basis), Shannon-Rényi entropy in a measurement local basis, and stabilizer Rényi entropy, which quantifies departure from stabilizer/Clifford structure. For free (Gaussian) fermion chains, I show an exact bridge: the stabilizer Rényi entropy of any Gaussian eigenstate equals the Shannon-Rényi entropy of a number-conserving free-fermion eigenstate on a doubled system in the computational basis -- collapsing two diagnostics into one and carrying over the same CFT signatures. I also prove that, for a broad class of critical closed free-fermion systems, the stabilizer Rényi entropy reduces to that of the transverse field Ising (TFI) chain, making TFI results widely applicable. I will present exact lattice/CFT results, supporting numerics, experimental results on shannon-Renyi entropy of critical chains and the first central charge measurement using IBM platform, and two analytic tools for free fermions: a Berezin-Grassmann representation and a perturbation theory around an exactly solvable Rényi index.