Linear Trace-Deficit Decay for Permanents of Positive Semidefinite Contractions

Every Hermitian positive semidefinite contraction $A$ satisfies $\operatorname{per}(A)\leq\exp(-\operatorname{tr}(I-A)/68)$. This proves the linear trace-deficit bound conjectured by Ebrahimnejad, Nagda, and Oveis Gharan, whose estimate had quadratic dependence on the normalized deficit. The proof combines a size-biased complex Gaussian representation, a block version of Hanson–Wright decoupling, and a rational Loewner majorant. The constant is uniform in the dimension and is not optimized.