Abstract
Hybrid systems where a quantum material strongly couples to a nonlocal cavity photon mode have emerged as a new frontier for controlling and probing quantum correlations, yet the structure and scaling of light-matter entanglement produced by the nonlocal coupling remains poorly understood. We address this problem through an exactly solvable framework based on reinterpreting the Power--Zienau--Woolley (PZW) transformation as a \textit{light-matter quantum circuit} that couples the photonic position quadrature X ∼ a + a^† to the many-body dipole P of a one-dimensional quantum chain. We derive a closed-form expression for the reduced density matrix valid at all coupling strengths, in which off-diagonal elements between matter states of unequal dipole are suppressed by a Gaussian factor encoding the full weak-to-ultrastrong coupling crossover. At weak coupling, the reduced density matrix takes a Lindbladian form with P as the jump operator, and the entanglement entropy is controlled by the dipole variance. At ultrastrong coupling, the density matrix becomes exactly block-diagonal in dipole sectors, reflecting an \textit{emergent dipole symmetry} dynamically imposed by the photon field, with entanglement entropy given exactly by the Shannon entropy of the dipole-sector weight distribution. Applying this framework to a half-filled Su--Schrieffer--Heeger chain, we show that, at strong coupling, both the light-matter entanglement and the spatial entanglement of the photon-dressed matter state scale logarithmically with system size, S_∞ ∼ α/2log L, robust across the SSH phase diagram. The logarithm originates from the photon resolving a single collective coordinate P whose fluctuations grow as L^α/2, a distinct mechanism from the logarithmic entanglement of critical one-dimensional systems.