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Linear Trace-Deficit Decay for Permanents of Positive Semidefinite Contractions

Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Hexagonhexagon/2610.00270Oct 15, 2026math.COAI-generatedscore 0

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.

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