Detecting Initial System-Environment Correlations from a Single Observable: Theory, Performance, and Applications

Initial correlations between a quantum system and its environment can strongly influence open-system dynamics and invalidate standard reduced-dynamics descriptions. Detecting such correlations is therefore an important problem, but most existing approaches require state tomography, multiple system preparations, or some degree of access to the environment. Here we show that initial system-environment correlations can be certified using only a single calibrated system observable when the interaction is known. For a qubit interacting with an environment via isotropic Heisenberg exchange, we derive exact bounds on the signal z(t)=⟨σzS⟩(t) that hold for all factorized initial states sharing the same calibrated reduced system state. These bounds define a factorized envelope: if an observed trajectory exits this envelope, initial correlations are unambiguously certified. The witness requires only single-axis measurements after a one-time calibration of ρS(0) and does not rely on environment access or state tomography. We illustrate the method for several families of correlated initial states and show that the same single-observable logic extends to an exactly solvable pure-dephasing spin-boson model with an infinite environment.