Persistent Does Not Mean Readable: A Readout-Rate Operator for Semistable Quantum Dynamics

Protected and peripheral Liouvillian modes identify which components of an open quantum system can survive, but they do not determine whether those components are readable by a specified detector, syndrome channel, or feedback record. This preprint introduces a readout-rate operator for finite-dimensional semistable quantum dynamics. The operator isolates the coefficient of the linear large-window growth of the finite-time readout Gramian and provides a detector-relative criterion for persistent readout visibility. For semisimple neutral spectrum, the readout-rate operator is expressed in terms of the Riesz projectors onto the neutral frequency blocks and the experimentally specified readout map. Thus the Liouvillian spectrum identifies persistent coordinates, whereas the readout-rate operator identifies the subset of those coordinates that produces a persistent measurement record. Under additive Gaussian readout noise, the same operator becomes the classical Fisher-information-rate matrix for estimating the persistent coordinate from the specified record. The construction is intended as a diagnostic and design principle for decoherence-free subspaces, quantum memories, syndrome records, feedback architectures, and stable auxiliary transduction paths in open quantum systems. It is detector-relative and does not claim optimization over all possible quantum measurements. The main theorem applies to finite-dimensional semistable generators with semisimple neutral spectrum. Keywords: open quantum systems; semistable Liouvillian dynamics; quantum readout; observability Gramian; decoherence-free subspaces; Fisher information; quantum feedback; syndrome extraction; Riesz projectors.