Complementary quantum and classical records of qubit decoherence
Decoherence is usually viewed as the loss of local coherence, but it also writes information into the environment. Here we show that the environment stores this information in two distinct forms. One is a recoverable quantum record: after a transverse qubit measurement, the bath is projected onto a Schr\"odinger-cat-like state in a mode-matched physical collective coordinate. The other is a redundant classical which-path record distributed across physical frequency-band fragments. Using tensor-network simulations of a spin--boson reservoir, we reconstruct the conditional bath Wigner function and find visible negativity. In the chain representation used for the simulations, one natural orbital carries more than $95\%$ of the bath one-body occupation associated with the record. A parity symmetry gives an exact nonperturbative identity between the remaining qubit coherence and the overlap of the two environmental branches, while the classical pointer information forms a Darwinian record across fragments. At finite temperature the quantum record is thermally smoothed. In the pure-dephasing limit it becomes an exact mixture of displaced cat states, and negativity remains visible over the simulated range. These results connect decoherence to phase-space tomography and outline how both records can be observed by qubit readout and collective-mode Wigner tomography.