Complex Operator Growth in Dissipative Quantum Systems

The universal operator-growth hypothesis (OGH) states that, in a closed chaotic system, the Lanczos coefficients grow linearly, bn≃αn. 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 pair of growth rates αC≡αR+iαI-defined as effective slopes of the sub-diagonal and diagonal Arnoldi coefficients over a pre-registered fit window-around two statements whose logical status we delimit precisely. First, whenever the dissipator acts as D=-2γG^ with G^, a Hermitian grading (all dephasing-type baths), the diagonal obeys the identity Rean=-2γ⟨G^⟩n: the imaginary rate measures how fast the growing operator accumulates weight in the dissipation channels. Second, we prove a conditional parity theorem: if the Hamiltonian, jump operators, and seeds can be made simultaneously real in some basis (an antiunitary condition), then bn is even, and Rean is odd in γ exactly, so αR is renormalized only at O(γ2), and αI=2κ0γ follows from closed-system data alone. We exhibit a one-qubit Lindbladian that satisfies the often-assumed generator symmetry G†(γ)=-G(-γ) yet violates parity (b1=|1-γ|), showing that the extra condition is essential; all models studied here satisfy it bit-exactly. For large-q SYK, these ingredients predict αC=J-2i(q-2)γ, whose imaginary part is fixed solely by the interaction range; the first ladder step is exact, and the multi-step increments approach q-2 with system size (1.92±0.04 at N=12, q=4). Under a common fit protocol, the closed-system rates saturate by N=10 (αR(0)→0.437, 2κ0→0.224). The imaginary rate is not an independent observable at leading order-its content is its sign, which resolves how the growing operator meets its environment (opposite for spin chains and SYK).