Resolved and Collective Loss in a Two-Excitation Oscillator–Qubit Code: Exact Recovery and Finite-Time Separation from a Hard-Core Realization

We derive an explicit test of whether a reduced quantum-memory model overstates recoverability. For channels with orthogonal logical outputs in distinguishable sectors, the minimum recovery error follows directly from differences between environmental record amplitudes, without numerical recovery optimization. Applying this test to two-excitation oscillator–qubit and four-qubit encodings reveals that matching single-loss responses need not preserve finite-time recoverability. Under collective loss, the four-qubit encoding is perfectly recoverable, whereas oscillator emission records reveal logical information. The resulting unavoidable error initially grows as the cube of storage time. The test quantifies omitted-level effects and benchmarks reduced models and recovery algorithms.