Complex Operator Growth in Dissipative Quantum Systems

The universal operator-growth hypothesis (OGH) states that in a closed chaotic system the Lanc- zos coeffcients grow linearly, bn ≃ αn, and that the rate α bounds the Lyapunov exponent, λL ≤ 2α. We ask how this structure is modified when the system is coupled to a Markovian environment, so that the generator becomes non-Hermitian. Applying the Arnoldi recursion to the vectorized Lindbladian in the infinite-temperature Wightman inner product, we organize the resulting com- plex growth rate αC = αR + iαI (the real sub-diagonal and complex diagonal Arnoldi coeffcients) around three exact statements. First, the diagonal’s real part is fixed exactly by Re an = −2γ⟨ ˆ G⟩n, identifying the imaginary rate as the rate at which the growing operator accumulates weight in the dissipation channels; its magnitude σG is non-universal (set by the jump operators), but the generator symmetry G† (γ) = −G(−γ) makes bn even and Re an odd in the dissipation strength γ, so universally αI = 2γσG turns on linearly and αR is renormalized only at O(γ2 ). Second, for the large-q SYK model we obtain the closed form αC = J − 2i(q − 2)γ, whose imaginary part is fixed solely by the interaction range. Third, we argue—on the same footing as the OGH itself—that the bound extends as λL ≤ 2αR(γ) with αR(γ) ≤ αR(0) + O(γ), because the dissipator is a local bounded superoperator that narrows the autocorrelation’s analyticity strip by at most O(γ). Rather than merely acquiring a phase, the imaginary rate is an exact, parity-odd measure of dissipative operator spreading, while the complexity-growth rate is protected to O(γ2 ): dissipation does not suppress operator growth.