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, $b_n\simeq \alpha n$. 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 ${\alpha }_{\mathbb{C}}\equiv {\alpha }_R+i{\alpha }_I$—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 $\mathcal{D}=-2\gamma \widehat{G}$ with $\widehat{G}$, a Hermitian grading (all dephasing-type baths), the diagonal obeys the identity $\mathrm{Re}{a}_n=-2\gamma {⟨\widehat{G}⟩}_n$: 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 $b_n$ is even, and $\mathrm{Re}{a}_n$ is odd in $\gamma$ exactly, so ${\alpha }_R$ is renormalized only at $O\left({\gamma }^2\right)$, and ${\alpha }_I=2{\kappa }_0\gamma$ follows from closed-system data alone. We exhibit a one-qubit Lindbladian that satisfies the often-assumed generator symmetry ${\mathcal{G}}^{†}\left(\gamma \right)=-\mathcal{G}(-\gamma )$ yet violates parity ($b_1=|1-\gamma |$), showing that the extra condition is essential; all models studied here satisfy it bit-exactly. For large-q SYK, these ingredients predict ${\alpha }_{\mathbb{C}}=J^{(-)}2i(q-2)\gamma$, 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 ($1.92\pm 0.04$ at $N=12$, $q=4$). Under a common fit protocol, the closed-system rates saturate by $N=10$ (${\alpha }_R\left(0\right)\to 0.437$, $2{\kappa }_0\to 0.224$). 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).