Detecting Initial System-Environment Correlations from a Single Observable
We address the problem of detecting initial system--environment correlations when the environment is not directly accessible. Most existing approaches rely on full state tomography or multiple system preparations, which can be experimentally demanding. We show that, for a known interaction, it can be sufficient to monitor a single expectation value of the system. Focusing on a qubit interacting with an environment via isotropic Heisenberg exchange, we derive exact bounds on the signal $z(t)=\langle\sigma_z^S\rangle(t)$ that hold for all factorized initial states. These bounds define a \emph{factorized envelope}: if an observed trajectory exits this envelope at any time, initial system--environment correlations are certified. From a reduced-dynamics perspective, the envelope admits a clear operational interpretation as the admissible region generated by the standard product assignment (embedding) map, which serves as a null model for uncorrelated preparations. Envelope violations therefore rule out the entire product-assignment class using only a single calibrated observable. We illustrate the method using three families of correlated initial states and observe clear envelope violations, including cases in which the reduced system state is maximally mixed. We further show that the same single-observable logic extends to an exactly solvable pure-dephasing spin--boson model with an infinite environment, where factorized initial states generate a simple coherence envelope whose violation certifies initial correlations. Overall, our results demonstrate that single-axis measurements, combined with a one-time calibration of $\rho_S(0)$, can certify initial system--environment correlations without tomography or environment access.