{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T09:23:46Z","timestamp":1787649826831,"version":"build-2736575974"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T00:00:00Z","timestamp":1787529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The universal operator-growth hypothesis (OGH) states that, in a closed chaotic system, the Lanczos coefficients grow linearly, bn\u2243\u03b1n. 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 \u03b1C\u2261\u03b1R+i\u03b1I\u2014defined as effective slopes of the sub-diagonal and diagonal Arnoldi coefficients over a pre-registered fit window\u2014around two statements whose logical status we delimit precisely. First, whenever the dissipator acts as D=\u22122\u03b3G^ with G^, a Hermitian grading (all dephasing-type baths), the diagonal obeys the identity Rean=\u22122\u03b3\u27e8G^\u27e9n: 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 \u03b3 exactly, so \u03b1R is renormalized only at O(\u03b32), and \u03b1I=2\u03ba0\u03b3 follows from closed-system data alone. We exhibit a one-qubit Lindbladian that satisfies the often-assumed generator symmetry G\u2020(\u03b3)=\u2212G(\u2212\u03b3) yet violates parity (b1=|1\u2212\u03b3|), showing that the extra condition is essential; all models studied here satisfy it bit-exactly. For large-q SYK, these ingredients predict \u03b1C=J\u22122i(q\u22122)\u03b3, whose imaginary part is fixed solely by the interaction range; the first ladder step is exact, and the multi-step increments approach q\u22122 with system size (1.92\u00b10.04 at N=12, q=4). Under a common fit protocol, the closed-system rates saturate by N=10 (\u03b1R(0)\u21920.437, 2\u03ba0\u21920.224). The imaginary rate is not an independent observable at leading order\u2014its content is its sign, which resolves how the growing operator meets its environment (opposite for spin chains and SYK).<\/jats:p>","DOI":"10.3390\/e28090953","type":"journal-article","created":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T17:20:28Z","timestamp":1787592028000},"page":"953","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Complex Operator Growth in Dissipative Quantum Systems"],"prefix":"10.3390","volume":"28","author":[{"given":"Hikaru","family":"Wakaura","sequence":"first","affiliation":[{"name":"QIRI (Quantum Integrated Research Institute Inc.), Tokyo 107-0061, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Taiki","family":"Tanimae","sequence":"additional","affiliation":[{"name":"QIRI (Quantum Integrated Research Institute Inc.), Tokyo 107-0061, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/00018732.2016.1198134","article-title":"From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics","volume":"65","author":"Kafri","year":"2016","journal-title":"Adv. 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